Chapter 5

The Time Value of Money

Time Value

• The process of expressing

–the present in the future (compounding)

–the future in the present (discounting)

Time Value

• Payments are either

–a single payment

–a series of equal payments (an annuity)

Time Value

• Time value of money problems may be solved by using

–Interest tables

–Financial calculators

–Software

Variables for Time Value of Money Problems

• PV = present value

• FV = future value

• PMT = annual payment

• N = number of time periods

• I = interest rate per period

Financial Calculators

• Express the cash inputs (PV, FV, and PMT) as cash inflows and cash outflows

• At least one of the cash variables must be

–an inflow (+)

–an outflow (-)

Future Value

• The future value of $1 takes a single payment in the present into the future

• The general equation for the future value of $1:

P0(1 + i)n = Pn

Future Value Illustrated

• PV = -100

• I = 8

• N = 20

• PMT = 0

• FV = ?

• = 466.10

Greater Terminal Values

• Higher interest rates

• Longer time periods

• Result in greater terminal values

Greater Terminal Values

Present Value

• The present value of $1 brings a single payment in the future back to the present

• The general equation for the present value of $1:P0 = Pn

(1+i)n

Present Value Illustrated

• FV = 100• I = 6• N = 5• PMT = 0• PV = ?

• = -74.73

Lower Present Values

• Higher interest rates

• Longer time periods

• Result in lower present values

Lower Present Values

Annuities - Future Sum

• The future sum of annuity takes a series of payments into the future

• Payments may be made–at the end of each time period

(ordinary annuity)–at the beginning of each time

period (annuity due)

FV Time Lines

•Ordinary annuity

Year Payment - $100 100 100

0 1 2 3

FV Time Lines

•Annuity due

Year Payment $100 100 100 -

0 1 2 3

Future Value of an Ordinary Annuity Illustrated

• PV = 0• PMT = -100• I = 5• N = 3• FV = ?

• = 315.25

Greater Terminal Values

• Higher interest rates

• Longer time periods

• Result in greater terminal values

Greater Terminal Values

Annuities - Present Value

• The present value of an annuity brings a series of payments in the future back to the present

Present Value of an Ordinary Annuity Illustrated

• FV = 0• PMT = 100• I = 6• N = 3• PV = ?

• = -267.30

Annuities - Present Value

• Higher interest rates result in lower present values

• But longer time periods increases the present value (because more payments are received)

Annuities - Present Value

Additional Time Value Illustrations

• The following is a series of problems or questions that use the time value of money.

Illustration 1

• You deposit $1,000 in an account at the end of each year for twenty years. What is the total amount in the account if you earn 6 percent annually?

Future Value of an Ordinary Annuity

• The unknown: FV

• The givens:

–PV = 0

–PMT = -1,000

–N = 20

–I = 6

The answer:$36,786

Interpretation

• For an annual cash payment of $1,000, you will have $36,786 after twenty years

• Of the $36,786

–$20,000 is the total cash outflow

–$16,786 is the earned interest

Illustration 2

• What is the present value of (or required cash outflow to purchase) an ordinary annuity of $1,000 for twenty years, if the rate of interest is 6 percent?

Present Value of an Annuity

• The unknown: PV

• The givens:

–FV = 0

–PMT = 1,000

–N = 20

–I = 6

The answer:$11,470

Interpretation

• For a present payment of $11,470, the individual will annually receive $1,000 for the next twenty years

• The $11,470 is an immediate cash outflow

• The $1,000 annual payment to be received is a cash inflow

Illustration 3

• What is the future value after ten years of a stock that cost $10 and appreciates at 9 percent annually?

Future Value of $1

• The unknown: FV

• The givens:

–PV = 10

–PMT = 0

–N = 10

–I = 9

The answer: $23.67

Interpretation

• A $10 stock will be worth $23.67 after 10 year if its price grows 9% annually.

Illustration 4

• What is the cost of a stock that was sold for $23.67, held for 10 years and whose value appreciated 9 percent annually?

Present Value of $1

• The unknown: PV

• The givens:

–FV = 23.67

–PMT = 0

–N = 10

–I = 9

The answer:$10

Interpretation

• $23.67 received after ten years is worth $10 today if the return rate is 9 percent.

Interpretation of Future and Present Values

These two problems are the same:

• In the first case the $10 is compounded into its future value ($23.67)

• In the second case the future value ($23.67) is discounted back to its present value ($10)

Illustration 5

• A stock was purchased for $10 and sold for $23.67 after 10 years. What was the return?

Future Value of I

• The unknown: I

• The givens:

–PV = 10

–PMT = 0

–N = 0

–FV = 23.67

The answer: 9%

Interpretation

• The yield on a $10 investment that was sold after 10 years for $23.67 is 9%.

Illustration 6

• If an investment pay $50 a year for 10 years and repays $1,000 after 10 years, what is this investment worth today if you can earn 6 percent?

Determination of Present Value

• The unknown: PV

• The givens:

–FV = 1,000

–PMT = 50

–I = 6

–N = 10

The answer: $926

Interpretation

• If you collect $50 a year for 10 years and receive $1,000 after 10 years, those cash inflows are currently worth $926 at 6 percent.

Illustration 7

• Time value is used to determine a loan repayment schedule such as a mortgage.

Loan Repayment Schedule

• Amount borrowed (PV) = $80,000

• Interest rate (I) = 8%

• Term of the loan (N) = 25 years

• No future value since loan is repaid

• Amount of the annual payment = $7,494.30

Loan Repayment Schedule

Principal Balance

Pmnt Interest Repayment Owed

1 $6,400.00 $1,094.15 $78,905.85

2 6,312.47 1,181.68 77,724.17

. . . .

. . . .

. . . .

25 555.13 6,939.17 .00

Illustration 8

• You have $115,000 and spend $24,000 a year. If you earn 8% annually, how long will your funds last?

Determination of Number of Years

• The unknown: N

• The givens:

–PV = 115,000

–I = 8

–FV = 0

–PMT = -24,000

The answer: 6.3 years

Interpretation

• If you have $115,000 and earn 8 percent annually, you can spend $24,000 per year for approximately 6 years and 4 months.

Non-annual Compounding

• More than one interest payment a year

• More frequent compounding

Non-annual Compounding

• Multiply number of years by frequency of compounding

• Divide interest rate by frequency of compounding

Periods less than One Year

• Same variables as in all time value problems except N < 1.

Illustration 9

• What is the return on an ivestment that costs $98,543 and pays $100,000 after 45 days?

Determination of Return

• The unknown: I

• The givens:

–PV = 98,543

–N = 0.1233

–FV = 100,000

–PMT = 0

The answer: 12.64%

Interpretation

• $98,543 invested for 45 days grows to $100,000 at 12.64 percent.