Valuation Under Certainty

• Investors must be concerned with:

- Time

- Uncertainty

• First, examine the effects of time for one-period assets.

• Money has time value. $100 in one year is not as attractive as $100 today.

• Rule 1: A dollar today is worth more than a dollar tomorrow, because it can be reinvested to earn more by tomorrow.

Session 1

• Topics to be covered:– Time value of money– Present value, Future value– Interest rates

• compounding intervals

– Bonds– Arbitrage

Present Value

• The value today of money received in the future is called the Present Value

• The present value represents the amount of money we would be prepared to pay today for something in the future.

• The interest rate, i is the price of credit in financial markets. – Interest rates are also known as discount rates.

Present Value

• The Present Value Factor or Discount Factor is the number we multiply by a future cash flow to calculate its present value.

• Present Value (PV)=Discount Factor*Future Value(FV)- Discount factor = 1/(1+i)

• Example (i=10%)- Discount factor = 1/(1+.10)

- The present value of $200 received in 1 year is

82.181$)200(9091.)200()10.1(

1

PV

Future Value

• Alternatively, we may use the interest rate, i, to convert dollars today to their value in the future.

• Suppose we borrow $50 today, and must repay this plus 5% interest in one year.

Future Value (FV) = Present Value (PV)(1+i)

FV = PV(1+i) = 50(1+.05) = $52.50

Bonds

• A bond is a promise from the issuer to pay the holder

- the principal, or face value, at maturity.

- Interest, or coupon payments, at intervals up to maturity.

• A $100 face value bond with a coupon rate of 7% pays $7.00 each year in interest, and $100 after a pre-specified length of time, called maturity.

Zero Coupon Bonds

• A zero coupon bond has no coupon payments.• The holder only receives the face value of the bond at

maturity.• Suppose the interest rate is 10%. A zero coupon bond

promises to pay the holder $1 in one year. Its price today is therefore

• The discount factor is just the price of a zero coupon bond with a face value of $1.

9091.0$)1()10.1(

1)1(

1

1

i

Net Present Value• The Net Present Value is the present value of the

payoffs minus the present value of the costs.• Suppose Treasury Bills yield 10%.• The present value of $110 in one year is

• Suppose we could guarantee this payoff by investing in a project that only costs $98 today.

• The NPV of this project is

110$100

1.10

100 98 $2NPV

NPV• The formula for calculating the NPV (one-period

case) is

• Note that C0 is usually negative, a cost or cash outflow.

• In the above example, C0 = -98 and C1 = 110.

)1(1

0 i

CCNPV

2)10.1(

11098

NPV

Rate of Return• The rate of return is the interest rate expected to be

earned by an investment.

• The rate of return for this project is

• We only want to invest in projects that return more than the opportunity cost of capital.

• The cost of capital in this case is 10%.

investment

profit

PV

PVFVi

122.98

98110

i

Decision Rules

• We know:1.) This project only costs $98 to guarantee $110 in one

year. In “the market”, it costs $100 to buy $110 in one year.

2.) This project returns 12.2%. In the market, our return is only 10%.

• This project looks good.

Decision Criteria

• We have equivalent decision rules for capital investment (with a ONE-PERIOD investment horizon):

- Net Present Value Rule: accept investments that have a positive NPV.

- Rate of Return Rule: accept investments that offer a return in excess of their opportunity cost of capital.

• These rules are equivalent for one period investments.• These rules are NOT equivalent in more complicated

settings.

Example: Market Value

• Continue to suppose you can borrow or lend money at 10%.

• Assume the price of a one-year zero-coupon bond with a FV of $110 is $98.

• The price of this bond is less than its present value.• We may use this example to illustrate the concept of

“arbitrage.”

110$100

(1 .10)PV

No Arbitrage

• Arbitrage is a “free lunch,” a way to make money for sure, with no risk and no net cost.

• For example:

- Buy something now for a low price and immediately sell it for a higher price.

- Buy something now and sell something else such that you have no net cash flows today, but will earn positive net cash flows in the future.

• Assets must be priced in financial markets to rule out arbitrage.

Example (cont.)

• To arbitrage this opportunity, we– 1.) buy the bond

– 2.) borrow $100 for one year.

• The cash flows from this strategy today and at the end of one year are:

Today One Year

Buy the bond -98 +110

Borrow $100 (1 yr) +100 -110

Net cash flow +2 0

Short Selling• Suppose the price of the zero-coupon bond were $102.• Our arbitrage strategy would be reversed.

- Lend $100 for one year.

- Short Sell the zero-coupon bond.

• The cash flows from this strategy would be

Today One Year

Sell the bond +102 -110

Lend $100 (1 yr) -100 +110

Net cash flow +2 0

Market Value

• As the above example illustrates, the only price for a bond which rules out arbitrage is $100.

• $100 is also the present value of the payoff of the bond.

• RULE 2: Assets must be priced in the market to rule out arbitrage (i.e., “no arbitrage”)– Therefore, the present value of an asset is its market

price.

Compound Interest Vs. Simple Interest• Next we consider assets that last more than one period.• How is multi-period interest paid?• Invest $100 in bonds earning 9% per year for two years:

- After one year: $100(1.09) = $109

- Reinvest $109 for the second year: $109(1.09) = 118.81

• We do NOT earn just 9% * 2 = 18% .• We earn “interest on our interest”, or COMPOUND• SIMPLE INTEREST: interest paid only on the initial investment• COMPOUND INTEREST: interest paid on the initial

investment and on prior interest.

Example: Simple Interest• $100 invested at 10% with no compounding becomes:

Year Start Interest End

1 $100 $10 $110

2 $110 $10 $120

3 $120 $10 $130

… …

10 $190 $10 $200

11 $200 $10 $210

… …

100 $1,090 $10 $1,100

Example: Compound Interest• $100 invested at 10% compounded annually becomes:

Year Start Interest End

1 $100 $10 $110

2 $110 $11 $121

3 $121 $12 $133

… …

10 $236 $24 $259

11 $259 $26 $285

… …

100 $1.25m $.125m $1.38m

Compound Interest• A present value $PV invested for n years at an interest rate of i

per year grows to a future value

• (1 + in)n is the Compound Amount Factor.

• Above, the FV of $100 compounded annually at 10% for 3 years is

• In principle, the interest rate in may vary with the length of the investment horizon, n. More later . . .

(1 )nnFV PV i

10.133)331.1(100)10.1)(100( 3 FV

Present Value• We may use the above relation to calculate the present

value of an n-period investment, with compound interest:

where is the discount factor, or present value factor.

• For example, the present value of $100 in 6 years at 10% per year with annual compounding:

nni

FVPV

)1(

nni )1(

1

45.56$)10.1(

1006

PV

Semi-Annual Compounding• So far, we assume cash flows occur at annual intervals.

- In Europe, most bonds pay interest annually.

- In the U.S., most bonds pay interest semiannually.• A $100 bond pays interest of 10% per year, but

payments are semi-annual.

- Half of the interest (5% or $50) is paid after 6 months.

- Reinvest this $50 for the second 6 months.• By the end of the year, we would have

50.102,1)05.1(1000)05.1)(501000( 2

Example• This return is as if we earned

if we had only received our payment at the end of the year.

• 10% compounded semiannually is equal to 10.25% compounded annually.

- 10% is called the nominal interest rate.

- 10.25% is called the effective interest rate.

(1,102.50 1000) 102.5010.25%

1000 1000

Example

• Suppose you buy $100 of a 7-year Treasury note that pays interest at a nominal rate of 10% per year, compounded semiannually.

• Define one period as 6 months:– The interest rate per period is 5%.

– There are 14 (6-month) periods until the 7-year maturity.

• So, we can use our general formula for future values to compute the value at maturity:

99.197)05.1(100)1( 14 nniPVFV

Extending the PV Formula• RECALL: For a project with one cash flow, C1, in one

year,

• If a project produces one cash flow, C2 after TWO years, then the present value is

• If a project produces one cash flow, C1 after one year, and a second cash flow C2 after two years , then

)1( 1

1

i

CPV

22

2

)1( i

CPV

22

2

1

1

)1()1( i

C

i

CPV

General Present Value

• By extension, the present value of an extended stream of cash flows is

• This is called the Discounted Cash Flow or Present Value formula:

• Similarly, the Net Present Value is given by

...)1()1()1( 3

3

32

2

2

1

1

i

C

i

C

i

CPV

...)1()1()1( 3

3

32

2

2

1

10

i

C

i

C

i

CCNPV

Example• Suppose a project will produce $50,000 after 1 year,

$10,000 after 2 years, and $210,000 after 4 years.• It costs $200,000 to invest.• We may earn 9% per year (compounded annually) on 1,

2, or 4 year zero-coupon bonds.• The present value of this project is

• The NPV of this project is

65.057,203)09.1(

000,2100

)09.1(

000,10

09.1

000,5042

PV

65 . 057 , 3 65 . 057 , 203 000 , 200 NPV

Net Present Value Rule

• In the last example, the PV of payoffs exceeded the PV of the costs, so the project is a good one.

• Investment Criterion (The NPV Rule): Accept a project if the NPV is greater than 0. – This criterion is a good general rule for all types of

projects.

– The NPV Rule can also be used to rank projects; a project with a larger (positive) NPV is better than one with a smaller NPV.