Chapter 5The Time Value of Money

Laurence Booth, Sean Cleary and Pamela Peterson Drake

Outline of the chapter

5.1 Time is money

• Simple versus compound interest

• Future value of an amount

• Present value of an amount

5.2 Annuities and

perpetuities• Ordinary

annuities• Annuity due• Deferred

annuities• Perpetuities

5.3 Nominal and effective

rates• APR• EAR• Solving for the

rate

5.4 Applications

• Savings plans• Loans and

mortgages• Saving for

retirement

5.1 Time value of money

Simple interestSimple interest is interest that is paid only on the principal amount.

Interest = rate × principal amount of loan

Simple interest: exampleA 2-year loan of $1,000 at 6% simple interest

At the end of the first year, interest = 6% × $1,000 = $60

At the end of the second year,interest = 6% × $1,000 = $60 and loan repayment of $1,000

Compound interestCompound interest is interest paid on both the principal and any accumulated interest.

Interest =

CompoundingCompounding is translating a present

value into a future value, using compound interest.

Future value interest factor is also referred to as the compound factor.

Terminology and notationTerm Notatio

nMeaning

Future value FV Value at some specified future point in time

Present value PV Value today

Interest i Compensation for the use of funds

Number of periods

n Number of periods between the present value and the future value

Compound factor (1 + i)n Translates a present value into a future value

Compare: simple versus compoundSuppose you deposit $5,000 in an account that pays 5% interest per year. What is the balance in the account at the end of four years if interest is:1. Simple interest?2. Compound interest?

Simple interestYear Beginning Add interest Ending

1 $5,000.00 + (5% × $5,000) = $5,250.00

2 $5,250.00 + (5% × $5,000) = $5,500.00

3 $5,500.00 + (5% × $5,000) = $5,750.00

4 $5,750.00 + (5% × $5,000) = $6,000.00

Compound interestYear Beginning Compounding Ending

1 $5,000.00 × 1.05 = $5,250.00

2 $5,250.00 × 1.05 = $5,512.50

3 $5,512.50 × 1.05 = $5,788.13

4 $5,788.13 × 1.05 = $6,077.53

Interest on interest

How much interest on interest?Interest on interest = FVcompound – FVsimple

Interest on interest = $6,077.53 – 6,000.00 = $77.53

Comparison

Year

End of year balance

Simple interest

Compound interest

0 $5,000.00 $5,000.00

1 $5,250.00 $5,250.00

2 $5,500.00 $5,512.50

3 $5,750.00 $5,788.13

4 $6,000.00 $6,077.53 0 1 2 3 4$0

$1,000

$2,000

$3,000

$4,000

$5,000

$6,000

$7,000

$5

,00

0

$5

,25

0

$5

,50

0

$5

,75

0

$6

,00

0

$5

,00

0

$5

,25

0

$5

,51

3

$5

,78

8

$6

,07

8

Simple interestCompound interest

Year in the future

Ba

lan

ce

in

th

e a

cco

un

t

Try it: Simple v. compoundSuppose you are comparing two accounts:The Bank A account pays 5.5% simple

interest. The Bank B account pays 5.4% compound

interest. If you were to deposit $10,000 in each, what balance would you have in each bank at the end of five years?

Try it: Answer1. Bank A: $12,750.002. Bank B: $13,007.78

A note about interestBecause compound interest is so common, assume that interest is compounded unless otherwise indicated.

Short-cutsExample:Consider $1,000 deposited for three years at 6% per year.

The long wayFV1 = $1,000.00 × (1.06) = $1,060.00

FV2 = $1,060.00 × (1.06) = $1,123.60

FV3 = $1,123.60 × (1.06) = $1,191.02

orFV3 = $1,000 × (1.06)3 = $1,191.02 orFV3 = $1,000 × 1.191016 = $1,191.02

Future value factor

Short-cut: CalculatorKnown values:

PV = 1,000n = 3i = 6%

Solve for: FV

Input three known values, solve for the one unknown

HP10B BAIIPlus HP12C TI83/84

1000 +/- PV3 N6 I/YRFV

1000 +/- PV3 N6 I/YRFV

1000 CHS PV3 n6 iFV

[APPS] [Finance] [TVM Solver]N =3I%=6PV = -1000FV [Alpha] [Solve}

Known: PV, i , nUnknown: FV

Short-cut: spreadsheetMicrosoft Excel or Google Docs

=FV(RATE,NPER,PMT,PV,TYPE)TYPE default is 0, end of period

=FV(.06,3,0,-1000)or

A

1 6%

2 3

3 -1000

4 =FV(A1,A2,0,A3)

Problems Set 1

23

Suppose you deposit $2,000 in an account that pays 3.5% interest annually. 1. How much will be in the account at the

end of three years?2. How much of the account balance is

interest on interest?

Problem 1.1

24

If you invest $100 today in an account that pays 7% each year for four years and 3% each year for five years, how much will you have in the account at the end of the nine years?

Problem 1.2

Discounting

DiscountingDiscounting is translating a future value

into a present value.The discount factor is the inverse of the

compound factor: To translate a future value into a present

value, PV=

ExampleSuppose you have a goal of saving $100,000 three years from today. If your funds earn 4% per year, what lump-sum would you have to deposit today to meet your goal?

Example, continuedKnown values:

FV = $100,000n = 3i = 4%

Unknown: PV

Example, continuedPV = =PV = $100,000 × 0.8889964PV = $88,899.64

Check: FV3 = $88,899.64 × (1 + 0.04)3 = $100,000

Short-cut: CalculatorHP10B BAIIPlus HP12C TI83/84

100000 +/- PV3 N4 I/YRPV

100000+/- PV3 N4 I/YRPV

100000 CHS PV3n4iPV

[APPS] [Finance] [TVM Solver]N =3I%=4FV = 100000PV [Alpha] [Solve]

Short-cut: spreadsheetMicrosoft Excel or Google Docs

=PV(RATE,NPER,PMT,PV,TYPE)TYPE default: end of period

=PV(.06,3,0,-1000)or

A

1 6%

2 3

3 100000

4 =PV(A1,A2,0,A3)

Try it: Present valueWhat is the today’s value of $10,000 promised ten years from now if the discount rate is 3.5%?

Try it: AnswerGiven:

FV = $10,000N = 10I = 3.5%Solve for PV

PV = = $7,089.19

Frequency of compounding If interest is compounded more than once

per year, we need to make an adjustment in our calculation.

The stated rate or nominal rate of interest is the annual percentage rate (APR).

The rate per period depends on the frequency of compounding.

Discrete compounding:AdjustmentsAdjust the number of periods and the rate

per period.Suppose the nominal rate is 10% and

compounding is quarterly: The rate per period is 10% 4 = 2.5% The number of periods is

number of years × 4

Continuous compounding:AdjustmentsThe compound factor is eAPR x n.The discount factor is .Suppose the nominal rate is 10%.

For five years, the continuous compounding factor is e0.10 x 5 = 1.6487

The continuous compounding discount factor for five years is 1 ÷ e0.10 x 5 = 0.60653

Try it: Frequency of compoundingIf you invest $1,000 in an investment that pays a nominal 5% per year, with interest compounded semi-annually, how much will you have at the end of 5 years?

Try it: AnswerGiven:

PV = $1,000 n = 5 × 2 = 10 i = 0.05 2 = 0.25Solve for FV

FV = $1,000 × (1 + 0.025)10 = $1,280.08

Problem Set 2

40

Suppose you set aside an amount today in an account that pays 5% interest per year, for five years. If your goal is to have $1,000 at the end of five years, what would you need to set aside today?

Problem 2.1

41

Suppose you set aside an amount today in an account that pays 5% interest per year, compounded quarterly, for five years. If your goal is to have $1,000 at the end of five years, what would you need to set aside today?

Problem 2.2

42

Suppose you set aside an amount today in an account that pays 5% interest per year, compounded continuously, for five years. If your goal is to have $1,000 at the end of five years, what would you need to set aside today?

Problem 2.3

5.2 Annuities and Perpetuities

0 1 2 3 4 5| | | | | |

CF CF CF CF CF

PV? FV?

What is an annuity?An annuity is a periodic cash flow.

Same amount each period Regular intervals of time

The different types depend on the timing of the first cash flow.

Type of annuitiesType First cash flow ExamplesOrdinary One period from

todayMortgage

Annuity due Immediately Lottery paymentsRent

Deferred annuity Beyond one period from today

Retirement savings

Time lines: 4-payment annuity

0 1 2 3 4 5| | | | | |

Ordinary PVCF CF CF CF

FV

Annuity due CFPV

CF CF CFFV

Deferred annuity PV

CF CF CF CFFV

Key to valuing annuitiesThe key to valuing annuities is to get the

timing of the cash flows correct.When in doubt, draw a time line.

Example: PV of an annuityWhat is the present value of a series of three cash flows of $4,000 each if the discount rate is 6%, with the first cash flow one year from today?

0 1 2 3 4| | | |

$4,000 $4,000 $4,000

Example: PV of an annuityThe long way

0 1 2 3 4| | | |

$4,000 $4,000 $4,000$3,773.58

3,559.99 3,358.48

$10,692.05

Example: PV of an annuityIn table form

Year Cash flowDiscount

factorPresent

value1 $4,000.00 0.94340 $3,773.582 $4,000.00 0.89000 3,559.99

3 $4,000.00 0.83962 3,358.48

2.67301$10,692.0

5

PV = $4,000.00 × 2.67301 = $10,692.05

Example: PV of an annuityFormula short-cuts

PV =

PV = $4,000 × 2.67301

PV = $10,692.05

Example: PV of an annuityCalculator short cutsGiven:

PMT = $4,000i = 6%N = 3

Solve for PV

Example: PV of an annuitySpreadsheet short-cuts=PV(RATE,NPER,PMT,FV,TYPE)=PV(.06,3,4000,0)

Note: Type is important for annuities• If Type is left out, it is assumed a 0• 0 is for an ordinary annuity• 1 is for an annuity due

Example: FV of an annuityWhat is the future value of a series of three cash flows of $4,000 each if the discount rate is 6%, with the first cash flow one year from today?

0 1 2 3 4| | | |

$4,000 $4,000 $4,000

Example: FV of an annuityThe long way

0 1 2 3 4| | | |

$4,000.00 $4,000.00 $4,000.00

4,240.00

4,494.40$12,734.40

Example: FV of an annuityIn table form

Year Cash flowCompound

factor Future value1 $4,000.00 1.1236 $4,494.402 $4,000.00 1.0600 4,240.00

3 $4,000.00 1.0000 4,000.00

3.1836 $12,734.40

PV = $4,000.00 × 3.1836 = $12,734.40

Example: FV of an annuityCalculator short cutsCALCULATORGiven:

PMT = $4,000i = 6%N = 3

Solve for FV

Example: FV of an annuitySpreadsheet short-cuts=FV(RATE,NPER,PMT,PV,type)=FV(.06,3,4000,0)

Annuity dueConsider a series of three cash flows of $4,000 each if the discount rate is 6%, with the first cash flow today.1. What is the present value of this annuity?2. What is the future value of this annuity?

The time line0 1 2 3| | | |

$4,000.00 $4,000.00 $4,000.00PV? FV?

This is an annuity due

Valuing an annuity due

Present value

YearEnd of year cash flow

Compound factor

Present value of cash flow

0 $4,000.00 1.00000 $4,000.001 $4,000.00 0.94340 3,773.582 $4,000.00 0.89000 3,559.99

2.83339 $11,333.57

Future value

YearEnd of year cash flow Factor Future value

0 $4,000.00 1.19102 $4,764.061 $4,000.00 1.12360 4,494.402 $4,000.00 1.06000 4,240.00

3.37462 $13,498.46

Valuing an annuity due: Using calculators

Present value

PMT = 4000N = 3I = 6%BEG modeSolve for PV

Future value

PMT = 4000N = 3I = 6%BEG modeSolve for FV

Valuing an annuity due: Using spreadsheetsPresent value=PV(RATE,NPER,PMT,FV,TYPE)=PV(0.06,3,4000,0,1)

Future value=PV(RATE,NPER,PMT,FV,TYPE)=PV(0.06,3,4000,0,1)

Any other way?There is one period difference between an ordinary annuity and an annuity due. Therefore:PVannuity due = PVordinary annuity × (1 + i)

andFVannuity due = FVordinary annuity × (1 + i)

Valuing a deferred annuityA deferred annuity is an annuity that

begins beyond one year from today. That means that it could begin 2, 3, 4, … years

from today, so each problem is unique.

Valuing a deferred annuity0 1 2 3 4 5

| | | |

CF CF CF CF

4-payment ordinary annuity,

then discount value one period

PV0 ←PV1

4-payment annuity due, then

discount value two periods

PV0 ←PV2

Example: Deferred annuityWhat is the value today of a series of five cash flows of $6,000 each, with the first cash flow received four years from today, if the discount rate is 8%?

0 1 2 3 4 5 6 7 8 9 10| | | | |

PV? CF CF CF CF CF

Example, cont.Using an ordinary annuity:PV3 = $23,956.26

PV0 = $19,017.25

Using an annuity due:PV4 = $25,872.76

PV0 = $19,017.25

Discount 3 periods at 8%

Discount 4 periods at 8%

Example: Deferred annuityCalculator solutionsHP10B BAIIPlus TI83/84

0 CF0 CF0 CF0 CF6000 CF6000 CF6000 CF6000 CF6000 CF8 iNPV

0 CF ↑ 1 0 CF ↑ 1 F10 CF ↑ 1 F20 CF ↑ 1 F36000 CF ↑ 1 F46000 CF ↑ 1 F56000 CF ↑ 1 F66000 CF ↑ 1 F76000 CF ↑ 1 F88 iNPV

[2nd] {0 0 0 6000 6000 6000 6000 6000}STO [2nd] L1[APPS] [Finance][ENTER] 7NPV(.08,0,L1)[ENTER]

Example: Deferred annuitySpreadsheet solutions

A B

Year Cash flow

1 1 $0

2 2 $0

3 3 $0

4 4 $6000

5 5 $6000

6 6 $6000

7 7 $6000

8 8 $6000

1. =PV(0.08,3,0,PV(0.08,5,6000,0))2. =PV(0.08,4,0,PV(0.08,5,6000,0,1))3. =NPV(0.08,A1:A9)

PerpetuitiesA perpetuity is an even cash flows that occurs at regular intervals of time, forever.The valuation of a perpetuity is simple:+++…

Problem Set 3

73

Which do you prefer if the appropriate discount rate is 6% per year:1. An annuity of $4,000 for four annual

payments starting today.2. An annuity of $4,100 for four annual

payments, starting one year from today.3. An annuity of $4,200 for four annual

payments, starting two years from today.

Problem 3.1

5.3 Nominal and effective rates

𝑬𝑨𝑹=𝒆

𝑨𝑷𝑹 −𝟏

𝑬𝑨𝑹=(𝟏+𝑨𝑷𝑹𝒎 )

𝒎

%i

APR & EARThe annual percentage rate (APR) is the

nominal or stated annual rate. The APR ignores compounding within a year. The APR understates the true, effective rate.

The effective annual rate (EAR) incorporates the effect of compounding within a year.

APR EAREAR = Suppose interest is stated as 10% per years, compounded quarterly. EAR = EAR = 10.3813%

77

EAR with continuous compoundingEAR = Suppose interest is stated as 10% per years, compounded continuously. EAR = EAR = 10.5171%

Frequency of compounding If interest is compounded more frequently

than annually, then this is considered in compounding and discounting.

There are two approaches1. Adjust the i and n; or2. Calculate the EAR and use this

Example: EAR & compoundingSuppose you invest $2,000 in an investment that pays 5% per year, compounded quarterly. How much will you have at the end of 4 years?

Example: EAR & compoundingMethod 1:

FV = $2,000 (1 + 0.0125)16 = $2,439.78

Method 2:EAR = (1 + )4 – 1 = 5.0945%FV = $2,000 (1 + 0.050945)4 = $2,439.78

Try it: APR & EARSuppose a loan has a stated rate of 9%, with interest compounded monthly. What is the effective annual rate of interest on this loan?

Try it: AnswerEAR = EAR EAR = 9.3807%

Problem Set 4

84

What is the effective interest rate that corresponds to a 6% APR when interest is compounded monthly?

Problem 4.1

85

What is the effective interest rate that corresponds to a 6% APR when interest is compounded continuously?

Problem 4.2

5.4 Applications

Saving for retirementSuppose you estimate that you will need $60,000 per year in retirement. You plan to make your first retirement withdrawal in 40 years, and figure that you will need 30 years of cash flow in retirement. You plan to deposit funds for your retirement starting next year, depositing until the year before retirement. You estimate that you will earn 3% on your funds. How much do you need to deposit each year to satisfy your plans?

Deferred annuity time line

0 1 2 3 4 5 6 7 8 … 39

40

41

42

43

… 79

| | | | | | | | | | | | | |

D D D D D D D D W W W W W W

D = Deposit (39 in total)W = Withdrawal (30 in total)

Deferred annuity time line0 1 2 3 4 5 6 7 8 … 39 40 41 42 43 … 79

| | | | | | | | | | | | | |

W W W W W

PV ← Ordinary annuity

↓

Ordinary annuity → FV

D D D D D D D D … D

Two stepsStep 1: Present value of ordinary annuity

N = 30; i = 3%; PMT = $60,000PV39 = $1,176,026.48

Step 2: Solve for payment in an ordinary annuity

N = 39; i = 3%; FV = $1,176,026.48PMT = $16,280.74

What does this mean?If there are 39 annual deposits of $16,280.74 each and the account earns 3%, there will be enough to allow for 30 withdrawals of $60,000 each, starting 40 years from today.

Balance in retirement account

1 7 13 19 25 31 37 43 49 55 61 67$0

$200,000

$400,000

$600,000

$800,000

$1,000,000

$1,200,000

$1,400,000

Year into the future

Balance in the re-

tirement account

Practice problems

$€

= ¥+£

Problem 1What is the future value of $2,000 invested for five years at 7% per year, with interest compounded annually?

Problem 2What is the value today of €10,000 promised in four years if the discount rate is 4%?

Problem 3What is the present value of a series of five end-of-year cash flows of $1,000 each if the discount rate is 4%?

Problem 4Suppose you plan to save $3,000 each year for ten years. If you earn 5% annual interest on your savings, how much more will you have at the end of ten years if you make your payments at the beginning of the year instead of the end of the year?

Problem 5Sue plans to deposit $5,000 in a savings account each year for thirty years, starting ten years from today. Yan plans to deposit $3,500 in a savings account each year for forty years, starting at the end of this year. If both Sue and Yan earn 3% on their savings, who will have the most saved at the end of forty years?

Problem 6Suppose you have two investment opportunities:Opportunity 1: APR of 12%, compounded

monthlyOpportunity 2: APR of 11.9%, compounded

continuouslyWhich opportunity provides the better return?

Problem 7If you can earn 5% per year, what would you have to deposit in an account today so that you have enough saved to allow withdrawals of $40,000 each year for twenty years, beginning thirty years from today?

Problem 8Suppose you deposit ¥50000 in an account that pays 4% interest, compounded continuously. How much will you have in the account at the end of ten years if you make no withdrawals?

The end