投资学10版习题答案15.docx

1、投资学10版习题答案15CHAPTER 15: THE TERM STRUCTURE OF INTEREST RATESPROBLEM SETS.1. In general, the forward rate can be viewed as the sum of the markets expectation of the future short rate plus a potential risk (or liquidity) premium. According to the expectations theory of the term structure of interest r

2、ates, the liquidity premium is zero so that the forward rate is equal to the markets expectation of the future short rate. Therefore, the markets expectation of future short rates (i.e., forward rates) can be derived from the yield curve, and there is no risk premium for longer maturities.The liquid

3、ity preference theory, on the other hand, specifies that the liquidity premium is positive so that the forward rate is greater than the markets expectation of the future short rate. This could result in an upward sloping term structure even if the market does not anticipate an increase in interest r

4、ates. The liquidity preference theory is based on the assumption that the financial markets are dominated by short-term investors who demand a premium in order to be induced to invest in long maturity securities.2. True. Under the expectations hypothesis, there are no risk premia built into bond pri

5、ces. The only reason for long-term yields to exceed short-term yields is an expectation of higher short-term rates in the future.3. Uncertain. Expectations of lower inflation will usually lead to lower nominal interest rates. Nevertheless, if the liquidity premium is sufficiently great, long-term yi

6、elds may exceed short-term yields despite expectations of falling short rates.4. The liquidity theory holds that investors demand a premium to compensate them for interest rate exposure and the premium increases with maturity. Add this premium to a flat curve and the result is an upward sloping yiel

7、d curve.5. The pure expectations theory, also referred to as the unbiased expectations theory, purports that forward rates are solely a function of expected future spot rates. Under the pure expectations theory, a yield curve that is upward (downward) sloping, means that short-term rates are expecte

8、d to rise (fall). A flat yield curve implies that the market expects short-term rates to remain constant.6. The yield curve slopes upward because short-term rates are lower than long-term rates. Since market rates are determined by supply and demand, it follows that investors (demand side) expect ra

9、tes to be higher in the future than in the near-term.7.MaturityPriceYTMForward Rate1$943.406.00%2$898.475.50%(1.0552/1.06) 1 = 5.0%3$847.625.67%(1.05673/1.0552) 1 = 6.0%4$792.166.00%(1.064/1.05673) 1 = 7.0%8. The expected price path of the 4-year zero coupon bond is shown below. (Note that we discou

10、nt the face value by the appropriate sequence of forward rates implied by this years yield curve.)Beginning of YearExpected PriceExpected Rate of Return1$792.16($839.69/$792.16) 1 = 6.00%2($881.68/$839.69) 1 = 5.00%3($934.58/$881.68) 1 = 6.00%4($1,000.00/$934.58) 1 = 7.00%9. If expectations theory h

11、olds, then the forward rate equals the short rate, and the one-year interest rate three years from now would be 10. a. A 3-year zero coupon bond with face value $100 will sell today at a yield of 6% and a price of:$100/1.063 =$83.96Next year, the bond will have a two-year maturity, and therefore a y

12、ield of 6% (from next years forecasted yield curve). The price will be $89, resulting in a holding period return of 6%.b. The forward rates based on todays yield curve are as follows:YearForward Rate2(1.052/1.04) 1 = 6.01%3(1.063/1.052) 1 = 8.03%Using the forward rates, the forecast for the yield cu

13、rve next year is:MaturityYTM16.01%2(1.0601 1.0803)1/2 1 = 7.02%The market forecast is for a higher YTM on 2-year bonds than your forecast. Thus, the market predicts a lower price and higher rate of return.11. a. b. The yield to maturity is the solution for y in the following equation:Using a financi

14、al calculator, enter n = 2; FV = 100; PMT = 9; PV = 101.86; Compute i YTM = 7.958%c. The forward rate for next year, derived from the zero-coupon yield curve, is the solution for f 2 in the following equation: f 2 = 0.0901 = 9.01%.Therefore, using an expected rate for next year of r2 = 9.01%, we fin

15、d that the forecast bond price is:d. If the liquidity premium is 1% then the forecast interest rate is:E(r2) = f2 liquidity premium = 9.01% 1.00% = 8.01%The forecast of the bond price is:12. a. The current bond price is:($85 0.94340) + ($85 0.87352) + ($1,085 0.81637) = $1,040.20This price implies a

16、 yield to maturity of 6.97%, as shown by the following:$85 Annuity factor (6.97%, 3) + $1,000 PV factor (6.97%, 3) = $1,040.17b. If one year from now y = 8%, then the bond price will be:$85 Annuity factor (8%, 2) + $1,000 PV factor (8%, 2) = $1,008.92The holding period rate of return is:$85 + ($1,00

17、8.92 $1,040.20)/$1,040.20 = 0.0516 = 5.16%13.YearForward RatePV of $1 received at period end1 5%$1/1.05 = $0.95242 71/(1.05 1.07) = $0.89013 81/(1.05 1.07 1.08) = $0.8241a. Price = ($60 0.9524) + ($60 0.8901) + ($1,060 0.8241) = $984.14b. To find the yield to maturity, solve for y in the following e

18、quation:$984.10 = $60 Annuity factor (y, 3) + $1,000 PV factor (y, 3)This can be solved using a financial calculator to show that y = 6.60%:PV = -$984.10; N = 3; FV = $1,000; PMT = $60. Solve for I = 6.60%.c.PeriodPayment Received at End of Period:Will Grow bya Factor of:To a FutureValue of:1$60.001

19、.07 1.08$69.34260.001.0864.8031,060.001.001,060.00$1,194.14$984.10 (1 + y realized)3 = $1,194.141 + y realized = y realized = 6.66%Alternatively, PV = -$984.10; N = 3; FV = $1,194.14; PMT = $0. Solve for I = 6.66%.d. Next year, the price of the bond will be:$60 Annuity factor (7%, 2) + $1,000 PV fac

20、tor (7%, 2) = $981.92Therefore, there will be a capital loss equal to: $984.10 $981.92 = $2.18The holding period return is: 14. a. The return on the one-year zero-coupon bond will be 6.1%.The price of the 4-year zero today is:$1,000/1.0644 = $780.25Next year, if the yield curve is unchanged, todays

21、4-year zero coupon bond will have a 3-year maturity, a YTM of 6.3%, and therefore the price will be:$1,000/1.0633 = $832.53The resulting one-year rate of return will be: 6.70%Therefore, in this case, the longer-term bond is expected to provide the higher return because its YTM is expected to decline

22、 during the holding period.b. If you believe in the expectations hypothesis, you would not expect that the yield curve next year will be the same as todays curve. The upward slope in todays curve would be evidence that expected short rates are rising and that the yield curve will shift upward, reduc

23、ing the holding period return on the four-year bond. Under the expectations hypothesis, all bonds have equal expected holding period returns. Therefore, you would predict that the HPR for the 4-year bond would be 6.1%, the same as for the 1-year bond.15. The price of the coupon bond, based on its yi

24、eld to maturity, is:$120 Annuity factor (5.8%, 2) + $1,000 PV factor (5.8%, 2) = $1,113.99If the coupons were stripped and sold separately as zeros, then, based on the yield to maturity of zeros with maturities of one and two years, respectively, the coupon payments could be sold separately for:The

25、arbitrage strategy is to buy zeros with face values of $120 and $1,120, and respective maturities of one year and two years, and simultaneously sell the coupon bond. The profit equals $2.91 on each bond.16. a. The one-year zero-coupon bond has a yield to maturity of 6%, as shown below: y1 = 0.06000

26、= 6.000%The yield on the two-year zero is 8.472%, as shown below: y2 = 0.08472 = 8.472%The price of the coupon bond is: Therefore: yield to maturity for the coupon bond = 8.333%On a financial calculator, enter: n = 2; PV = 106.51; FV = 100; PMT = 12b. c. Expected price(Note that next year, the coupo

27、n bond will have one payment left.)Expected holding period return = This holding period return is the same as the return on the one-year zero.d. If there is a liquidity premium, then: E(r2) 6%17. a. We obtain forward rates from the following table:MaturityYTMForward RatePrice (for parts c, d)1 year1

28、0%$1,000/1.10 = $909.092 years11%(1.112/1.10) 1 = 12.01%$1,000/1.112 = $811.623 years12%(1.123/1.112) 1 = 14.03%$1,000/1.123 = $711.78b. We obtain next years prices and yields by discounting each zeros face value at the forward rates for next year that we derived in part (a):MaturityPriceYTM1 year$1

29、,000/1.1201 = $892.7812.01%2 years$1,000/(1.1201 1.1403) = $782.9313.02%Note that this years upward sloping yield curve implies, according to the expectations hypothesis, a shift upward in next years curve.c. Next year, the 2-year zero will be a 1-year zero, and will therefore sell at a price of: $1

30、,000/1.1201 = $892.78Similarly, the current 3-year zero will be a 2-year zero and will sell for: $782.93Expected total rate of return:2-year bond: 3-year bond: d. The current price of the bond should equal the value of each payment times the present value of $1 to be received at the “maturity” of that payment. The present value schedule can be taken directly from the prices of zero-coupon bonds calculated above.Current price = ($120 0.90909) + ($120 0.81162) + ($1,120 0.71178) = $109.0908 + $97.3944 + $797.1936 = $1,003.68Similarly, the expected prices of zeros one year from now can be used