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Chapter Four

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?

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Chapter Four

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  1. Chapter Four Present Value

  2. 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.

  3. Present Value(cont’d) Present value is based in two ideas • You can determine how much money you have available at different times • Interest is earned on past interest (compounding)

  4. 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

  5. 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

  6. 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

  7. 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!!!

  8. 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?

  9. 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.

  10. Why Discount? Key question What is your rate of discount? What would you do with $10,000 today? What is the expected return?

  11. 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

  12. 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

  13. Different Types of Securities • A way of describing the payments promised by a financial security Example: N = 1 year F = $10,000

  14. Present Value of a Perpetuity 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

  15. 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.

  16. 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

  17. 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

  18. Payment Timeline Fixed Payment Securities (cont’d) To calculate present value:

  19. Present Value of a Coupon Bond Coupon Bond: Pays a regular interest payment until maturity, when face value is repaid (e.g. most corporate & government bonds)

  20. Present Value of a Coupon Bond (cont’d) To calculate present value: interest payments are present value of fixed-payment security face value

  21. 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

  22. 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% • N = 360 (12 months x 30 years) • i = 0.0075 (0.09/12 months)

  23. 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

  24. 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.

  25. 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!

  26. 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?

  27. 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

  28. 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

  29. 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!

  30. 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)

  31. so Using Present Value Formula to Calculate Payments Example (cont.):

  32. 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

  33. 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)

  34. 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

  35. 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

  36. 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 Looking Forward or Backward at Returns (cont’d)

  37. 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

  38. 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

  39. 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.

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