Chapter 7

7

Time Value of Money

C H A P T E R

Chapter Objectives

• Understand the concept of future value, including its calculation

for a single amount of cash received today.

• Understand the concept of present value, including calculation of

the present value of a single payment at a particular time in the

future.

• Calculate the present value of multiple payments received in the

future, including perpetuities and annuities.

• Describe basic concepts of risk and return and their relationship

to computing the present value of future payments.

• Use risk statistics to define risk, including those relating the

performance of financial assets to risk.

The Value of Money

• How much is money worth?

– The value of the dollar one receives today is worth more

than the value of the dollar one receives in the future.

The Time Value of Money

• Represents the concept that the value of money in

today’s dollars decreases in value the further out

into the future it is expected to be received.

Future Value and Present Value

Future Value

• The value of an initial lump sum of money after it is invested over one or more periods of time

• Opposite of present value

Present Value

• The current value of a future amount of money

• PV = CX / (1 + r)

– PV = present value

– CX = cash flow received at the end of year X

– r = appropriate interest rate

Discount Rate

• Discount rate is the interest rate or percentage return that can be earned on an initial amount of money.

– Discount rate = variable r in PV equation

• Example: If a 5% annual return on a one-year CD would yield $102,500, then a payment of $102,500 earned in one year is valued at $97,619 today.

– In other words, a company would find $97,619 paid today or $102,500 paid one year from today as equally acceptable alternatives.

– If paid $97,619 today, they could place the money in the one-year CD at 5% and receive $102,500 in one year.

Compounding

• Compounding: The process of holding the stock and accruing a further return over the second year.

• Compound interest: With many loans, interest is added to the principal loaned amount over the loan’s life.

– Each payment of interest or return that is reinvested earns a return also.

– Example: If you borrowed $100 at 10% interest compounded annually, you would owe $110 after the first year, $121 for the second year ($110 x 110%), $133 for the third year ($121 x 110%), and so on throughout the life of the loan (if you were making 0 payments).

Calculating Future Value

The following formula can be used to calculate future value:

• FV = C0 x (1 + r)t

– C0 = the initial amount of cash that is invested today

– r = the interest rate or rate of return

– t = the number of years over which the cash is invested.

• Note: Table A.1 in appendix A presents future values of $1 at the end of t periods, which means a variable number of time periods into the future over which the $1 is earning interest.

– To use the table, locate the appropriate interest rate or return on the horizontal axis and the appropriate number of periods on the vertical axis.

• If you started out with $500 and earned an 8% annual return over two years, then the future value at the end of two years would be:

FV = 500 x 1.1664 = $583.20

Calculating Present ValueThe following formula can be used to calculate present value over more than one period in the future:

PV x (1 + r)t = FV

• FV = future value

• PV = present value

• t = number of years

• r = annual rate of return

Example: Assume that Swish James, the best shooter on the team, has signed a contract including $2 million in deferred compensation that will be paid at the end of two years.

• What is the present value of this contract if he could earn a 4% annual return on his money?

Example: PV x (1.04)2 = $2,000,000

• PV = $2,000,000 / (1.04)2 = $2,000,000 / 1.0816 = $1,849,112

• The present value of $2 million received in two years is $1,849,200.

(continued)

Calculating Present Value (continued)

• An alternative way to calculate the results obtained is to use table A.2 in appendix A in order to obtain a present value factor.

– To use the table, locate the appropriate number of periods on the vertical axis and the interest rate on the horizontal axis.

• Example: There are two periods and a 4% interest rate.

– Thus, the present value factor, or discount factor, is 0.9246.

• This process of obtaining a present value is known as discounting.

– If we multiply $2 million by the present value factor (0.9246), we find that the present value of $2 million received in two years is $1,849,200.

Perpetuities and Annuities

• There are shortcuts for calculating present values.

• In particular, we discuss shortcut methods for two

types of cash flow streams:

1. Perpetuities

2. Annuities

Perpetuities

Perpetuity: A single cash flow per year forever into the

future.

• The concept of a perpetuity comes into play when valuing

preferred stock since the price of a preferred share is equal to

the present value of expected constant dividends that will be

paid each year forever into the future.

(continued)

Perpetuities (continued)

• The present value of a perpetuity is equal to

PV = C / (1 + r) + C / (1 + r)2 + C(1 + r)3 + . . .

o C = a constant annual cash flow

o r = the interest rate or rate of return

• Which results in PV = C / r

Present Value of Perpetuities

• Example: Assume the perpetuity pays $100 per

year at an interest rate of 8%; then the present

value of the perpetuity is

– PV = $100 / 0.08 = $1,250

• Example: If interest rates fell to 6%, the present

value of the perpetuity would be

– PV = $100 / 0.06 = $1,666.67

Annuities

• Annuity: A constant stream of payments that is received for a fixed number of periods.

– Annuities are common in the real world.

• Examples: home mortgages, leases, and pensions paid at retirement

• When evaluating annuities, we utilize the following formula:

PV = C / (1 + r) + C / (1 + r)2 + C / (1 + r)3 + . . . + C / (1 + r)T

PV = present value

C = constant cash flow per period

r = annual rate of return

T = number of periods during which the cash flow will be received

Present Value of Annuities

• Like the perpetuity formula, the annuity formula can be simplified to yield the following formula for an annuity that is paid over t periods (Brigham & Ehrhardt, 2010):

PV = C{1 – [1 / (1 + r) n]} / r

PV = present value

C = constant cash flow per period

r = annual rate of return

n = number of periods during which the cash flow will be received

Example: Present Value of Annuities• In August of 2009, Nick Saban, the head football coach of the

Alabama Crimson Tide, signed a new contract that would pay him

approximately $4,000,000 in salary for nine years (Low, 2009).

• For our purposes, this is treated as a nine-year annuity, effective

August 1, 2006. If the appropriate discount rate is 7%, then

PV = $4,000,000 x {1-[1/(1.07)9]} / 0.07

o = periodic payment x annuity factor

o = $4,000,000 x 6.5152 = $26,060,800

• The present value of Saban’s contract, as of August 1, 2009, when

he signed it (and assuming a 7% discount rate), equaled

$26,060,800. The numbers in braces in the equation are the present

value factor for an annuity. In this example involving the value of

Saban’s salary, the present value factor for the annuity equals

6.5152.

Risk• When we calculated the present value of future cash

flows, we did not discuss how to choose a discount rate.

– The choice of discount rates is tied to the risk of the future cash flows.

Risk: Uncertainty concerning the future cash flows.

• In the previous example, we calculated the cash flows without making any allowances for future uncertainty about whether Saban would actually receive all of that compensation.

• As it happens, the job tenure of head coaches in college football is tenuous at best.

• To measure risk, we need to examine the behavior of returns on different types of financial assets.

Capital Gains and Losses

• Capital gains: The total gains in value of an investment excluding dividends.

– However, dividends can be included in the capital gains if the dividends are reinvested in additional shares of the same stock.

• Capital losses: All realized losses from an investment.

• Long-term capital gains or losses: Investments that have matured for over one year and result in capital gains or losses that must be reported to the IRS.

• Short-term capital gains or losses: Applies to investments held for less than one year.

Rate of ReturnDividend yield = Dt+1 / P0

– Dt+1 = dividend paid per share in a given year

– P0 = price when the stock was initially purchased

Rate of return = (P1 – P0) / P0

– P1 = stock price at the end of year 1

– P0 = price when the stock was initially purchased

Average rate of return (R) = (R1 + R2 + . . . + Rt) / t

– R1 = rate of return in year 1

– R2 = rate of return in year 2

– Rt = rate of return in year t

– t = number of years

Questions for Class Discussion

1. Evaluate the following statement: “Because a well-diversified portfolio of

stocks tends to outperform U.S. Treasury bills, there is no reason to invest

in Treasury bills.”

2. Increased criticism has been directed to corporate executives for being too

short-term oriented and not focused enough on the long-run performance

of their companies. Should companies place an emphasis on maximizing

long-term profits?

3. What happens to the present value of an annuity if the discount rate is

increased? What happens to the future value if the discount rate is

increased?

(continued)

Questions for Class Discussion (continued)

4. Assume that two athletes sign 10-year contracts that pay out a total of

$100 million over the life of the contracts. One contract will pay the $100

million in equal installments over the 10 years. The other contract will pay

the $100 million in installments, but the installments increase 5% per

year. Which athlete received the better deal?

5. A professional sport athlete, let’s call her Sue, has a nonguaranteed

(meaning that she receives no salary if she is cut from the team or

injured) contract that pays her $5 million per year for the next five years.

Sue has been severely injured in an automobile accident and will no

longer be able to work, let alone play for her team. When Sue sues the

driver of the other vehicle in court, the jury awards her $25 million to

cover the loss of earnings over the next five seasons. Given what we

have learned in this chapter, was the jury correct?

(continued)

Questions for Class Discussion (continued)

6. The returns in table 7.3 in your book (page 135) have not been adjusted

for inflation. What would happen to the expected return if we adjusted the

annual returns for inflation?

7. Table 7.3 demonstrated that stocks outperform Treasury bills over long

periods. If that is the case, why do some people avoid the stock market

and invest only in Treasury bills over long periods?

8. What kind of asset would have a beta of zero?