Chapter 4 The Time Value of Money. Essentials of Chapter 4 Why is it important to understand and...

Chapter 4The Time Value of Money

Essentials ofChapter 4

Why is it important to understand and apply time value to money concepts?

What is the difference between a present value amount and a future value amount?

What is an annuity?

What is the difference between the Annual Percentage Rate and the Effective Annual Rate?

What is an amortized loan?

How is the return on an investment determined?

Time Value of Money

The most important concept in finance

Used in nearly every financial decisionBusiness decisions

Personal finance decisions

Time 0 is todayTime 1 is the end of Period 1 or the beginning of Period 2.

Graphical representations used to show timing of cash flows

Cash Flow Time Lines

Time:

FV = ?

0 1 2 36%

PV=100Cash Flows:

Future Value

The amount to which a cash flow or series of cash flows will grow over a period of time when compounded at a given interest rate.

PV = Present value, or the beginning amount that is invested

r = Interest rate the bank pays on the account each year

INT = Dollars of interest you earn during the year

FVn = Future value of the account at the end of n periods

n = number of period interest is earned

Terms Used in Calculating Future Value

Future Value

In general, FVn = PV (1 + r)n

Four Ways to Solve Time Value of Money Problems

Use Cash Flow Time Line

Use Equations

Use Financial Calculator

Use Electronic Spreadsheet

Time(t):

133.10

0 1 2 310%

100.00Account balance: x (1.10) x (1.10) x (1.10)

110.00 121.00

The Future Value of $100 invested at 10% per year for 3 years

Time Line Solution

Numerical (Equation) Solution

FVn PV(1 r)n

FV3 =$100(1.10)3

=$100(1.331)=133.10

Financial Calculator Solution

INPUTS

OUTPUT

3 10 -100 0 ? N I/YR PV PMT FV

133.10

Spreadsheet Solution

Click on Function Wizard and choose Financial/FV

Set up Problem

Spreadsheet SolutionReference cells:

Rate = interest rate, r

Nper = number of periods interest is earned

Pmt = periodic payment

PV = present value of the amount

Present Value

Present value is the value today of a future cash flow or series of cash flows.

Discounting is the process of finding the present value of a future cash flow or series of future cash flows; it is the reverse of compounding.

What is the PV of $100 due in 3 years if r = 10%?

Time(t):

100.00

0 1 2 310%

75.13Account balance: x (1.10) x (1.10) x (1.10)

82.64 90.90

What is the PV of $100 due in 3 years if r = 10%?

$75.13 =0.7513$100 =

PV = FVn 1

(1+r)n

PV = $100 1

(1.10)3

Financial Calculator Solution

INPUTS

OUTPUT

3 10 ? 0100

N I/YR PV PMT FV

-75.13

Spreadsheet Solution

What interest rate would cause $100 to grow to $125.97 in 3 years?

100 (1 + r )3 = 125.97

100.00 = FV

0 1 2 3

r = ?

PV = 100 100 = 125.97/ (1 + r )3 ] [

$100 (1 + r )3 = $125.97

INPUTS

OUTPUT

3 ? -100 0 125.97

8%

N I/YR PV PMT FV

What interest rate would cause $100 to grow to $125.97 in 3 years?

Spreadsheet Solution

How many years will it take for $68.30 to grow to $100 at an interest rate of 10%?

68.30 (1.10 )n = 100.00

100.00 = FV

0 1 2 n-1

10%

PV = 68.30 68.30 = $100.00 (1.10 )n ] [

. . .

n = ?

How many years will it take for $68.30 to grow to $100 if interest of 10% is paid each year?

FVn = PV(1+r)n

$100.00 = $68.30 (1.10)n

FV4 = 68.30(1.10)4 = $100.00

INPUTS

OUTPUT

? 10 -68.30 0 100.00

4.0

N I/YR PV PMT FV

How many years will it take for $68.30 to grow to $100 if interest of 10% is paid each year?

Spreadsheet Solution

Future Value of an Annuity

Annuity: A series of payments of equal amounts at fixed intervals for a specified number of periods.

Ordinary (deferred) Annuity: An annuity whose payments occur at the end of each period.

Annuity Due: An annuity whose payments occur at the beginning of each period.

PMT PMTPMT

0 1 2 3r%

PMT PMT

0 1 2 3r%

PMT

Ordinary Annuity

Annuity Due

Ordinary Annuity Versus Annuity Due

100 100100

0 1 2 35%

105

110.25

FV = 315.25

What’s the FV of a 3-year Ordinary Annuity of $100 at 5%?

FVAn PMT (1 r)n

t0

n1

PMT

(1 r)n 1r

FVA3 $100(1.05)3 1

0.05

$100(3.15250)$315.25

Numerical Solution:

Financial Calculator Solution

INPUTS

OUTPUT

3 5 0 -100 ?

N I/YR PV PMT FV

315.25

Spreadsheet Solution

Find the FV of an Annuity Due

100 100

0 1 2 35%

100

105.00

110.25

115.76

331.01

x[(1.05)0](1.05)

x[(1.05)1](1.05)

x[(1.05)2](1.05)

FVA(DUE)3 =

Numerical Solution

FVA(DUE)n PMT (1 r)t

t1

n

PMT

(1+r)n 1

r

x(1 r)

FVA(DUE)3 $100(1.05)3 1

0.05

x(1.05)

= $100[3.15250x1.05]

= $100[3.310125] = 331.01

Financial Calculator Solution

Switch from “End” to “Begin”.

INPUTS

OUTPUT

3 5 0 -100 ? N I/YR PV PMT FV

331.01

Spreadsheet Solution

Present Value of an Annuity

PVAn = the present value of an annuity with n payments.

Each payment is discounted, and the sum of the discounted payments is the present value of the annuity.

248.69 = PV

100 100100

0 1 2 310%

90.91

82.64

75.13

What is the PV of this Ordinary Annuity?

PVAn PMT1

(1 r)tt1

n

PMT

1- 1(1r)n

r

$248.6985)$100(2.486

0.10

-1$100PVA

3(1.10)1

3

Numerical Solution

Financial Calculator Solution

INPUTS

OUTPUT

3 10 ? -100 0 N I/YR PV PMT FV

-248.69

Spreadsheet Solution

100 100

0 1 2 35%

100

100.00

95.24

90.70

285.94

(1.05)x[1/(1.05)1]x

PVA(DUE)3=

(1.05)x[1/(1.05)2]x

(1.05)x[1/(1.05)3]x

Find the PV of an Annuity Due

Financial Calculator Solution

INPUTS

OUTPUT

3 5 ? -100 0 N I/YR PV PMT FV

285.94

250 250

0 1 2 3r = ?

- 864.80

4

250 250

You pay $864.80 for an investment that promises to pay you $250 per year for the next four years, with payments made at the end of each year. What interest rate will you earn on this investment?

Solving for Interest Rates with Annuities

Use trial-and-error by substituting different values of r into the following equation until the right side equals $864.80.

Numerical Solution

$864.80$250

1- 1(1r)4

r

Financial Calculator Solution

INPUTS

OUTPUT

4 ? -846.80 250 0 N I/YR PV PMT FV

7.0

Spreadsheet Solution

Annuities that go on indefinitely

PVP Payment

Interest rate

PMTr

Perpetuities

Uneven Cash Flow Streams

A series of cash flows in which the amount varies from one period to the next:Payment (PMT) designates constant cash

flows—that is, an annuity stream.

Cash flow (CF) designates cash flows in general, both constant cash flows and uneven cash flows.

0

100

1

300

2

300

310%

-50

4

90.91

247.93

225.39

-34.15530.08 = PV

What is the PV of this Uneven Cash Flow Stream?

Numerical Solution

PVCF1

1(1 r)1

CF2

1(1 r)2

...CFn

1(1 r)n

4321 (1.10)

150)(

(1.10)

1300

(1.10)

1300

(1.10)

1100PV

$100(0.90909)$300(0.82645)$300(0.75131) ( $50)(0.68301)

$530.09

Financial Calculator Solution

Input in “CF” register:CF0 = 0CF1 = 100CF2 = 300CF3 = 300CF4 = -50

Enter I = 10%, then press NPV button to get NPV = 530.09. (Here NPV = PV)

Spreadsheet Solution

Semiannual and Other Compounding Periods

Annual compounding is the process of determining the future value of a cash flow or series of cash flows when interest is added once a year.

Semiannual compounding is the process of determining the future value of a cash flow or series of cash flows when interest is added twice a year.

If compounding is more frequent than once a year—for example, semi-annually, quarterly, or daily—interest is earned on interest—that is, compounded—more often.

LARGER!

Will the FV of a lump sum be larger or smaller if we compound more often, holding the stated r constant? Why?

0 1 2 310%

100133.10

Annually: FV3 = 100(1.10)3 = 133.100 1 2 3 4 5 6

5%

134.01100

Semi-annually: FV6/2 = 100(1.05)6 = 134.01

Compounding Annually vs. Semi-Annually

rSIMPLE = Simple (Quoted) RateSimple (Quoted) Rate used to compute the interest paid per period rEAR = Effective Annual RateEffective Annual Ratethe annual rate of interest actually being earned

APR = Annual Percentage RateAnnual Percentage Rate = rSIMPLE periodic rate X the number of periods per year

Distinguishing Between Different Interest Rates

Comparison of Different Types of Interest Rates

rSIMPLE: Written into contracts, quoted by

banks and brokers. Not used in calculations or shown on time lines.

rPER: Used in calculations, shown on time

lines.

rEAR: Used to compare returns on

investments with different payments per year.

Simple (Quoted) Rate

rSIMPLE is stated in contracts Periods per year (m) must also be given

Examples:8%, compounded quarterly8%, compounded daily (365 days)

Periodic Rate

Periodic rate = rPER = rSIMPLE/m, where m is number of compounding periods per year. m = 4 for quarterly, 12 for monthly, and 360 or 365 for daily compounding.

Examples:8% quarterly: rPER = 8/4 = 2%

8% daily (365): rPER = 8/365 = 0.021918%

The annual rate that causes PV to grow to the same FV as under multi-period compounding.

Example: 10%, compounded semiannually:

rEAR = (1 + rSIMPLE/m)m - 1.0

= (1.05)2 - 1.0 = 0.1025 = 10.25%

Effective Annual Rate

1 - m

r + 1 = r

mSIMPLE

EAR

10.25% = 0.1025 =1.0 - 1.05 =

1.0 - 2

0.10 +1 =

2

2

How do we find rEAR for a simple rate of 10%, compounded semi-annually?

FVn = PV 1 + rSIMPLE

m

mn

$134.0110)$100(1.3402

0.10 + 1$100 = FV

32

23

$134.4989)$100(1.3444

0.10 + 1$100 = FV

34

43

FV of $100 after 3 years if interest is 10% compounded semi-annual? Quarterly?

0 0.25 0.50 0.7510%

- 100

1.00

FV = ?

Example: $100 deposited in a bank at EAR = 10% for 0.75 of the year

INPUTS

OUTPUT

0.75 10 -100 0 ?

107.41

N I/YR PV PMT FV

Fractional Time Periods

Spreadsheet Solution

Amortized Loans

Amortized Loan: A loan that is repaid in equal payments over its life

Amortization tables are widely used for home mortgages, auto loans, business loans, retirement plans, and so forth to determine how much of each payment represents principal repayment and how much represents interest

PMT PMTPMT

0 1 2 310%

-1,000

Construct an amortization schedule for a $1,000, 10 percent loan that requiresthree equal annual payments.

PMT PMTPMT

0 1 2 310%

-1000

Step 1: Determine the required payments

INPUTS

OUTPUT

3 10 -1000 ? 0 N I/YR PV PMT FV

402.11

INTt = Beginning balancet (r)

INT1 = 1,000(0.10) = $100.00

Step 2: Find interest charge for Year 1

Repayment = PMT - INT= $402.11 - $100.00= $302.11.

Step 3: Find repayment of principal in Year 1

Ending bal. = Beginning bal. - Repayment

= $1,000 - $302.11 = $697.89.

Repeat these steps for the remainder of the payments (Years 2 and 3 in this case) to complete the amortization table.

Step 4: Find ending balance after Year 1

Spreadsheet Solution

Loan Amortization Table10 Percent Interest Rate

YR Beg Bal PMT INT Prin PMT End Bal

1 $1000.00 $402.11 $100.00 $302.11 $697.89

2 697.89 402.11 69.79 332.32 365.57

3 365.57 402.11 36.56 365.55 0.02

Total 1,206.33 206.35 999.98 *

* Rounding difference

Chapter 4 EssentialsWhy is it important to understand and apply time value to money concepts?To be able to compare various investments

What is the difference between a present value amount and a future value amount?Future value adds interest - present value subtracts

interest

What is an annuity?A series of equal payments that occur at equal time

intervals

Chapter 4 EssentialsWhat is the difference between the Annual Percentage Rate and the Effective Annual Rate?APR is a simple interest rate quoted on loans. EAR is the

actual interest rate or rate of return.

What is an amortized loan?A loan paid off in equal payments over a specified period

How is the return on an investment determined?The amount to which the investment will grow in the

future minus the cost of the investment