McGraw-Hill/Irwin Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.

Chapter 07 Interest

Rates and Present Value

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Chapter Outline

• Interest Rates• Present Value• Future Value• Kick It Up a Notch: Risk and

Reward

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Interest Rates

The Market for Money

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Interest Rate

• The interest rate is the percentage, usually expressed in annual terms, of a balance that is paid by a borrower to a lender that is in addition to the original amount borrowed or lent.

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Figure 1 The Market for Money

Supply

Demand

r*

$*

Interest rate (r)

Money ($) Borrowed/Saved

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Nominal vs. Real Interest Rates

• Nominal Interest Rate: the advertised rate of interest

• Real Interest Rate: the rate of interest after inflation expectations are accounted for; the compensation for waiting on consumption

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Present Value• Present Value is the interest

adjusted value of future payment streams.

• Mathematically, the present value of a payment is =(payment)/(1+r)n Where

r is the interest raten is the number of years until the payment is received/made.

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The Amount Payable for Every Dollar Borrowed (For several interest rates

and loan durations)

Interest rate -> Years

20% 10% 5% 2% 1%

30 237.38

17.45

4.32 1.81 1.35

10 6.19 2.59 1.63 1.22 1.10

5 2.49 1.61 1.28 1.10 1.05

1 1.20 1.10 1.05 1.02 1.01

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Examples From This Table• If you borrow $1 and promise to pay

it back in 5 years at 5% interest you will owe $1.28 which is the original $1 plus 28 cents in interest.

• If you borrow $1 and promise to pay it back in 30 years at 20% interest you will owe $237.38 which is the original $1 plus $236.38 in interest.

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Mortgages, Car Payments, and other Multiple-Payment Examples• Mortgages are loans taken out to

buy homes. Typically you borrow a large sum of money and promise to pay it back in even amounts each month for 10, 15, or 30 years.

• Car loans are similar to mortgages in that you borrow a large sum but the loan duration is usually two to six years.

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A Multiple Year Example @ 5%Year Cost Benefit PV Cost

@5%PV Benefit @5%

1 100 100.00

2 100 95.24

3 100 90.70

4 100 86.38

5 100 82.27

6 100 78.35

7 100 74.62

8 100 71.07

9 100 67.68

10 100 64.46

11 100 61.39

12 100 58.47

500 700 454.60 476.05

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A Multiple Year Example @ 8%Year Cost Benefit

PV Cost @8%

PV Benefit @8%

1 100 100.00

2 100 92.59

3 100 85.73

4 100 79.38

5 100 73.50

6 100 68.06

7 100 63.02

8 100 58.35

9 100 54.03

10 100 50.02

11 100 46.32

12 100 42.89

500 700 431.21 382.68

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A Multiple Year Example @ 10%Year Cost Benefit

PV Cost @10%

PV Benefit @10%

1 100 100.00

2 100 90.91

3 100 82.64

4 100 75.13

5 100 68.30

6 100 62.09

7 100 56.45

8 100 51.32

9 100 46.65

10 100 42.41

11 100 38.55

12 100 35.05

500 700 416.99 332.52

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Internal rate of return

• Internal rate of return : The interest rate where the present value of costs and benefits are equal.

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Monthly Payments Required on per $1000 of loan (For Several Interest

Rates and Loan Durations)

Interest

rate ->

Years

20% 10% 5% 2% 1%

30 16.71

8.78 5.37 3.70 3.22

10 19.33

13.22

10.61

9.20 8.76

5 26.49

21.25

18.87

17.53

17.09

1 92.63

87.92

85.61

84.24

83.79

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Examples From This Table• If you borrow $1000 and

promise to pay it back monthly over 5 years at 5% interest you will owe $18.87 per month.

• If you borrow $1000 and promise to pay it back monthly over 10 years at 20% interest you will owe $19.33 per month.

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Future Value

• Future value: the interest-adjusted value of past payments.

nrpaymentValueFuture 1

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Rule of 72

• Rule of 72: A short cut that allows you to estimate the time it would take for an investment to double by dividing 72 by the annual interest rate.• For example: How long would it take to

double your money ($10,000) at 4% interest?• FV formula: $10,000x(1.04)^18=$20,258.17 (so

a little less than 18 years is the answer).• Rule of 72: 72/4=18 years

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Kick It Up A Notch:

Risk and Reward

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Kick It Up A Notch: Risk and Reward

• Risk: the possibility that the investor will not get those anticipated payoffs• Default Risk: the risk to the investor that

the borrower will not pay • Market Risk: the risk that the market

value of an asset will change in an unanticipated manner

• Reward• Risk Premium the reward investors

receive for taking greater risk

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The Yield Curve• Yield Curve: the relationship between reward and

the time until the reward is received

2.00

2.50

3.00

3.50

4.00

4.50

5.00

Year

Inte

rest

Rat

e

US Treasury Yield Curve (January 2005)