©2008 Professor Rui Yao All Rights Reserved CHAPTER3CHAPTER3 CHAPTER3CHAPTER3 The Interest Factor in Financing

3-2 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

3-3 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?

3-4 Future Value of a Lump 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

3-5 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, 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

3-6 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

3-7 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, Q: What happens if i=0%? Q: What if n goes to infinity?

3-8 Sinking Fund Payment Example: assume Astute investor wants to accumulate $6, 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

3-9 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%

3-10 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?

3-11 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)?

3-12 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 Bond Pricing

3-13 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 = n = 20 × 2 = 40 P = $50/0.055 × [1- 1/(1.055) 40 ] + $1,000/( ) 40 = $50 × $1,000/ = $ $ = $  Note that P<F, i.e., bond trades at a discount.  Q: what is the yield to maturity if P=$900?

3-14 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):

3-15 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):

3-16 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%

3-17 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%

3-18  Other conventional yield measures Current yield = (annual coupon)/(current price)  E.g.: Face is $100, current price is $80, coupon rate is 8%: y c = 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

3-19 Useful Excel Functions FV PV Rate PMT  IPMT  PPMT NPER NPV IRR GOAL SEEK / SOLVER