1

Lecture 1 Outline

Chapter 31 Valuing Costs and Benefits2 The Time Value of Money3 The NPV Decision Rule4 Arbitrage (Law of One Price)5 No-Arbitrage and Risky Securities

Chapter 4 (More on the Time Value of Money)9 The Timeline10 Time Travel (the Three Rules of Time Travel)11 Net Present Value of a Stream of Cash Flows12 Perpetuities and Annuities

2

1 Valuing Costs and Benefits

• Using market prices to determine cash values

• When competitive market prices are not available

3

Example 1: Calculating cash values using market prices

4

Example 1 (continued): Calculating cash values using market prices

Example 1.a: Calculating cash values using market prices

Textbook Example 1.a (continued)

7

Example 2: When competitive prices are not available (and value depends on

preferences)

8

Example 2 (continued): When value depends on preferences

9

2 The Time Value of Money

We will see that interest rates are exchange rates across time. They allow us to convert $1 (say) today into $’s tomorrow, and vice-versa.

Interest Rates and the Time Value of Money

• Time Value of Money

– Consider an investment opportunity with the following certain cash flows.

• Cost: $100,000 today

• Benefit: $105,000 in one year

– The difference in value between money today and money in the future is due to the time value of money.

The Interest Rate: An Exchange Rate Across Time

• Value of Investment in One Year

– If the interest rate is 7%, then we can express our costs as:

Cost = ($100,000 today) × (1.07 $ in one year/$ today)

= $107,000 in one year

The Interest Rate: An Exchange Rate Across Time (cont'd)

• Value of Investment in One Year

– Both costs and benefits are now in terms of “dollars in one year,” so we can compare them and compute the investment’s net value:

$105,000 − $107,000 = −$2000 in one year

– In other words, we could earn $2000 more in one year by putting our $100,000 in the bank rather than making this investment. We should reject the investment.

The Interest Rate: An Exchange Rate Across Time (cont'd)

• Value of Investment Today

– Consider the benefit of $105,000 in one year. What is the equivalent amount in terms of dollars today?

Benefit = ($105,000 in one year) ÷ (1.07 $ in one year/$ today)

= ($105,000 in one year) × 1/1.07 = $98,130.84 today

– This is the amount the bank would lend to us today if we promised to repay $105,000 in one year.

The Interest Rate: An Exchange Rate Across Time (cont'd)

• Value of Investment Today

– Now we are ready to compute the net value of the investment:

$98,130.84 − $100,000 = −$1869.16 today

– Once again, the negative result indicates that we should reject the investment.

The Interest Rate: An Exchange Rate Across Time (cont'd)

• Present Versus Future Value

– This demonstrates that our decision is the same whether we express the value of the investment in terms of dollars in one year or dollars today. If we convert from dollars today to dollars in one year,

(−$1869.16 today) × (1.07 $ in one year/$ today) = −$2000 in one year.

– The two results are equivalent, but expressed as values at different points in time.

The Interest Rate: An Exchange Rate Across Time (cont'd)

• Present Versus Future Value

– When we express the value in terms of dollars today, we call it the present value (PV) of the investment. If we express it in terms of dollars in the future, we call it the future value of the investment.

17

Example 3: Comparing costs at different points in time

18

Example 3 (continued): Comparing costs at different points in time

19

Converting Between Dollars Today and Gold, Euros, or Dollars in the Future

We can convert dollars today to different goods, currencies, or dollars in the future by using the competitive market price, exchange rate, or interest rate.

20

3 The NPV Decision Rule

• Net Present Value

• The NPV Decision Rule – Accepting or Rejecting a Project

– Choosing Among Projects

• NPV and Individual Preferences for cash today or tomorrow

21

Net Present Value

(All project cash flows)=NPV PV

When making an investment decision, take the alternative with the highest NPV. Choosing this alternative is equivalent to receiving its NPV in cash today

The NPV Decision Rule

• Accepting or Rejecting a Project

– Accept those projects with positive NPV because accepting them is equivalent to receiving their NPV in cash today.

– Reject those projects with negative NPV because accepting them would reduce the wealth of investors.

23

Example 4: The NPV is equivalent to cash today

24

Example 4 (continued): The NPV is equivalent to cash today

Choosing Among Alternatives

• We can also use the NPV decision rule to choose among projects. To do so, we must compute the NPV of each alternative, and then select the one with the highest NPV. This alternative is the one which will lead to the largest increase in the value of the firm.

26

Example 5: Cash flows of alternative possible projects/plans

27

Example 5 (continued): Computing the NPV of each project/plan

28

Example 5 (continued): Cash flows and NPV

29

Example 5 (continued): Cash flows and NPV

30

4 Arbitrage (Law of One Price)

• The notion of arbitrage – The practice of buying and selling equivalent

goods in different markets to take advantage of a price difference. An arbitrage opportunity occurs when it is possible to make a profit without taking any risk or making any investment

• Normal Market

– A competitive market in which there are no arbitrage opportunities.

4 Arbitrage (Law of One Price) continued

• Law of One Price

– If equivalent investment opportunities trade simultaneously in different competitive markets, then they must trade for the same price in both markets.

32

Example 6: Net cash flows from buying the bond and borrowing

Consider a bond which promises to pay $1,000 in one year’stime. Assume the risk-free rate is 5%. Assume the price of the bond is $940.

I could borrow from the bank, buy the bond, and this wouldbe an arbitrage (a profit without any risk).

33

Example 6 (continued): Net cash flows from selling the bond and investing

Consider a bond which promises to pay $1,000 in one year’stime. Assume the risk-free rate is 5%. Assume the price of the bond is $960.

I could sell short the bond, put the money in the bank, and this would be an arbitrage.

When there is no-arbitrage?

34

When P(bond) = $952.38.

In other words, when the price of the bond is equal to the discounted present value of its future cash flows.

In other words, when:

P(bond) = ($1,000 in one year)/(1.05$ in one year/$ today) = $952.38.

Law of One Price

35

Notice that we have two alternative ways to receive the same cash flow: (1) buy the bond and (2) put $952.38 in the bank for a 5% risk-free rate. Since they are equivalent transactions, absence of arbitrage implies the Law of One Price, namely the price of the bond should be $952.38.

36

No Arbitrage Price of a Security

Differently put, the NPV from security trading should be zero.

37

Example 6.a: Computing the no-arbitrage price

38

Example 6.a (continued): Computing the no-arbitrage price

39

Example 7: Separating investment and financing

40

Example 7 (continued): Separating investment and financing

The Separation Principle

41

Security transactions in a normal market neither create nor destroy value on their own. The NPV from buying or sellingsecurities should be zero.

Therefore, we can evaluate the NPV of an investment decision a firm makes separately from the decision the firm makes regarding how to finance this investment.

42

5 No-arbitrage and Risky Securities

• Risky versus risk-free cash flows

• Risk aversion and the risk premium

• The no-arbitrage price of a risky security

43

Example 8: Cash flows and market prices (in $) of a risk-free bond and an investment in the market

portfolio

Suppose that the risk-free rate is 4%. The no-arbitrage price of the bond below is $1,058.

What about the market index? Let’s calculate the corresponding expected return. Is it larger than 4%?

44

Expected return from a risky investment

Hence, in the case of the market index:

Expected return =

Of course, if the economy is strong the realized return is

If the economy is weak, the realized return is

1 1($800) ($1,400) 1,000

2 2 10%1,000

+ −=

$1, 400 $1,00040%

$1,000

− =

$800 $1,00020%

$1,000

− = −

Valuing risky securities

45

Investors in the market index earn an expected return of 10%rather than the risk-free rate of 4% on their investments. Thedifference between 10% and 4% is called risk premium. Therisk premium is the additional return that investors expectto earn to compensate them for the security’s risk.

When a cash-flow is risky, to compute its present value we must discount the cash-flow we expect on average at a rate which equals the risk-free rate plus an appropriaterisk premium.

46

Example 9: Determining the market price of security A (cash flows in $)

By the Law of One price, the no-arbitrage price of Security Ahas to be $1,000 – $769 = $231. Hence, the expected return on this security will bewhich is higher than that of the market.

1 1($0) ($600) 231

2 2 30%231

+ −=

47

Example 10: A negative risk premium

48

Example 10 (continued): A negative risk premium

Why can we get a negative risk-premium from a risky asset?

49

The risk of a security is evaluated with respect to fluctuationsin the overall economy. A security’s risk premium will be higherthe more the returns vary with the overall economy and the market index. If the security’s returns vary in the opposite direction, the security offers insurance (i.e., it is a hedge) and, therefore, will have a low or negative risk premium.

50

Returns on risky assets

51

Example 11: Using the risk premium to compute a price

52

Example 11 (continued): Using the risk premium to compute a price

53

6 The Timeline

A timeline is a linear representation of the timing of potential cash flows.

Drawing a timeline of the cash flows will help you visualize the financial problem.

54

Example 12: Constructing a Timeline

55

Example 12 (continued): Constructing a Timeline

56

7 Time Travel

• Moving cash flows forward in time

• Moving cash flows back in time

• The three rules of time travel

57

Future value of a cash flow

Figure 4.1 The Composition of Interest Over Time

Example 13: Compounding

Example 13: Compounding (continued)

61

Present value of a cash flow

To move a cash flow backward in time, we must discount it.

62

Example 14: Present value of a single future cash flow

63

Example 14 (continued): Present value of a single future cash flow

64

The Three Rules of Time Travel

65

Present value of a cash flow stream

66

Example 15: Present value of a stream of cash flows

67

Example 15 (continued): Present value of a stream of cash flows

Example 16: Present value of a stream of cash flows

Example 16: Present value of a stream of cash flows (continued)

70

Future value of a cash flow stream with a present value of PV

Example 17: Future value of a cash flow stream

• Problem– What is the future value in three years of the

following cash flows if the compounding rate is 5%? 0 321

$2,000 $2,000 $2,000

Example 17: Future value of a cash flow stream (continued)

• Solution

• Or

0 321

$2,000

$2,000x 1.05 x 1.05

$2,315x 1.05

$2,205

$2,000x 1.05 x 1.05

$2,100$6,620

x 1.05

0 321

$2,000

x 1.05$4,100$2,100

$4,305

$2,000 $2,000

x 1.05$6,305

x 1.05$6,620

73

8 The Net Present Value of a Stream of Cash Flows

• Calculating the NPV of future cash flows allows us to evaluate an investment decision.

• Net Present Value compares the present value of cash inflows (benefits) to the present value of cash outflows (costs).

74

Example 18: Net Present Value of an investment opportunity

75

Example 18 (continued): Net Present Value of an investment opportunity

76

9 Perpetuities and Annuities

• Perpetuities (cash flow C from period 1 to the infinite future)

• Annuities (cash flow C from period 1 to period N)

• Growing Cash Flows– Growing Perpetuity

– Growing Annuity

77

Present Value (and Price) of a Perpetuity

Why is this the price of a Perpetuity?

Suppose you can invest $100 in a bank account paying $5 (or 5%) forever.

Consider the following strategy:At time 0 you invest $ 100 (-$100)At time 1 you get $105. You collect $5 and reinvest ($100).At time 2 you get $105. You collect $5 and reinvest ($100).At time 3 you get $105. You collect $5 and reinvest ($100).…

Using a bank, we have created a perpetuity C = $5 with r = 5% and an initial investment of $100. Hence, the value of the perpetuity is $100.

More generally, using a bank, we can create a perpetuity C = r*P given an initial investment of P. Rearranging, P = C/r.

78

Example 19: Endowing a perpetuity

79

Example 19 (continued): Endowing a perpetuity

80

Present Value (and Price) of an Annuity

Why is this the price of an annuity?

Suppose you can invest $100 in a bank account paying $5 (or 5%) for N periods.

Consider the following strategy:At time 0 you invest $ 100 (-$100)At time 1 you get $105. You collect $5 and reinvest ($100).At time 2 you get $105. You collect $5 and reinvest ($100).At time 3 you get $105. You collect $5 and reinvest ($100).…At time N you get $105.

Using a bank, we have created an annuity C = $5 for N periods + a payment of $100 at period N. Clearly, $100 = PV(Annuity of $5 for N periods) + 100/(1.05)N .

More generally, using a bank, we can create an annuity C = r*P plus a payment of P dollars at period N.Hence, P = PV(Annuity of C for N periods) + P/(1+r)N .

Rearranging now, we obtain PV(Annuity of C for N periods) = (C/r)*(1-1/(1+r)N)

81

Example 20: Present value of a lottery prize annuity

82

Example 20 (continued): Present value of a lottery Prize annuity

Future Value of an Annuity

• Future Value of an Annuity

( )

(annuity) V (1 )

1 1 (1 )

(1 )

1 (1 ) 1

= × +

= − × + +

= × + −

N

NN

N

FV P r

Cr

r r

C rr

Example 21: Future value of annuity

Example 21: Future value of annuity (continued)

86

Present value of a growing perpetuity and a growing annuity

Show it for yourself (same as what we did before)

Show it for yourself (same as what we did before)

Example 22: Present value of growing perpetuity

Example 22: Present value of growing perpetuity (continued)