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CHAPTER 3

CHAPTER 3. The Interest Factor in Financing. Chapter Objectives. Future value of a lump sum Present value of a lump sum Future value of an annuity Present value of an annuity Price and yield relationships Internal rate of return / yield to maturity. Future Value of a Lump Sum.

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CHAPTER 3

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  1. CHAPTER3 The Interest Factor in Financing

  2. Chapter Objectives • Future value of a lump sum • Present value of a lump sum • Future value of an annuity • Present value of an annuity • Price and yield relationships • Internal rate of return / yield to maturity

  3. Future Value of a Lump Sum • FV = PV (1+i)n • FV = future values; PV = present value • i = interest rate, discount rate, rate of return • The principle of compounding, or interest on interest: if we know 1. An initial deposit - PV 2. An interest rate - i 3. Time period - n We can compute the values at some specified future time period. Q: What happens with simple interests?

  4. Future Value of aLump Sum: An Example • Example: assume Astute investor invests $1,000 today which pays 10 percent, compounded annually. What is the expected future value of that deposit in five years? • Solution= $1,610.51

  5. Present Value of a Future Sum • The discounting process is the opposite of compounding PV = FV /(1+i)n • Example: assume Astute investor has an opportunity that provides $1,610.51 at the end of five years. If Ms. Investor requires a 10 percent annual return, how much can astute pay today for this future sum? • Solution= $1,000

  6. Annuities • Ordinary Annuity • Payment due at the end of the period • e.g., mortgage payment • Annuity Due • Payment due at the beginning of the period • e.g., a monthly rental payment

  7. Future Value of an Annuity • FVA = PMT (1+i )n-1+PMT (1+i )n-2 …+ PMT = PMT [1/i ( (1+i )n-1)] • Example: assume Astute investor invests $1,000 at the end of each year in an investment which pays 10 percent, compounded annually. What is the expected future value of that investment in five years? • Solution = $6,105.10 • Q: What happens if i=0%? • Q: What if n goes to infinity?

  8. Sinking Fund Payment • Example: assume Astute investor wants to accumulate $6,105.10 in five years. Assume Ms. Investor can earn 10 percent, compounded annually. How much must be invested each year to obtain the goal? • Solution= $1,000.00

  9. Present Value of an Annuity • PVA = PMT /(1+i)1 + PMT /(1+i)2…+ PMT /(1+i)n = PMT [1/i (1-1/(1+i)n)] Special cases: Q: What happens if i = 0 % ? Q: What happens if n goes to infinity? Example: What is the PV of 8-period annuity with pmt of $1,000, and discount rate of 10%

  10. Investment Yields / Internal Rate of Return • The discount rate that sets the present value of future investment cash returns equal to the initial investment costs today • Example: What is the investment yield if you will receive $400 monthly payment for the next 20 years for an initial investment of $51,593?

  11. Present Value of an Annuity • What if compounding frequency is not annual? • Adjust i and n to reflect compounding frequency Q: What happens if m goes to infinity (continuous compounding)?

  12. Bond Pricing • Bond is exchange of CF now (the PV, or price) for a pattern of cash flows later (coupons + par) • Bond price = PV(coupon payments) + PV(par value) • Requires determination of • Expected cash flows (coupons and par) • “Required” discount rate, or required yield

  13. Combine our PV for annuity and lump sum • Example • Semiannual, 10%, fixed rate 20-year bond with a par of $1,000. No credit risk, not callable, etc. Required yield is 11% • C = c × F = 0.1/2 × $1,000 = $50 • r = 0.11/2 = 0.055 • n = 20 × 2 = 40 • P = $50/0.055 × [1- 1/(1.055)40] + $1,000/(1+0.055)40 = $50 × 16.04613 + $1,000/8.51332 = $802.31 + $117.46 = $919.77 • Note that P<F, i.e., bond trades at a discount. • Q: what is the yield to maturity if P=$900?

  14. More on Bond Pricing • Yield = Internal Rate of Return (IRR) • IRR sets the NPV to zero for a bond investment • Solve using financial calculator, Excel function RATE or IRR • Special case of one future cash flow (zero-coupon bond):

  15. More on Bond Pricing • The required yield or discount rate can be thought of simply as another way of quoting the price. • Special case of one future cash flow (zero-coupon bond):

  16. Price-yield relationship: • Decreasing • For non-callable bonds, convex • Callable bond and Yield to call • Ex: Using Excel (or other) show this • For the previous example, vary bond yield to maturity from 5% to 15%

  17. More on Bond Pricing • If required discount rate remains unchanged, a par bond’s price will remain unchanged, but a discount bond will appreciate and a premium bond will depreciate over time. • Why? • Show with a spreadsheet • Example: What is your return if you buy a 10% semi-annual coupon bond at 11% yield to maturity and hold for one year while the yield to maturity • stays the same • goes up to 12% • goes down to 10%

  18. Other conventional yield measures • Current yield = (annual coupon)/(current price) • E.g.: Face is $100, current price is $80, coupon rate is 8%: yc = 0.08 × 100/80 = 8/80 = 0.1 = 10% • Ignores capital gains (losses) and reinvestment income • Yield to maturity: yield (IRR) if bond is held to maturity • Q: what is the “current yield” for a stock? • Q: What is the ranking of a. couple rate; b. current yield; c. yield to maturity for a. par bond; b. discount bond; c. premium bond

  19. Useful Excel Functions • FV • PV • Rate • PMT • IPMT • PPMT • NPER • NPV • IRR • GOAL SEEK / SOLVER

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