Chapter Four Present Value The Present Value of

Chapter Four Present Value

The Present Value of One Future Payment • 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. Copyright © Houghton Mifflin Company. All rights reserved. 4|2

Present Value (cont’d) 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) Copyright © Houghton Mifflin Company. All rights reserved. 4|3

The Power of Compounding • 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 Copyright © Houghton Mifflin Company. All rights reserved. 4|4

Compounding (cont’d) 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 Copyright © Houghton Mifflin Company. All rights reserved. 4|5

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 Copyright © Houghton Mifflin Company. All rights reserved. 4|6

Compounding (cont’d) 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!!! Copyright © Houghton Mifflin Company. All rights reserved. 4|7

Discounting • 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? Copyright © Houghton Mifflin Company. All rights reserved. 4|8

Discounting (cont’d) 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. Copyright © Houghton Mifflin Company. All rights reserved. 4|9

Why Discount? Key question What is your rate of discount? What would you do with $10, 000 today? What is the expected return? Copyright © Houghton Mifflin Company. All rights reserved. 4 | 10

Discounting (cont’d) • 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 Copyright © Houghton Mifflin Company. All rights reserved. 4 | 11

The General Form of the Present-Value Formula • 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 Copyright © Houghton Mifflin Company. All rights reserved. 4 | 12

Different Types of Securities • A way of describing the payments promised by a financial security Example: N = 1 year F = $10, 000 Copyright © Houghton Mifflin Company. All rights reserved. 4 | 13

Present Value of a 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 Copyright © Houghton Mifflin Company. All rights reserved. 4 | 14

Perpetuities (cont’d) 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. Copyright © Houghton Mifflin Company. All rights reserved. 4 | 15

Perpetuities (cont’d) Present value of perpetuities are also affected by the rate of discount. Compare the sensitivity of P to i where F = $1, 000 i =. 08 P = $12, 500 i =. 05 P = $20, 000 i =. 02 P = $50, 000 Copyright © Houghton Mifflin Company. All rights reserved. 4 | 16

Fixed Payment Securities • 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 Copyright © Houghton Mifflin Company. All rights reserved. 4 | 17

Fixed Payment Securities (cont’d) Payment Timeline To calculate present value: Copyright © Houghton Mifflin Company. All rights reserved. 4 | 18

Present Value of a Coupon Bond: Pays a regular interest payment until maturity, when face value is repaid (e. g. most corporate & government bonds) Copyright © Houghton Mifflin Company. All rights reserved. 4 | 19

Present Value of a Coupon Bond (cont’d) To calculate present value: interest payments are fixed-payment security Copyright © Houghton Mifflin Company. All rights reserved. present value of face value 4 | 20

Payments More Than Once Per Year • 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 Copyright © Houghton Mifflin Company. All rights reserved. 4 | 21

Payments More Than Once Per Year (cont’d) • 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% q N = 360 (12 months x 30 years) q i = 0. 0075 (0. 09/12 months) Copyright © Houghton Mifflin Company. All rights reserved. 4 | 22

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 Copyright © Houghton Mifflin Company. All rights reserved. 4 | 23

Buying or Leasing a Car 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. Copyright © Houghton Mifflin Company. All rights reserved. 4 | 24

Buying or Leasing a Car (cont’d) 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! Copyright © Houghton Mifflin Company. All rights reserved. 4 | 25

Interest-Rate Risk • 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? Copyright © Houghton Mifflin Company. All rights reserved. 4 | 26

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 Copyright © Houghton Mifflin Company. All rights reserved. 4 | 27

Interest-Rate Risk (cont’d) 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 Copyright © Houghton Mifflin Company. All rights reserved. 4 | 28

Interest-Rate Risk (cont’d) 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! Copyright © Houghton Mifflin Company. All rights reserved. 4 | 29

Using Present Value Formula to Calculate Payments • 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) Copyright © Houghton Mifflin Company. All rights reserved. 4 | 30

Using Present Value Formula to Calculate Payments Example (cont. ): so Copyright © Houghton Mifflin Company. All rights reserved. 4 | 31

Looking Forward or Backward at Returns 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 Copyright © Houghton Mifflin Company. All rights reserved. 4 | 32

Looking Forward or Backward at Returns (cont’d) 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) Copyright © Houghton Mifflin Company. All rights reserved. 4 | 33

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 Copyright © Houghton Mifflin Company. All rights reserved. 4 | 34

Looking Forward or Backward at Returns (cont’d) One payment in more than one year (N years) in the future: (1 + i)N × P = F, so i = (F/P)1/N – 1 Copyright © Houghton Mifflin Company. All rights reserved. 4 | 35

Looking Forward or Backward at Returns (cont’d) • 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 Copyright © Houghton Mifflin Company. All rights reserved. 4 | 36

Looking Forward or Backward at Returns (cont’d) • 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 Copyright © Houghton Mifflin Company. All rights reserved. 4 | 37

Policy Insight: Annual Percentage Yield (APY) • 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 annually B: 7. 95% compounded monthly Copyright © Houghton Mifflin Company. All rights reserved. 4 | 38

Policy Insight: Annual Percentage Yield (APY) (cont’d) Example (cont. ) A: $1000 × 1. 081 = $1080 B: $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. Copyright © Houghton Mifflin Company. All rights reserved. 4 | 39