© 2004 Pearson Addison-Wesley. All rights reserved 4-1 Present Value: Learn It!!! Suppose you are promised $100 at the end of each year for the next ten years: $ What is this cash flow worth today? Surely not $1,000. Suppose you can earn a 10% annual interest rate on money you have in hand today. Then (1+.10) x PV $100 next year = $100 next year PV $100 next year = $100 / 1.10 = $90.91 Need only $90.91 now to put out to interest to have $100 next year. The Present Value of $100 next year is $90.91.

© 2004 Pearson Addison-Wesley. All rights reserved 4-2 Present Value What’s the Present Value of the $100 you’ll have in two years when you can earn 10% annually on money you have now? From before, you know that you’d only need $90.91 next year to have $100 the year after: 1.10 x $90.91 next yr = $100 in 2 yrs. And you only need $90.91/1.10 now to have $90.91 next year. PV $100 in 2 yrs = $90.91/1.10 = [$100/1.10]/1.10 = $100/ PV $100 in 2 yrs = $82.64 In general, PV $100 received n years from now = $100/(1+i) n where i = annual interest rate = rate at which future cash flows are discounted.

© 2004 Pearson Addison-Wesley. All rights reserved 4-3 Present Value Using the Present Value calculator on the course webpage, PV $100 received 1 year from now = $100/(1.10) 1 = $90.91 PV $100 received 2 years from now = $100/(1.10) 2 = $82.64 PV $100 received 3 years from now = $100/(1.10) 3 = $75.13 : = : = : PV $100 received 10 years from now = $100/(1.10) 10 = $38.55 PV of $100 rec’d in each of next 10 yrs =$ In other words, $ put out to interest at 10% now would generate a cash flow of $100 per year for the next ten years. If the interest rate were lower, say 5%, PV $100 rec’d in each of next 10 yrs would be higher ( = $772.17) –Each future cash inflow would be discounted to a lesser extent by (1/1.05) n = n rather than (1/1.10) n =.9091 n –You would need $ rather than $ to generate a cash flow of $100 per year for each of the next ten years.

© 2004 Pearson Addison-Wesley. All rights reserved 4-4 Types of Credit Instruments 1.Discount (zero coupon) bond: pay PV now, get face value after n years 2.Simple loan: lend PV now. Get PVx(1 + i) n after n years 3.Coupon bond: pay PV now. Get coupon of $C each year until bond matures; get face value of bond ($F) as well when bond matures. 4.Fixed-payment loan: lend PV now.Get fixed payment ($FP) for n yrs. Concept of Present Value Simple loan of $100 at 10% interest held for n years Years123n $110$121$133$100x(1 + i) n $1 PV of future $1 = (1 + i) n Present Value

© 2004 Pearson Addison-Wesley. All rights reserved 4-5 Yield to Maturity: We know the price of a bond now (its PV) and the cash flow it generates over time. What is its yield to maturity? Yield to maturity = interest rate that equates today’s value with the discounted present value of all future payments 1.Simple Loan of $100 for 1 year (i = 10%) $100 = $110/(1 + i)  $110 – $100 $10 i = == 0.10 = 10% $100 2.Fixed Payment Loan for 25 years (i = 12%) $126$126$126 $126 $1000 = (1+i) (1+i) 2 (1+i) 3 (1+i) 25 FP FP FP FP PV = (1+i) (1+i) 2 (1+i) 3 (1+i) n

© 2004 Pearson Addison-Wesley. All rights reserved 4-6 Yield to Maturity: Bonds  4. One year discount bond whose current price is P (P = $900, F = $1000) $1000 $900 = (1+i) $1000 – $900 i == = 11.1% $900 F – P i = P 3.Ten year coupon bond whose current price is P (Coupon rate = 10% = C/F) $100$100$100$100$1000 P = (1+i) (1+i) 2 (1+i) 3 (1+i) 10 (1+i) 10 C C C C F P = (1+i) (1+i) 2 (1+i) 3 (1+i) n (1+i) n Consol: Fixed coupon payments of $C foreverC P = i = iP

© 2004 Pearson Addison-Wesley. All rights reserved 4-7 Relationship Between Price and Yield to Maturity Three Facts in Table 1 1.When bond is at par, yield equals coupon rate 2.Price and yield are negatively related 3.Yield is greater than coupon rate when bond price is below par value

© 2004 Pearson Addison-Wesley. All rights reserved 4-8 Current Yield i c = C/P 1. Current yield is an approximation to the true yield to maturity 2.Current yield is a better approximation to the true yield to maturity the closer the current price is to par and the longer is the time till the bond matures 2.Change in current yield always signals change in same direction as yield to maturity

Bond Page of the Newspaper

© 2004 Pearson Addison-Wesley. All rights reserved 4-10 Distinction Between Interest Rates and Returns Rate of Return C + P t+1 – P t RET == i c + g P t C where: i c = = current yield P t P t+1 – P t g == capital gain P t

© 2004 Pearson Addison-Wesley. All rights reserved 4-11 Interest Rates and Returns

© 2004 Pearson Addison-Wesley. All rights reserved 4-12 Maturity and Volatility of Bond Returns 1. The only bond whose return equals its yield to maturity is the one whose maturity equals the holding period 2.For bonds with maturity > holding period, i  P  … a capital loss 3.The longer the time till a bond matures, the greater the percentage price change for a given change in market interest rate, i The longer the time to maturity, the more return changes with a change in interest rate 4. A bond with high initial interest rate can still have negative return if i  Conclusions 1. Prices and returns are more volatile for long-term bonds … they have higher interest-rate risk 2. There is no interest-rate risk for a bond whose maturity equals your holding period. You’re sure to get the bond’s face value when the bond matures.

© 2004 Pearson Addison-Wesley. All rights reserved 4-13 Distinction Between Real and Nominal Interest Rates Real Interest Rate Interest rate adjusted for expected changes in the price level i r = i –  e where  e = expected rate of inflation 1.Real interest rate more accurately reflects true cost of borrowing 2.When real rate is low, there are greater incentives to borrow and less to lend if i = 5% and  e = 3% i r = 5% – 3% = 2% if i = 8% and  e = 10% i r = 8% – 10% = –2% … in real terms, you expect to pay back less than you borrowed

© 2004 Pearson Addison-Wesley. All rights reserved 4-14 U.S. Real and Nominal Interest Rates