Chapter 2Time Value of Money

• Future value• Present value• Rates of return• Amortization

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Time Value of Money• What is the price of an asset?

– Estimation of expected future cash flows– Discounting expected future cash flows to present

value• What is the value of a series of cash flows at a

time in the future?– Compounding expected cash flows to future value

• What is a discount rate? – Cost of capital– Opportunity cost of capital– Required return

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Why does money have time value?

• You can invest your money and earn interest on it.

• $1 today ≠ $1 tomorrow• Would you prefer to get $1 today or to get

$1 in a week? In a year? In 10 years?

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Time lines show timing of cash flows

CF0 CF1 CF3CF2

0 1 2 3r%

Tick marks at ends of periods, so Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2.

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Future Values: General Formula

• FV = PV(1 + r)t

FV = future value PV = present value r = period interest rate, expressed as a decimal t = number of periods

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Future Values• Suppose you leave the money in a savings

account with 5% return for two years. How much will you have two years from now?

FV = 1050(1.05) = 1102.50

FV = 1000(1.05)(1.05) = 1102.50

FV = 1000(1.05)2 = 1102.50

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Effects of Compounding• Simple interest • Compound interest• Consider the previous example

FV with simple interest = 1000 + 50 + 50 = 1100FV with compound interest = 1102.50The extra 2.50 comes from the interest

of .05(50) = 2.50 earned on the first interest payment

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Future Values – Example 2• Suppose you invest the $1000 from the

previous example for 5 years. How much would you have? FV = 1000(1.05)5 = 1276.28

• The effect of compounding is small for a small number of periods, but increases as the number of periods increases. (Simple interest would have a future value of $1250, for a difference of $26.28.)

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Future Value as a General Growth Formula

• Suppose your company expects to increase unit sales of goods by 15% per year for the next 5 years. If you currently sell 3 million goods in one year, how many goods do you expect to sell in 5 years?

FV = 3,000,000(1.15)5 = 6,034,072

2nd [CLR TVM]3000000 PV, 15 I/Y, 5 N, CPT FV => -6034072

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Present Values• You want to begin saving for you daughter’s

college education and you estimate that she will need $150,000 in 17 years. If you feel confident that you can earn 8% per year, how much do you need to invest today?

PV = 150,000 / (1.08)17 = 40,540.34

2nd [CLR TVM]150000 FV, 8 I/Y, 17 N, CPT PV => -40540.34

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Discount Rate – Example 1• You are looking at an investment that will pay

$1200 in 5 years if you invest $1000 today. What is the implied rate of interest?

r = (1200 / 1000)1/5 – 1 = .03714 = 3.71% (Basis Points?) Calculator – the sign convention matters!!!

2nd [CLR TVM]-1000 PV (you pay $1,000 today) 1200 FV (you receive $1,200 in 5 years) 5 N, CPT I/Y => 3.71%

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PV – Important Relationship I

1

1 tPV FVr

0PVt

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PV – Important Relationship I

• For a given interest rate – the longer the time period, the lower the present value

• What is the present value of $500 to be received in 5 years? 10 years? The discount rate is 10%

5 years: PV = 500 / (1.1)5 = 310.4610 years: PV = 500 / (1.1)10 = 192.77

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Number of Periods – Example 1

• You want to purchase a new car and you are willing to pay $20,000. If you can invest at 10% per year and you currently have $15,000, how long will it be before you have enough money to pay cash for the car?

t = ln(20,000 / 15,000) / ln(1.1) = 3.02 years

2nd [CLR TVM]-15000 PV, 20000 FV, 10 I/Y, CPT N => 3.02

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PV – Important Relationship II

1

1 tPV FVr

0PVr

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PV – Important Relationship II• For a given time period – the higher the

interest rate, the smaller the present value• What is the present value of $500 received

in 5 years if the interest rate is 10%? 15%?

Rate = 10%: PV = 500 / (1.1)5 = 310.46 Rate = 15%; PV = 500 / (1.15)5 = 248.58

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The Basic FV Equation - Refresher

• Financial calculators solve this equation: FV + PV (1+i)n = 0

• There are four variables to this equation– PV, FV, r and t– If we know any three, we can solve for the fourth

• If you are using a financial calculator, be sure and remember the sign convention or you will receive an error when solving for r or t

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Multiple CFs – FV Example 2• Suppose you invest $500 in a mutual fund

today and $600 in one year, and $700 in three years. If the fund pays 9% annually, how much will you have in four years? FV = 500(1.09)4 + 600(1.09)3 + 700(1.09) FV = 705.79 + 777.02 + 763.00 FV = 2,245.81

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Multiple CFs – FV Example 2

• How much will you have in 40 years (from today) if you make no further deposits?

FV = 2,245.81(1.09)36 = 49,972.02

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Multiple CFs – PV Example 2

• You are offered an investment that costs $5000. It will pay you $1000 in one year, $2000 in two years and $3000 in three years. If you want to earn 10% on your money should you do it? PV = 1000 / (1.1)1 = 909.09PV = 2000 / (1.1)2 = 1652.89PV = 3000 / (1.1)3 = 2253.94PV = 909.09 + 1652.89 + 2253.94 = 4815.93

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Multiple CFs – PV & FV Example 1

• Suppose you are looking at the following possible cash flows: Year 1 CF = $100; Years 2 and 3 CFs = $200; nothing in Year 4, Year 5 CF = $100. The required discount rate is 7%.

1. What is the value of the cash flows today?2. What is the value of the cash flows at year 6?

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PV Annuity – Example 1• Suppose you win $10 million in the Texas

Lotto. The money is paid in equal annual installments over 25 years. Assume the first payment occurs in one year. If the appropriate discount rate is 4%, how much are your winnings actually worth today?

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Finding the Payment

Suppose you want to borrow $20,000 for a new car. You can borrow at 9% per year, compounded monthly. If you take a 4 year loan, what is your monthly payment?

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Finding the Number of Payments

Suppose you borrow $2000 at 5% compounded annually and you are going to make annual payments of $734.42. How long before you pay off the loan?

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Amortized Loan with Fixed Payment - Example

Rate Periods Principal Pmnt

0.08 5 20,000 5009.13

Year Start Bal. Interest Payment Principal End Bal.

1 20,000 1600.00 5009.13 3409.13 16,590.87

2 16,590.87 1327.27 5009.13 3681.86 12,909.01

3 12,909.01 1032.72 5009.13 3976.41 8,932.60

4 8,932.60 714.61 5009.13 4294.52 4,638.08

5 4,638.08 371.05 5009.13 4638.08 0.00

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Present Value of Perpetuity

0 1 2 * * * 0 1 2 * * * |--------|--------|--------|--------|--------|--------| C C C C C C PV

10CPVr

¥

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Perpetuity Example 1Stock Price is $35 and quarterly dividend is

$.80, what is the current required return?35 = .8 / rr = .0229 or 2.29% per quarter

Stock Price is $50 and your required return is 2.3% per quarter, what dividend do you expect?50 = C / .023C = $1.15 per quarter

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Perpetuity Example 2You are considering a preferred stock that

will start paying a quarterly dividend of $1.50 three years from today. If your desired return is 3% per quarter, how much would you be willing to pay today?

PV = 1.50/.03 = $50

PV = 50/(1.03)11 = $36.12

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Computing EAR (Equivalent Annual Rate)

• You are looking at two savings accounts. One pays 5.25%, with daily compounding. The other pays 5.35% with semiannual compounding. Which account should you use? First account:

• EAR = (1 + (.0525/365))365 – 1 = .0539 or 5.39%

Second account:• EAR = (1 + (.0535/2))2 – 1 = .0542 or 5.42%

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