Lecture 2 Present Values

Corporate FinanceLecturer: Quan, Qi

Winter 2010

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Topics Covered

Introduction to present value Opportunity cost of capital Net present value How to value long-lived assets Shortcuts that make PV calculation easy Compound interest vs. simple interest Nominal and real rates of interest

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Introduction to present value 1)

The basics about understanding how assets are valued– How to find real assets that are worth more than they cost?

The bridge between ‘present’ and ‘future’ Some examples:

– Advance sale of Expo tickets

– Do you want to buy a pair of shoes now or later? Additional note: PV is a way of calculating the price of

“waiting”

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Introduction to present value 2)

A dollar today is worth more than a dollar tomorrow– The dollar today can be invested to earn interest– The reward for patience (the costs of impatience)

If you invest 100 at 5% interest, what do you have a year from now?

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Introduction to present value 3)

Present value (PV) = discount factor * C1

Present Value

Value today of a future cash flow

Discount rate

Interest rate used to compute present values of future cash flows

Discount Factor

Present value of 1 yuan future payment

r1

1

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Opportunity cost of capital

What is the opportunity cost of capital– The return forgone by investing in the project rather than

investing in comparable securities

Risk– The uncertainty– A second basic financial principle: A safe dollar is worth more

than a risky one

A common source of confusion– The opportunity cost of capital is decided by the risk of the

project, not by the cost of debt

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Net present value 1)

NPV=PV-required investment Return = Profit/investment Two equivalent rules for capital investment

– Net present value rule: Accept investments that have positive net present values

– Rate of return rule: Accept investments that offer rates of return in excess of their opportunity cost of capital

r

CC

1

10

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Net present value 2)

Example:The cost of the building an office building is 370,000;Your real estate

agent tells you that the building would be worth 420,000 1 year from now

– Scenario 1 (if the project is completely safe): PV=420,000/(1+5%)=400,000;NPV=-370,000+400,000=30,000

– Scenario 2 (if the project is risky; using 12% cost of capital)– PV=420,000/(1+12%)=375,000; NPV=-370,000+375,000=5000

Expected Return=Expected profit/investment=(420,000-370,000)/370,000=13.5%

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Net present value 3)

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How to value long-lived assets 1)

We use a discounted cash flow (DCF) formula to calculate the present value of any cash flow

For one period

For multi-period

With this formula, you are allowed to add up all the cash flows from different time periods

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How to value long-lived assets 2)

Example: Calculating NPV for multi-period (using the present value table):

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How to value long-lived assets 3)

Term structure of interest rate: the relationship between the interest rate and maturity

Is there a money machine?– Suppose r1 is 20% and r2 is 7%, what is wrong?– DF1=0.83 whereas DF2=0.87– What is the arbitrage strategy?

There is no such thing as a money machine, especially in a well-functioning capital market

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Shortcuts that make PV calculation easy 1)

How to value perpetuities– The bonds that the government is under no obligation to repay but that

offer a fixed income for each year to perpetuity – Return=cashflow/present value

How to value growing perpetuities

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How to value annuities

Shortcuts that make PV calculation easy 2)

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Shortcuts that make PV calculation easy 3)

Example 1: If you lease a car for 1 year at 1000 yuan per month. No additional payments required. If your opportunity cost of capital is 1% per month, what is the cost of lease?

Lease cost=1000(1/0.01-1/0.01*(1+0.01)12)=1000*11.26=11,260

Example 2: You take out a $250,000 house mortgage, with equal monthly installments over the next 30 years. The interest rate is 1% a monthPV=mortgage payment *360-month annuity factor=250,000

Mortgage payment=$250,000/360-month annuity factor=250,000/97.218=$2,572

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Compound interest vs. simple interest 1)

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Compound interest vs. simple interest

By compounding, you get interest on your interest Problems in finance almost always involve compound

interest rather than simple interest Discounting is a process of compound interest Different compounding intervals

– If an investment is compounded m times a year, the equivalent annually compounded rate of interest is [1+(r/m)]m-1

Continuous compounding: – 1 yuan invested at a continuously compounded rate of r will grow to

ert=(2.718)r

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Nominal and real rates of interest

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Assignments

Practice question 8 and 24 in 8th edition As winner of a breakfast cereal competition, you can choose one of

the following prizes: a. $100,000 now. b. $180,000 at the end of five years. c. $11,400 a year forever. d. $19,000 for each of 10 years. e. $6,500 next year and increasing thereafter by 5 percent a year

forever.

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Assignments

You are considering the purchase of an apartment complex that will generate a net cash flow of $400,000 per year. You normally demand a 10 percent rate of return on such investments. Future cash flows are expected to grow with inflation at 4 percent per year. How much would you be willing to pay for the complex if it:

a. Will produce cash flows forever?b. Will have to be torn down in 20 years? Assume that the site will be worth$5 million at that time net of demolition costs. (The $5 million includes 20 years’inflation.)Now calculate the real discount rate corresponding to the 10 percent nominal

rate.Redo the calculations for parts (a) and (b) using real cash flows. (Your answers

shouldnot change.)