Finance SimiKedia Combined

RUTGERS BUSINESS SCHOOL

Professor Simi Kedia Financial Management Office: Levin Building, 143 W 6.40- 9.40 Phone: 973-353-1145 22:390:587:61, Fall 2007 Email: [email protected] New Brunswick, LCB-109

Financial Management This course is an introductory course in finance. After a brief introduction on financial markets and institutions, we will look at the decision criteria a firms uses to evaluate which projects to undertake, how it raises money from capital markets and how it decides on which financial instruments, i.e., debt or equity to use. We will examine in depth how firm’s make real investment decisions to maximize shareholder value. Textbook and Readings Required Text: Corporate Finance, by Ross, Westerfield, and Jaffe, 8th Edition. Supplemental material will be added in Lecture notes that will be available on Blackboard. Office Hours The usual office hours will be on Wednesday evening from 5.00 – 6.30 pm. My office is Levin 143. If you are not able to make it of office hours and have been attending class regularly please contact me and we will schedule another time. It is easiest to reach me by email. Grading: Your grade will be determined as follows: homework assignments (10%), mid-term examination (40%), and final examination (50%). Problem sets are meant to be individual work and to help you to practice and consolidate concepts before we move to new subjects. All problems set are to be handed to me on the appointed day or before. I will not accept problem set by email. If you are not able to make it to class that day, you can drop the problem set in my mail box anytime before the class. The midterm is scheduled for October 17th. I would encourage all students to ensure that they take the midterm at the appointed day. Topics and their approximate order: These topics do not correspond to class sessions for each week. Some topics might take longer than anticipated and some shorter, but by and large we will follow the order of these topics. Some modifications will develop as the course develops.

Classes Chapters Topics 5th September 1,4 Introduction, Basics, Time Value of Money 12th September 5 Bond Valuation 19th September 5,6 Stock Valuation, Capital Budgeting 26th September 6 Making Decisions Using NPV 3rd October 7 Discounted Cash Flow 10th October 8, 9 DCF, Real Options, Risk First Problem Set due and discussed 17th October Midterm 24th October 9, 10 Risk and Return 31st October 11 Cost of Capital 7th November 15 Capital Structure 14th November 16 Limits to the use of Debt 21st November No Classes 28th November 17, 13 Valuation for levered firm, Market Efficiency, 5th December Summary, Overview 2nd Problem Set due and discussed

IntroductionClass of September 5th

Prof. Simi KediaFinancial ManagementRutgers Business School

Information

Professor: Simi Kedia

Phone: 973-353-1145 Email: [email protected] Hours: 5.00 – 6.15 Wednesday

Levin 113

Textbook and Readings

Required Text: Corporate Finance by Ross, Westerfield and Jaffe, 8th Edition.

Supplemental material will be added in Lecture notes that will be available on Blackboard.

Grading Problem Sets

There are two problem setsTogether they account for 10% of the grade

ExamsThere are two examsMidterm: 40% Final: 50%

Exam attendanceI would really like you to take the exam at the appointed day and timeIf you have to miss

Need verifiable and acceptable reasonMake up exam will be harder

The CourseThis is the required course in Finance

We will spend most of the time mastering basic and core conceptsWe will not spend much time on the cutting edge of financeFor those of you that have done this before – it may seem slow. For those of you that have not seen it before – it may seem fastMost of the course and the exams will be quantitative and number drivenPlease ask questions and stop me if you have questions.

What will we do?

The course is about giving you an understanding of how firms or corporations make their financial decisions

ContextWhat is a firm or corporation?

What is the role of finance in a firm?

What is the environment in which firms make and implement financial decision?

Financial Markets, Instruments and Institutions

Corporations Examples of corporations?

MicrosoftProcter and GambleGeneral Electric

What else do we see that is not a corporation?

PartnershipsConsulting firms: Mckenzie, Ernst and Young, e.tc

Sole ProprietorCorner Deli, Dry Cleaning store

Characteristics of Corporations

Several Shareholder (owners)Partnerships are owned by small group of partnersSole proprietors are owned by a individual

Limited LiabilityProfessional Management

Separation of ownership and control

Board of directorsLegal Entity: Pay taxes, can be sued

Role of FinanceWhat Projects?

Applications in all areasMarketing: Invest in advertising campaign?Operations: Invest in new machinery or use the old one at higher operating cost

Capital Budgeting Decision

How to Finance the projects?Financial markets, instrumentsIssue Equity or DebtCapital Structure Decision

Decision criteriaMaximize Firm Value

What is Debt?When companies borrow moneyThey promise to repay the money back after a fixed number of years --- maturity of debtThey pay interest on the amount borrowedAdhere to some guidelines…Debt Covenants What if they cannot pay the money back?

What if they owe a million dollars but all the assets in the firm are worth only $100,000Bankruptcy

Examples:Verizon borrows 5 million dollars from Citi Bank for a new plantPfizer issues 20 years bonds paying 5%6 month treasury bills

What is Equity?

This is selling ownershipCorporations can sell shares to raise moneyWhat happens if the firm is not doing well and your shares are not worth the purchase price?

NOTHING. The firm is not obliged to compensate you for the loss in your investmentIf the firm has some money, it has to pay the debt before

Financial Markets

Capital MarketsLong term financing: Debt and Equity

Debt: Public debt and Bank debtEquity

Money marketsShort term financing

Commercial paperTrade Credit

Financial Institutions

Mutual FundsRaises money from investors (by selling its shares) and invests in the shares of firms

Pension FundsGets money from employees and invests in the shares of the firms

BanksGet deposits and makes loans to firmsIn the US not allowed to invest in equity

Insurance CompanyGets money when consumers buy insurance policy

Financial Statements

This is how the firm communicates with the market and investors

Summary of firm’s activity and performance

They are prepared on a routine basis

They have been standardized so that we can follow clearly what they do.

What are the main financial statements of the firmsBalance Sheet

Income Statement

Cash flow Statement

Financial Statements

Balance SheetSnapshot of assets and liabilities at a point in timePrepared by firms at fiscal year end

Income StatementSummary of firm’s activity over the yearTotal Revenues, total costs and net profits

Cash Flow Statement Shows the firm’s cash receipts and cash paymentsFor e.g., a store can sell furniture for cash or for creditImportant to see what the net cash flow vs. profits

Classes Chapters Topics 5th September 1,4 Introduction, Basics, Time Value of Money 12th September 5 Bond Valuation 19th September 5,6 Stock Valuation, Capital Budgeting 26th September 6 Making Decisions Using NPV 3rd October 7 Discounted Cash Flow 10th October 8, 9 DCF, Real Options, Risk First Problem Set due and discussed 17th October Midterm 24th October 9, 10 Risk and Return 31st October 11 Cost of Capital 7th November 15 Capital Structure 14th November 16 Limits to the use of Debt 21st November No Classes 28th November 17, 13 Valuation for levered firm, Market Efficiency, 5th December Summary, Overview 2nd Problem Set due and discussed

Time Value of MoneyChapter 4

Future Value: One Year

What is the Value of $100 invested for 1-year if the interest rate is 9%?

After 1-year you earn:Interest = .09(100) = $9.00Total value = $100 + $9.00 = $109Total Value = $100 (1+.09) = $100(1.09)

This is also called Future Value (FV)

Future Value: Two Years

What is the future value of $100 in 2 years if the interest rate is 9%.

After 1-year we have $109.Second year interest = .09(109) = $9.81Future Value = $100 + $9 + $9.81 = 118.81Future Value = $109 + 9.81FV = $109*(1+.09)FV = $100 (1.09)2

General Future Value Formula

In general the future value of $P invested today for n years is:

FV = P(1+r)…(1+r) = P(1+r)n

TerminologyFV = Future Value. PV = Present Value. The value of a cash flow right now. It is also P the principal

r = the interest rate (sometimes i). Unless other wise stated itis annual interest

N = time period (sometimes t). Unless specified, number of years.

Example

Ex: P = $1000, r = 8%

How much will you have at the end of 10 years?

At the end of 10 years you have:FV = $1000(1+.08)10 = $2,158.925

How much will you have in 5 years if r = 10%FV = $1000(1+.10)5 = $1,610.51

Ways to calculate Future Value

Ex: P = $1000, N = 10, r = 8%:

1. Regular calculator, use yx key.FV = $1000(1+.08)10 = $2,158.925

2. FV tables – Future Value Interest Factor or FVIF - (Table A.3)

FVIF(8%,10) = 2.1589FV = $1000*2.1589 = $2,158.90

Period 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15%1 1.010 1.020 1.030 1.040 1.050 1.060 1.070 1.080 1.090 1.100 1.110 1.120 1.130 1.140 1.1502 1.020 1.040 1.061 1.082 1.103 1.124 1.145 1.166 1.188 1.210 1.232 1.254 1.277 1.300 1.3233 1.030 1.061 1.093 1.125 1.158 1.191 1.225 1.260 1.295 1.331 1.368 1.405 1.443 1.482 1.5214 1.041 1.082 1.126 1.170 1.216 1.262 1.311 1.360 1.412 1.464 1.518 1.574 1.630 1.689 1.7495 1.051 1.104 1.159 1.217 1.276 1.338 1.403 1.469 1.539 1.611 1.685 1.762 1.842 1.925 2.0116 1.062 1.126 1.194 1.265 1.340 1.419 1.501 1.587 1.677 1.772 1.870 1.974 2.082 2.195 2.3137 1.072 1.149 1.230 1.316 1.407 1.504 1.606 1.714 1.828 1.949 2.076 2.211 2.353 2.502 2.6608 1.083 1.172 1.267 1.369 1.477 1.594 1.718 1.851 1.993 2.144 2.305 2.476 2.658 2.853 3.0599 1.094 1.195 1.305 1.423 1.551 1.689 1.838 1.999 2.172 2.358 2.558 2.773 3.004 3.252 3.518

10 1.105 1.219 1.344 1.480 1.629 1.791 1.967 2.159 2.367 2.594 2.839 3.106 3.395 3.707 4.04611 1.116 1.243 1.384 1.539 1.710 1.898 2.105 2.332 2.580 2.853 3.152 3.479 3.836 4.226 4.65212 1.127 1.268 1.426 1.601 1.796 2.012 2.252 2.518 2.813 3.138 3.498 3.896 4.335 4.818 5.35013 1.138 1.294 1.469 1.665 1.886 2.133 2.410 2.720 3.066 3.452 3.883 4.363 4.898 5.492 6.15314 1.149 1.319 1.513 1.732 1.980 2.261 2.579 2.937 3.342 3.797 4.310 4.887 5.535 6.261 7.07615 1.161 1.346 1.558 1.801 2.079 2.397 2.759 3.172 3.642 4.177 4.785 5.474 6.254 7.138 8.13716 1.173 1.373 1.605 1.873 2.183 2.540 2.952 3.426 3.970 4.595 5.311 6.130 7.067 8.137 9.35817 1.184 1.400 1.653 1.948 2.292 2.693 3.159 3.700 4.328 5.054 5.895 6.866 7.986 9.276 10.76118 1.196 1.428 1.702 2.026 2.407 2.854 3.380 3.996 4.717 5.560 6.544 7.690 9.024 10.575 12.37519 1.208 1.457 1.754 2.107 2.527 3.026 3.617 4.316 5.142 6.116 7.263 8.613 10.197 12.056 14.23220 1.220 1.486 1.806 2.191 2.653 3.207 3.870 4.661 5.604 6.727 8.062 9.646 11.523 13.743 16.36725 1.282 1.641 2.094 2.666 3.386 4.292 5.427 6.848 8.623 10.835 13.585 17.000 21.231 26.462 32.91930 1.348 1.811 2.427 3.243 4.322 5.743 7.612 10.063 13.268 17.449 22.892 29.960 39.116 50.950 66.21235 1.417 2.000 2.814 3.946 5.516 7.686 10.677 14.785 20.414 28.102 38.575 52.800 72.069 98.100 133.17640 1.489 2.208 3.262 4.801 7.040 10.286 14.974 21.725 31.409 45.259 65.001 93.051 132.782 188.884 267.86450 1.645 2.692 4.384 7.107 11.467 18.420 29.457 46.902 74.358 117.391 184.565 289.002 450.736 700.233 1,083.657

Future value interest factor of $1 per period at i% for n periods, FVIF(i,n).

Observation on FV

Which of the following are true

FV factors are always greater the 1

Future value of a 1$ for 8% is higher at 3 years than at 5years.

Future value 1$ for 5 years is higher at 10% than at 8%

Simple vs. Compound Interest

Simple Interest:Interest is paid just on original principle

Compound InterestInterest is paid on principle plus past earned interest.

Simple Interest Example

Interest is paid only on the principle.

Ex. Invest $100 for 2 years with 10% simple interest.

Interest earned in year 1 = 100*.10 = $10Interest earned in year 2 = 100*.10 = $10Total payout (after 2 years) = $120

Compound Interest: FV = P(1+r)n

Compound Interest

What if the compounding interval is less than a year?

For example: What is the FV of $100 invested for 2 years at 10% annual interest compounded semi-annually?

FV = $100(1+(.10/2))4 = $121.55FV = $100(1+.05)4 = $121.55

Compound Interest II

In general:

FV = $P(1+r/k)nk

r = annual interest ratek = compounding intervals per yearn = number of years

Compounding Interval

What is the future value if you invest $100 for 2-years at 10% and receive quarterly compounding?

K = 4FV =$100(1+.(10/4))8 = $121.84

What is the future value if you invest $100 for 2-years at 10% and receive monthly compounding?

K = 12FV = $100(1+.(10/12))24 = $122.04

Example

Suppose I invest $100 for 1-year at 10% compounded semi-annually.

How much do I have in one year?FV = $100 (1.05)2=$110.25

What is your total return for the year?Return=(110.25-100)/100 = 10.25/100= 10.25%

In this example the stated rate is 10%. The effective annual rate (EAR) is 10.25%

Effective Annual Rate

Effective Annual Rate (EAR)This is the effective yield you receive over a 1-year period

Stated Annual RateThis is the rate stated as the annual rate.

Why would these be different?Compounding

EAR with more compoundingConsider the same $100 investment at a 10% rate for one year. Compounded:

Quarterly: FV = 100 (1+.10/4)4 = 110.38EAR = 10.38%

Monthly: FV = 100(1+.10/12)12=110.47EAR = 10.47%

Daily: FV = 100(1+.10/365)365=110.52EAR = 10.52%

EAR with a 2-year problem

How do I calculate the EAR if I’m investing for 2-years? Consider Investing $150 for 2-years at 8% compounded quarterly. What is my EAR?

Answer1. Calculate the FV:

FV = $150(1+.02)8=175.75

2. Determine what interest rate (compounded annually) will give you FV = 175.75

FV = 150(1+EAR)2= 175.75(1+EAR)2 = 175.75/150 = 1.1717(1+EAR) = SQ Root(1.1717)=1.0825EAR = .0825 = 8.25%

Present Value

Present Value (PV) is the value of future cash flows right now, today.

We saw earlier that the future value of $Pinvested today for n years is:

So Present Value of FV is $P

FV = P(1+r)…(1+r) = P(1+r)n

Present Value: Example

What is the PV of $1000 received 3 years from now if the interest rate is 8%?

If we have X dollars today and invest it for 3-years at 8%, what is the FV?FV = $X(1+.08)3 = $1000$X = $1000/(1.08)3 = $793.83So, PV of $1000 3-years from now at 8% interest is $793.83

Present Value

The present value of $F paid in n years is:PV = F/(1+r)n = F x [1/(1+r)n]

F is the Future Valuer is the interest raten is the number of years

Example: Present Value

What is the present value of $ 1213 received In 9 yearsIf interest rate is 7%?

Present Value is PV = $1213/(1.07)9 = 659.79Can use the Present Value Tables (A1) PVIF(7%,9) = 0.5439PV = $1213 * 0.5439 = $659.7

Period 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15%1 0.990 0.980 0.971 0.962 0.952 0.943 0.935 0.926 0.917 0.909 0.901 0.893 0.885 0.877 0.8702 0.980 0.961 0.943 0.925 0.907 0.890 0.873 0.857 0.842 0.826 0.812 0.797 0.783 0.769 0.7563 0.971 0.942 0.915 0.889 0.864 0.840 0.816 0.794 0.772 0.751 0.731 0.712 0.693 0.675 0.6584 0.961 0.924 0.888 0.855 0.823 0.792 0.763 0.735 0.708 0.683 0.659 0.636 0.613 0.592 0.5725 0.951 0.906 0.863 0.822 0.784 0.747 0.713 0.681 0.650 0.621 0.593 0.567 0.543 0.519 0.4976 0.942 0.888 0.837 0.790 0.746 0.705 0.666 0.630 0.596 0.564 0.535 0.507 0.480 0.456 0.4327 0.933 0.871 0.813 0.760 0.711 0.665 0.623 0.583 0.547 0.513 0.482 0.452 0.425 0.400 0.3768 0.923 0.853 0.789 0.731 0.677 0.627 0.582 0.540 0.502 0.467 0.434 0.404 0.376 0.351 0.3279 0.914 0.837 0.766 0.703 0.645 0.592 0.544 0.500 0.460 0.424 0.391 0.361 0.333 0.308 0.284

10 0.905 0.820 0.744 0.676 0.614 0.558 0.508 0.463 0.422 0.386 0.352 0.322 0.295 0.270 0.24711 0.896 0.804 0.722 0.650 0.585 0.527 0.475 0.429 0.388 0.350 0.317 0.287 0.261 0.237 0.21512 0.887 0.788 0.701 0.625 0.557 0.497 0.444 0.397 0.356 0.319 0.286 0.257 0.231 0.208 0.18713 0.879 0.773 0.681 0.601 0.530 0.469 0.415 0.368 0.326 0.290 0.258 0.229 0.204 0.182 0.16314 0.870 0.758 0.661 0.577 0.505 0.442 0.388 0.340 0.299 0.263 0.232 0.205 0.181 0.160 0.14115 0.861 0.743 0.642 0.555 0.481 0.417 0.362 0.315 0.275 0.239 0.209 0.183 0.160 0.140 0.12316 0.853 0.728 0.623 0.534 0.458 0.394 0.339 0.292 0.252 0.218 0.188 0.163 0.141 0.123 0.10717 0.844 0.714 0.605 0.513 0.436 0.371 0.317 0.270 0.231 0.198 0.170 0.146 0.125 0.108 0.09318 0.836 0.700 0.587 0.494 0.416 0.350 0.296 0.250 0.212 0.180 0.153 0.130 0.111 0.095 0.08119 0.828 0.686 0.570 0.475 0.396 0.331 0.277 0.232 0.194 0.164 0.138 0.116 0.098 0.083 0.07020 0.820 0.673 0.554 0.456 0.377 0.312 0.258 0.215 0.178 0.149 0.124 0.104 0.087 0.073 0.06125 0.780 0.610 0.478 0.375 0.295 0.233 0.184 0.146 0.116 0.092 0.074 0.059 0.047 0.038 0.03030 0.742 0.552 0.412 0.308 0.231 0.174 0.131 0.099 0.075 0.057 0.044 0.033 0.026 0.020 0.01535 0.706 0.500 0.355 0.253 0.181 0.130 0.094 0.068 0.049 0.036 0.026 0.019 0.014 0.010 0.00840 0.672 0.453 0.307 0.208 0.142 0.097 0.067 0.046 0.032 0.022 0.015 0.011 0.008 0.005 0.00450 0.608 0.372 0.228 0.141 0.087 0.054 0.034 0.021 0.013 0.009 0.005 0.003 0.002 0.001 0.001

Present value interest factor of $1 per period at i% for n periods, PVIF(i,n).

Interest Rates and Time Changes

PV of $1000 3-years from now at 8% interest is $793.83

What is the PV of $1000 received in 3-years if the interest rate is 10%? Less than or greater than $793.83?

PV = $1000/(1.10)3 = 751.31 < $793.83

What is the PV of $1000 received in 5 years if the interest rate is 10%? Less than or greater than $793.83?

PV = $1000/(1.10)5 = $620.92 < $751.31

Example: Comparing Cash Flows

You can receive either:1. $100 now2. $110 in 1-year3. $115 in 2 years. The discount rate is 8%

Which option will you choose? Why?

Converting to Present Value

To compare cash that come over different times, you need to change them to the same year. The most obvious is to convert them to year zero or now or present value

1. PV = 1002. PV = 110/1.08 = 101.853. PV = 115/(1.08)2 = 98.59

⇒ Choose (b)

Converting to Future Value

You can also convert to future value.Convert to the largest number of years. In this example it is two years

1. FV = 100(1.08)2 = 116.642. FV = 110 (1.08)1 = 118.83. FV = 115

⇒ Choose (b)

Can you convert to any other year?

You can convert to any year you want and compare. The answer will be the same

Say convert to the end of year one

1. FV = 100(1.08)1 = 1082. FV = 110 = 1103. FV = 115/(1.08)1 = 106.48

⇒ Choose (b)

Example: PV of multiple CF

Suppose you are offered an investment that gives you $200 in 1-year, $400 in 2-years, and $600 in 3-years. The discount rate, r is 12%. What is this investment worth today?

Present Value

We can calculate the PV of each cash flow individually and then add them up.

Year PV of Cash Flow1 $200 200/1.12 = 178.572 $400 400/(1.12)2= 318.883 $600 600/(1.12)3=427.07

Total(PV)= $924.52

PV Additivity

Suppose you have a stream of cashflows for the next three years:

Year 0 1 2 3Cashflow C0 C1 C2 C3

PV = C0 + C1/(1+r) + C2/(1+r)2 + C3/(1+r)3

FV of multiple cash flows

What if you will receive $100 in 1-year, 200 in 2-years, and $150 in 3-years. The interest rate is 10%.

You are planning to buy a car in three years. How much will have from the above at the end of 3 years?

FV of multiple cash flows

FV is additive too:FV = $100(1.10)2 +$200(1.1)+150FV = 121 + 220 + 150 = 491.00

Year 0 1 2 3

Cashflow C0 C1 C2 C3

FV = C0 (1+r)3 + C1(1+r)2 + C2(1+r) + C3

Example2: Value of multiple CF

Suppose you are offered an investment that gives you $220 in 1-year, $440 in 2-years, and $500 in 3-years. The discount rate, r is 12%.

What is the PV of these cash flows?What is the FV of these cash flows?

Value of multiple cash flowsYear PV of Cash Flow

1 $220 220/(1+.12) = 196.422 $440 440/(1.12)2= 350.753 $500 500/(1.12)3= 355.89

Total(PV)= $903.08

Year FV of Cash Flow1 $220 220*(1.12)2 = 275.972 $440 440*(1.12) = 492.83 $500 500 = 500

Total(FV) = $1268.77

What is the PV of $1268.77?

Annuities and Perpetuities

Next we turn to regular cash flow streams, Annuities and Perpetuities

Annuity

An annuity is a regular stream of payments for a specified period of time.

For Example: An annuity of C dollars for T years will payout C dollars, beginning in one year, every year for the next T years.

Example: Annuity

Consider an annuity of $150 per year for 3 years.

Year Cash Flow1 $1502 $1503 $150

What is the PV of the annuity if the interest rate is 10%?

PV Annuity

We can just take the PV of the individual cash flows and then add them up.

Year Cash flow PV1 150 150/1.1 = 136.362 150 150/(1.1)2 = 123.973 150 150/(1.1)3 = 112.70

Total (PV annuity) = 373.03

PV Annuity

We can also calculate the PV of the annuity using the tables. The three PVIF at 10% are 0.9091, 0.8264, and 0.7513

PV = $150(.9091+.8264+.7513)PV = $150(2.4868) = 373.02

We can also use the PVIFA table (A.3) and find PVIFA(10%,3) = 2.4869

PV = $150(2.4869)=373.04

Period 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% 16% 17% 18% 19% 20%1 0.990 0.980 0.971 0.962 0.952 0.943 0.935 0.926 0.917 0.909 0.901 0.893 0.885 0.877 0.870 0.862 0.855 0.847 0.840 0.8332 1.970 1.942 1.913 1.886 1.859 1.833 1.808 1.783 1.759 1.736 1.713 1.690 1.668 1.647 1.626 1.605 1.585 1.566 1.547 1.5283 2.941 2.884 2.829 2.775 2.723 2.673 2.624 2.577 2.531 2.487 2.444 2.402 2.361 2.322 2.283 2.246 2.210 2.174 2.140 2.1064 3.902 3.808 3.717 3.630 3.546 3.465 3.387 3.312 3.240 3.170 3.102 3.037 2.974 2.914 2.855 2.798 2.743 2.690 2.639 2.5895 4.853 4.713 4.580 4.452 4.329 4.212 4.100 3.993 3.890 3.791 3.696 3.605 3.517 3.433 3.352 3.274 3.199 3.127 3.058 2.9916 5.795 5.601 5.417 5.242 5.076 4.917 4.767 4.623 4.486 4.355 4.231 4.111 3.998 3.889 3.784 3.685 3.589 3.498 3.410 3.3267 6.728 6.472 6.230 6.002 5.786 5.582 5.389 5.206 5.033 4.868 4.712 4.564 4.423 4.288 4.160 4.039 3.922 3.812 3.706 3.6058 7.652 7.325 7.020 6.733 6.463 6.210 5.971 5.747 5.535 5.335 5.146 4.968 4.799 4.639 4.487 4.344 4.207 4.078 3.954 3.8379 8.566 8.162 7.786 7.435 7.108 6.802 6.515 6.247 5.995 5.759 5.537 5.328 5.132 4.946 4.772 4.607 4.451 4.303 4.163 4.031

10 9.471 8.983 8.530 8.111 7.722 7.360 7.024 6.710 6.418 6.145 5.889 5.650 5.426 5.216 5.019 4.833 4.659 4.494 4.339 4.19211 10.368 9.787 9.253 8.760 8.306 7.887 7.499 7.139 6.805 6.495 6.207 5.938 5.687 5.453 5.234 5.029 4.836 4.656 4.486 4.32712 11.255 10.575 9.954 9.385 8.863 8.384 7.943 7.536 7.161 6.814 6.492 6.194 5.918 5.660 5.421 5.197 4.988 4.793 4.611 4.43913 12.134 11.348 10.635 9.986 9.394 8.853 8.358 7.904 7.487 7.103 6.750 6.424 6.122 5.842 5.583 5.342 5.118 4.910 4.715 4.53314 13.004 12.106 11.296 10.563 9.899 9.295 8.745 8.244 7.786 7.367 6.982 6.628 6.302 6.002 5.724 5.468 5.229 5.008 4.802 4.61115 13.865 12.849 11.938 11.118 10.380 9.712 9.108 8.559 8.061 7.606 7.191 6.811 6.462 6.142 5.847 5.575 5.324 5.092 4.876 4.67516 14.718 13.578 12.561 11.652 10.838 10.106 9.447 8.851 8.313 7.824 7.379 6.974 6.604 6.265 5.954 5.668 5.405 5.162 4.938 4.73017 15.562 14.292 13.166 12.166 11.274 10.477 9.763 9.122 8.544 8.022 7.549 7.120 6.729 6.373 6.047 5.749 5.475 5.222 4.990 4.77518 16.398 14.992 13.754 12.659 11.690 10.828 10.059 9.372 8.756 8.201 7.702 7.250 6.840 6.467 6.128 5.818 5.534 5.273 5.033 4.81219 17.226 15.678 14.324 13.134 12.085 11.158 10.336 9.604 8.950 8.365 7.839 7.366 6.938 6.550 6.198 5.877 5.584 5.316 5.070 4.84320 18.046 16.351 14.877 13.590 12.462 11.470 10.594 9.818 9.129 8.514 7.963 7.469 7.025 6.623 6.259 5.929 5.628 5.353 5.101 4.87025 22.023 19.523 17.413 15.622 14.094 12.783 11.654 10.675 9.823 9.077 8.422 7.843 7.330 6.873 6.464 6.097 5.766 5.467 5.195 4.94830 25.808 22.396 19.600 17.292 15.372 13.765 12.409 11.258 10.274 9.427 8.694 8.055 7.496 7.003 6.566 6.177 5.829 5.517 5.235 4.97935 29.409 24.999 21.487 18.665 16.374 14.498 12.948 11.655 10.567 9.644 8.855 8.176 7.586 7.070 6.617 6.215 5.858 5.539 5.251 4.99240 32.835 27.355 23.115 19.793 17.159 15.046 13.332 11.925 10.757 9.779 8.951 8.244 7.634 7.105 6.642 6.233 5.871 5.548 5.258 4.99750 39.196 31.424 25.730 21.482 18.256 15.762 13.801 12.233 10.962 9.915 9.042 8.304 7.675 7.133 6.661 6.246 5.880 5.554 5.262 4.999

Present value interest factor of an (ordinary) annuity of $1 per period at i% for n periods, PVIFA(i,n).

PV of an Annuity/Formula

The PV of an annuity is:

PV =

This simplifies to:r = discount rate

PV = C = cashflowT = years/periods

C(1 r)

C(1 r)

C(1 r)

C(1 r)2 3 T+

++

++

+ ++

...

C 1r

1r(1 r)T−

+⎡⎣⎢

⎤⎦⎥

Example: Annuity

Consider the same annuity of $150 per year for 3 years. However, you also get a payment today.

Year Cash Flow0 $1501 $1502 $1503 $150

What is the PV of this set of cash flow if the interest rate is 10%?

PV Annuity

We know that from PVIFA table (A.3) that PVIFA(10%,3) = 2.4869

PV of year 1 to 3 payment = $150(2.4869)=373.04

PV of $ 150 today is $150PV of cash flows is = 150 + 373.04 = 523.04

Note: PV of Annuity

The formula as well as the tables give you the value of a set of cash flows beginning one year from now, i.e., year 1.

The annuity formula is applicable only if the cash flows are the same, i.e., C every year.

Future Value: Annuity

Consider our 3-year, $150 per year annuity. What is the future value (at year 3) if the interest rate is 10%?

Year Cash flow FV1 150 150(1.1)2 = 181.502 150 150(1.1) = 1653 150 150 = 150Total FV = 496.50

Calculating Future Value of Annuity

Use the Future Value Tables150 x (1.210 + 1.1 + 1)= $ 496.50

Use FVIFA (table A.4)FVIFA(3,10%) = 3.31Future Value = $150 x 3.31 = 496.50

Period 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% 16% 17% 18% 19% 20%1 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.0002 2.010 2.020 2.030 2.040 2.050 2.060 2.070 2.080 2.090 2.100 2.110 2.120 2.130 2.140 2.150 2.160 2.170 2.180 2.190 2.2003 3.030 3.060 3.091 3.122 3.153 3.184 3.215 3.246 3.278 3.310 3.342 3.374 3.407 3.440 3.473 3.506 3.539 3.572 3.606 3.6404 4.060 4.122 4.184 4.246 4.310 4.375 4.440 4.506 4.573 4.641 4.710 4.779 4.850 4.921 4.993 5.066 5.141 5.215 5.291 5.3685 5.101 5.204 5.309 5.416 5.526 5.637 5.751 5.867 5.985 6.105 6.228 6.353 6.480 6.610 6.742 6.877 7.014 7.154 7.297 7.4426 6.152 6.308 6.468 6.633 6.802 6.975 7.153 7.336 7.523 7.716 7.913 8.115 8.323 8.536 8.754 8.977 9.207 9.442 9.683 9.9307 7.214 7.434 7.662 7.898 8.142 8.394 8.654 8.923 9.200 9.487 9.783 10.089 10.405 10.730 11.067 11.414 11.772 12.142 12.523 12.9168 8.286 8.583 8.892 9.214 9.549 9.897 10.260 10.637 11.028 11.436 11.859 12.300 12.757 13.233 13.727 14.240 14.773 15.327 15.902 16.4999 9.369 9.755 10.159 10.583 11.027 11.491 11.978 12.488 13.021 13.579 14.164 14.776 15.416 16.085 16.786 17.519 18.285 19.086 19.923 20.799

10 10.462 10.950 11.464 12.006 12.578 13.181 13.816 14.487 15.193 15.937 16.722 17.549 18.420 19.337 20.304 21.321 22.393 23.521 24.709 25.95911 11.567 12.169 12.808 13.486 14.207 14.972 15.784 16.645 17.560 18.531 19.561 20.655 21.814 23.045 24.349 25.733 27.200 28.755 30.404 32.15012 12.683 13.412 14.192 15.026 15.917 16.870 17.888 18.977 20.141 21.384 22.713 24.133 25.650 27.271 29.002 30.850 32.824 34.931 37.180 39.58113 13.809 14.680 15.618 16.627 17.713 18.882 20.141 21.495 22.953 24.523 26.212 28.029 29.985 32.089 34.352 36.786 39.404 42.219 45.244 48.49714 14.947 15.974 17.086 18.292 19.599 21.015 22.550 24.215 26.019 27.975 30.095 32.393 34.883 37.581 40.505 43.672 47.103 50.818 54.841 59.19615 16.097 17.293 18.599 20.024 21.579 23.276 25.129 27.152 29.361 31.772 34.405 37.280 40.417 43.842 47.580 51.660 56.110 60.965 66.261 72.03516 17.258 18.639 20.157 21.825 23.657 25.673 27.888 30.324 33.003 35.950 39.190 42.753 46.672 50.980 55.717 60.925 66.649 72.939 79.850 87.44217 18.430 20.012 21.762 23.698 25.840 28.213 30.840 33.750 36.974 40.545 44.501 48.884 53.739 59.118 65.075 71.673 78.979 87.068 96.022 105.9318 19.615 21.412 23.414 25.645 28.132 30.906 33.999 37.450 41.301 45.599 50.396 55.750 61.725 68.394 75.836 84.141 93.406 103.74 115.27 128.1219 20.811 22.841 25.117 27.671 30.539 33.760 37.379 41.446 46.018 51.159 56.939 63.440 70.749 78.969 88.212 98.603 110.28 123.41 138.17 154.7420 22.019 24.297 26.870 29.778 33.066 36.786 40.995 45.762 51.160 57.275 64.203 72.052 80.947 91.025 102.44 115.38 130.03 146.63 165.42 186.6925 28.243 32.030 36.459 41.646 47.727 54.865 63.249 73.106 84.701 98.347 114.41 133.33 155.62 181.87 212.79 249.21 292.10 342.60 402.04 471.9830 34.785 40.568 47.575 56.085 66.439 79.058 94.461 113.28 136.31 164.49 199.02 241.33 293.20 356.79 434.75 530.31 647.44 790.95 966.71 1,181.935 41.660 49.994 60.462 73.652 90.320 111.43 138.24 172.32 215.71 271.02 341.59 431.66 546.68 693.57 881.17 1,120.7 1,426.5 1,816.7 2,314.2 2,948.340 48.886 60.402 75.401 95.026 120.80 154.76 199.64 259.06 337.88 442.59 581.83 767.09 1,013.7 1,342.0 1,779.1 2,360.8 3,134.5 4,163.2 5,529.8 7,343.950 64.463 84.579 112.80 152.67 209.35 290.34 406.53 573.77 815.08 1,163.9 1,668.8 2,400.0 3,459.5 4,994.5 7,217.7 10,436 15,090 21,813 31,515 45,497

Future value interest factor of an ordinary annuity of $1 per period at i% for n periods, FVIFA(i,n).

Calculating Future Value of Annuity

Future Value is given by

FV = C = cash flowT = years/periodsr = discount rate

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −+r

1Tr)1(C

Annuity with different timing

What is the PV of annuity that pays $250 per year for 4 years where the first payment occurs 3-years from today?

Year: 0 1 2 3 4 5 6 7CF 0 0 0 250 250 250 250 0Assume the interest rate is 8%

Annuity with different timing (2)

Year: 0 1 2 3 4 5 6 7CF 0 0 0 250 250 250 250 0

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+−= Tr)r(1

1r1C PV(a)

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+= 2r)(1

PV(a) PV(final)

Timing1. First find the value of the annuity at year 2

(this is what the formula, or our tables give us)

PVIFA(8%,4) = 3.3121PV (year 2) = $250(3.3121) = 828.03

2. Now this is a year 2 payment, just find the PV of this at year 0.

PV = 828.03/(1.08)2= 709.90

Timing: Future Value

Year: 0 1 2 3 4 5 6 7CF 0 0 0 250 250 250 250 0Assume the interest rate is 8%What is the value of this annuity in year 7?

Future Value

We know the present value is $709.90FV in year 7 = 709.9 (1.08)7=1216.6

orYear CF FV(7) FV(7)

0 01 02 03 250 250(1.08 4̂) 340.12224 250 250(1.08^3) 314.9285 250 250(1.08 2̂) 291.66 250 250(1.08 1̂) 2707 0 0 0

Total FV 1216.65

Using the Formula

Value of 250 for six years at the end of 6 year

= 1126.53

Value of CF in 7th year = 1126.53 (1.08) = 1216.65

⎥⎦

⎤⎢⎣

⎡ +=

.081-.08)(1250 year)6th of FV(end

4

Different Cash Flows

What is the PV of the following cash flows:

Year: 0 1 2 3 4 5 6 CF 0 250 250 250 300 300 300

Assume the interest rate is 8%

Different Cash Flows

PV at year 3 of first annuity = $ 300 * PVIFA (8%,3) = 300 x 2.577 = 773.1PV at year 0 = 773.1/(1.08)3 = 613.71

PV at year 0 of second annuity$ 250 x PVIFA(8%,3) = 250 x 2.577 = 644.25

Total PV of cash flows: 613.71 + 644.25 = 1258

End Example 1:

You have the following three options1. $ 132 in one year2. $ 139 in two years3. $ 150 in four years

The interest rate is 7%

Which option will you choose?Convert to year 0Convert to year 3

End Example 1

Converting to Year 01. PV = 132/(1.07)1 = 123.362. PV = 139/(1.07)2 = 121.43. FV = 150/(1.07)4 = 114.4Choose (1)

Converting to Year 31. FV = 132(1.07)2 = 151.132. FV = 139(1.07)1 = 148.73. PV = 150/(1.07)1 = 140.19

End Example 2: 3 year EARConsider Investing $100 for 3-years at 8%, with semi annual compounding. What is the EAR?

1. Calculate the FV:FV = $100(1+.04)6= 126.53

2. Determine what interest rate (compounded annually) will give you FV = 126.53

FV = 100(1+EAR)3= 126.53(1+EAR)3 = 126.53/100 = 1.2653(1+EAR) = 1.0816EAR = .0816 = 8.16%

Time Value of Money, Chapter 4Bond Valuation, Chapter 5

Class of September 12th

Prof. Simi KediaFinancial Management

Rutgers Business School

Annuity Formulas

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −+r

1Tr)1(C

C 1r

1r(1 r)T−

+⎡⎣⎢

⎤⎦⎥

PV of Annuity

FV of Annuity

Example: Annuity

Suppose you win the lottery. It will pay you 1 million dollars, but in yearly payments of $50,000 a year for 20 years. Your first payment will be in 1 year. What is the present value of your prize if the interest rate is 8%?

Example: Annuity

The value of your prize is:

PV =

= $490,907.37Or use the PVIFA table. PVIFA(20,8%)=9.8181

PV = 50,000(9.8181) = $490,905

What if your payments start immediately instead of in 1 year?

50,000 1.08

1.08(1 .08)20−

+⎡⎣⎢

⎤⎦⎥

Example: Annuity Cont.

What if your payments start immediately instead of in 1 year?

The value of your prize is:

PV = 50,000 +

From PVIFA table, PVIFA(19,8%)=9.6036PV =50,000 + 50,000 (9.6036)

= 50,000 + $480,180 = 530,180

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+− 19.08).08(1

1.08150,000

Retirement Example

Suppose for your retirementSave and Invest $3000 per year retire at 70 r = 10%n = 30 years (start saving at 40 years old)

How much money will you have when you retire?

Retirement Example

Using Formula

FV =

Using Table: FVIFA(10%,30) = 164.494

FV = 3000*(164.494) = 493,482

482,493.10

130.10)1(0003 =⎥⎥⎦

⎤

⎢⎢⎣

⎡ −+

Retirement Example Cont.

What if you start saving at 30 (so you have 40 years of saving)?

FV=

FVIFA (10%,40 years) = 442.5926FV = $3000(442.5926) = 1,327,778See the power of compounding over many years?

778,327,1.10

140.10)1(0003 =⎥⎥⎦

⎤

⎢⎢⎣

⎡ −+

Retirement Example Cont.

How much do we save every year if I want to retire at age 70 with 1,000,000. Interest rate is 10% and I have 40 years to save.

Here FVIFA(10%,40) =442.59.So if I save $X per year:FV = $X(442.59) = 1,000,000$X = 1,000,000/(442.59)$X = 2,259.43 per year

000,000,1.10

140.10)1(=

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −+X

Determining n

Suppose I can save $2000 per year. How many years do I have to save at 16% to get $500,000?

FV = $2000(FVIFA) = 500,000FVIFA = 500,000/2000 = 250 at 16%FVIFA(n,16%) = 250, FVIFA(25,16%) = 249.2140Need to save for 25 years

000,500.16

1n.16)1(2000 =⎥⎥⎦

⎤

⎢⎢⎣

⎡ −+

Perpetuity

A perpetuity of $A, is the payment of $A a year forever, beginning in one year.

Time 0 1 2 3 4...Cashflow 0 A A A A...

The PV of a perpetuity of $A, when the discount rate is r, is:

PV = A/r

Example: What is the PV of receiving $120 a year forever, beginning next year if the interest rate is 10%?

PV = $120/.10 = $1,200

Example: What is the PV of receiving $120 a year forever, beginning today if the interest rate is 10%?

PV = $120 + $120/.10 = 1,320

Perpetuity

Example 1

To complete last year of business school and go through law school you will need $10,000 per year for 4 years, starting next year. Your rich uncle offers to put you through school. He will deposit in the bank that pays interest at 7%, compounded annually, a sum of money that is sufficient to provide the 4 payments. His deposit will be made today.

How large must the deposit be?How much will be in the bank immediately after you make the first withdrawal?

Example 1

Present value of the cost of going to school

872,33.07).07(1

1.07110,000 PV(a) 4 =

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+−=

Year 1 Year 2 Year 3 Year 4Beginning Bank Balance 33872.11 26243.16 18080.18 9345.79Interest earned 2371.05 1837.02 1265.61 654.21Withdrawal 10000.00 10000.00 10000.00 10000.00Ending Bank Balance 26243.16 18080.18 9345.79 0.00

Example 2Ernie wants to save money for two objectives:

1. He would like to retire 30 years from today, with a annual retirement income of $300,000 for 20 years. The first $300,000 will be exactly 31 years from today.

2. He would like to purchase a cabin in the mountains 10 years from today at an estimated cost of $350,000.

He can afford to save only $40,000 per year for the first 10 years. He expects to earn 7% per year from investments. Assuming he saves the same amount each year (for years 11-30), what must Ernie save annually from years 11 to 30 to meet his objectives?

Example 2First figure out how much he needs at year 30, i.e., present value of retirement income

The present value of the stream of his retirement benefits is $3,178,204 at year 30.

Second: He need to save enough, such that the future value of all his savings in year 30 is worth $3,178,204

204,178,3)07.1(07.

107.1000,300 20 =

⎥⎥⎦

⎤

⎢⎢⎣

⎡−

Example 2FV of savings in year 30 is $3,178,204 Savings for the first 10 years:He saves 40,000 for the next 10 years. At the end of 10 years this is worth

He buys a house for $350,000 in year 10. This leave him with 552,658 – 350000 = 202658 in year 10

At the end of year 30, this is worth = 202,658(1.07)20 = $784,223

658,55207.

1)07.1(000,4010

=⎥⎥⎦

⎤

⎢⎢⎣

⎡ −+

Example 2 cont.

How much should he save over years 11-30?

His savings over years 11-30 should be worth the following in year 30

3,178,204 - 784,223 = 2,393,981

Use the FVA formula

yearper 396,58

07.1)07.1(

981,393,2 X07.

1)07.1(X981,393,220

20=

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −+=⇒

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −+=

Amortized Loans

Loans that are paid off in installments over time. For e.g. automobile loans, home mortgage loans, student loans

Example: Consider a firm that borrows $1000To be repaid in three equal payments at the end of each of the next three yearsInterest rate of 6%

Amortized Loans: Example 3

1000 = C(2.673) C = 1000/2.673 = 374.11

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+−= Tr)r(1

1r1C PV

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+−= 30.06)0.06(1

10.06

1C $1,000

Example 4

You need to borrow $23,000 to buy a truck. The current loan rate is 7.9% compounded monthly and you want to pay the loan off in equal monthly payments over 5 years. What is the size of your monthly payment?

Example 4

The monthly interest rate = 7.9%/12 = 0.66%Set the PV of a 60-month annuity equal to the $23,000 loan.

X = 23,000/49.4117 = $465.48

X[49.4117])0066.1(0066.

1.0066

1X000,23 60 =⎥⎥⎦

⎤

⎢⎢⎣

⎡−=

Example 5

A rookie quarter back has the following three offers. The money is guaranteed in all three contracts. The interest rate is 10%. Which of the three contracts offers him the most money? All payments are in millions.

Year 1 Year 2 Year 3 Year 4Contract 1 3.00 3.00 3.00 3.00Contract 2 2.00 3.00 4.00 5.00Contract 3 7.00 1.00 1.00 1.00

Example 5 cont.

The present value of the three contracts are

Contract 2 offers him the most money

Year 1 Year 2 Year 3 Year 4 PV Contract 1 3.00 3.00 3.00 3.00 9.51 mContract 2 2.00 3.00 4.00 5.00 10.72 mContract 3 7.00 1.00 1.00 1.00 8.62 m

Conclusion: Time Value of Money

What did we do: Simple Interest vs. Compound InterestCompounding Interval and Effective Annual Rate (EAR)Present Value (PV) and Future Value (FV)Annuities: PV and FVPerpetuitySimple application to lotteries, retirement, amortization schedules

Valuing BondsChapter 5

Bonds

What is a Bond?

A bond is just a promissory note. The bond issuer, or borrower agrees to pay the holder, or lender a specified amount of interest each year, as well as repaying the original principal.

Example of Bond

A 30-Year treasury bond with face value of $1000 and coupon of 7%.

Face Value or Par Value = $ 1000Time to maturity = 30 YearsInterest rate or coupon = 7%

Interest payments = 7% x 1000 = $ 70 every year

Issued by government

Bond Characteristics

Coupon interest rate:Percentage of par that will be paid out annually in interestE.g., 9% coupon bond pays $90 annually

Maturity:Length of time until bondholder receives principalSometimes will be referred to as the year in which the principal is due. For e.g., maturity 2034

Quotation of Coupon Bonds

The annual coupon payment is typically quoted as a percentage of the face value. The face value for US bonds is typically $1000

For example, what are the payments of a 15% coupon bond with a face value of $1000 that matures in 2 years with semi-annual coupon payments?

Annual Interest = 15% x 1000 = $ 150

Year 0 1/2 1 1 1/2 20 $75 $75 $75 $1000 + $75

Bond Characteristics

SecurityCollateral – secured by financial securitiesMortgage – secured by real property, normally land or buildingsDebentures (sr.) – unsecuredSubordinated (jr.) debentures

Fall behind secured debt and senior debentures in case of default.

Notes – unsecured debt with original maturity less than 10 years

Have to be paid before shareholders

Cash Flows to Firm from Bonds

Ex: Bond with face value $1000 pays $I each year in interest. Let the price of the bond today be $P.What are the cash flows to the issuing firm?

Year Cash Flow0 + $P1 - $I2 - $I3 - $I…15 - $ (I+1000)

Note

The cash flows are negative when the firm needs to make a cash payment

The cash flows are positive when the firm receives cash

At maturity, in this case 15 years, the firm has to pay the par value back

The price at which the bond sells need not be equal to the par value or face value.

Cash Flows to Investors from Bonds

Ex: Bond with face value $1000 pays $I each year in interest. Let the price of the bond today be $P.What are the cash flows to the investor?

Year Cash Flow0 - $P1 + $I2 + $I3 + $I…15 + $ (I+1000)

Valuation of Coupon Bonds

The value of a coupon bond, or its price, is the present value of the bonds cash flows.

If the bond has a face value of F (usually $1000), a coupon payment of C dollars, and matures in n periods, what is the value of the bond?

Valuation of Coupon Bond

Year: 1 2 3 …………… nCF C C C F+C

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+−= nn r)(1

Fr)r(1

1r1C PV

n32 r)(1CF...

r)(1C

r)(1C

r)(1C

++

+++

++

++

=PV

What is the discount rate r?

The discount rate is The current interest rate in the marketIt is not the coupon rate on the bond

Why not?The bond could have been issued in the past when the interest rates were different

For e.g. firm Smarts issued bonds in 1999 at 9%. The market interest rate was 9%. Today the market interest rate is 5%.

Coupon rate = 9%Discount rate or r = 5%.

Ex: Coupon Bond

Consider the following bond25 years10% annual coupon rateFace Value $1000Interest rate, r = 10%

What is this bonds current price?

Determine Cash flows

Annual coupon payments = 10%*1000 = 100

Year 1 2 3…… 25100 100 100 1100

How can we value this cash flow?

Bond Value

Use the annuity formular = discount rate

PV of coupon = C = cash flowT = periods

PV =

C 1r

1r(1 r)T−

+⎡⎣⎢

⎤⎦⎥

1000)10.1(

1000.10).10(1

1.101100 2525 =+

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+−

Ex: Coupon Bond


What is this bonds current price?


Annual coupon payments = 10%*1000 = 100

Year 1 2 3…… 25100 100 100 1100

How can we value this cash flow?

Bond Value



PV =

PVIFA(8%,25) = 10.6748, PVIF(8%,25)=0.1460

PV = 100*10.6748 + 1000*0.1460 = 1213.48

C 1r

1r(1 r)T−

+⎡⎣⎢

⎤⎦⎥

48.1213)08.1(

1000.08).08(1

1.081100 2525 =+

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+−

Ex: Coupon Bond (2)

Consider the following bond25 years10% annual coupon rateFace Value $1000Interest rate, r = 8%Semi-annual coupon payments

What is this bond’s current price?


Annual coupon payments = 10%*1000 = 100Semi-annual coupon payments = 100/2 = $50

Year 0.5 1 1.5 2 2.5 3…… 2550 50 50 50 50 50 1050

Convert to 6 month time periodsTime 1 2 3 4 5 6 50

50 50 50 50 50 50 1050How can we value this cash flow?

Bond Value



PV =

PVIFA(4%,50) = 21.4822, PVIF(4%,50)=0.1407

PV = 50*21.4822+1000*0.1407 = 1214.81

C 1r

1r(1 r)T−

+⎡⎣⎢

⎤⎦⎥

82.1214)04.1(

1000.04).04(1

1.04150 5050 =+⎥⎦

⎤⎢⎣

⎡+

−

Ex: Coupon Bond (3)


What is this bond’s current price?

Bond Value



PV =

C 1r

1r(1 r)T−

+⎡⎣⎢

⎤⎦⎥

13.843)12.1(

1000.12).12(1

1.121100 2525 =+

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+−

NoteWhen coupon rate = 10% and r = 10%

Value of bond = 1000

When coupon rate = 10% and r = 8%Value of bond = 1213.48

When coupon rate = 10% and r = 12%Value of bond = 843.13

When the coupon = r, value of bond = parWhen the coupon < r, value of bond < parWhen the coupon > r, value of bond > par

Example

Your investment advisor suggests you buy bonds of firm Smarts. These have a face value of $ 1000, maturity of 3 years and a coupon of 10%. They are selling for $975.

What is the return I am earning? Like the EAR (effective annual return) problem

Example cont.Let the return be y

Y = 11%

This y is also called the yield to maturity in the context of bond valuations

Should you buy the bonds? If the market interest rate r <= y, then buy the bonds

32 y)(11001000

y)(1100

y)(1100975

++

++

++

=

Yield to Maturity

The discount rate that makes the price of the bond equal the discounted cash flows it produces is call the Yield to Maturity(YTM)of the bond.

The YTM tell you the annual return the bond holder is receiving over the life of the bond (assuming they buy the bond at the current price).

Calculating YTM

Consider a coupon bond, that pays annual coupons, C, for T years. If the current price of the bond is P, then the yield to maturity is the interest rate (YTM) such that the following equation holds:

P = C/(1+YTM) + C/(1+YTM)2 + … +

C/(1+YTM)T + FV/(1+YTM)T

Example: Calculating YTM

Consider a bond that pays annual coupons of 8.5% on a face value of $1000 that matures in 3 year. If the current price of the bond is $981.10 then what is the YTM?

$981.10 = 85/(1+YTM) + 85/(1+YTM)2

+1085/(1+YTM)3

⇒ YTM = 9.25%

When is it easy to calculate?

Bond Types: Coupon Rates

Coupon BondPays coupon (quoted as annual % of face value). Can be annual, semi-annul, or any regular intervalFixed or floating

Zero Coupon bondPays no interest, sells below Face Value (discount bond)

Valuation of Discount BondsConsider a discount bond (or zero coupon bond) with a face value of F dollars that matures in n years.

Bond Cash flows:Year 0 1 2 3 …. n

0 0 0 0 FIf the annual interest rate is r, then the present value of the bond’s CF is:

PV = F/(1+r)n

YTM of a discount bond

Consider the following discount bond:Face Value = $1000Matures in 10 yearsCurrent price = $422.41

What is the YTM?Can use calculator or PVIF table

YTM of discount bond

Calculator: (solve the following)$422.41 = $1000/(1+YTM)10

$422.41(1+YTM)10 = $1000(1+YTM)10 = $1000/$422.41 = 2.3674(1+YTM) = (2.3674)1/10 = 1.09 ⇒ YTM = 9%

With Tables$422.42 = $1000*PVIF(YTM,10)PVIF(YTM,10) = 422.42/1000 = 0.4224PVIF(9%,10) = 0.4224 from table

Interest Rates and Bond Prices:Example 1

Consider a 2-year with par value of $1000 and coupon rate of 5%, paid annually.

What are the cash flows for these bonds?What are the prices for these bonds, if the interest rate is 8%?

Year 1 Year 2CF(2-year) 50 1050

As the interest rate is greater than coupon, price is less than par

PV(2-year) = 50/(1.08) + 1050/(1.08)2 = $946.50

Bond Prices and Interest Rate

What happens if the interest rate increases from 8% to 9%? PV(2-year) = 50/(1.09) + 1050/(1.09)2 = $929.64When interest rate was 8% price was 946.50Interest rate increases and the bond price falls. Interest rates and bond prices are inversely related.

Interest Rates and Bond PricesWhen interest rates changes

The coupon payments do not changeThe face value or par value does not changeThe maturity does not changeOnly the discount rate changes

When Interest rate increases, discount rate increases PV falls, price of bond falls

When Interest rate decreases, discount rate decreases PV increases, price of bond increases

n32 r)(1CF...

r)(1C

r)(1C

r)(1C

++

+++

++

++

=PV

Interest Rates and Bond Prices

When interest rates increase, prices of all bonds fall

Let there be a 2 year bond and a 3 year bond. Should there be a difference between these two?

2-year: (929.64 - 946.50)/ 946.50 = -1.78%3-year: (898.75 - 922.69)/ 922.69 = -2.59%

The bond with the longer maturity is effected more by the change in interest rates.

Interest Decrease and Value: Example 1

Suppose the interest rate decreases to 7%. For the same two bonds, what is the value of the bonds?

The percent changes in bond value are:2-year: (963.84 - 946.50)/ 946.50 = 1.83%3-year: (947.51 - 922.69)/ 922.69 = 2.69%

The bond with the longer maturity is effected more by the change in interest rates.

Interest Rates and Bond Price: Example 2

Ex: Consider two bonds which both mature in 2 years. Bond A has a 5% coupon and Bond B has a 15% coupon. Each have a face value of $1000 and pay annual coupons.

What are the bond prices if interest rate is 8.5%?

Example Continued

Bond Payments Year 1 Year 2Bond A CF(5%) 50 1050Bond B CF(15%) 150 1150

If interest rate is 8.5%, value of the bonds are:A = 50/(1.085) + 1050/(1.085)2 = $938.01B = 150/(1.085) + 1150/(1.085)2 = $1115.12

Increase in Interest Rates: Example 2

Suppose the interest rate increases to 9%, now the bond values are:

A = 50/(1.09) + 1050/(1.09)2 = $929.64B = 150/(1.09) + 1150/(1.09)2 = $1105.55

Percent changes in price:A = (929.64 - 938.01)/938.01 = -0.89%B = (1105.55 - 1115.12)/1115.12 = -0.86%

The bond with the lower coupon is effected more by the change in interest rates.

Determination of interest rates

Bond prices are readily available. Bond payments are fixed based on the bond contracts.From this data we can determine the current interest rate.

One Year Interest Rate

Suppose we have the following One-year zero coupon government bond:

Face Value = $1000Current Price = $966.18

What is the current one year interest rate?

One Year Interest Rate

The current one year interest rate:

$966.18 = $1000/(1+r1) ⇒ (1+r1) = $1000/966.18(1+r1) = 1.035 or r1 = 3.5%

Two Year Interest Rate

Suppose we have the following Two-year zero coupon government bond:

Face Value = $1000Current Price = $924.56

What is the current two year interest rate?

Two Year Interest Rate

The current two year interest rate:

$924.56 = $1000/(1+r2)2

⇒ (1+r2)2 = $1000/924.56(1+r2)2 = 1.0816

⇒(1+ r2) = (1.04)r2 = 4%

Summary

One year interest rate is 3.5%Two year interest rate (annual): 4%

How can this be?

Term Structure The relation between short and long term interest rates is known as term structure of interest rates. It is also called the yield curve.

When long term interest rates are higher than short term rates, then the term structure is upward sloping. This is usually the case.

When the long term interest rates are lower than short term rates, the term structure is downward sloping.

Term Structure

In the above case, one year rate is 3.5% and the 2 year rate is 4%. The term structure is upward sloping

What determines the term structureTime value of moneyInflation premiumInterest rate risk premium

Term Structure

Time Value of money: Real RateDetermines the overall level, not shape of term structure

Inflation PremiumCompensation for Inflation

Interest Rate Risk PremiumLonger term bonds have greater risk from changes in interest rateInvestors want a higher interest rate to take this risk

Price of a 2-year Coupon Bond

Given the above term structure of interest rates (r1 = 3.5%, r2=4%) what is the current price of the following 2-year bond:

Face Value $1000Coupons of 10% paid annually

PV = 100/(1.035) + 1100/(1.04)2 = $1,113.63

What is the YTM of this bond?

Yield to Maturity vs. Term Structure

What is the YTM of the above 2-year coupon bond?$1,113.63 = 100/(1+YTM) + 1100/(1+YTM)2

⇒ YTM = 3.98%

YTM is between the 1-year and 2-year interest rates.

Bond Valuation, Chapter 5

Class of September 19th



Valuation of Coupon Bond

Year: 1 2 3 …………… nCF C C C F+C

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+−= nn r)(1

Fr)r(1

1r1C PV

n32 r)(1CF...

r)(1C

r)(1C

r)(1C

++

+++

++

++

=PV

Yield To Maturity (YTM)The discount rate that makes the price of the bond equal the discounted cash flows it produces is call the Yield to Maturity(YTM) of the bond.

Consider a coupon bond, that pays annual coupons, C, for T years. If the current price of the bond is P, then the yield to maturity is the interest rate (YTM) such that the following equation holds:

P = C/(1+YTM) + C/(1+YTM)2 + … + C/(1+YTM)T +

FV/(1+YTM)T

Term Structure The relation between short and long term interest rates is known as term structure of interest rates. It is also called the yield curve.

When long term interest rates are higher than short term rates, then the term structure is upward sloping. This is usually the case.

When the long term interest rates are lower than short term rates, the term structure is downward sloping.

Price of a 2-year Coupon Bond

Given the above term structure of interest rates (r1 = 3.5%, r2=4%) what is the current price of the following 2-year bond:

Face Value $1000Coupons of 10% paid annually

PV = 100/(1.035) + 1100/(1.04)2 = $1,113.63

What is the YTM of this bond?

Yield to Maturity vs. Term Structure

What is the YTM of the above 2-year coupon bond?$1,113.63 = 100/(1+YTM) + 1100/(1+YTM)2

⇒ YTM = 3.98%

YTM is between the 1-year and 2-year interest rates.

Example

Consider a 2-year bond with FV=$1000 and a coupon of 25% paid annually.

What is the price of this bond?What is the YTM of this bond?The one year interest rate is 3.5% and the two year interest rate is 4%.

Example

What is price and YTM of a 2-year bond with FV=$1000 and a coupon of 25% paid annually?

Price = $250/(1.035) + 1250/(1.04)2

= 1397.24YTM:$1397.24 = 250/(1+YTM) + 1250/(1+YTM)2

⇒ YTM = 3.95%

Another Term Structure Problem

Suppose we have the following two bonds:1. FV = $1000, maturity in 1-year, 5% annual

coupons, current price = $972.222. FV = $1000, maturity in 2-years 15% annual

coupons, current price = $1089.30

Given this information, what is the value of a 2-year annuity with a payment of $100?

Term Structure Example

The first year interest rate972.22 = 1050/(1+r)

Interest rate r1 = (1050/972.22) - 1 = 8%

The second year interest rate1089.3 = 150/(1+.08) + 1150/(1+r)2

(1+r)2 = 1150/950.41=1.21

Interest rate r2 = 10%

Term Structure Example

What is the value of a 2-year annuity with a payment of $100?

PV of annuity = 100/(1.08) + 100/(1.1)2

=92.59 + 82.65 = 175.24

Bond Price MovementsInterest Rates changes

Interest rate increases – bond prices dropInterest rate decreases – bond prices increase

Does the price of the Bond change if there are no interest rate changes? Bond A:

Coupon: 8%, semi-annualYTM: 6%13 years to maturityWhat is the price today? In one year? In 8 years?

Bond Price Changes

Price today =

Price one year later =

Price five years later =

77.1178)03.1(

1000)03.1(03.0

103.0140 2626 =+⎟⎟

⎠

⎞⎜⎜⎝

⎛−

36.1169)03.1(

1000)03.1(03.0

103.0140 2424 =+⎟⎟

⎠

⎞⎜⎜⎝

⎛−

30.1085)03.1(

1000)03.1(03.0

103.0140 1010 =+⎟⎟

⎠

⎞⎜⎜⎝

⎛−

Bond Price Movements

900

950

1000

1050

1100

1150

1200

0 1 3 8 12 13

Years From Today

Bon

d Pr

ices

Bond Price


Bond B: Coupon: 6%, semi-annualYTM: 8%13 years to maturityWhat is the price today? In one year? In 5 years?

Bond Price Changes

Price today =

Price one year later =

Price five years later =

17.840)04.1(

1000)04.1(04.0

104.0130 2626 =+⎟⎟

⎠

⎞⎜⎜⎝

⎛−

53.847)04.1(

1000)04.1(04.0

104.0130 2424 =+⎟⎟

⎠

⎞⎜⎜⎝

⎛−

89.918)04.1(

1000)04.1(04.0

104.0130 1010 =+⎟⎟

⎠

⎞⎜⎜⎝

⎛−


0

200

400

600

800

1000

1200

1400

0 1 3 8 12 13

Years from Today

Bon

d Pr

ices

Price - Bond APrice Bond B

Inflation: Real vs. Nominal Interest

Nominal Interest Rate: Actual stated interest rate.

Real Interest Rate: Interest earned adjusted for inflation (i.e., adjusted for change in purchasing power of money)

Example: Real vs. Nominal

Suppose I have $100 today. I like baseball and can buy Baseball Weekly (BW) magazine for $1 each. Today I can buy $100 BW

Suppose I invest my $100 for 1-year.

The one-year nominal interest rate = 8%, and inflation is 3%.

How many BW can I buy in one year?

Example: Real vs. Nominal

In one year my $100 grows to:$100*(1.08) = $108.00BW now costs $1(1.03) = $1.03I can $108/(1.03) = 104.85 BW

Real Return:rr = (104.85-100)/100=4.85%

Nominal ReturnrN = (108-100)/100 = 8%

Determining the real rate

General Formula:(1+rr)=(1+rn)/(1+i)

Where:rr = real interest raterN = nominal interest ratei = inflation rate

Quick estimate: rr ≈ rN - i Here rr ≈ 8% - 3% ≈ 5% vs. 4.85 actual rr

What rate should you use?

Be consistent:

Discount nominal cash flows with the nominal rate.Discount real cash flows with the real rate.Note: we almost always will be working with nominal numbers. However, there are cases where real cash flows are easier to use.

Rates

Suppose the nominal interest rate is 6%.The inflation rate is 2%What is the real rate?(1+rr) = (1.06)/(1.02) = 1.0392rr = 3.92%

Note: Estimate rr = 6%-2% = 4%

Example: Real and Nominal

Consider the following Cash Flows:Year Cash Flow (Nominal)

0 01 1002 1503 2504 200What is the PV of these cash flows if the interest rate is 6%?

What is the PV? (nominal)

Discount the nominal cash flows by the nominal interest rate:

Year Cash Flow Present Value0 0 01 100 = 100/1.06 = 94.342 150 = 150/(1.06)2 = 133.503 250 = 250/(1.06)3 = 209.904 200 = 200/(1.06)4 = 158.42

Total PV = 596.16How will you convert these cash flows to real cash flows?

Example Cont.

The real cash flows:Year Cash Flow Real Cash Flow

0 0 01 100 = 100/(1.02) = 98.042 150 = 150/(1.02)2 = 144.183 250 = 250/(1.02)3 = 235.584 200 = 200/(1.02)4 = 184.77

What is the present value of these real cash flows?

What is the PV (real)?

Discount the real cash flows at the real interest rate.

Year Real Cash Flow Present Value0 0 01 98.04 = 98.04/1.0392 = 94.342 144.18 = 144.18/(1.0392)2 = 133.513 235.58 = 235.58/(1.0392)3 = 209.914 184.77 = 184.77/(1.0392)4 = 158.43

Total PV = 596.19

Default Risk

Bonds are issued by GovernmentsCorporations

Most governments bonds are considered riskless. Governments do not default, they print more money

But firms could run into trouble be unable to pay the interest and/or face value

Lenders demand a premium (higher interest rate) from firms in comparison to governments. This is the default premium

Bond Types: Rating

Ratings attempt to convey two factorsThe likelihood of defaultAmount of recovery if default occurs

Investment Grade (AAA – BBB)

Junk (BB and below)Historically Junk bond were “fallen Angles”More recently firms issue junk bonds (since the mid 80s)

Issued by independent firmsMoody’s and S&P, measure of default risk

Bond Valuation

Bond Terminology

Bond Valuation: What is the price of the bond?

YTM

Effect of interest rates on bond prices

Term Structure of Interest Rates

Real vs. NominalCash flows and interest rates

Valuing Stocks

Prof. Simi KediaIntroduction to Financial Management


Key Concepts

Preferred SockCommon StockStock Characteristics (features)Stock Valuation

Dividend Growth Model

Types of Stock

Two basic types of stockCommon StockPreferred Stock

One firm can have many different issues of each type of stock

Class A common stockClass B common stock

Preferred Stock

Preferred stock has features of debt and equity.

Like equity:No maturity dateCompany does not “have” to pay dividendDividend not tax deductible

Like DebtDividends are fixed

Preferred Stock Features

Claims on AssetsBetween bond holders and common stock holders. Preferred dividend must be paid before common stock dividends

Cumulative Dividends are common

ConvertibilityMany issues are convertible to common stock

Adjustable ratesometimes the dividend is linked to current interest rates

Does not have voting rights

Preferred Stock Valuation

Discount the cash flowsWhat are the cash flows for preferred stock?

Periodic fixed payment that never matures

How do we calculate the value of preferred stock?

Preferred Stock Valuation

Discount the cash flowsWhat are the cash flows for preferred stock?

Periodic fixed payment that never matures Perpetuity value = C/r

Let D = dividend per year and rps = required rate of return for the preferred stock

Stock Value = D/rps

Example

Consider a company’s preferred stock with a dividend of $5 per year (beginning is one year). If the required rate of return is 8% what is the value of the preferred stock?Value = $5/0.08 = $62.50

Determining the required returnOften times we know the current price of preferred stock and the dividend, but do not know the required rate of return. In this case we can solve for the expected rate of return.

What is the expected rate of return if a preferred stock sells for $50 and pays a $5 dividend per year.

Value = Div/rps⇒ $50 = $5/rps

⇒ $50rps = $5rps = $5/$50 = .10 or 10%

Features of Common Stock

Claim on Assets (after bondholders and preferred stock holders)

Voting RightsElect board of directors, vote on changes in charter.Proxy voting, Classes of stock (different voting rights)

. Other RightsShare proportionally in declared dividendsShare proportionally in remaining assets during liquidationPreemptive right – first shot at new stock issue to maintain proportional ownership if desired

Dividend Characteristics

Dividends are not a liability of the firm until a dividend has been declared by the Board

Consequently, a firm cannot go bankrupt for not declaring dividends

Dividends and TaxesDividend payments are not considered a business expense, therefore, they are not tax deductibleDividends received by individuals are taxed as ordinary income

Stock Valuation

Stocks are valued by determining the present value of the cashflows produced.

What are the Cashflows?What discount rate should be used?

1-Year Stock Return

If you purchase stock for 1 year and then sell it you receive all dividends the stock pays during the year plus the stock price at the end of one year.

Assume all dividends are paid at the end of the year. The present value of owning a share of stock is:PV = Div1/(1+r) + P1/(1+r) = P0

Example: GM

Suppose I think GM will sell for $37.50 one-year from now. I also know that GM will pay $2.00 Div one year from now.I require a 10% return.What should the price of GM be today?

Price (GM) = 2.00/(1.1) + 37.50/1.1Price (GM) = $35.91

Stock Valuation Problems

This method of stock valuation has several shortfalls.

(1) We have to assume a holding period

(2) We have to assume a price at the end of the holding period

(3) Different holding periods can give different current valuations

What is P1?

If someone buys the stock 1 year from now, their 1 year return will be Div2 plus price at year 2.

P1= Div2/(1+r) + P2/(1+r)

Substitute this equation in for P1 yields:P0 = Div1/(1+r) + Div2/(1+r)2 + P2/(1+r)2

Dividend Discount Model

If we continue the above substitution pattern (next substitute for P2, then P3, and so on) the resulting current price can be expressed as follows:

P0 = Div1/(1+r) + Div2/(1+r)2 + Div3/(1+r)3+…

The current price is just the present value of the expected dividend payments.

Constant Dividend Stock

Consider a stock with a constant dividend. If we expect the dividend to remain constant indefinitely, then the current value of the stock is:

P0 = Div/(1+r) + Div/(1+r)2 + Div/(1+r)3+…= Div/r

This is just a perpetuity of Dividends.

Constant Growth Dividend Stock

Consider a stock where the dividend is expected to grow at a constant rate, g. The dividends are:

Year 1 2 3 4...Div1 Div1(1+g) Div1(1+g)2 Div1(1+g)3...

This is a growing perpetuity. The present value is:P0 = Div1/(r-g)

Example

Suppose GM is expected to pay a dividend of $2.08 at the end of the year. You expect the dividend to grow at a 4% rate forever. The required rate of return on GM stock is 10%. What is the price today?

Price (GM) = 2.08/(0.10-.04)Price (GM) = 2.08/0.06 = $34.67

Change in growth rate

What if new news comes out today.The 0% financing has hurt current and future profits. New Growth rate is expected to be 3% What is the new Price?Price(GM) = 2.08/(.10-.03) = $29.71

%Price Drop = (34.67-29.71)/34.67 = 14%

Big Price change from small change in growth rate.

Example

GM paid a dividend of $2.08 today. You expect the dividend to grow at a 4% rate forever. The required rate of return on GM stock is 10%. What is the price today?

Price (GM) = 2.08(1.04)/(0.10-.04)Price (GM) = 2.16/0.06 = $36.05

Determining Expected Return

If we know the current price, dividend, and growth rate we can calculate the expected return of the stock.

Price = D1/(rcs-g) ⇒ P(rcs-g) = D1

⇒ rcs-g = D1/P⇒ rcs = (D1/P) + g

Example: Expected Return

General Electric closed at $22.70 and expected to pay a dividend of $0.76. Suppose the growth rate, g = 6%. What is the expected return on GE stock?

22.7 = 0.76/(r-0.6)rcs = 0.76/22.70 + .06 = 0.0934

or 9.34%

Differential Growth Stock

Consider a stock whose dividend is expected to grow at a rate gh for the next T years, and then grow at the rate gl from year T into perpetuity.How would you value this stock?

Differential Growth Stock

How would you value this stock?

First, value the next T years cash flows.

Second, value the remaining years as a growing perpetuity.

Example: Differential Growth

Consider the following stock

Expected to pay a dividend of $0.45 in one year.The dividend is expected to grow at a rate of 20% for following 4 years.Following this rapid growth period the dividend is expected to grow at 15% into perpetuity.The discount rate for this stock is 22%

Dividend Payments

Year Growth Dividend1 0.452 0.20 0.54 = .45(1+.20)3 0.20 0.648 = .54(1+.20)4 0.20 0.7776 = .648(1+.20)5 0.20 0.9331 = .7776(1+.20)6 0.15 1.073 = .9331(1+.15)7 0.15 1.234 = .9331(1+.15)2

all additional years at 15%

Value of Growing Annuity

The value of the first five years of dividends with a discount rate of 22%.

PV = 0.45/(1.22) + 0.54/(1.22)2 + 0.648/(1.22)3

+0.7776/(1.22)4 + 0.9331/(1.22)5

= 1.7848

Value of Growing Perpetuity

The dividend payments for year 6 on can be valued as a growing perpetuity. The year 6 dividend is 1.073 and the growth rate is 15% forever (same discount rate of 22%). The year 5 value of this perpetuity is:PV(at year 5) = 1.073/(.22-.15) = $15.3286This is a year 5 value, so the year 0 value is:PV = $15.3286/(1+.22)5 = $5.6716

Determining Value

Year: 0 1 2 3 4 5 6 7CF 0 D1 D2 D3 D4 D5 D6 D7

3.15$0.15-0.22

1.073g-r

D6 PV(a) ===

67.5$.22)(1

15.3r)(1

PV(a) PV(final) 55 =+

=⎥⎥⎦

⎤

⎢⎢⎣

⎡

+=

Present Value of Stock

The time 0 stock price should be the sum of the annuity value (of dividends 1 - 5) and the perpetuity value from year 6 on.

P0 = 1.7848 + 5.6716 = $7.4564

Determining gDividends

seldom constantIf growing: the price of the stock is very sensitive to the assumption regarding growth rate g

How do we determine g? Can use historical dividends to determine g

Example: Determining g

The following are the historical dividends paid by Coke. What is the growth rate of dividends?

Dividends2000 1.982001 2.042002 2.12003 2.172004 2.24

Example

What are the problems with this approach? Future growth rates are not likely to be the same as past.Young companies grow at a rapid pace yearly on in their life and then growth rates dropVery difficult to predict future growth rates

Dividends Annual Growth Rates2000 1.982001 2.04 (2.04 - 1.98) / 1.98 0.0302002 2.1 (2.1 - 2/4)/ 2.04 0.0292003 2.17 (2.17 - 2.1)/ 2.1 0.0332004 2.24 (2.24 - 2.17)/2.17 0.032

Average growth rate of dividends 0.031

Other Approaches

What happens when the firm does not pay any dividends?The P/E Approach

The is the price to earning ratioIt is quite simpleEssentially, price is a multiple of some important attributeFor e.g., the retail price of the book for our course is $140. Your friend bought a used copy for $70, i.e., at half the priceYou want to buy a used copy for Marketing Management. The retail price of the book is $90. What price would you be willing to pay for the used copy?

Multiple ApproachNeed to also keep in mind

Is it the same edition?How many people have owned the book before? Is it in good condition

Are there markingsDog ears

Similarly, firms could be priced as a multiple of earnings.

Coke annual earnings per share were $8. It is trading at a P/E ratio of 12. The share price is ? If Pepsi had earnings of $6 a share, what would its share price be?

P/E Ratios

Earnings of Kawasaki motorcycles is $ 5 a share. What is its price?

Before you use the P/E ratio need to be comfortable that stocks are similarDifficult to translate it across industriesDifficult to translate it across firms with different risks

What is an example of a market where multiples are used commonly?

Other Approaches

The fundamental approach is Discounted Cash FlowWhat does a firm do with its earnings

Pays it out as dividendsOr keeps it in the firm and invests it in other projectsThese other projects generate their own profits

If the firm does not pay out any cash, it can be valued by asking how much cash it will generate

This is the true value. As a shareholder you can get it by receiving dividends or increase in stock price from the new projects undertakenNext module, we will spend time on how to estimate the firms cash flows.Once we have the cash flows, these can be discounted to get the value of the firm.

1

Net Prevent ValueSept 26th



2

Concepts/Introduction

Capital Budgeting (CB) CriteriaHow Firms decide what to do?How much value do these activities generate?

NPVPaybackDiscounted PaybackInternal Rate of ReturnProfitability Index

3

Value Projects

Determine the Cash Flows

Calculate the Present Value

Decision Criteria: Should we do this project or not?

4

Example: Project Begin

You are looking at a new project, Project Begin. The details of the project are as follows

The Project requires investment in plant and machinery of $ 165,000 nowThe project will generate cash flows of $63,120 at the end of one year. It will generate cash flows of $70,800 in the second year and cash flow of $ 91,080 in the third year.The required return on the project of this risk is 12%.

Question:Should you undertake this project?How much will you gain from it?

5

Project Begin: PV of Cash Inflows

The Cash inflows from Project Begin

The cash inflows are:Year 1: CF = 63,120; Year 2: CF = 70,800; Year 3: CF = 91,080;

Your required return for assets of this risk is 12%.

PV = 63,120/(1.12) + 70,800/(1.12)2 + 91,080/(1.12)3

PV = 177,627.42

6

Should we do the Project?

Present Value of cash inflows $ 177,627.42Cost of doing the project: $ 165,000Net Present Value or NPV = $12,627.42

Do the Project if NPV>0

7

Project Begin: PV of Cash Flows

Alternatively, we could look at all cash flows from Project Begin. These are

Year 0: CF = -165,000Year 1: CF = 63,120; Year 2: CF = 70,800; Year 3: CF = 91,080;

Net Present Value or NPV is:

63,120/(1.12) + 70,800/(1.12)2 + 91,080/(1.12)3 – 165,000= $12,627.42

8

Net Present Value (NPV)

The difference between the market value (or present value) of a project and its cost

If the NPV is positive, accept the project

A positive NPV: project adds value to the firm and will increase the wealth of the owners.Since our goal is to increase owner wealth, NPV is a direct measure of how well this project will meet our goal.

9

Payback Period

How long does it take to get the initial cost back in a nominal sense?

ComputationEstimate the cash flowsSubtract the future cash flows from the initial cost until the initial investment has been recovered

Decision Rule – Accept if the payback period is less than some preset limit

10


Let us look at Project Begin againThe Project requires investment in plant and machinery of $ 165,000 nowThe project will generate cash flows of $63,120 at the end of one year. It will generate cash flows of $70,800 in the second year and cash flow of $ 91,080 in the third year.The required return on the project of this risk is 12%.

Question:What is the payback period?If the criteria is to accept projects with payback period of twoyears, should we accept the project?

11

Payback Period for Project Begin

Assume we will accept the project if it pays back within two years.Year 1: 165,000 – 63,120 = $101,880 still to recoverYear 2: 101,880 – 70,800 = $31,080 still to recoverYear 3: 31,080 – 91,080 = -60,000

Project pays back in year 3

We reject the project.

12

Problems with Payback Criteria

Year Cashflow A Cashflow B0 $-1000 $-10001 $ 900 $ 3002 $ 0 $ 03 $ 300 $ 900

Payback = 3 years for both A and BAre the projects worth the same?

13

Problems with Payback Criteria

Year Cashflow A Cashflow B0 $-1000 $-10001 $ 900 $ 3002 $ 0 $ 03 $ 300 $ 900

Payback = 3 years for both A and B

At a discount rate of 10%NPV(A) = $43.58NPV(B) = $ - 51.09

14

Advantages and Disadvantages of Payback

AdvantagesEasy to understandAdjusts for uncertainty of later cash flowsBiased towards liquidity

DisadvantagesIgnores the time value of moneyRequires an arbitrary cutoff pointIgnores cash flows beyond the cutoff dateBiased against long-term projects, such as research and development, and new projects

15

Discounted Payback Period

Compute the present value of each cash flow and then determine how long it takes to payback on a discounted basis

Compare to a specified required period

Decision Rule - Accept the project if it pays back on a discounted basis within the specified time

16



Question:What is the discounted payback period?If the criteria is to accept projects with payback period of twoyears, should we accept the project?

17

Discounted Payback for the Project Begin

Assume we will accept the project if it pays back on a discounted basis in 2 years.

Compute the PV for each cash flow and determine the payback period using discounted cash flows

Year PV of CF Amount to RecoverYear 1: 63,120/1.121 = 56,357 165,000 – 56,357 = 108,643Year 2: 70,800/1.122 = 56441 108,643 – 56,441 = 52,202Year 3: 91,080/1.123 = 64829 52,202 – 64,829 = -12,627

Project pays back in year 3

We reject the project

18

Advantages and Disadvantages of Discounted Payback

AdvantagesIncludes time value of moneyEasy to understandDoes not accept negative estimated NPV investmentsBiased towards liquidity

DisadvantagesMay reject positive NPV investmentsRequires an arbitrary cutoff pointIgnores cash flows beyond the cutoff pointBiased against long-term projects, such as R&D and new products

19

Internal Rate of Return

This is the most important alternative to NPVIt is often used in practice and is intuitively appealingIt is based entirely on the estimated cash flows and is independent of interest rates found elsewhere

20

IRR – Definition and Decision Rule

Definition: IRR is the return that makes the NPV = 0IRR solves the following equation:

Decision Rule: Accept the project if the IRR is greater than the required return

NPV = C0C1IRR

C2IRR

CnIRR n+

++

++ +

+=

( ) ( )...

( )1 1 2 10

21



Question:What is the IRR?Should we accept the project?

22

Example: Computing IRR

Continuing our example, IRR is obtained by solving.

Look at following table:r NPV(project)0 $60,00010 $19,32412 $12,62716 $38116.132 $0.717 -$2,463

3)IRR1(91,080

2)IRR1(70,800

)IRR1(63,120-165,0000 = NPV

++

++

++=

23

Decision Criteria with IRR

The estimated IRR = 16.132%As the required rate of return is 12%, and IRR = 16.132% > 12%

Accept the Project

24

Advantages of IRR

Knowing a return is intuitively appealingIt is a simple way to communicate the value of a project to someone who doesn’t know all the estimation detailsIf the IRR is high enough, you may not need to estimate a required return, which is often a difficult taskTakes into account the time value of money

25

Summary of Criteria

For the exampleNPV method: We accept the projectBy IRR method: We accept the projectBy payback method (2 years): We reject projectBy discounted payback method (2 years): We reject the project

If conflict: Always use NPV rule

26

ExampleFirm Dark Ages has the following project:

Year Cash Flow0 -15001 3002 8003 6004 500

Dark Ages uses a payback period of 3 years as its criteria.

a) What is the payback period (in years)?b) Will they accept of reject this project?

27

Payback Period

Year Cash Flow Amount Left0 -15001 300 12002 800 4003 600 -2004 500Payback period is 3 yearsAccept the projectIf discount rate is 15%, What is the discounted payback period? Should you accept the project?

28

Discounted Payback Period

YearCash Flows

PV of Cash Flows

Amount Left to Recover

0 -1500 -15001 300 260.87 1239.132 800 604.91 634.223 600 394.51 239.714 500 285.88 -46.17

Discounted Payback period is 4 yearsReject the project

Is this a good or bad project?

29

NPV

YearCash Flows

PV of Cash Flows

0 -1500 -15001 300 260.872 800 604.913 600 394.514 500 285.88

NPV 46.17

The NPV is positiveAccept the project

30

NPV Vs. IRR

NPV and IRR will generally give us the same decisionExceptions

Non-conventional cash flows – cash flow signs change more than onceMutually exclusive projects – When only one of two or more projects can be chosen

31

Mutually Exclusive Projects

Two or more projects that cannot be pursued simultaneouslyIf you can choose only one project:

The NPV rule: Choose the project with the highest positive NPV

The natural IRR ruleAccept the project with the highest IRR.

32

Example

Consider the following two projects

If the discount rate is 5% should we do project S? Project L?

Year Project S Project L0 -1000 -10001 500 1002 400 3003 300 4004 100 600

33

Example Cont..

Year Project SDiscounted Cash Flows at 10%

0 -1000 -10001 500 476.192 400 362.813 300 259.154 100 82.27

NPV 180.42

As NPV = 180>0, Accept the project

34

Example Cont..

Year Project LDiscounted Cash Flows at 10%

0 -1000 -10001 100 95.242 300 272.113 400 345.544 600 493.62

NPV 206.50

As NPV = 206.5>0, Accept the projectWhat are the IRR of the two projects?

35

Example: IRR

What are the IRR of the two projects?For Project S

IRR for project S is 14.5%

The IRR for project L is

The IRR for Project L is 11.8%

04)IRR1(100...3)IRR1(

3002)IRR1(

400)IRR1(

5001000- = 0 =+

+++

++

++

+

04)IRR1(600...3)IRR1(

4002)IRR1(

300)IRR1(

1001000- = 0 =+

+++

++

++

+

36

NPV Profiles

Cost of Capital NPV of Project S NPV of Project L0 300 4005 180.42 206.5

10 78.8 49.1815 -8.33 -80.14

37

Net Present Value Profiles

-200

-100

0

100

200

300

400

500

0 5 10 15

Cost of Capital

NPV NPV of Project S

NPV of Project L

IRR(L) = 11.8%

IRR (S) = 14.5%

Cross Over Rate = 7.2%

38

Project S and L

If you can choose only Project S or Project L which one will you choose?NPV Rule

If the cost of capital is less than 7.2% (the cross over rate): Choose Project LIf the cost of capital is greater than 7.2%: Choose Project SWhy? Project S cash flows come sooner. It is better if the discount rate is high

39

Mutually Exclusive Projects: Another Example

Consider projects A and B with r =15%Year Cash Flow(A) Cash Flow (B)0 -$1,000 -$10,0001 $1,500 $12,500

IRR 50% 25%NPV $304 $870

NPV rule: Chose Project BIRR rule: Choose Project A

A has higher IRR, but B will make the firm richer

40


Two basic conditions can cause the answers from NPV and IRR to conflict with mutually exclusive projects

When timing differences exist (first example)When scale differences exist (second example)

Always use the NPV method when there is a conflict

41

Evaluating Incremental CF

One way around this problem is by evaluating the incremental cash flows.

Start with the highest IRR project (Project A). Decide whether you want to undertake it.

If yes, then ask if taking the incremental investment required for B is a good idea

Incremental investment is the difference in the investment of Project A and Project B

B is equivalent to A + (B-A) = B

42

Ex: Continued

Consider projects A and B with r =15%

Yr CF(A) CF(B) CF(B-A) Incremental project0 -$1,000 -$10,000 -$9,0001 $1,500 $12,500 $11,000

IRR 50% 25% 22%

IRR for incremental Project = 22% > 15% so we should take the incremental project, i.e, take Project BNotice: could just use NPV to begin with

43

Second Problem with IRR

IRR and NPV give different results if there are unconventional cash flows i.e., multiple sign changesIf the cash flows are not all negative followed by all positive there are two possibilities:

There are multiple IRRsThere is no IRR in relevant range (I.e..., positive real numbers)

44

Example: Multiple Sign Changes

Consider the following projectYear Cash Flow0 -$4,0001 $25,0002 -$25,000

IRR 25% and 400%NPV at 10% = -1934

45

Discount Rates vs. PV

-5000-4000-3000-2000-1000

0100020003000

0 10 25 100 200 300 400

Discount Rate

PV

46

Example: No IRR

Consider the following projectYear Cash Flow0 $1,0001 -$3,0002 $2,500

IRR noneNPV at 10% = 339

47

0

100

200

300

400

500

600

0 10 20 30 40 50 100

Discount Rate

PV

48

Conflicts Between NPV and IRR

IRR is unreliable in the following situationsNon-conventional cash flows

Two or more IRRNo IRR

Mutually exclusive projectsMay make you choose the wrong project

NPV measures the increase in value to the firmWhenever there is a conflict between NPV and another decision rule, you should always use NPV

49

The Verdict on IRR

IRR can be useful.However, only useful when it is equivalent to the NPV rule.With information we need to compute IRR we can compute NPV.NPV gives us what we want: the addition to firm value of the project.Should use NPV or incremental IRR.

50


Interesting Issues that arise when we have to choose between projects

Capital Rationing: When capital in limitedProfitability Index

Choosing between projects with different durationsInvestment Timing: When to undertake the project

51

Profitability Index (PI)

PI = Net Present Value (NPV)Initial Investment

Measures the benefit per unit cost, based on the time value of moneyNotice that:

(NPV)/(Inv) > 0 when NPV > 0

Decision Rule: Take projects with profitability index greater than 0.

52

Example

Suppose you can choose between two mutually exclusive projects.

Project Year 0 Year 1 Year 21 -100 200 1502 -500 350 1000

Assume that r = 10%

53

Example Continued

We can calculate the PI for each project.Project 1: NPV= -100 + 200/1.1 + 150/1.12 = $205.79Initial Investment = 100PI = 205.79/100 = 2.06Project 2: NPV = -500 + 350/1.1 + 1000/1.12 = 644.63Initial Investment = 500PI = 644.63/500 = 1.29

54

Example ContinuedDecision Criteria: Accept Project with the highest Profitability Index, i.e., accept project 1Wrong:The above decision criteria does not take into account the scale of the project (like IRR)

Ex: A project with a cost benefit-ratio of 0.5 and initial investment of $100,000,000 is better than a project with a cost-benefit ration of 1 and an initial investment of $100.

This problem can be solved like the IRR problemUse NPVOr calculate the PI of incremental project

55

Example Continued

Project Year 0 Year 1 Year 2 PI1 -100 200 150 2.062 -500 350 1000 1.29

IP (2-1) -400 150 850 1.10

If profitability index of incremental project is greater than zero, then take project #2.

56

Capital Rationing

Profitability index is useful when capital is limited. This may happen when Manager of division given a limited capital budget.For e.g., consider the following

Project Capital NPV A 100 200 B 100 100C 200 100

Which projects should you choose if you have only $200 to invest?

57

Capital Rationing (2)Project Capital NPV PI Rank(PI)A 100 200 2 1B 100 100 1 2C 200 100 0.5 3

Choices are 1) Project C: NPV = $ 1002) Project A and B: NPV = $ 200 + $ 100 = $ 300

Use Profitability IndexRank projects By Profitability IndexTake projects with highest PI until capital is used up

or PI < 0.

58

Ex. Capital Rationing

Pjct Cost NPV

A 200,000 100,000B 500,000 120,000C 400,000 300,000D 200,000 75,000E 100,000 30,000F 100,000 40,000

Capital Budget 1Million.

59

Ex. Capital Rationing

Pjct Cost NPV PI RANK(PI)A 200,000 100,000 0.50 2B 500,000 120,000 0.24 6C 400,000 300,000 0.75 1D 200,000 75,000 0.38 4E 100,000 30,000 0.30 5F 100,000 40,000 0.40 3

Capital Budget 1Million.Take C,A,F,D, and E.

60

Advantages and Disadvantages of Profitability Index

AdvantagesClosely related to NPV, generally leading to identical decisionsEasy to understand and communicateMay be useful when available investment funds are limited

DisadvantagesMay lead to incorrect decisions in comparisons of mutually exclusive investments. Does not account for the scale of the project

61

Choosing Between Projects of Different Durations

Many times a firm is faced with a choice between two projects with different lives:For example, a firm needs to choose

between two machines.One is more expensive, but lasts longerOne is cheaper, but less durable

We need a framework to compare the merits of the two different investments

62

Ex. Projects with different durations

Consider the choice between the following machines:

Initial Cost Annual Operating Cost LifeA:15M 2M 2 yearsB:20M 1M 3 years

Revenue generated per year is the same.r = 5%

63

Take PV of CostsAs the revenues from both machines are the same, choosing projects with maximum NPV is the same as choosing projects with lowest Present value of Cost

PV of costs for the two projects are

A: 15 + 2/(1.05) + 2/(1.05)2 = 18.72 M

B: 20 + 1/(1.05) + 1/(1.05)2 + 1/(1.05)3 = 22.72 M

64

Example: continuedProject A has lower PV of costsShould we choose Project A?

But Project B lasts longer? How do we take this into account?

Two techniques:1: Compare over a common duration (Replacement

Cost) 2: Compare Equivalent Annual Cost

65

Common Duration

Suppose firm needs to produce this product for 6 years. Machine A needs to be purchased 3 times

Yr 1st Time 2nd Time 3rd Time Total Costs

0 15M 15M1 2M 2M2 2M 15M 17M3 2M 2M4 2M 15M 17M5 2M 2M6 2M 2M

66

Common Duration

Machine B needs to be purchased 2 times. Total Costs for both machines are

Yr Cost of A Cost of B0 15M 20M1 2M 1M PV cost A = 51.10M2 17M 1M3 2M 21M PV cost B = 42.35M4 17M 1M5 2M 1M Buy B6 2M 1M

67

Equivalent Annual Cost (EAC)

We could also estimate the equivalent annual cost. This is equivalent to a rental rate which would allow a firm renting the machine to break even.

The annualized costs for each machine are such that the present value of these costs equals the present value of the actual costs of the machines.

68

Computing EAC

Machine A:

Year Annual Cost EAC0 15M 0.00M1 2.00M X M2 2.00M X MPV 18.72M 18.72M

X is the 2-year annuity payment that has a PV of 18.72 at the 5% cost of capital.

Firm is indifferent between buying A or renting A for $X M per year.

69

Computing EAC

EAC is determined by:EAC = (PV of Costs)/(Annuity Factor,r,n)

n = life of machine, r = discount factor

EAC =

Here: PV = 18.72Mr = .05n = 2 years (life of the machine)

EAC for Machine A = 10.07M

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+− nr)r(1

1r1

PV

70

EAC continued

Buying machine A 3 times over the 6 years is equivalent to paying out 10.07M per year.Notice that:

PV of a 6 year annuity of 10.07M is 51.10M. This is the same as the PV of buying machine A 3 times over 6 years.

71

EAC for Machine B

Machine B:EAC = 22.72 / (Annuity Factor, 5%, 3 years)

= 8.34MMachine B has lower EAC and should be purchased over machine A.

72

Projects with different duration

1. Compare over a common duration (Replacement Cost)

Choose the project with the higher NPV over the common duration. If revenues are same, this is similar to choosing project with lower PV of costs over the common durations

2. Compare Equivalent Annual CostChoose project with lower equivalent annual cost

73

Example: Different DurationFirm is considering two project C and F with the following costs and Cash flows

Project C: NPV (at 11.5%) = 7,165, IRR = 17.5%Project F: NPV (at 11.5%) = 5,391, IRR = 25.2%

Project C Project F0 -40,000 -20,0001 8000 7,0002 14,000 13,0003 13,000 12,0004 12,0005 11,0006 10,000

74

ExampleThough NPV for project C is higher, the analysis is incomplete due to different duration

Method 1: Same Duration Analysis Project F will have to be done twice. The NPV of the project the second time is also 5391 but in year 3. Total NPV5391 + 5391/(1.115)3 = 9280The NPV of project C = 7165Choose Project F.

75

Different DurationOur methods of comparing projects with different durations are valid only if

The firm actually intends to operate the project for the extended time period.

If there is inflation, then cost of replacing the machinery may be different

If replacements incorporate new technology then how do we incorporate these

It is not easy to estimate the lives of projects anyway. Is there a big difference between projects with lives of 8 years or 10 years?

76

Economic life vs. Physical Life

Projects are normally evaluated with the assumption that they will run for their full physical life

This need not be true

Salvage value is the value the firm will get if it terminates the project at that time

Usually the sale of the physical plant and equipment

77

Example

Consider the firm with the following cash flows and salvage values. The discount rate is 10%

How long should you run the project?

Year Cash Flows Salvage Value0 -4,800 48001 2000 30002 2000 16503 1750 0

78

Example cont.

If you run the project for three yearsNPV = -4,800 + 2000/(1.1)1 + 2000/(1.1)2 + 1750/(1.1)3

= -$14.12

If you run the project for two years then NPV = -4,800 + 2000/(1.1)1 + 2000/(1.1)2 + 1650/(1.1)2

= $34.71

If you run the project for one years then NPV = -4,800 + 2000/(1.1)1 + 3000/(1.1)1

= -$254.55

79

Example cont.

The project should be run for two years

2 years is the economic life of the project, as opposed to the physical or engineering life which is 3 years.

80

Investment TimingYou are considering buying a cell phone.It has a life of 2 years. The discount rate is 10% .Other details are as follows:

Year Price PV of gains 0 50 1001 45 1002 33 1003 30 100

Should you buy it today, next year, or even later?

81

Investment Timing

Year Price PV (gains) NPV NPV (Purchase) (today)

0 50 100 50 50 = 501 45 100 55 55/(1.1)1 = 502 33 100 67 67/(1.1)2 = 553 30 100 70 70/(1.1)3 = 53

Buy the phone in year 2

82

Conclusion for NPVNPV tells us what we want to know

Whether the project should be done or notHow much will be gain by doing the project

Other criteria can be useful:Payback, Discounted Payback: Emphasis on LiquidityProfitability Index: Helps when capital rationingIRR: IntuitiveMust know the possible pitfalls of these methods.

If conflict , always use NPV rule.

1

Discounted Cash FlowsOct 3rd



2

Announcements

The problem set is due on Oct 10th.

We will discuss it on Oct 10th. If there are any other questions we will also discuss that on Oct 10th

The Midterm is on Oct 17th. It will be mostly multiple choice problemsThere will be no class after the exam

3

Introduction To DCF

We have learntThe mechanics of calculating present valuesThe different criteria used to decide whether to do projects or not

Now we will studyHow to estimate Cash flowsWhat to include and what to excludeDetermine Cash Flows from accounting numbersUse the above cash flows to value a whole firm

4

Cash Flows from Financial Statements

We have been given cash flows till nowNow we will study how to come up with cash flows from the financial statements of firms

We need cash flows not profitsWhy?

5

Our Typical Problem

Firm buys machine A. The cash flows from the machine are:

Year Cash Flows0 -30M1 +16M2 +16M3 + 16M

Let us understand how we will put this transaction on our books….account for it.

What is Depreciation?

6

DepreciationInitial Value = $ 30,000Life of the equipment = 3 yearsStraight Line Depreciation = $30,000/3 = 10,000

Year 1 Year 2 Year 3Begin value 30k 20k 10k Depreciation 10k 10k 10kEnd value 20k 10k 0k

7

Depreciation

This is a non cash expenseIt is there for accounting.The corresponding cash expense was undertaken at the beginning of the projectOther non cash expenses are amortization.

As it is not a cash outflow, and accounting profits deduct this we have to

Add this back to get cash flows

8

Example

A firm spends $30,000 in cash to purchase a machine today that it plans to depreciate on a straight line basis over three years. With this machine, the firm can produce 10,000 units that cost $1 to make and sell for $3. Taxes are 40%

What is the income statement?What are the cash flows from this project?

9

Example: Income Statement

Sales 30,000 less: COGS 10,000Gross Profits 20,000 Less: Depreciation 10,000EBIT 10,000 Less: Taxes (40%) 4,000Net Income 6,000

How much cash does this project generate every year?

Net Income + Depreciation: $6,000 + $10,000 = $16,000

10

Could we do the following?

Sales 30,000 less: COGS 10,000Gross Profits 20,000 Less: DepreciationEBIT 20,000 Less: Taxes (40%) 8,000Net Income 12,000

What is the problem with this?

11

Depreciation Tax Shield

Depreciation is deductible to calculate taxes payableWith deprecation: Taxes were 4000If we do not deduct depreciation: Taxes are 8000The 4,000 is the difference we see

If we ignore depreciation in calculating net income, then need to add the above benefit on the sideDepreciation tax shield = DT

D = depreciation expenseT = marginal tax rate10,000*.4 = 4000 every yearCash flows are: 12000 + 4000 = 16,000

12

Example: Income Statement

Sales 30,000 less: COGS 10,000Gross Profits 20,000 Less: Taxes (40%) 8,000After Tax Income (1) 12,000

Depreciation 10,000Depreciation Tax Shield (Dxt) (2) 4000

Total Cash Flow ( 1 + 2 ) 16,000

13

Cash Flow

Cash Flows from the project

Year 0 Year 1 Year 2 Year 3Cash Flow from Operations $16,000 $16,000 $16,000Cash Flow from Investment -30,000Total Cash Flows from Projec -30,000 $16,000 $16,000 $16,000

14

Computing Depreciation

Straight-line depreciationD = Initial cost / number of yearsVery few assets are depreciated straight-line for tax purposes

MACRS (Modified Accelerated Cost Recovery System)Need to know which asset class is appropriate for tax purposesMultiply percentage given in table by the initial costDepreciate to zero

15

Example: Depreciation

You purchase equipment for $110,000.Other information

The company’s marginal tax rate is 40%.Life is six yearsWhat is the depreciation expense each year

With straight line depreciationWith MACRS?

What is the book value of the asset at the end of six years?

16

Example: Depreciation


The company’s marginal tax rate is 40%.Life is six yearsWhat is the depreciation expense each year

With straight line depreciationWith MACRS?

What is the book value of the asset at the end of six years?

With straight-line depreciationAnnual Depreciation = 110,000 / 6 = 18,333.33Book Value in year 6 = 110,000 – 6(18,333.33) = 0

17

MACRS

Was put forth by the Tax Reform Act of 1986This sets out annual depreciation deductions for various classes of assets

Automobiles for business are in the three-year classComputer equipment is part of the five year classMost manufacturing equipment is in the seven year class

18

Example: Three-year MACRS

Year MACRS percent

Dep.

1 .3333 .3333(110,000) = 36,663

2 .4444 .4444(110,000) = 48,884

3 .1482 .1482(110,000) = 16,302

4 .0741 .0741(110,000) = 8,151

BV in year 6 = 110,000 – 36,663 –48,884 – 16,302 –8,151 = 0

19

Example: 7-Year MACRSYear MACRS

PercentDep.

1 0.1429

2 0.24493 0.1749

4 0.12495 0.0893

678

0.08930.08930.0445

What is the book value at the end of six years

20

Example: 7-Year MACRS

Year MACRS Percent

Dep.

1 .1429 .1429(110,000) = 15,719

2 .2449 .2449(110,000) = 26,939

3 .1749 .1749(110,000) = 19,239

4 .1249 .1249(110,000) = 13,739

5 .0893 .0893(110,000) = 9,823

6 .0893 .0893(110,000) = 9,823

BV in year 6 = 110,000 – 15,719 –26,939 – 19,239 –13,739 – 9,823 –9,823 = 14,718

21

Salvage Value

This is the value the asset can be sold for at the end of the project. There are two effectsThe amount of depreciation

Straight-line depreciationD = (Initial cost – salvage) / number of years

MACRS: It does not effect the depreciation

If the salvage value is different from the book value of the asset, then there is a tax effect

22

Salvage Value and Tax EffectThis is cash inflow, it should be included in cash flowsHowever, the gains on selling the machine are taxed by the government. These taxes paid are a cash outflow. We have to subtract the taxes from cash flows.

Gain from selling the machine = Value at which the machine is sold – book value of machineBook value = initial cost – accumulated depreciationTaxes on the profits: T(salvage – book value).

After-tax impact on cash flow = salvage – T(salvage –book value)

23

Continuing Example


The company’s marginal tax rate is 40%.Life is six yearsSalvage Value at the end of six years is $ 17,000With both depreciation methods (3yr and 7 yr MACRS)

What is the annual depreciation expense?What is the after tax salvage value?

24

Example: Straight-line Depreciation

With straight-line depreciationAnnual Depreciation = (110,000 – 17,000) / 6 = 15,500Book Value in year 6 = 110,000 – 6(15,500) = 17,000Salvage value = 17,000Profit/ Loss on selling the machine = 0After-tax salvage = 17,000 - .4(17,000 – 17,000) = 17,000

25

Example: Three-year MACRS

Year MACRS percent

Dep.

1 .3333 .3333(110,000) = 36,663

2 .4444 .4444(110,000) = 48,884

3 .1482 .1482(110,000) = 16,302

4 .0741 .0741(110,000) = 8,151

BV in year 6 = 110,000 – 36,663 –48,884 – 16,302 –8,151 = 0

After-tax salvage = 17,000 -.4(17,000 – 0) = $10,200

26

Example: 7-Year MACRS

Year MACRS Percent

Dep.

1 .1429 .1429(110,000) = 15,719

2 .2449 .2449(110,000) = 26,939

3 .1749 .1749(110,000) = 19,239

4 .1249 .1249(110,000) = 13,739

5 .0893 .0893(110,000) = 9,823

6 .0893 .0893(110,000) = 9,823

BV in year 6 = 110,000 – 15,719 –26,939 – 19,239 –13,739 – 9,823 –9,823 = 14,718

After-tax salvage = 17,000 -.4(17,000 –14,718) = 16,087.20

27

Net Working Capital

Net working Capital = A/c Receivables + Inventory – A/c Payables

Accounts Receivables: Having receivables means that the company has made the sale but has yet to collect the money from the purchaser.

Accounts Payables: Money that is owed to suppliers. The company has got its raw materials but not paid for them yet

Inventory: The value of the goods not sold yet

28

More on NWC

Why do we have to consider changes in NWC separately?

GAAP requires that sales be recorded on the income statement when made, not when cash is received

GAAP also requires that we record cost of goods sold when the corresponding sales are made, regardless of whether we have actually paid our suppliers yet

Finally, we have to buy inventory to support sales although we haven’t collected cash yet

29

Working Capital

The difference in timing between accounts receivable and payable plus inventory costs is accounted for in changes of Working Capital.This working capital can be absorbed as a cash inflow when the project is concluded.

30

More on Working CapitalNet working Capital = A/c Receivables + Inventory – A/c

payables

Year 0 Year 1 Year 2 Year 3Sales 10M 12M 8MCost 8M 9M 7MNI 2M 3M 1M

A/C Receivable 800k 1.2M 0A/c Payable 300k 500k 0Inventory 2M 2.5M 0NWC 2.5M 3.2M 0∆NWC -2.5M -0.7M +3.2M

31

Why does WC change over time?

What would cause working capital to increase?

If inventories are increasingIf accounts receivable are increasesIf accounts payable are decreasing

What would cause working capital to decrease?

If inventories are decreasingIf accounts receivable are decreasingIf accounts payable are increasing

32

Pro Forma Statements and Cash Flows

Cash inflows or outflows can come fromFrom operations: If you are a manufacturing firm, this is the cash from selling the machinery producedFrom the investments required to run the business

Cash flow from assets: For e.g. buying machinery and equipmentInvestment in Working Capital: This is the cash buffer you need to run the business

33

Relevant Cash Flows

The cash flows that should be included in a capital budgeting analysis are those that will only occur if the project is acceptedThese cash flows are called incremental cash flowsThe stand-alone principle allows us to analyze each project in isolation from the firm simply by focusing on incremental cash flows

34

Asking the Right Question

You should always ask yourself “Will this cash flow occur ONLY if we accept the project?”

If the answer is “yes”, it should be included in the analysis because it is incrementalIf the answer is “no”, it should not be included in the analysis because it will occur anywayIf the answer is “part of it”, then we should include the part that occurs because of the project

35

Sunk Costs

These are costs that have been incurred. For e.g., Smart Inc is considering launching a

new line of fashion clothing. It paid $30,000 to Greatads Inc. to study the success of this fashion line. In deciding whether to launch this line, should Smart Inc include the payment to GreatadsInc?

36

Opportunity Costs

These are not out-of-pocket expenses. This requires us to give up a benefit.An advantage Smart Inc has in launching its new fashion line is a fortunate real estate deal. 10 years back it bought office space in the garment district in mid town Manhattan for 1M dollar. Now it is worth 10M dollar. As office costs will only be 1M the project is likely to be positive NPVDo you agree with Smart Inc.?

37

Side Effects

These are effects on other areas of the firm.The new fashion line with be integrated with the existing accessories which Smart Inc sells. Smart Inc. expects the sales of its accessories to increase by 25% with the launch of its new line. This is another advantage associated with the new line of clothing.Do you agree with Smart Inc.?

38

Summary: Relevant Cash Flows

Sunk costsCosts already incurredDo not included

Opportunity costsCost of lost options. Included

Side effectsPositive side effects – benefits to other projectsNegative side effects – costs to other projectsIncluded

39

Example of Relevant Cash Flows

Which of these cash flows should be included

1. A reduction in the sale of companies other products caused by the investment

2. An expense on equipment that will be made only if the project is undertaken

3. Cost of research and development undertaken in the past three years to develop the product. You are considering whether to manufacture the product or not

40

Example of Relevant CF cont.

Which of these cash flows should be included

Annual depreciation expense from the investment in the projectSome of the people working on the project will be from other departments in the firm. The salary and medical costs associated with these people.

41

Another Example

Golfco is considering the following projectIntroducing a brightly colored ballThey spend $250k to investigate the potential of selling these ballsIf they decide to manufacture and sell this ball, they will require an investment of $100K in equipment to manufacture a new line of balls. The cash flow from selling the machine at the end of year 5 is $ 21.76They plan to use a building owned by the firm which is currently vacant. The building is valued at $150K and expected to keep its value in five yearsThe tax rate = 34%, required rate of return = 10%Years of project = 5 Years

42

Income Statement

Year 0 Year 1 Year 2 Year 3 Year 4 Year 5Revenue 100 163.2 249.72 212.2 129.9Costs 50 88 145.2 133.1 87.84Depreciation 20 32 19.2 11.52 11.52EBIT 30 43.2 85.32 67.58 30.54Taxes @ 34% 10.2 14.688 29.0088 22.9772 10.3836Net inocme 19.8 28.512 56.3112 44.6028 20.1564

NWC 10 10 16.32 24.97 21.22

43

Salvage Value

Accumulated Depreciation: 20 + 32 + 19.2 + 11.52 + 11.52 = 94.24Book Value at the end of five years

100 – 94.24 = 5.76

Gain from selling the machine: 21.76 – 5.76 = 16Taxes from selling it = 0.34 x 16 = 5.44After Tax cash flow

21.76 – 5.44 = 16.32

44

Cash Flows

EBIT 30.00 43.20 85.32 67.58 30.54Taxes @ 34% 10.20 14.69 29.01 22.98 10.38Depreciation 20.00 32.00 19.20 11.52 11.52Operting CF (1) 39.80 60.51 75.51 56.12 31.68

NWC 10 10.00 16.32 24.97 21.22Change in NWC (2) -10 0.00 -6.32 -8.65 3.75 21.22

Equipment (3) -100 16.32Building (4) -150 150.00Test Marketing CF ( 1+2+3+4) -260 39.80 54.19 66.86 59.87 219.22

45

NPV

NPV at 10% = $ 51kGolfco should accept the project

Note:Did not include test marketing costs. This is sunk costsIncluded the building cost: This is the opportunity costUsed Cash flows not accounting earnings / net income

46

What if?

What if the firm is not able to sell the number of Balls it plans to?What if there is a lot of competition and the firm has to reduce its price to sell?What if the are costs are higher than anticipated? Before accepting the project it is a good idea to get a sense of how bad or good things might be?

47

Scenario Analysis

One way to handle this is to construct the pessimistic and optimistic caseConsider the following example. After analyzing the various possibilities you come up with the following

Base Case Pessimistic OptimisticSales 6,000 5,500 6,500Price $80 $75 $85Variable. Cost $60 $62 $58Fixed Cost $50,000 $55,000 $45,000

48

Scenario Analysis

Base Case Pessimistic Optimistic

Net Income $19,800 -15,510 59,730Cash Flow $59,800 24,490 99,730NPV $15,567 -111,719 159,504IRR 15.1% -14.4% 40.9%

This can help us get a sense of what kind of disasters can happenIt does not give us a “rule” for whether to take the project or not

49

Sensitivity Analysis

There can be a problems in forecastingUnits soldFuture Price at which the units are soldThe variable costsThe fixed costs

Sensitivity analysis helps to figure out which errors are likely to have the biggest impact on the decisionThe firm can then spend more resources to examine in detail this “sensitive” variable

50


We keep all the variables, except one fixed.We calculate NPV for different values of the one variable which is not fixedWith this we get a sense of how much this variable will effect the NPV

51


Let us consider how sensitive our analysis is to the number of units sold

Units Sold Cash Flow NPV5000 46,000 -32,0175400 51,880 -12,2986000 59,800 15,5666600 67,720 44,1157000 73,000 63,149

52


-40,000

-20,000

0

20,000

40,000

60,000

80,000

5000 5400 6000 6600 7000

Units Sold

NPV NPV

53


Unit Costs Cash Flow NPV70 20,200 -127,18466 36,040 -70,08460 59,800 15,56654 83,560 101,21550 99,400 158,315

54


Decrease in units sold by 10% (to 5400) reduce NPV by 16.6%

Increase in unit costs by 10% (to $66) reduces NPV by more than 500%. NPV falls from $15,566 to -$70,084.

55

Example: Replacement Problem

Original MachineInitial cost = 100,000Annual depreciation = 9000Purchased 5 years agoBook Value = 55,000Salvage today = 65,000Salvage in 5 years = 10,000

New MachineInitial cost = 150,0005-year lifeSalvage in 5 years = 0Cost savings = 50,000 per year3-year MACRS depreciation

Required return = 10%Tax rate = 40%

1

Discounted Cash FlowsOct 10th



2

DCF: Discounted Cash FlowsThree sources of cash flows

Operating cash flowsDepreciation: Add Depreciation to Net income

Straight line MACRS: 3 year, 5 year or 7 year schedule

Capital ExpendituresUpfront ExpenseSalvage Value: SV – Tax Rate (SV – BV)

New Working CapitalChange in NWC

Add all three sources to get total cash flowsRemember to exclude cash flows that were not relevant – for e.g. sunk costs

3

Example: Operating CF

ABC Inc. has the gross profits of $4000, $5000 and $6000 for the three years of the project. Its annual depreciation is $ 5000 and tax rate is 40%. What are the operating cash flows for the three years?

4

Example: Operating CF

1 2 3Gross Profits 4000 5000 6000Depreciation (1) 5000 5000 5000Earnings Before Interest and Taxes (EBIT) -1000 0 1000Taxes -400 0 400Net Income (2) -600 0 600

Operating Cash Flows (1+2) 4400 5000 5600

5

Example: Salvage Value

Continuing with the previous example ABC Inc can sell its plant at the end of three years for $3000. It had bought the plant for $20,000. What are the cash flows from selling the plant?

6

Example: Salvage Value

Accumulated Depreciation: $5000 + $5000 + $5000 = $ 15,000Book Value of plant at time of selling

= $20,000 - $ 15,000 = $ 5000Profit/Loss on selling is = 3000 – 5000 = -2000Cash Flows = 3000 - 40%(-2000) = $3800

7

Example: Replacement Problem

Original MachineInitial cost = 100,000Annual depreciation = 9000Purchased 5 years agoBook Value = 55,000Salvage today = 65,000Salvage in 5 years = 10,000

New MachineInitial cost = 150,0005-year lifeSalvage in 5 years = 0Cost savings = 50,000 per year3-year MACRS depreciation

Required return = 10%Tax rate = 40%

8

Example

We have two optionsOption 1: Buy the new machine and sell the oldOption 2: Keep the old machine

Two ways to tackle the problemFind the NPV of Option 1 and of Option 2 and choose the one which is higherWe will ask ourselves, what is different between situation 1 and 2 i.e., what is incremental. Then evaluate those cash flows

9

Buy the New and Sell the Old

Capital Expenditure Cost of new machine = 150,000

After-tax salvage on old machine = 65,000 - .4(65,000 –55,000) = 61,000 Net capital spending = 150,000 – 61,000 = 89,000

Year 5: No Salvage valueOperating Cash Flows

Savings of 50,000 every yearDepreciation

10

Cash Flows of Option 1

0 1 2 3 4 5Capital Expenditure -150000Salvage Value 61000

Cost Saving 50000 50000 50000 50000 50000MACRS 0.3333 0.4444 0.1482 0.0741Depreciation 49995 66660 22230 11115EBIT 5 -16660 27770 38885 50000Taxes 2 -6664 11108 15554 20000Net Income 3 -9996 16662 23331 30000

Operating Cash Flows 49998 56664 38892 34446 30000

Total CF -89000 49998 56664 38892 34446 30000PV at 10% -89000 45452.73 46829.75 29220.14 23527.08 18627.64NPV 74657.34

11

Option 2: Keep the Old

Capital ExpenditureYear 0: No expenseYear 5: After-tax salvage on old machine

=10,000 - .4(10,000 – 10,000) = 10,000Operating Cash flows

No Cost SavingsDepreciation of 9000 every year

12

Cash Flows for Option 2

0 1 2 3 4 5Capital Spending Salvage Value 10000

Cost Saving 0 0 0 0 0Dep 9000 9000 9000 9000 9000EBIT -9000 -9000 -9000 -9000 -9000Taxes -3600 -3600 -3600 -3600 -3600Net Income -5400 -5400 -5400 -5400 -5400

Operating Cash Flows 3600 3600 3600 3600 3600Total Cash Flows 3600 3600 3600 3600 13600PV at 10% 3272.727 2975.207 2704.733 2458.848 8444.53NPV 19856.05

13

Incremental Cash Flows

To get to incremental cash flows we will compare two situations

Situation 1: Buy the new machine and sell the oldSituation 2: Keep the old machine

We will ask ourselves, what is different between situation 1 and 2 i.e., what is incrementalThen we will evaluate those cash flows

14

Operating Cash Flows

Things that are different between Situation1 and Situation2

Cost saving of 50,000 in situation 1 relative to situation 2Depreciation

Situation 1: 3 year MACRS (33.33%, 44.44%, 14.82%, 7.41%) Situation 2: annual deprecation of 9000

Capital Expenditure and Salvage Value

15


1 2 3 4 50.3333 0.4444 0.1482 0.0741

Dep: Situation 1 49,995 66,660 22,230 11,115 0Dep: Situation 2 9,000 9,000 9,000 9,000 9,000Incremental Deprecation (a) 40,995 57,660 13,230 2,115 -9,000

Cost Savings 50000 50000 50000 50000 50000EBIT 9,005 -7,660 36,770 47,885 59,000 Less Taxes at 40% 3602 -3064 14708 19154 23600Net Income (b) 5,403 -4,596 22,062 28,731 35,400

Operating Cash Flows 46,398 53,064 35,292 30,846 26,400

16

Incremental Net Capital Spending

Situation 1: Buy new and sell oldYear 0

Cost of new machine = 150,000After-tax salvage on old machine = 65,000 - .4(65,000 – 55,000) = 61,000 Net capital spending = 150,000 – 61,000 = 89,000

Year 5: No Salvage value

Situation 2: Keep the old machineYear 0: No expenseYear 5: After-tax salvage on old machine = 10,000 - .4(10,000 –10,000) = 10,000

Incremental Net Capital SpendingYear 0: -89,000 Year 5: 0 – 10,000 = -10,000

17

Cash Flow From Assets

Year 0 1 2 3 4 5

OCF 46,398 53,064 35,292 30,846 26,400

NCS -89,000 -10,000

∆ In NWC

0 0

CFFA -89,000 46,398 53,064 35,292 30,846 16,400

18

Analyzing the Cash Flows

Now that we have the cash flows, we can compute the NPV

Compute NPV (at 10%) = 54801.10Should the company replace the equipment?

19

Lemon Juice ProjectEquipment costs 240,000. Depreciation is 3 years MACRS with 33%, 45%, 15% and 7%Salvage value is $25,000 at end of life of 4 years.Average inventory is $25,000 and accounts payable is $ 5,000.Unit sales are 100,000 cans per year at $2 a can. Cash operating costs are expected to be 60% of dollar sales.Tax rate is 40%Cost of capital is 10%

20


1 2 3 4Revenue (100,000 x 2) 200,000 200,000 200,000 200,000Cost of Goods Sold 120000 120000 120000 120000Gross Profit 80,000 80,000 80,000 80,000 Depreciation 79200 108000 36000 16800EBIT 800 -28,000 44,000 63,200 Taxes at 40% 320 -11200 17600 25280Net Income 480 -16,800 26,400 37,920

Operating Cash Flows 79,680 91,200 62,400 54,720

21

Other Cash Flows

Fixed Assets -240000Salvage Value 25000Book Value 0Taxes on Gain 10000Cash flow from Sale 15000Total Cash Flow -240000 15000

Net Working Capital 20,000 20,000 20,000 20,000Changes in NWC -20,000 20000

22

NPV Analysis

0 1 2 3 4

Operating Cash Flows 79,680 91,200 62,400 54,720Fixed Assets -240000Cash from Salvage Value 15000Changes in NWC -20,000 20000

Total Cash Flow -260,000 79,680 91,200 62,400 89,720NPV -4030

Risk and ReturnOct 24th



Dollar Returns

Total dollar return = income from investment + capital gain (loss) due to change in priceExample:

You bought a bond for $950 1 year ago. You have received two coupons of $30 each. You can sell the bond for $975 today.What is your total dollar return?What is your percentage return?

Dollar Returns

Total dollar return = income from investment + capital gain (loss) due to change in priceExample:

You bought a bond for $950 1 year ago. You have received two coupons of $30 each. You can sell the bond for $975 today. What is your total dollar return?

Income = 30 + 30 = 60Capital gain = 975 – 950 = 25Total dollar return = 60 + 25 = $85

Percentage Returns

It is generally more intuitive to think in terms of percentages than dollar returnsIn last example:

Percentage return = ($Return)/Price PaidPercentage return = 85/950 = .0895 = 8.95%

Example – Calculating Returns

You bought a stock for $35 and you received dividends of $1.25. The stock is now selling for $40.

What is your dollar return?Dollar return = 1.25 + (40 – 35) = $6.25

What is your percentage return?Percentage return = $6.25/$35 = .1786 or 17.86%

Percentage Return for Stocks

Dividend yield = dividend / price paidCapital gains yield = (selling price – price paid) / price paidTotal percentage return = dividend yield + capital gains yieldIn above example:

Dividend yield = 1.25 / 35 = 3.57%Capital gains yield = (40 – 35) / 35 = 14.29%Total percentage return = 3.57 + 14.29 = 17.86%

Figure 12.4

Average Returns (1925-2000)

Investment Average ReturnLarge stocks 13.0%

Small Stocks 17.3%

Long-term Corporate Bonds 6.0%

Long-term Government Bonds 5.7%

U.S. Treasury Bills 3.9%

Inflation 3.2%

Risk Premiums

The “extra” return earned for taking on risk

Treasury bills are considered to be risk-free

The risk premium is the return over and above the risk-free rate

Historical Risk Premiums

Large stocks: 13.0 – 3.9 = 9.1%Small stocks: 17.3 – 3.9 = 13.4%Long-term corporate bonds: 6.0 – 3.9 = 2.1%Long-term government bonds: 5.7 – 3.9 = 1.8%

Risk, Return and Financial Markets

Lesson from capital market historyThere is a reward for bearing riskThe greater the risk the greater the required returnThis is called the risk-return trade-off

How do we measure Risk? What is the price of Risk?

Frequency distribution, Large-Stocks

Variance and Standard Deviation

Variance and standard deviation measure the volatility of asset returnsThe greater the volatility the greater the uncertaintyHistorical variance = sum of squared deviations from the mean / (number of observations – 1)Standard deviation = square root of the variance

ExampleConsider a stock with the following returns for four years

Year Return1 0.152 0.093 0.064 0.12

Average Returns = 0.15 + 0.09 + 0.06 + 0.12 = 0.105

4

Example – Variance and Standard Deviation

Year Actual Return

Average Return

Deviation from the Mean

Squared Deviation

1 .15 .105 .045 .002025

2 .09 .105 -.015 .000225

3 .06 .105 -.045 .002025

4 .12 .105 .015 .000225

Totals .42 .00 .0045

Variance = .0045 / (4-1) = .0015 Standard Deviation = .03873

Figures

Comments

Another Lesson from HistoryOn average you are rewarded handsomely for bearing risk. However, in any one year there is a significant chance of loss in value

Do we have a measure or Risk?Use Standard deviation to measure riskHigher the standard deviation the greater is the risk

Portfolios

A portfolio is a collection of assetsThe risk-return trade-off for a portfolio is measured by the portfolio expected return and standard deviation, just as with individual assets

Portfolio Weights

Investors hold more than one stock.Consider a portfolio of 2 stocks.$50 in stock A and $150 in stock BPortfolio weight:wA = $50/($200) = .25wB = $150/($200) = .75

Note: weights need to sum to 1.

Expected Return of Portfolio

The expected return on a portfolio is just the weighted average of the returns of each stock in the portfolio:

E[r] = w1 µ1 + w2 µ2

wi = portfolio weight of asset iµi = expected return of asset i

Ex: Portfolio Expected Return

You have $3000 to invest today. You decide to buy $2000 of stock A, $500 of stock B, and $500 of stock C.

If E[rA] = 10%, E[rB] = 6%, E[rC] = 12%What is the expected return of your portfolio?

Expected Return

First determine the portfolio weights.wA = 2000/3000 = 2/3wB = 500/3000 = 1/6wC = 500/3000 = 1/6

E[rPortfolio] = (2/3)(.10)+(1/6)(.06)+(1/6)(.12) = .0967 or 9.67%

Example: Umbrella Maker

Consider an umbrella maker, if it could either rain or shineState Prob. Umbrella MakerRain .50 70%Shine .50 -20%

Expected Return = .5(70%) + .5(*-20%) = 25%

Variance and Standard deviation

Variance = Expected squared deviation from the meanStandard deviation = SQRT(Var.)

Prob. Return Deviation Prob.X Squared Deviation from mean

0.5 .70 .70-.25 = .45 .5 (.45)2 = 0.10130.5 -.20 -.20-.25 = -.45 .5(-.45)2 = 0.1013

Sum = 0.2025

Var. = .2025 Standard Deviation (STD) = (.2025)1/2 =0.45 or 45%

Example: Ice Cream Maker

Consider an ice cream maker, if it could either rain or shineState Prob. Ice Cream MakerRain .50 -10%Shine .50 50%

Expected Return = .5(-10%) + .5(*50%) = 20%

Variance and Standard deviation



0.5 -.10 -.10-.20 = .30 .5(-.30)2 = 0.0450.5 .50 .50-.20 = .30 .5 (.30)2 = 0.045

Sum = 0.09

Var. = .09Standard Deviation (STD) = (.09)1/2 =0.30 or 30%

Example: Portfolio

If you own both the umbrella and ice cream maker (50% of money in each)

State Prob. Umbrella Ice Cream PortfolioRain .5 70% -10% 30%Shine .5 -20% 50% 15%

Expected Return (rain) = .5(30%) + .5(15%) = 22.5%

Standard deviation of Portfolio



0.5 .30 .30-.225 = .075 .5(.075)2 = 0.00280.5 .15 .15-.225 = -.075 .5 (-.075)2 = 0.0028

Sum = 0.0056

Var. = .0056Standard Deviation (STD) = (.0056)1/2 =0.075 or 7.5%

Summary

Return Standard DevUmbrella 25% 45%Ice Cream 20% 30%Portfolio 22.5% 7.5%

So what happened?

Portfolio Variance

Portfolio variance is given by

Where are the portfolio weights

are the variance of stock A and B

is the covariance between stock A and B

bababbaap wwwwVAR ,2222 2 σσσ ++=

ba ww ,22 , ba σσ

ba,σ

Covariance

The variance of the portfolio depends onThe variance of individual stocksThe covariance between the two stocks

If the covariance is negativeThen it reduces the variance of the portfolio.

Weighted average of STD

If we used the weighted average of the standard deviation: (DON’T DO THIS)

Negative Covariance

Umbrella Maker

Ice Cream Maker

Positive Covariance

The Principle of Diversification

Portfolio diversification is the investment in several different asset classes or sectorsDiversification can substantially reduce the variability of returns without an equivalent reduction in expected returnsThis reduction in risk arises because worse than expected returns from one asset are offset by better than expected returns from anotherHowever, there is a minimum level of risk that cannot be diversified away and that is the systematic portion

Diversification and Risk

Diversification and Risk

Unsystematic Risk

The risk that can be eliminated by combining assets into a portfolio.

It is also called unique, asset-specific risk or idiosyncratic risk

Risk factors that affect a limited number of assets. Includes such things as labor strikes, part shortages, etc.

Systematic Risk

The risk that cannot be eliminated

Also known as non-diversifiable risk or market risk

Risk factors that affect a large number of assets . Includes such things as changes in GDP, inflation, interest rates, etc.

Total Risk

Total risk = systematic risk + unsystematic riskThe standard deviation of returns is a measure of total riskFor well diversified portfolios, unsystematic risk is very smallConsequently, the total risk for a diversified portfolio is essentially equivalent to the systematic risk

Systematic Risk Principle

There is a reward for bearing riskThere is not a reward for bearing risk unnecessarilyThe expected return on a risky asset depends only on that asset’s systematic risk since unsystematic risk can be diversified awayThe measure of risk for any stock is the level of systematic risk it bears

Measuring Systematic Risk

How do we measure systematic risk?We use the sensitivity of a stock’s return to macroeconomic events, i.e., the sensitivity of the stock return to the fluctuations in the returns of the market portfolio.The market portfolio should contain all the assets in the economy:

Stock, Bonds, Foreign securities, Real EstateIn reality, most people use stock portfolio (S&P 500 index)

The measure of sensitivity is called the Beta of a stock.

Beta

What does beta tell us?A beta of 1 implies the asset has the same systematic risk as the overall marketA beta < 1 implies the asset has less systematic risk than the overall marketA beta > 1 implies the asset has more systematic risk than the overall market

Stocks with Beta greater than one are risky. They will earn a higher return.

Example of Betas

Firm BetaExxon .75IBM .95General Motors 1.05Harley Davidson 1.20AOL – Time Warner 1.75

Total versus Systematic Risk

Consider the following information:Standard Deviation Beta

Security C 20% 1.25Security K 30% 0.95

Which security has more total risk?Which security has more systematic risk?Which security should have the higher expected return?

Example: Portfolio Betas

Consider this example with the following four securitiesSecurity Weight BetaDCLK 0.133 3.69KO 0.2 0.64INTC 0.167 1.64KEI 0.5 1.79

What is the portfolio beta?.133(3.69) + .2(.64) + .167(1.64) + .5(1.79) = 1.78

Example

You put $300 in the equity of firm A. The expected return for firm A is 12% and its beta is 1.1. The remaining money of $ 500 you invest in security B. B has expected return of 15% and a beta of 1.4. What is the portfolio’s expected return?What is the portfolio’s beta?

Example

Total investment = 300 + 500 = 800Wa = 300/ 800 = 0.375Wb = 500/ 800 = 0.625

Expected return = 0.375x12% + 0.625 x 15%= 13.875%

Beta of portfolio = .375 x 1.1 + 0.625 x 1.4= 1.2875

Next…

Measure of Risk we will useBeta

What is the Price of Risk?Risk premium = expected return – risk-free rateHigher risk implies higher return. So higher the beta, the greater should be the risk premium. Relation between beta and returns formalized by Capital Asset Pricing Model

The Capital Asset Pricing Model (CAPM)

The capital asset pricing model defines the relationship between risk and returnE(RA) = Rf + βA(E(RM) – Rf)If we know an asset’s systematic risk, we can use the CAPM to determine its expected returnThis is true whether we are talking about financial assets or physical assets

Factors Affecting Expected Return

Pure time value of money – measured by the risk-free rateReward for bearing systematic risk –measured by the market risk premiumAmount of systematic risk – measured by beta

Example - CAPM

Consider the betas for each of the assets given earlier. If the risk-free rate is 4.5% and the market risk premium is 8.5%, what is the expected return for each?

Security Beta Expected ReturnDCLK 3.69 4.5 + 3.69(8.5) = 35.865%KO .64 4.5 + .64(8.5) = 9.940%INTC 1.64 4.5 + 1.64(8.5) = 18.440%KEI 1.79 4.5 + 1.79(8.5) = 19.715%


The capital asset pricing model defines the relationship between risk and returnE(RA) = Rf + βA(E(RM) – Rf)

βA is the beta and captures riskE(RM) – Rf is the market risk premium

and captures the price of riskRf is the risk free rate todayE(RA) is the expected rate or return for A

Inputs to the CAPM

Risk Free Rate: Most people will use the short term treasury bill rateMarket Risk Premium;

Historical average return earned by the market portfolio over the risk free rateProblem is that using the past to predict the futureIs sensitive to how long back the data goes

For e.g. in the depression of 1927 included or not?This number is manipulated a lot

In 2000, prestigious analysts used market risk premiums from 2% to 8%

Beta

Beta can be estimated from past return dataUse regression analysis to see how much the equity returns change when the market changesThese are also a function of how long back you go to get returnsAs beta could change over time

This suggests you take a shorter time period of say 3 yearsThis introduces noise in the estimation

Betas can be obtained from several sources: Bloomberg, Value Line

Advantages and Disadvantages of CAPM

AdvantagesExplicitly adjusts for systematic riskApplicable to all companies, as long as we can compute beta

DisadvantagesHave to estimate the expected market risk premium, which does vary over timeHave to estimate beta, which also varies over timeWe are relying on the past to predict the future, which is not always reliable

Does the CAPM work?

Do stock returns essentially fit in with CAPM in real life? Several studies over the years have tried to test this:

There is broad support for CAPM.Risk is based on covariance rather than varianceThere is little reward for unsystematic risk

Does the CAPM work?

However,The Security market line appears to be too flat. Characteristics other than the beta appear to explain stock returns

Small companies have higher returns than large companies (called the size effect)Stock with high ratio of book-to-market have higher returns than those with low book-to-market.Stock Returns are higher in January (January effect)Stock that have higher returns in the past continue to do well (momentum effect)

The CAPM Approach

The CAPM gives usExpected Return on the StockThis is the same as the required rate of returnThis is the same as the cost of equity capital to the firm

We can use the CAPM to get required returns on Bonds or any physical asset

Example – CAPM

Consider the following firm

Equity beta = 1.02 Current risk-free rate = 1.8%Expected market risk premium = 9.1%What is your cost of equity capital?

RE = 1.8 + 1.02(9.1) = 11.08%

Expected versus Unexpected Returns

CAPM gives you the Expected Returns These are returns which you expect to earn when you buyThis is not the realized returnsRealized Returns are what you actually make

Realized Return = expected return (CAPM) + unexpected return

Unexpected Returns:At any point in time, the unexpected return can be either positive or negativeOver time, the average of the unexpected component is zero

Expected versus Unexpected Returns

Can Expected Returns be negative?

Can Realized Returns be negative?

When you buy a stock do you know its expected return? Realized Return?

Example 1

Stock ABC has a beta of 1.1 and an expected return of 15%. The risk-free asset has an return of 5%.

What is the expected return on a portfolio that is equally invested in stock ABC and the risk free asset? What is the portfolio’s beta?

Example 1

Equally Invested:50% in stock ABC50% in the risk free asset

ReturnsExpected Return for ABC is 15%Expected Return for risk free asset is 5%

E[rp] = .5(15%) + .5(5%) = 10%Beta of portfolio = 0.5(1.1) + 0.5(0) = 0.55

Example 2Stock ABC has a beta of 1.1 and an expected return of 15%. The risk-free asset has an return of 5%.

If a portfolio of stock ABC and the risk free asset has a beta of 0.6, what are the portfolio weights?

0.6 = x(1.1) + (1-x)(0)X = 0.6/1.1 = 0.5455

Stock ABC accounts for 54.55% of the portfolio and the riskfree asset accounts for 45.45%


What is the market risk premium?Information for ABC stock:

Beta of 1.1Expected return of 15%Using the CAPM

15% = 5% + 1.1(market risk premium)Market risk premium = 10%/1.1 = 9.09%

Example 4Stock ABC has a beta of 1.1 and an expected return of 15%. The risk-free asset has an return of 5%. The market risk premium is 9.09%

If a portfolio of stock ABC and the risk free asset has an expected return of 9%, what is the portfolio’s beta?

Using the CAPM 9% = 5% + Beta(9.09)Beta = 0.44


If a portfolio of stock ABC and the risk free asset has a beta of 2.20, what are the portfolio weights? How do you interpret the weights for the two assets in this case?

2.2 = x(1.1) + (1-x)(0)x = 2.2/1.1 = 2(1-x) = 1-2 = -1

The portfolio weight of –1 means that you sell short or in this case you borrow.

Risk and ReturnOct 31st




The capital asset pricing model defines the relationship between risk and returnE(RA) = Rf + βA(E(RM) – Rf)

βA is the beta and captures riskE(RM) – Rf is the market risk premium

and captures the price of riskRf is the risk free rate todayE(RA) is the expected rate or return for A

Cost of Capital

However, the firm can issue several different kinds of securities

EquityDebtPreferred Stock

What is the total required rate of return or the cost of capital for the firm?

Cost of Equity

The cost of equity is the return required by equity investors given the risk of the cash flows from equity

There are two major methods for determining the cost of equity

Capital Asset Pricing Model (CAPM)Dividend growth model

The Dividend Growth Model Approach

Start with the dividend growth model formula and rearrange to solve for RE

gPDR

gRDP

E

E

+=

−=

0

1

10

Dividend Growth Model Example

Consider the following exampleExpected dividend next year: $1.50 Growth rate of dividends: 5.1% The current stock price is $25What is the cost of equity?

111.051.2550.1

=+=ER

Estimating the Dividend Growth Rate

One method for estimating the growth rate is to use the historical average

Year Dividend Percent Change1995 1.231996 1.301997 1.361998 1.431999 1.50

(1.30 – 1.23) / 1.23 = 5.7%(1.36 – 1.30) / 1.30 = 4.6%(1.43 – 1.36) / 1.36 = 5.1%(1.50 – 1.43) / 1.43 = 4.9%

Average = (5.7 + 4.6 + 5.1 + 4.9) / 4 = 5.1%

Advantages and Disadvantages of Dividend Growth Model

Advantage – easy to understand and useDisadvantages

Only applicable to companies currently paying dividendsNot applicable if dividends aren’t growing at a reasonably constant rateExtremely sensitive to the estimated growth rate – an increase in g of 1% increases the cost of equity by 1%Assumes that the market has considered risk and has already come up with a current stock price

Another Example – Cost of Equity

Consider the firmBeta = 1.5Market risk premium = 9%Current risk-free rate is 6%Expected dividend growth rate = 6% Dividend yesterday = $2Current Stock price = $15.65What is our cost of equity?

Using CAPM Approach?Using the Dividend Growth Model?

Another Example – Cost of Equity

Using CAPMRE = 6% + 1.5(9%) = 19.5%

Using Dividend Growth ModelRE = [2(1.06) / 15.65] + .06 = 19.55%

As the two estimates are close, it should give you confidence in your estimate

Cost of DebtThe cost of debt is the required return on our company’s debtWe usually focus on the cost of long-term debt or bondsRequired return on long-term debt

Compensates the investor for riskWe can use the CAPM to get the expected return for debtWhen debt is very risky, it makes sense to calculate the beta for debtHowever, when debt is not very risky, it can be estimated by computing the yield-to-maturity on the existing debtThe cost of debt is NOT the coupon rate

Example: Cost of Debt

Consider the following zero coupon bond:Maturity: 10 yearsCurrent price: $385.54Face Value: $1000 bondWhat is the cost of debt?

Price = $1000/(1+YTM)10 = $385.54YTM = (1000/385.54)1/10 - 1= 0.10 or 10%

Cost of Debt, Preferred Stock

Generally, firms issue coupon bonds. Can use the YTM on existing bonds to get the required return on bonds

Firms also issue preferred stockPreferred generally pays a constant dividend every periodDividends are expected to be paid every period foreverPreferred stock is a perpetuity, so we take the perpetuity formula, rearrange and solve for the required return or RP

RP = D / P0

Example: Cost of Preferred Stock

Your company has preferred stock that has an annual dividend of $3. If the current price is $25, what is the cost of preferred stock?RP = 3 / 25 = 12%

Example

Firm A has an equity beta of 1.5. The risk free rate is 3% and the market risk premium is 7%. What is the expected return on Firm A equity?

E[r] = 3% + 1.5(7) = 13.5%

Example

Company A has preferred stock that has an annual dividend of $5. If the current price is $50, what is the cost of preferred stock?RP = 5 / 50 = 10%

Example

Consider the following zero coupon bond:Maturity: 5 yearsCurrent price: $585Face Value: $1000 bondWhat is the cost of debt?

Price = $1000/(1+YTM)5 = $585YTM = (1000/585)1/5 – 1 = 0.113 or 11.3%

The Average Cost of Capital

We can use the individual costs of capital that we have computed to get our “average” cost of capital for the firm.This “average” is the required return on our assets, based on the market’s perception of the risk of those assetsThe weights are determined by how much of each type of financing that we use

Capital Structure Weights

NotationE = market value of equity = # outstanding shares times price per shareD = market value of debt = # outstanding bonds times bond priceV = market value of the firm = D + E

WeightswE = E/V = percent financed with equitywD = D/V = percent financed with debt

Example: Capital Structure Weights

Suppose you have a market value of equity equal to $500 million and a market value of debt = $475 million.

What are the capital structure weights?V = 500 million + 475 million = 975 millionwE = E/D = 500 / 975 = .5128 = 51.28%wD = D/V = 475 / 975 = .4872 = 48.72%

Average Cost of Capital

Now we can combine all the components

ACC = wERE + wDRDwE and wD : market weights of debt and equity

RE is the cost of equityRDcost of debt

This would be the cost of capital for the firm. It is the required rate of return we need to discount the cash flows of the firm

Example: Cost of Capital

In1998, Pinnacle’s stock accounted for 80% of firm value and debt accounted for 20% of firm value.

Beta of equity = 1.5Market Risk Premium = 7%Risk free rate = 4%Cost of debt = 5%

What is the firm’s average cost of capital?


Cost of equity = 4% + 1.5(7%) = 14.5%

Firm Cost of Capital is= .2 (5%) + .8 (14.5%) = 12.6%

This is only in a world with no taxes

Taxes and Cost of Debt

There is an asymmetry between debt and equity

Payments to Debt holders, i.e., interest is tax deductiblePayments to Equity holders, i.e., dividends are not tax deductibleThis implies that the cost of debt is effectively lower than the required rate we calculated

Tax Benefit of Debt

Equity Debt

Dividend Payments 50 Interest Payments 50

EBIT 100 100Interest 50Earnings Before Taxes 100 50 Taxes Paid @ 40% 40 20Net Income 60 30

Dividends to Shareholder 50Retained Earnings 10 30Tax Savings 20Tax Savings Tax rate x Interest PaymentsCost of Debt (1-tax rate) x Interest Payments

Weighted Average Cost of Capital (WACC)

Due to this tax benefit, the effective cost of debt is actually

After-tax cost of debt = RD(1-TC)Where RD = required return on debtTC is the corporate tax rate

In a world of taxes, the cost of capital for the firms isWACC = wERE + wDRD(1-TC)

This is the required rate of return we need to discount the cash flows of the firm

WACC Example

Firm A required return on equity is 15%. The cost of debt is 8%. Firm A has 20% debt and 80% equity. The tax rate is 40%. What is the firm’s WACC?WACC = .8x15% + (1-.4)x0.2x8%

= 12.96%

Extended Example – WACC - I

Equity Information50 million shares$80 per shareBeta = 1.15Market risk premium = 9%Risk-free rate = 5%

Debt Information$1 billion in outstanding zero coupon debt (face value)Current quote = 338.00 per 1000 face value15 years to maturity

Tax rate = 40%

WACC Example Cont…

What is the cost of equity?RE = 5 + 1.15(9) = 15.35%

What is the cost of debt?Price = 1000/(1+RD)15 = 338.00RD = (1000/338)1/15 –1 = 7.5%

What is the after-tax cost of debt?RD(1-TC) = 7.5(1-.4) = 4.5%

WACC Example Cont..

What are the capital structure weights?E = 50 million x 80 = 4 billionD = 1 billion x (.338) = 0.338 billionV = 4 + 0.338 = 4.338 billionwE = E/V = 4 / 4.338 = 0.9221wD = D/V = .338 / 4.338 = 0.0779

What is the WACC?WACC = .9221(15.35%) + .0779(4.5%) = 14.50%

WACC: Example 2

Financial Data for Dixon2001 2002

Sales ($ millions) $4.0 $4.3Net income ($ millions) 0.71 0.79Earnings per share ($) $1.42 $1.58Dividends per share ($) $0.35 $0.40Dividend yield (c) 2.9% 3.5%Common stock - high (c) 13 14.25Common stock - low (c) 7.50 9Common stock - close (c) 12 11.50Closing P/E 8.5 7.3Total capitalization(Book values) 2.60 3.20

Debt 0.49 0.48Preferred stock 0.00 0.00Common stock 2.11 2.72

Number of shares (million) 0.50 0.50

WACC: Example 2

Further information for Dixon is Market Risk Premium = 7%Risk Free rate = 4%Beta: 1.2Cost of debt of 6% Tax rate of 35%

What is the Dixon’s WACC?

Market Weights

Financial Data for Dixon2001 2002

Total capitalization(Book values) 2.60 3.20Debt (1) 0.49 0.48Preferred stock 0.00 0.00Common stock 2.11 2.72

Number of shares (million) 0.50 0.50Market value of equity ($ million) (2) 6.00 5.75Common stock - close 12 11.50

Market Value of Firm (1+ 2) 6.49 6.23E/V 0.924 0.923D/V 0.076 0.077AverageE/V 0.923D/V 0.077

WACC: Dixon

Cost of equity = 4% + 1.2(7) = 12.4%

WACC = wERE + wDRD(1-TC)= 0.923 x 12.4% + 0.077 x 6% x (1-.35)= 11.75%

More than two sources of Capital

Reactive Industries has the following capital structureSecurity Market Value Required Return

Debt $ 20 million 6%Preferred Stock $ 10 million 8%Common Stock $ 50 million 12%

Corporate tax rate is 35%What is the firm’s WACC?

More than two sources of Capital

Market Value of Firm: 20+10+ 50 = $ 80 millionE/V = 50/80 = 62.5% D/V = 20/80 = 25%P/V = 10/80 = 12.5%

=0.625(12%) + 0.25(6%)(1-0.35) + 0.125(8%)= 9.475%

pde RVPTR

VDR

VEWACC +−+= )1(

Appropriate Cost of Capital

The cost of capital used to discount cash flows from the project

Should capture the risk of those cash flowsThe risk of cash flows are captured by the beta of the projectUsually, beta of firms in a similar activity are similar. If the fraction of debt and equity are the same, then firms in the same industry have similar risk, similar beta, and the resulting cost of capitalFor e.g. Gold mining is riskier than manufacturing cheese. Gold mining firms will have higher expected returns on average than cheese manufacturers.

Appropriate Cost of Capital

Honda, the Japanese automaker is considering manufacturing a new line of mini cars. Honda cost of capital is 13%.

What is the appropriate cost of capital for the mini car project?

Honda is also considering manufacturing a new motorcycle.

What is the appropriate cost of the motorcycle project?The average cost of capital for Harley-Davidson and Kawasaki is 15.2%.

Using WACC for All Projects -Example

Consider the following projects in the firm.

Project Required Return IRRfor this project

A 20% 17%B 15% 18%C 10% 12%

Which Projects should we take?


Consider the following projects in the firm. Project Required Return IRR Take?A 20% 17% NoB 15% 18% YesC 10% 12% Yes


Assume the WACC = 15%What happens if we use the WACC for all projects regardless of risk?

Project Required Return IRRA 20% 17%B 15% 18%C 10% 12%

Divisional Costs of Capital

If a firms has different divisions with different risk, using one cost of capital is not appropriate

Note:Using the WACC as our discount rate is only appropriate for projects that are the same risk as the firm’s current operationsIf we are looking at a project that is NOT the same risk as the firm, then we need to determine the appropriate discount rate for that project

Divisional Cost of Capital

Seagrams, a premium manufacturer of alcoholic beverages is considering acquiring Columbia Tristar, a motion picture studio. Seagrams cost of capital is 13%. The average cost of capital for motion picture studios is 20%. What is the discount rate used to evaluate the decision?

Market vs. Book Weights

The following is the summary balance sheet of Firm A.

The Market Value of Firm A’s Equity is 700.What are the Capital structure weights?

Assets Liabilities and Shareholder Equity %

Assets 647 Debt 194 30%Shareholders Equity 453 70%

Total Value 647 Total Value 647

Market vs. Book Value

Use Market Value weights rather than Book Value weights.Book value of debt will be different from market value if the interest rates have changed. Usually, this is not large. So market value weights are

Total value = 194 + 700 = 894Weight of Equity = 700/894 = 78.3Weight of Debt = 194/894 = 21.7

Floatation Costs

These are costs incurred in issuing securities.When the firm raises debt or issues equity, the funds it finally gets

Amount raised – Floatation Costs

How do we account for these costs? Basic Approach

Compute the weighted average flotation costUse the target weights because the firm will issue securities in these percentages over the long term

Example – Floatation CostsYour company is considering a project that will cost $1 million. The project will generate after-tax cash flows of $156,000 for ever. The cost of capital is 15%. The firm has 62.5% equity and 37.5% debt. The flotation cost for equity is 5% and the flotation cost for debt is 3%. What is the NPV for the project after adjusting for flotation costs?

Average Floatation costs fA = (.375)(3%) + (.625)(5%) = 4.25%If need $ 1 million, then need to raise

1,000,000/(1-.0425) = 1,044,386

Example – Floatation CostsPresent Value of Cash flows = 156,000/.15 =1,040,000NPV = 1,040,000 – 1,044,386 = -4386The project has a negative NPV with floatation costs so do not undertake it

If we had ignored floatation costs thenNPV = 1,040,000 – 1,000,000 = 40,000

For projects with small NPV this is important. Many firms deal with floatation costs by increasing the cost of capital. That is not the correct way to deal with them

Summary

CAPM can be used to get cost of equity

Cost of Capital for a project is the weighted average cost of equity and debt

As debt is tax advantaged, need to take the after tax cost of debt

The cost of capital for a firm is there given by WACC = wERE + wDRD(1-TC)

Summary

WACC is used to discount the cash flows of the project

Takes care of the risk of the cash flowIt takes care of the tax benefit to debt

Capital Structure

The division between debt and equity is called capital structure .

The firm with no debt is called an all equity firm. When it has debt, the firm is said to be levered

Two Questions:Does it matter what capital structure the firm has?If it does, what is the optimal capital structure

Answer it in two stages?Simple world: no taxesIn a world with taxes and other frictions

Cost of Capital

When there were no taxes:Cost of capital was referred to as the Average Cost of CapitalACC = wERE + wDRD

In a world with taxesCost of Capital was referred to as the Weighted Average Cost of Capital (WACC)WACC = wERE + wDRD(1-TC)

Average Cost of Capital

Let us go back to a simple world with no taxes.

ACC = wERE + wDRDwE and wD : market weights of debt and equity

RE is the cost of equityRDcost of debt

This would be the cost of capital for the firm. It is the required rate of return we need to discount the cash flows of the firm


Pinnacle’s stock accounted for 80% of firm value and debt accounted for 20% of firm value.

Beta of equity = 1.5Market Risk Premium = 7%Risk free rate = 4%Cost of debt = 5%

Cost of equity = 4% + 1.5(7%) = 14.5%Firm Cost of Capital is= .2 (5%) + .8 (14.5%) = 12.6%

QuestionsFacts:

Cost of debt = 5%Cost of equity = 14.5%Total cost of capital =.2 (5%) + .8 (14.5%) = 12.6%

Why is Debt Cheaper? Can the firm reduce its cost of capital by issuing more debt?What happens when you issue more debt?

Cost of Debt

Note:

Cost of debt is 5% while the cost of equity is 14.5%

Debt is cheaper than equity. Debt is less risky. It gets paid before equity gets paid.

Does this mean that the firm can issue more debt and reduce its cost of capital?

A lower cost of capital means a higher NPV from projects, and higher firm value.The answer is NO.

Capital StructureNov 7th





Where � RD = required return on debt� TC is the corporate tax rate� wE,, wD) are the weights of equity and debt� RE required return on equity

Capital Structure

The division between debt and equity is called capital structure . � The firm with no debt is called an all equity firm.

When it has debt, the firm is said to be levered

Two Questions:� Does it matter what capital structure the firm has?� If it does, what is the optimal capital structure

Answer it in two stages?� Simple world: no taxes� In a world with taxes and other frictions

Cost of Capital

When there were no taxes:� Cost of capital was referred to as the Average

Cost of Capital� ACC = wERE + wDRD

In a world with taxes� Cost of Capital was referred to as the Weighted

Average Cost of Capital (WACC)� WACC = wERE + wDRD(1-TC)

Example: Average Cost of Capital

Pinnacle’s stock accounted for 80% of firm value and debt accounted for 20% of firm value.� Beta of equity = 1.5� Market Risk Premium = 7%� Risk free rate = 4%� Cost of debt = 5%

Cost of equity = 4% + 1.5(7%) = 14.5%Firm Cost of Capital is= .2 (5%) + .8 (14.5%) = 12.6%

QuestionsFacts:� Cost of debt = 5%� Cost of equity = 14.5%� Total cost of capital =.2 (5%) + .8 (14.5%) = 12.6%

Why is Debt Cheaper? Can the firm reduce its cost of capital by issuing more debt?What happens when you issue more debt?

CoC and Capital Structure

Say the firm buys back some of its stock and issues more debt such that� Equity is now 60% (instead of 80%)� Debt is now 40%

When you increase debt� What happens to the risk of debt ? � What happens to the risk of equity?


Say the firm buys back some of its stock and issues more debt such that� Equity is now 60% (instead of 80%)� Debt is now 40%

When you increase debt� What happens to the risk of debt ? It increases� What happens to the risk of equity? It increases

too.

Capital Restructuring

We are going to look at how changes in capital structure affect the value of the firm, all else equalCapital restructuring involves changing the amount of leverage a firm has without changing the firm’s assetsIncrease leverage by issuing debt and repurchasing outstanding sharesDecrease leverage by issuing new shares and retiring outstanding debt

Example: Trans Am Corp.

Assets have a market value of $8M.Currently Debt = 0Shares outstanding = 400,000Share price = 8M/400k = 20.00Proposed capital restructuring:� Issue 4M Debt at 10% interest� Buy back 4M/20 = 200,000 shares� For now assume stock price stays at $20.00

Example Continued..

10%10%Interest Rate

200,000400,000# Shares

2020Share Price

10D/E Ratio

4,000,0008,000,000Equity

4,000,0000Debt

8,000,0008,000,000Assets

ProposedCurrent

Current Capital Structure

3.752.501.25EPS

18.75%12.5%6.25%ROE

1,500,0001,000,000500,000NI

000Interest

1,500,0001,000,000500,000EBIT

ExpansionExpectedRecession

Proposed Capital Structure ($4M Debt)

5.503.000.50EPS

27.50%15.00%2.50%ROE

1,100,000600,000100,000NI

400,000400,000400,000Interest

1,500,0001,000,000500,000EBIT

ExpansionExpectedRecession

Summary

Current Capital structure (All Equity)� ROE is : 6.25% 12.5% 18.75%� EPS is: 1.25 2.50 3.75

Proposed Capital Structure (50% equity)� ROE is : 2.5% 15% 27.5%� EPS is: 0.5 3.00 5.50

Break-Even EBIT

Find EBIT where EPS is the same under both the current and proposed capital structuresIf EBIT is less than the Break-Even EBIT then adding leverage reduces EPSIf EBIT is greater than Break-Even EBIT then adding leverage increases EPS

Example: Break-Even EBIT

( )

$2.00400,000800,000

EPS

$800,000EBIT800,0002EBITEBIT

400,000EBIT200,000400,000

EBIT

200,000400,000EBIT

400,000EBIT

==

=−=

−��

��

�=

−=

Financial Leverage, EPS and ROE

Variability in ROE� Current: ROE ranges from 6.25% to 18.75%� Proposed: ROE ranges from 2.50% to 27.50%

Variability in EPS� Current: EPS ranges from $1.25 to $3.75� Proposed: EPS ranges from $0.50 to $5.50

The variability in both ROE and EPS increases when financial leverage is increased

As leverage increases, both debt and equity become more risky

The Effect of Leverage

When we increase the amount of debt financing, we increase the fixed interest expense

If we have a really good year, then we pay our fixed cost and we have more left over for our stockholders

If we have a really bad year, we still have to pay our fixed costs and we have less left over for our stockholders

Leverage amplifies the variation in both EPS and ROE


You cannot change the overall cost of capital of the firm by increasing debt. As you increase debt� Debt gets riskier and its cost goes up� Equity gets riskier and its cost goes up� Such that total cost of capital stays the same

� This is the irrelevance of Capital Structure in a perfect world � Modigliani and Miller won the Nobel Prize to show this

� Note: The above holds only in a world with no tax asymmetries

QuestionsWhy is Debt Cheaper? � Because it is less risky

What happens when you issue more debt?� Debt becomes riskier, Cost of Debt increases� Equity becomes riskier, cost of equity increase

Can the firm reduce its cost of capital by issuing more debt?� No� Without Taxes, cost of capital stays the same

Theory of Capital Structure?

The division between debt and equity is called capital structure . � The firm with no debt is called an all equity firm.

When it has debt, the firm is said to be leveredDoes it matter what the capital structure of the firm is?Modigliani and Miller� With no taxes: Capital structure does not matter � With taxes: It matters. Theory of how much debt

the firm should have

Start with a Simple World

In this world

� There are no taxes

� There is no costs of Financial distress or cost of bankruptcy

� Firms Assets/ Investments are fixed

� Perfect Markets - Complete information

� No transactions costs � These are the assumptions of M&M world

Financial Claims on the FirmWe will assume there are 2 claims on the firm: debt (D) and equity (E).� Total value of D + E = firm value V� D is the market value of debt� E is the market value of equity� V is the market value of the firm

Debt has a promised interest and principal paymentsEquity has rights to the residual cash flows and has control rights.Does how you “slice up” the firm into debt and equity change the value of the firm, V?

Goal of Management

If the goal of management is to maximize firm value, then they should pick a debt/equity ratio that produces the largest V.

Changing the debt/equity ratio, changes how cash is paid out from the firm.� For example, more debt produces larger interest

payments and less dividend payments.

What Determines Firm Value?

A new rate, , this is the discount rate of an all equity firm or the return on the asset.

rA

Assets Cash FlowsProduce

TotalFirmValue

Discount

rA



TotalFirmValue

CFD

Discount

ValueDebt

ValueEquityCFE

=+

Discount

Discount

rA

rD

rE



TotalFirmValue

CFD

Discount

ValueDebt

Value EquityCFE

=+

Discount

Discount

rA

rD

rE

Conservation of Value

You can split up claims to cash flows anyway you want, but as long as you don’t change the total payout to the claims, you don’t change the total value of the claims.� Or, $100 dollars is $100 dollars.

Conservation of value is the concept that underlies capital structure irrelevance.

MM Propositions

Modigliani & Miller MM Proposition I

= Value of an all equity firm= Value of equity= Value of debt= Value of a levered firm

Statement about conservation of firm value

LDEu VVVV =+=

uV

DV

LV

EV

MM Propositions

How come this is true? � We know that debt is cheaper than equity. Can’t we

reduce the cost of capital by increasing debt. � We saw earlier that as Debt increased

� Debt got riskier, i.e., cost of debt increased� Equity got riskier. Cost of equity also increased� These increased such that ACC = (E/V)RE + (D/V)RD = Ra

= stayed the same

MM Proposition II )( DaaE rr

ED

rr −+=

Example

Firm A has 100% equity. Its cost of capital is 16%. If it changed its capital structure to 45% debt and 55% equity, the cost of debt = 10%. What is the cost of equity? (there are no taxes)

Using MM Proposition II

� RE = .16 + (.16 - .10)(.45/.55) = .2091 = 20.91%� The cost of equity is going to rise such that the total cost

of capital stays the same

)( DaaE rrED

rr −+=

Example

Firm A has 100% equity. Its cost of capital is 16%. Under a new proposed capital structure the cost of equity will be 25% and the cost of debt = 10%. What is the debt-to-equity ratio in the new capital structure? (there are no taxes)

Example

Firm A has 100% equity. Its cost of capital is 16%. Under a new proposed capital structure the cost of equity will be 25% and the cost of debt = 10%. What is the debt-to-equity ratio in the new capital structure? (there are no taxes)

Using the MM proposition II

� .25 = .16 + (.16 - .10)(D/E)� D/E = (.25 - .16) / (.16 - .10) = 1.5

Based on this information, what is the percent of equity in the firm?� E/V = 1 / 2.5 = 40%

)( DaaE rrED

rr −+=

Example 2

Firm A’s old capital structure is� Debt: 20% at cost of 6%� Equity: 80% at cost of 13%

The new proposed capital structure is� Debt: 50% at cost of 8%� Equity: 50%

What will be the cost of equity in the new proposed capital structure?

Example 2

Total cost of capital in the old structure is = .2 (6%) + .8(13%) = 11.6%

Let the cost of equity in the new proposed capital structure be r

11.6% = 0.5r +0.5(8%) r = 15.2%

Graphical representation

D/E

Value

VL= Vura

rE

rD

Proposition I Proposition II

Summary of M&M

If the assumptions of M&M hold then � Capital structure choice is irrelevant� Shifting between debt and equity does not change firm value

or cost of capital. It does change the distributions of risk (and return) between debt and equity

M&M assumptions never hold

As we relax the assumptions, one at a time, capital structure will become important. This gives us a way to handle the complexity of the real world.

In a world with Taxes

There is an asymmetry between debt and equity� Payments to Debt holders, i.e., interest is tax deductible� Payments to Equity holders, i.e., dividends are not tax

deductible� This implies that the cost of debt is effectively lower than the

required rate we calculated

Due to this tax benefit, the effective cost of debt is actually� After-tax cost of debt = RD(1-TC)� Where RD = required return on debt we calculated � TC is the corporate tax rate



� wE and wD : market weights of debt and equity� RE is the cost of equity� RD(1-TC) is the after tax cost of debt

Now, as you increase debt the firm’s overall cost of capital or WACC will fall.

Tax Advantage of Debt

Consider two identical firms (except for capital structure). � Firm U is unlevered or all equity firm� Firm L has $100 of interest payments.

Each firm has $1000 of pretax earnings.

What is the total payout to each firm’s claimants (shareholders and debtholders)?

Assume corporate tax rate is 34%

What is the Tax Advantage of Debt?Firm U Firm L

Earnings 1000 1000interest paid 0 100pretax income 1000 900 34% tax (340) (306)Net Income 660 594

Payout to Debt Holders 0 100Payout to Equity Holders 660 594Total Firm value 660 694Payment to Govt. 340 306

Tax Saving or the Tax advantage of debt = $34This tax shield = Interest Payment x Tax rate = 100 x 34% = $34

Valuing the Tax ShieldLet D = face value of perpetual debt.

Let rD = borrowing rate for the firm.

Interest paid per year = rD D

Annual corporate tax shield = rD D x τcorp

� Τcorp = corporate tax shield

� Earn this tax shield every year

Value of this shield is � Present value of all future annual tax shields

Valuing the Tax Shield

Every year the firm saves = rD D x τcorp

What is the risk of these savings?� What discount rate should be used to get the present value

of these tax savings?

� The discount rate used is the cost of debt or rD

If the firm and its borrowing of D is perpetual then can use the perpetuity formula

� PV(tax shield) = [rD Dτcorp]/ rD = Dτcorp

� less if debt is not perpetual, or if tax shield will be eliminated sometime in the future.

Example: Value of Tax Shield

Forever Inc has Debt outstanding of 1.5 million dollars. The cost of debt is 6% and the corporate tax rate is 35%. What is the value of the tax shield from Debt?

Annual Interest payments=1.5m x 6%= 90000

Annual Tax shield = 90,000 x 34% = 31,500

PV of Tax Shield = 31,500/.06 = $525,000or

Use the formula = Dτcorp = 1.5m x 35% = $525,000

Example 2: Value of Tax Shield

Forever Inc has Debt outstanding of 1.5 million dollars. The cost of debt is 6% and the corporate tax rate is 35%. They plan to repay 1 million in the beginning of the fifth year and another .5 million at the beginning of 10th year. What is the value of the tax shield from Debt?

Valuing the Tax Shield

YearDebt oustanding

Interest Payments

Annual Tax Shield

PV of tax shield

( in Millions)1 1.5 90 31.5 29.7172 1.5 90 31.5 28.0353 1.5 90 31.5 27.2114 1.5 90 31.5 24.9515 0.5 30 10.5 7.8466 0.5 30 10.5 7.4027 0.5 30 10.5 6.9838 0.5 30 10.5 6.5889 0.5 30 10.5 6.215

10 0 0 0

Total Debt Tax Shield (thousands) 144.948

(in thousands)

Tax Advantage of Debt: Another Example

Firm A has 500 thousand dollar of debt at 8%. They are planning to maintain this level of debt in the future. The corporate tax rate is 34%. What is the tax advantage of debt?

Annual Interest payments=500,000 x 8%= 40,000

Annual Tax shield = 40,000x 34% = 13,600

PV of Tax Shield = 13,600/.08 = $170,000or

Use the formula = Dτcorp = .5m x 34% = $170,000

Tax Advantage of Debt: Another Example

If firm A has 10,000 shares outstanding, What is the change in stock price when it announces it will issue debt.

The tax advantage of debt = $170,000Per share value = 170,000/ 1000 = $ 17Share price will increase by $ 17 dollars

Value of a Levered Firm with Corporate Taxes

Value of a levered firm is MM Proposition I with taxes

VL = VU + PVTS (Present Value Tax Shield)

As the value of the levered firm > value of an unlevered firm

Capital Structure with Taxes

D/E

Total Firm Value

MM: Vu=VL

VL=Vu+ PVTS

Vu

PVTS = Present Value of Tax Shields

= t x D

Expected Return and Leverage (With Taxes)

MM Proposition II with taxes

Where Ra is the return to an all equity firm or to assetsD and E is the market value of debt and equity

)1)(( TcrrED

rr DaaE −−+=

ExampleThe EXES company is assessing its present capital structure. It is currently financed entirely with common stock, of which, 1000 share are outstanding. Given the risk of the underlying cash flows, investors currently require 20% return on its stock. The company pays out all its earnings as dividends. EXES expects to generate cash flows of $ 3000 in perpetuity. Assume that taxes are zero.

What is the value of EXES company?The Value of the firm = $3000/0.20 = $15,000

Example cont..

The president of the firm has decided that shareholders will be better off if the company had equal proportions of debt and equity. He therefore proposes to issue $7,500 of debt at an interest rate of 10%. He will use the proceeds to repurchase 500 shares of common stock.

• What will the new value of the firm be?• What will the value of the EXES debt be?• What will the value of EXES equity be?

Example cont..

The value of the firm stays the same at $15,000. Equity price� The share price before was 15,000/ 1000 = $15. � The new share price will be $ 15. The value of

equity is 500 x $15 = $7500. � The value of debt is $7500.

ExampleSuppose the president’s proposal is implemented

• What is the required return on equity?• What is firm’s overall cost of capital?

The total cost of capital for the firm does not change with leverage (note: there are no taxes).

20% = 0.5 x 10% + 0.5 x re. re = 30%.

Another Example

Firm ABC earns EBIT of 25 million per year forever. It has debt of $75 million at a cost of 9%. Nolever, an identical firm with no leverage has cost of capital of 12%. The corporate tax rate is 35%� What is the value of Nolever� What is the value of the tax shields for firm ABC� What is the value of ABC? What is the value of

ABC’s equity?

Another Example

Unlevered cost of capital = 12%The value of Nolever is � Vu = 25(1-.35)/ .12 = $135.42

The value of the tax shields is � 75x.35 = 26.25m

The value of ABC is � VL = 135.42 + 26.25 = 161.67� Value of equity = 161.67 – 75 = 86.67

Example Continued

What is the rate of return on ABC equity?

� .12 + 75/86.67 (1-.35)(.12 - .09)� = 13.69%

What is ABC’s WACC?� =10.05%� 86.67/ 161.67 (13.9%) + 75/161.67 (1-.35) x.09

Example 3

The Holland company expects perpetual EBIT of 4m per year. The firm all equity rate of return is 15% and tax rate is 35%. The cost of debt is 10% and the firm has 10m of debtWhat is Holland Value?What is Holland’s cost of equity?What is Holland’s WACC?

Example 3 cont..

Value of all equity firm = 4 (1-.35)/.15 = 17.33mValue of tax shield = Dxt = 10x.35 = 3.5 mValue of Holland = 17.33 + 3.5 = 20.83

Value of equity = 20.83 – 10 = 10.83Cost of equity = 15% + 10/10.83(1-.35)(.15 -.10) = 18%

WACC = 10.83/20.83 (18%) + 10/20.83 (1-.35)10% = 12.48%

Trade-off Theory

Should firms move towards 100% debt?No, firms should not go to 100% debt

Benefits of debt (assumption 1)� Tax benefits:

Costs of debt: Costs of financial distress � When firms cannot pay back debt they are taken to

bankruptcy court or are said to be in financial distress� Relaxation of M&M assumption 2

Trade-off between the benefits of debt (tax shield) and the cost of debt (costs of financial distress)

Conclusions

As you increase leverage� Equity gets riskier� Debt get riskier

With no Taxes: � Total cost of capital stays constant� ACC = wERE + wDRD

With Taxes� Debt has a tax advantage� WACC = wERE + wDRD(1-TC)

Optimal Capital Structure is determined by the tradeoff between benefit of debt (tax advantage) and cost of debt (financial distress)




Value of a Levered Firm with Corporate Taxes

Value of a levered firm is MM Proposition I with taxes

VL = VU + PVTS (Present Value Tax Shield)

As the value of the levered firm > value of an unlevered firm

Capital Structure with Taxes

Total Firm Value

MM: Vu=VL

VL=Vu+ PVTS

Vu

D/EPVTS = Present Value of Tax Shields

= t x D

Expected Return and Leverage (With Taxes)

MM Proposition II with taxes

Where Ra is the return to an all equity firm or to assetsD and E is the market value of debt and equity

)1)(( TcrrEDrr DaaE −−+=

Example 3

The Holland company expects perpetual EBIT of 4m per year. The firm all equity rate of return is 15% and tax rate is 35%. The cost of debt is 10% and the firm has 10m of debtWhat is Holland Value?What is Holland’s cost of equity?What is Holland’s WACC?

Example 3 cont..

Value of all equity firm = 4 (1-.35)/.15 = 17.33mValue of tax shield = Dxt = 10x.35 = 3.5 mValue of Holland = 17.33 + 3.5 = 20.83

Value of equity = 20.83 – 10 = 10.83Cost of equity = 15% + 10/10.83(1-.35)(.15 -.10) = 18%

WACC = 10.83/20.83 (18%) + 10/20.83 (1-.35)10% = 12.48%

Trade-off Theory Should firms move towards 100% debt?No, firms should not go to 100% debt

Benefits of debt (assumption 1)Tax benefits:

Costs of debt: Costs of financial distress When firms cannot pay back debt they are taken to bankruptcy court or are said to be in financial distressRelaxation of M&M assumption 2

Trade-off between the benefits of debt (tax shield) and the cost of debt (costs of financial distress)

Limits to the Use of DebtChapter 16

Costs of Financial Distress

As debt increases, is become more likely that the firm will not be able to pay its debt obligations In this case, the debtholders can take the firm to bankruptcyTill now we have assumed zero costs of bankruptcy

In reality, there are costs to financial distress

Example: Company in Distress

Assets BV MV Liabilities BV MVCash $200 $200 LT bonds $300Fixed Asset $400 $0 Equity $300Total $600 $200 Total $600 $200

What happens if the firm is liquidated today?

$200$0

The bondholders get $200; the shareholders get nothing.

What do we mean by costs of Financial Distress?

Two firms which are similarFirms Nodistress has to make interest payments of $50Firm Distress has more debt and has to make interest payments of $150We will assume investors are risk neutral

Special assumption to simplify thingsThe do not ask a higher rate for more risk - same rate of return for debt and equity – say 10%

Example cont..

This is MM Proposition I without taxes. There is financial distress or bankruptcy but no costs of distressSay the cost of distress – when you are not able to pay interest are $40. What are the debt, equity and firm values?

Recession Boom Recession BoomCash Flow 100 500 100 500Interest 50 50 100 150

Flows to Equity 50 450 0 350

Value of an all equity firm = 0.5(100) + 0.5(500) / 1.1 = 272.72

Nodistress Distress

Debt Value = 0.5(50) + 0.5(50) / 1.1 = $45.45 D = 0.5(100) + 0.5(150) / 1.1 = $113.64Equity Value = 0.5(50) +0-.5(450) / 1.1 = $227.72 E = 0.5(0) + 0.5(350) / 1.1 = $159.09

Total value = 45.45 + 227.27 = 272.27 Total value = 113.64 + 159.09 = 272.72

With Distress Costs

Recession Boom Recession BoomCash Flow 100 500 100 500Interest 50 50 60 150

Flows to Equity 50 450 0 350

Value of an all equity firm = 0.5(100) + 0.5(500) / 1.1 = 272.72PV of costs of financial distress = 0.5(40)/1.1 = 18.18

Nodistress Distress

Debt Value = 0.5(50) + 0.5(50) / 1.1 = $45.45 D = 0.5(60) + 0.5(150) / 1.1 = $95.45Equity Value = 0.5(50) +0-.5(450) / 1.1 = $227.27 E = 0.5(0) + 0.5(350) / 1.1 = $159.09

Total value = 45.45 + 227.27 = 272.72 Total value = 113.64 + 159.09 = 254.54

What happens with Financial Distress

What triggers financial distressWhen the firm is unable to make a promised payment -interest or principal Sometimes when the firm is in violation of covenants

Who can file for BankruptcyVoluntary: Managers realize the firm is insolvent and file for bankruptcyInvoluntary: Creditors file for Bankruptcy

Types of BankruptcyWorkout: consensual restructuring of liabilitiesChapter 11: ReorganizationChapter 7: Liquidation by court appointed trustee

Bankruptcy Law Overview (US)

Automatic Stay: Creditors are put to bay, managers get to stay

Restructuring: Management proposes plan, creditors vote by class, majority rule within class, all classes must approve

Debtor in Possession (DIP) Financing: New debtors are allowed seniority

Recontracting: Firm can break executorycontracts

Bankruptcy Law in other Countries

France: Court appointed official helps with reorganization plan

U.K.Administration: Accountant or lawyer runs the firmAdministrative receivership: Secured Creditors run firm

JapanInformal rescues more common than formal bankruptcy

Germany:Liquidations more frequent

Sweden:Court appointed official auctions the firm

Costs of Financial Distress Actual costs of resolution or liquidation. Also called direct costs. Includes things like lawyer fees e.t.c

Loss of competitive position Many companies loose market shareMany firms get caught up in the proceeding and miss opportunitiesA large fraction of firms that emerge from chapter 11 reenter chapter 11 (US Airways)

How large are these costs?

Estimates of Costs of Financial Distress

Study of 31 highly levered transactions that became distressed was about 10% to 20%

Study of 3000 firms: In industries that experience downturns, highly levered firms experience operating income, sales and stock returns that are 7%, 17% and 16% lower than industry average

Estimates of Costs of Financial Distress

Fire sales: Distressed airlines sell aircraft at significant discounts (upto 30%) especially when industry is depressed and buyer is a financial buyer

Direct costs of bankruptcy: 3.1% of value

Capital Structure with COFD

Total Firm Value

Vu

MM: Vu=VL

VL=Vu- COFD

D/E

COFD = Costs of Financial Distress

Agency Costs

Risk ShiftingUnder-investmentMilking the property

Agency Cost 1: Risk Shifting

Consider a firm with some assets and $150 of interest payments. The firm is choosing between two projects A and B. A is the safer project and B is the risky project

Risk Neutral – discount rate is 10%

Which Project should they do?

Recession Boom Recession BoomValue of Firm with Project 150 400 100 440Interest Payment 150 150 150 150

Project A Project B

Risk Shifting

If they do project AValue of the firm: 0.5 * (150) + 0.5*(400)/1.1 = 250Value of Debt = 150/1.1 = 136.36Value of Equity = 250 – 136.36 = 113.63

If they do Project BValue of the firm = 0.5 (100) + 0.5(440)/1.1 = 245.45Value of Debt = 0.5(100) + 0.5(150)/1.1 = 113.63Value of Equity = 245.45 – 113.64 = 131.82

Risk Shifting

Which project should be taken?

If there was only equity, which project will be taken?

In the current scenario, which project will be taken?

Agency Cost 2: Under-investment

Consider a firm with some assets and $150 of interest payments. The firm has a project that will pay $50 in both states and cost $40.

Risk Neutral – discount rate is 10%

Recession Boom Recession BoomOld Cash Flow 100 300 100 300New Project 50 50Total Cash flows 100 300 150 350

Interest Payment 150 150 150 150

Firm Without Project Firm With Project

Under InvestmentShould the project be done?

Value of project = 0.5 (50) + 0.5(50)/1.1 = 45.45NPV = $45.45 – $40 = $5.45Yes it should be done?

Will it be done?Bond Value without project = 0.5(100) + 0.5(150)/ 1.1 = 113.63With the project = 150/1.1 = 136.36Value of Equity without project = 0.5 (150)/ 1.1 = 68.18Value of Equity with Project = 0.5(200)/1.1 = 90.90

EquityRaise $40 to do the projectValue increases by 90.90 – 68.18 = 22.72Debt value increases by = 136.35 – 113.63 = 22.72Total increase in value = 22.72 + 22.72 = 45.44

Agency Cost 3: Milking the property

Firms close to distress or doing badly mayStart paying hefty dividendsSell out the profitable assets and pay out the profits as dividendsGenerally cannot do these

All these agency costs happenWhen the firm is close to bankruptcy Too much debt payments relative to cash flow Not seen in healthy companies

Covenants: Reducing the Costs of Debt

Negative Covenants: Things shareholder promise not to do

How much dividends that can be paid

Cannot issue senior debt

Cannot sell assets / merge with other firms

Positive Covenants: Things they promise to doMaintain Interest coverage ratios, working capital

Furnish financial statements

Costs of Debt

Costs of Financial DistressDirect CostsIndirect Costs

Agency CostsRisk ShiftingUnder-investmentMilking the property

Tax Effects and Financial Distress

There is a trade-off between the tax advantage of debt and the costs of financial distress.The value of the firm is now:

VL = VU + PVTS - PV of financial distress

PV of financial distress is determined by:No Precise Formula Probability of financial distress

How much debt does the firm haveHow much does its income vary

Costs of FDAsset tangibility: get more from liquidationsGrowth opportunities

Trade Off Theory

The value of a firm can now be written:ValueL = ValueU + PVTS - PVFD

Trade off theory says that firms should increase debt until it will cause the PVFD to increase more than the PVTS.

Tax Effects and Financial Distress

Debt (B)

Value of firm (V)Present value of tax

shield on debt

Value of firm underMM with corporatetaxes and debt

V = Actual value of firmVU = Value of firm with no debt

VL = VU + TCB

Maximumfirm value

Present value offinancial distress costs

0B*

Optimal amount of debt

Summary of Cases

Case I – no taxes or bankruptcy costsNo optimal capital structure

Case II – corporate taxes but no bankruptcy costsOptimal capital structure is 100% debtEach additional dollar of debt increases the cash flow of the firm

Case III – corporate taxes and bankruptcy costsOptimal capital structure is part debt and part equityOccurs where the benefit from an additional dollar of debt is just offset by the increase in expected bankruptcy costs

Consider the following situation

The CEO of a public firm wants more capital. The share price is trading at $10. He knows that the firm has a great R&D project that will work out great. When it does the share price should be at least $20.

Would be like to issue Equity now?

Say he knows that the firm will be sued for a problematic product. Would he like to issue equity now?As Investors, when you see that a firm is issuing equity what do you think?

Stock Price Reaction On Equity Issues

On average the stock price declines when the firm announces that it is issuing equity

Investors infer that on average the firm does not have good newsIf it had good news, it would be issuing debtThis means that it on average firms would rather not issue equity if they can help it.

This gives rise to the Pecking Order Theory of Capital Structure

The Pecking-Order Theory

Theory states that firms prefer to issue debt rather than equity if internal financing is insufficient.

Rule 1Use internal financing first.

Rule 2Issue debt next, new equity last.

The pecking-order theory is at odds with the tradeoff theory:

There is no target D/E ratio.Profitable firms use less debt.Companies like financial slack.

Observed Capital Structure

Most corporations have low Debt-Value ratios.These were 48% in USHigher in other countries:

Japan 72%France and Italy: 60%

There are many firms who have no debtCoca Cola, Microsoft (negative debt)These tend to very profitable firmsGenerating a lot of internal cash flows: enough to finance growth and more

Observed Capital Structure

Capital structure does differ by industriesDifferences according to Cost of Capital 2000 Yearbook by Ibbotson Associates, Inc.

Lowest levels of debtDrugs with 2.75% debtComputers with 6.91% debt

Highest levels of debtSteel with 55.84% debtDepartment stores with 50.53% debt

Why vary with Industry?

Costs of financial distressProbability of getting distressedCosts when distressed

Asset TangibilityMore tangible: sell easily and recover moreCosts when distressed are low

Growth options and R&DWhen distressed loose all these optionsCosts are higher

Business is very cyclical or volatileHigher probability of getting distressed

Matched Capital Structures

FIRM Industry Debt/Total Capital (MV)

CATERPILLAR INC Construction machinery 0.37 EDISON INTERNATIONAL Electric utility 0.62 FREEMARKETS INC Internet business services 0.00 HOST MARRIOTT CORP Hotels and real estate 0.75 MICROSOFT CORP Software 0.00 NAVIGANT CONSULTING INC Management consulting 0.00 PHARMACYLICS INC Drugs (ethical) 0.00 RENT-A-CENTER INC Equipment rental and leasing 0.62 RJ REYNOLDS Tobacco 0.48 UAL CORP Air Transport 0.55 YOUNG AND RUBICAM INC. Advertising 0.03

How Firms Establish Capital Structure

There is some evidence that firms behave as if they had a target Debt-Equity ratio.

They also behave in accordance with Pecking order theory

Some combination of the two

Many firms value financial slack a lotIn case they need to borrow quickly

Keep low debt levels

Valuation and Capital Budgeting for the Levered Firm

Chapter 17Very Briefly….

Introduction

We have seen that issuing debt is valuable as it generates tax savings.What is the tax advantage?This is also the Adjusted Present Value Method or the APVValue of levered firm = Value of all Equity

firm +Value of all effects of Debt

tBVV UL +=

Adjusted Present Value

Suppose PMM, Inc. has an investment that costs $10,000,000 with expected EBIT of $3,030,303 per year forever. The investment can be financed either with $10,000,000 in equity or with $5,000,000 of 10% debt and $5,000,000 of equity. The discount rate on an all-equity-financed project in this risk class is 20%. The firm's marginal tax rate is 34%.What is the value if financed with equity?What is the value if financed with debt and equity?

Example cont.

Value of All equity firm: Annual after-tax cash flows: (EBIT)(1- tc) = ($3,030,303)(1-.34) = $2,000,000Value = ($2,000,000 / .2) = $10,000,000

Tax Subsidy: txD = .34 * 5m = 1.7m

Value of a levered firm:= 10m + 1.7m = 11.7m

Example Continued

What is the cost of levered equity?What is the WACC?How do you use the WACC to get firm value of the levered firm?

WACC Approach

V = D + E = 11.7mE = V – D = 11.7m – 5m = 6.7m

Value of Firm: 3,030,303(1-0.34)/ 17.094% = 11.7m

)()1( 00 BCS RRTSBRR −×−×+=

%925.24%)10%20()34.1(7.6

5%20 =−×−×+=SR

DCDee RTwRwWACC )1( −+=

%094.17%10)34.1)(7.11/5(%925.24)7.11/7.6( =−+=WACC

Other Ways to Value Levered Firm

WACC Approach:Value of Levered Firm = Cash flow to an All

equity Firm / WACCWith the tax advantage of debt WACC< ro the required rate for all equity firmIf the CF are in perpetuity

A third method: Flow to equity: Value of Levered Equity ( we will not cover this)

Comparing the Methods

APV WACCCash Flows All Equity All Equity

CF CFDiscount Rates R0 RWACC

APV has the advantage that can easily put in costs of distress and other costs if wantedAll methods should give the same value

Another Example

Zipper Inc. has after tax cash flows of $1000 forever. Its all equity cost of capital is 15%. It has $2000 of debt at a cost of 5%. If the tax rate is 40% what is the value of Zipper Inc.?

Using the APV method?Using the WACC method?

Example Cont.

Using APV:Value of all equity firm = 1000/.15 = $6666.67Value of Tax shields = 2000 x .40 = $ 800Value of Zipper Inc = 6666.67 + 800 = 7466.67

Using WACCValue of Zipper equity = 7466.67 – 2000 = 5466.67Return on equity = .15 + 2000/5466.67 (.15 - .05)(1-.4) = 17.195%WACC = 5466.67/7466.67 (.17195) + 2000/7466.67(.05)(1-.4) = 13.39%Value of Zipper = 1000/.1384 = 7468

Note: Do not know the weights on equity and debt and need to calculate that for WACC

Example cont

Zipper Inc. has after tax cash flows of $1000 forever. Its all equity cost of capital is 15%. It has 24% debt at a cost of 5%. If the tax rate is 40% what is the value of Zipper Inc.?

Using the APV method?Using the WACC method?

Note: Instead of having $2000 of debt now we have a debt ratio?

Example Cont.

Using APV:Value of all equity firm = 1000/.15 = $6666.67How much Debt do we have:

24% of firm value (is actually levered value)Approximate by all equity value24% x 6666.67 = 1600

Value of Tax shields = 1600 x .40 = $ 640Value of Zipper Inc = 6666.67 + 640 = 7306.67

Note: Do not know the amount of debt and need to calculate that for APV

Example Cont.

Using WACCReturn on equity = .15 + .24/.76 (.15 - .05)(1-.4) = 16.89%WACC = .76 (.1689) +.24(.05)(1-.4) = 13.55%Value of Zipper = 1000/.1355 = 7380

WACC and APVAPV

Need the level of debt for e.g. $ 4m of debtUse this to calculate the present value of tax shieldsValue of Levered firm = Value of all equity firm + tax shields

WACCNeed the fraction of the firm that is debt i.e. D/VUse this to get the WACCValue of levered firms = After Tax Cash Flows/ WACC

APV is the natural way if you know the level of debt and WACC is the natural way if you know the percentage of debt




Introduction

We have seen that issuing debt is valuable as it generates tax savings.What is the tax advantage?This is also the Adjusted Present Value Method or the APVValue of levered firm = Value of all Equity

firm +Value of all effects of Debt

tBVV UL +=

Comparing the Methods

APV WACCCash Flows All Equity All Equity

CF CFDiscount Rates R0 RWACC

APV has the advantage that can easily put in costs of distress and other costs if wantedAll methods should give the same value

WACC and APVAPV � Need the level of debt for e.g. $ 4m of debt� Use this to calculate the present value of tax shields� Value of Levered firm = Value of all equity firm + tax shields

WACC� Need the fraction of the firm that is debt i.e. D/V� Use this to get the WACC� Value of levered firms = After Tax Cash Flows/ WACC

APV is the natural way if you know the level of debt and WACC is the natural way if you know the percentage of debt

Example 1

Beecorp has after tax cash flows of $5000 in perpetuity. The all equity cost for the firm is 14%. It has $3000 of debt at 5%. The tax rate is 40%.

What is the value of Beecorp using APV?Using WACC?

Example 1

Value of all equity firm: 5000/.14 = 35,714Tax shields: t x D = .4 x 3000 = 1200Value (APV) = 35714 + 1200 = 36,914

Cost of levered equity:� Value of equity = 36,914 – 3000 = 33,914

= .14 + 3000/33914 (.14 - .05)(1-.4) = 14.48%Wacc = 33914/36914 x 14.48% + 3000/36914 x 5%(1-.4) = 13.54%Value is 5000/.1354 = 36,927

Example 2

Austin Inc is planning on having 30% debt at a cost of 7%. It all equity cost of capital is 16%. It expects an after tax cash flow of $3000 in perpetuity. If the tax rate is 40% what is the value of Austin using WACC? Using APV?

Example 2

Cost of levered equity:= .16 + .3/.7 x (.16 - .07) (1-.4) = 18.31%

Wacc = .7 x 18.31% + .3 x 7% x (1-.4) = 14.08%Value = 3000/.1408 = $21,311

Using APV: Value of all equity firm = 3000/.16 = 18,750Level of debt (approx) = 30% x 18750 = 5625Tax shields = Dx t = 5625x .4 = 2250Value of firm: 18,750 + 2250 = 21,000

Example 3

Happy cards has 30% debt at 7% and a WACC of 15%. It earns after tax $4000 in perpetuity. If the tax rate is 40% what would the value of the firm be if it became an all equity firm (issued more shares to retire its debt)?

Example 3

From the WACC we can get its levered cost of equity: 0.15 = .7 Re + .3 x 7% x (1-.4)Re = 19.63%

Now using the MM Prop II with taxes we can get the all equity cost 0.1963 = Ra + .3/.7(Ra-.07)x (1-.4)Ra = 17.05%

Value of an all equity firm = 4000/.1705 = 23460

Same Example – with Betas

Zipper Inc. has after tax cash flows of $1000 forever. It has 24% debt at a cost of 5%. If the tax rate is 40% what is the value of Zipper Inc.? The market risk premium is 8%, all equity beta is 1.5 and the risk free rate is 3%?� What is the value of Zipper?

Example cont…

Note: The cost of all equity firm is not given but the details to calculate are. Use that to get to all equity valueAll equity cost of capital is = 3% + 1.5(8%) = 15%The you can proceed as before

When Discount Rates have to be Estimated

Eric Inc. is planning to start a business producing Sodium Chlorate. They have determined that they would like to have 10% debt at interest rate of 8%. There is only one other firm in the industry, Dominant Inc. Dominant Inc has 40% debt at interest rate of 10% and its equity has a beta of 2. The market risk premium is 7% and risk free rate is 4%. The corporate tax rate is 34%. What is the WACC for Eric Inc.?

Example

Step 1: Find Dominant’s cost of equity (using the CAPM) = 4% + 2(7%) = 18%

Step 2: Find the cost of capital for the all equity or unlevered firm (using MM II with taxes)

%)10()34.1(6.04.0

%18 00 −×−×+= RR

%56.150 =R

Example

Step 3: Get Eric Inc. cost of equity

Step 4: Get Eric Inc. WACC

DCDee RTwRwWACC )1( −+=

%)856.15()34.1(9.01.0

%56.1511.16 −×−×+=

%03.15%8)34.1(1.0%)11.16(9.0 =−+=WACC

Another Example

Experiment Inc is planning a new project in manufacturing pen. It is planning to have 20% debt at an interest rate of 6%. Greatpens, a pen manufacturer, with 10% debt at cost of 5%, and an equity beta of 1.8. The riskfree rate is 4% and the risk premium is 7%. If the tax rate is 34%, what is Experiment’s WACC?

Example

Step 1: Cost of levered equity for Greatpens� = 4% + 1.8 (7%) = 16.6%

Step 2: Cost of all equity pen manufacturer(Using MMII with taxes)16.6% = ra + .1/.9(ra-5%)(1-.34)Ra = 15.81%

Step 3: Cost of equity for experiment� Re = 15.81% + .2/.8(15.81% - 6%) (1-.34) = 17.43%

Step 4: WACC is WACC = .8(17.43%) + .2(6%)(1-.34) = 14.73%

Example 2

J. Lowes Corp currently manufactures stapes. It has debt of 20% and its WACC is 13.4%. It is considering a $1 million investment in the aircraft adhesives industry. The project generates EBIT of $300,000 into perpetuity. The firm plans to have 10% debt which is riskless� The three competitors in the new industry are currently

unlevered, with betas of 1.2, 1.3 and 1.4. Assuming a risk free rate of 5%, a market risk premium of 9%, and tax rate of 34%. What is the NPV of the project?

Example 2 cont..Average Un-levered beta (all equity):

(1.2 + 1.3 + 1.4)/3 = 1.3 All equity cost of capital� = 5% + 1.3(9%) = 16.7%. (this is ra)

Cost of levered equity = � = 16.7% + .1/.9(16.7% - 5%)(1-.34) = 17.56%

WACC = .9(17.56%) + .1(5%)(1-.34) = 16.13%Value = 300,000 (1-.34)/.1613 = 1,227,359NPV = 1,227,359 – 1m = 227,359

Note

Do not consider the firms WACC if it is not in the same businessUse the average of all competitorsMake sure they are all unlevered or all equity betas

Example 3

Golfco is currently unlevered with a cost of capital of 12%. It is planning to put on 30% debt at 8% for the new project. What is the relevant WACC for the new project?

Example 3:

Golfco is currently unlevered with a cost of capital of 12%. It is planning to put on 30% debt at 8% for the new project. What is the relevant WACC for the new project? What is the NPV of the project?

Cost of levered equity

WACC = 0.7(13.13%)+0.3(1-.34)8% = 10.78%

%13.13%)8%12()34.1(7.03.0

%12 =−×−×+=SR

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