chapter four present value. copyright © houghton mifflin company. all rights reserved.4 | 2 would...

Chapter Four

Present Value

Copyright © Houghton Mifflin Company. All rights reserved. 4 | 2

• Would you rather have $100 today or $105 in one year?

• What does your answer depend on?

• What happens to your choice as the interest rate rises? As the interest rate falls?

• Present value is the amount of money you would need to invest today to yield a given future return.

The Present Value of One Future Payment


Present value is based in two ideas

1. You can determine how much money you have available at different times

2. Interest is earned on past interest (compounding)

Present Value (cont’d)


• Compare $300 today vs. $350 in two years• What does it depend on?

– Today, your principal (P) represents your entire investment

– What rate of interest will you earn?

• In one year: (1 + i) × P– Where i is annual interest rate

• In two years: (1 + i)2 × P– Compounding is earning interest on interest that

was earned in prior years

The Power of Compounding


Extend as many years as you would like

Value after N years = (1 + i)N × P

Compounding enables your money to keep working for you, even without additions to the principal

Compounding (cont’d)


Compounding makes a huge difference


P = $1,000, i = 8%

1 year: $1,080

5 years: $1,469

10 years: $2,159

25 years: $46,902

100 years: $2,199,761


How much does the interest rate matter?


$1,000 × 1.08100 = $2,199,761

$1,000 × 1.07100 = $867,716

1% makes a BIG difference!!!


• In discounting, we consider an amount to be received in the future and ask how much it is worth today.

• Example: Would you rather have $350 in one year, or $300 today?

• What does it depend on? It is the same question as “compounding”… only in reverse!

• How much would you be willing to give up today to receive $350 in the future?

Discounting


F = (desired) future value

What P today is worth F in one year?

P = F/(1 + i) because (1 + i) × P = F

The term (1 + i) is the discount factor.

The term i is the rate of discount.

Discounting (cont’d)


Key question

What is your rate of discount?

What would you do with $10,000 today?

What is the expected return?

Why Discount?


• For a given F, what happens to P as i rises?

• Is $350 in one year worth more or less today as i rises? As i falls?

Key results• Present value is inversely related to the

rate of discount• Present value is inversely related to the

discount factor

Discounting (cont’d)


• Used to calculate the present value of almost any financial security

• Higher future amounts will yield a higher present value• Higher rates of discount or discount factors will yield a

lower present value

The General Form of the Present-Value Formula

N

N

i

F

i

F

i

FP

)1(...

)1()1( 2

2

1

1

.


• A way of describing the payments promised by a financial security

Example: N = 1 year

F = $10,000 0 N Payment F

Time (years)

Different Types of Securities


Perpetuity = financial security that never matures. It pays interest forever, does not repay principal (eg. a share of stock in a corporation)

Example: N = 1 year

F = $10,000

Perpetuity 0 1 2 3 4 . . . Payment F F F F . . .

Time (years)

Present Value of a Perpetuity


Would you ever want to own a perpetuity?

Is the present value really infinite?

How much would you pay for it?

To calculate: P = F / i

The present value of each successive payment is less than the last…Therefore, even the present

value of a perpetuity is finite.

Perpetuities (cont’d)


Present value of perpetuities are also affected by the

rate of discount.

Compare the sensitivity of P to iwhere F = $1,000

i = .08 P = $12,500i = .05 P = $20,000i = .02 P = $50,000

Perpetuities (cont’d)


• Fixed-payment security: The dollar payments are the same every year so that the principal is amortized

• Amortization: The process of repaying a loan’s principal gradually over time

Fixed Payment Securities


P = F × i

iN)

1

1(1

Fixed Payment Securities (cont’d)

0 1 2 3 4 . . . N Payment F F F F . . . F

Time (years)

Payment Timeline

To calculate present value:


Coupon Bond: Pays a regular interest payment until maturity, when face value is repaid (e.g. most corporate & government bonds) 0 1 2 3 4 . . . N Payments Interest F F F F . . . F Face Value V

Time (years)

Present Value of a Coupon Bond


To calculate present value:

P = [F × i

iN)

1

1(1

] + Ni

V

)1(

Present Value of a Coupon Bond (cont’d)

interest payments are present value of

fixed-payment security face value


• Many securities require payments more frequently

• Semi-annually: Government & corporate bonds

• Quarterly: Many stock dividends

• Monthly: Consumer & business loans

• Because of compounding, this frequency must be accounted for in calculating present value

Payments More Than Once Per Year


• Time period needs to be adjusted to account for payment frequency

• Assume that interest compounds each period and N = number of periods to maturity

Example: 30 year mortgage at 9% N = 360 (12 months x 30 years) i = 0.0075 (0.09/12 months)

Payments More Than Once Per Year (cont’d)


Present Value & Decision Making

Comparing alternative offers• A magazine subscription costs $50 for 1

year or $95 for 2 years. Which is better?• Comparing coupon bonds: use one as

an alternative for the other; use the interest rate on one bond as the rate of discount on other bonds in the secondary market


Question: Given a choice, would you buy or lease a car?

Answer: Financially there is not much difference between buying and leasing. Money paid today is worth more than money paid in the future.

Buying or Leasing a Car


Other factors matter in deciding to lease

Pros – option value (check car out & see how used car prices

change)– avoid transactions costs of selling

Cons – added transactions if you keep car; – limited mileage; – car dealers love them because of higher turnover, so

it must be bad for you!

Buying or Leasing a Car (cont’d)


• Why does the price of a security change when the market interest rate changes?

• This uncertainty is interest-rate risk

• Reflects a change in opportunities… suppose you buy a bond paying 6% but market rates rise to 8%. Does your bond price rise or fall?

Interest-Rate Risk


Interest-Rate Risk (cont’d)

• Bond price = present value of bond

• Present value of bond inversely related to i

• Bond price inversely related to i


Interest Rate Change Example

• A $1000 bond matures in one year and pays $50 interest. Other 1-year bonds also have interest rates of 5%.

P = $1050/1.05 = $1000

• What if market interest rate fell (just after you bought it) to 4%?

P = $1050/1.04 = $1009.62



Example (continued)

• What if the market interest rate rose (just after you bought it) to 10%?

P = $1050/1.10 = $954.55


Why did P change?

Market opportunity!


• Sometimes we already know present value, but are concerned with size of payments to be made to the lender…

Example: Mortgage loan• P = $100,000 • monthly payments for 30 years (N = 360)• annual interest rate = 6% (therefore i

= .06/12 = 0.005)

Using Present Value Formula to Calculate Payments


Example (cont.):P = F ×

ii

N)1

1(1

F = $100,000 × 360)

005.1

1(1

005.0

= $599.55

Using Present Value Formula to Calculate Payments

F = P × N

i

i

)1

1(1

so


Why would investors look forward or backward at security returns?– To calculate an expected return based on a

forecast they have received– To evaluate different loan offers– To determine the interest rate one will

receive on an annuity in retirement– To compare a stock’s return to the average

(historical) market return

Looking Forward or Backward at Returns


To calculate, still solve for i in the formula

• Thus far, i = rate of discount

• Backward looking: i = past return

• Forward looking

(1) i = expected return (accounts for probability of default)

(2) i = yield to maturity (average annual return if held to maturity without default)

Looking Forward or Backward at Returns (cont’d)


Payments made by security on right-hand side of equation, price on left-hand side, solve for i

One payment in one year:

(1 + i) × P = F, so i = (F/P) – 1



One payment in more than one year (N years) in the future:

(1 + i)N × P = F, so i = (F/P)1/N – 1




• With a fixed-payment security, you know the payment amount & price of security, and are looking for the implied interest rate

• Cannot determine i as a function of just P and F, so we must “guess, test & revise”

• Same holds true for coupon bonds

P = F × i

iN)

1

1(1


• The same method can be used when there are multiple payments per year

• In solving for i, note that i is not at an annual rate, so you must multiply by number of periods per year



• Annual Percentage Yield = The annual interest rate that would give you the same amount you would earn with more frequent compounding than with the stated annual interest rate

• The U.S. government requires banks to report APY on savings.

• APY offers a way to compare investment with different periods of compounding.

• Example: Which is better to invest in?A: 8.0% compounded annuallyB: 7.95% compounded monthly

Policy Insight: Annual Percentage Yield (APY)


Example (cont.)A: $1000 × 1.081 = $1080B: $1000 × [1+(.0795/12)]12 = $1082.46

Option B is a better investment

To compare easily, define:APY = [1 + (i/x)]x – 1

where compounding occurs x times per year

APY(A) = .08; APY(B) = .08246.

Policy Insight: Annual Percentage Yield (APY) (cont’d)

chapter four present value. copyright © houghton mifflin company. all rights reserved.4 | 2 would...

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