Chapter 5 The Time Value of Money Laurence

Chapter 5 The 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 interest is interest that is paid only on the principal amount. Interest = rate × principal amount of loan

Simple interest: example A 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 interest

Compounding

Terminology and notation Term Notation Meaning 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 compound Suppose 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 interest Year 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 interest Year 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 $5, 000. 00 1 $5, 250. 00 2 $5, 500. 00 $5, 512. 50 3 $5, 750. 00 $5, 788. 13 4 $6, 000. 00 $6, 077. 53 $5, 000 $6, 078 $5, 750 $5, 788 $5, 500 $5, 513 Compound interest $5, 250 $6, 000 $5, 000 0 $7, 000 Balance in the account Year End of year balance Simple Compound interest Simple interest $4, 000 $3, 000 $2, 000 $1, 000 $0 0 1 2 3 Year in the future 4

Try it: Simple v. compound Suppose 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: Answer 1. 2. Bank A: $12, 750. 00 Bank B: $13, 007. 78

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

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

The long way FV 1 = $1, 000. 00 × (1. 06) = $1, 060. 00 FV 2 = $1, 060. 00 × (1. 06) = $1, 123. 60 FV 3 = $1, 123. 60 × (1. 06) = $1, 191. 02 or FV 3 = $1, 000 × (1. 06)3 = $1, 191. 02 or FV 3 = $1, 000 × 1. 191016 = $1, 191. 02 Future value factor

Short-cut: Calculator Known values: PV = 1, 000 n = 3 i = 6% Solve for: FV

Input three known values, solve for the one unknown Known: Unknown: PV, i , n FV HP 10 B BAIIPlus HP 12 C TI 83/84 1000 +/- PV 3 N 6 I/YR FV 1000 CHS PV 3 n 6 i FV [APPS] [Finance] [TVM Solver] N =3 I%=6 PV = -1000 FV [Alpha] [Solve}

Short-cut: spreadsheet Microsoft 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(A 1, A 2, 0, A 3)

Problems Set 1

Problem 1. 1 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? 23

Problem 1. 2 If you invest $100 today in an account that pays 7% each year four years and 3% each year for five years, how much will you have in the account at the end of the nine years? 24

Discounting

Discounting

Example Suppose 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, continued Known values: FV = $100, 000 n = 3 i = 4% Unknown: PV

Example, continued

Short-cut: Calculator HP 10 B BAIIPlus HP 12 C TI 83/84 100000 +/- PV 3 N 4 I/YR PV 100000 CHS PV 3 n 4 i PV [APPS] [Finance] [TVM Solver] N =3 I%=4 FV = 100000 PV [Alpha] [Solve]

Short-cut: spreadsheet Microsoft 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(A 1, A 2, 0, A 3)

Try it: Present value What is the today’s value of $10, 000 promised ten years from now if the discount rate is 3. 5%?

Try it: Answer

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 period depends on the frequency of compounding.

Discrete compounding: Adjustments Adjust the number of periods and the rate period. Suppose the nominal rate is 10% and compounding is quarterly: The rate period is 10% 4 = 2. 5% The number of periods is number of years × 4

Continuous compounding: Adjustments

Try it: Frequency of compounding If 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: Answer Given: PV = $1, 000 n = 5 × 2 = 10 i = 0. 05 2 = 0. 25 Solve for FV FV = $1, 000 × (1 + 0. 025)10 = $1, 280. 08

Problem Set 2

Problem 2. 1 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? 40

Problem 2. 2 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? 41

Problem 2. 3 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? 42

0 | 1 2 3 4 5 CF CF CF | | | PV? 5. 2 Annuities and Perpetuities | | 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 annuities Type First cash flow Examples Ordinary One period from today Mortgage Annuity due Immediately Lottery payments Rent Deferred annuity Beyond one period from today Retirement savings

Time lines: 4 -payment annuity Ordinary Annuity due Deferred annuity 0 1 2 3 4 5 | | | CF CF FV CF CF CF PV FV CF PV CF CF CF FV

Key to valuing annuities The 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 annuity What 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, 000 4

Example: PV of an annuity The long way 0 1 2 3 | | $4, 000 $3, 773. 58 3, 559. 99 3, 358. 48 $10, 692. 05 4

Example: PV of an annuity In table form Year 1 2 3 Discount Present Cash flow factor value $4, 000. 00 0. 94340 $3, 773. 58 $4, 000. 00 0. 89000 3, 559. 99 $4, 000. 00 0. 83962 3, 358. 48 2. 67301 $10, 692. 05 PV = $4, 000. 00 × 2. 67301 = $10, 692. 05

Example: PV of an annuity Formula short-cuts

Example: PV of an annuity Calculator short cuts Given: PMT = $4, 000 i = 6% N = 3 Solve for PV

Example: PV of an annuity Spreadsheet 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 annuity What 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, 000 4

Example: FV of an annuity The long way 0 1 2 3 | | $4, 000. 00 4, 240. 00 4, 494. 40 $12, 734. 40 4

Example: FV of an annuity In table form Year 1 2 3 Cash flow $4, 000. 00 Compound factor Future value 1. 1236 $4, 494. 40 1. 0600 4, 240. 00 1. 0000 4, 000. 00 3. 1836 $12, 734. 40 PV = $4, 000. 00 × 3. 1836 = $12, 734. 40

Example: FV of an annuity Calculator short cuts CALCULATOR Given: PMT = $4, 000 i = 6% N = 3 Solve for FV

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

Annuity due Consider 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 line 0 1 2 3 | | $4, 000. 00 PV? This is an annuity due FV?

Valuing an annuity due Present value End of year Compoun Present value of Year cash flow d factor cash flow 0 $4, 000. 00 1. 00000 $4, 000. 00 1 $4, 000. 00 0. 94340 3, 773. 58 2 $4, 000. 00 0. 89000 3, 559. 99 2. 83339 $11, 333. 57 Future value End of year Year cash flow Factor Future value 0 $4, 000. 00 1. 19102 $4, 764. 06 1 $4, 000. 00 1. 12360 4, 494. 40 2 $4, 000. 00 1. 06000 4, 240. 00 3. 37462 $13, 498. 46

Valuing an annuity due: Using calculators Present value Future value PMT = 4000 N = 3 I = 6% BEG mode Solve for PV PMT = 4000 N = 3 I = 6% BEG mode Solve for FV

Valuing an annuity due: Using spreadsheets Present 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) and FVannuity due = FVordinary annuity × (1 + i)

Valuing a deferred annuity A 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 annuity 0 4 -payment ordinary annuity, then discount value one period PV 0 4 -payment annuity due, then discount value two periods PV 0 1 2 3 4 5 | | CF CF ←PV 1 ←PV 2

Example: Deferred annuity What 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 PV? 1 2 3 4 5 6 7 8 | | | CF CF CF 9 10

Example, cont. Using an ordinary annuity: PV 3 = $23, 956. 26 Discount 3 periods at 8% PV 0 = $19, 017. 25 Using an annuity due: PV 4 = $25, 872. 76 Discount 4 periods at 8% PV 0 = $19, 017. 25

Example: Deferred annuity Calculator solutions HP 10 B BAIIPlus TI 83/84 0 CF 6000 CF 6000 CF 8 i NPV 0 CF ↑ 1 F 1 0 CF ↑ 1 F 2 0 CF ↑ 1 F 3 6000 CF ↑ 1 F 4 6000 CF ↑ 1 F 5 6000 CF ↑ 1 F 6 6000 CF ↑ 1 F 7 6000 CF ↑ 1 F 8 8 i NPV [2 nd] { 0 0 0 6000 6000} STO [2 nd] L 1 [APPS] [Finance] [ENTER] 7 NPV(. 08, 0, L 1) [ENTER]

Example: Deferred annuity Spreadsheet 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, A 1: A 9)

Perpetuities

Problem Set 3

Problem 3. 1 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. 73

5. 3 Nominal and effective rates

APR & EAR The 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 EAR

EAR with continuous compounding 77

Frequency of compounding If interest is compounded more frequently than annually, then this is considered in compounding and discounting. There are two approaches 1. Adjust the i and n; or 2. Calculate the EAR and use this

Example: EAR & compounding Suppose 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 & compounding

Try it: APR & EAR Suppose 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: Answer

Problem Set 4

Problem 4. 1 What is the effective interest rate that corresponds to a 6% APR when interest is compounded monthly? 84

Problem 4. 2 What is the effective interest rate that corresponds to a 6% APR when interest is compounded continuously? 85

5. 4 Applications

Saving for retirement Suppose 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 | | | | D D D D D = Deposit (39 in total) W = Withdrawal (30 in total) … 39 40 41 42 43 | | | W W W … 79 | W

Deferred annuity time line 0 1 2 3 4 5 6 7 8 | | | | … 39 40 41 42 43 | | | W W W PV ↓ Ordinary annuity → FV D D D D … D … ← Ordinary annuity 79

Two steps Step 1: Present value of ordinary annuity N = 30; i = 3%; PMT = $60, 000 PV 39 = $1, 176, 026. 48 Step 2: Solve for payment in an ordinary annuity N = 39; i = 3%; FV = $1, 176, 026. 48 PMT = $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 Balance in the retirement account Year into the future

£ + =$ ¥ € Practice problems

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

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

Problem 3 What 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 4 Suppose 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 5 Sue 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 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 6 Suppose you have two investment opportunities: Opportunity 1: APR of 12%, compounded monthly Opportunity 2: APR of 11. 9%, compounded continuously Which opportunity provides the better return?

Problem 7 If 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 8 Suppose 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