Yield measures, spot rates and forward rates (Reading 58)


Exercise Problems: 

1. Consider a $1,000 par value bond, with an annual paid coupon of 7%, maturing in 10 years. If the bond is currently selling for $980.74, the YTM is closest to: A. 8.28% B. 7.28% C. 6.28%


Ans: B; Use the calculator to calculate YTM: N=10, PMT=70, FV=1000, PV=980.74 CPT > 1/Y=7.28 

2. Consider the three bonds in the following table. Which of the three bonds is most likely to have the greatest reinvestment risk?
A. Bond A B. Bond B C. Bond C


Ans: C; The yield to maturity assumes the coupon payments are reinvested at the yield to maturity and the bond will be held until maturity. The bond selling at a premium has the highest coupon rate and thus is expected to earn the most reinvestment income. If the reinvestment rate falls, this bond will suffer the greatest loss. Therefore Bond C, which is currently selling at premium, is most likely to have the greatest reinvestment risk. 

3. Using the U.S. Treasury forward provided in the following table, the value of a 2 year, 100 par value Treasury bond with a 4% coupon rate is closes to:
A. $104.20 B. $100 C. $98.74


Ans: A; According to the definition of the forward rate, the value of the bond= _{}+_{}+_{}+_{} =$104.20 

4. Using the BEY (bondequivalent yield) spot rates for U.S. Treasury yields provided in the following table, the 6month forward rate one year from now on a bondequivalent yield basis is closest to:
A. 4.41% B. 2.20% C. 2.30%


Ans: A; Assume: _{x}f_{y} represents xperiod forward rate yperiod from now; Z _{x+y} represents (x+y)period spot rate; Z _{y} represents yperiod spot rate. We have (1+Z _{x+y})^{x+y}=(1+Z_{y})^{y} (1+_{x}f_{y})^{x} 6month forward rate one year from now in this case is 1 period forward rate 2period from now. All spot rates are given on a BEY basis and must be divided by 2 in the calculation: (1+_{1}f _{2})^{1 }(1+0.023/2)^{2}=(1+0.03/2)^{3} _{1}f _{2}=0.022038 On a BEY basis, the forward rate is 0.022038*2=4.41% ^{} ^{} 

5. Elaine Wong has purchased an 8% A. 8% B. 6.5% C. 5%


Ans: C; C is correct. Yieldtomaturity measure assumes that the coupon payments can be reinvested at the yieldtomaturity. In this case, it’s 5%. C is the correct answer. 

6. The yield of a 3year bond issue quoted on an annualpay basis is 7.84%. The yieldtomaturity on a bondequivalent basis is closest to: A. 3.85% B. 7.69% C. 7.84%


Ans: B; (1+bondequivalent yield/2)^{ 2} =1+annualpay yield In this case, (1+bondequivalent yield/2)^{ 2} =1+0.0784 Therefore, bondequivalent yield=7.69% B is the correct answer. 

7. The U.S. Treasury spot rates are provided in the following table:
Given a consistent corporate spread of 0.50%, what will be the most likely price of a 4% coupon corporate bond with 2 years to maturity? A. $100.61 B. $102.96
C. $98.92


Ans: A; The current price should be calculated using cash flows discounted at appropriate spot rate plus corporate spread: Current Price = _{}+_{}+_{}+_{} =_{}+_{} +_{}+_{} =$100.61 

8. Tina Mo, a fixed income analyst, is asked to value a single, defaultfree cash flow of $60,000. She is given the information in the following table:
The value of this single cash flow at the end of Period 4 is closest to: A. $56,427 B. $56,309
C. $56,276


Ans: C; The theoretical spot rate for Treasury securities represent the appropriate set of interest rates that should be used to value single, defaultfree cash flows. Therefore: $60,000/(1+0.0323/2)^{4}=$56,276 

9. The zerovolatility spread is a measure of the spread off: A. one point on the Treasury yield curve. B. all points on the Treasury yield curve. C. all points on the Treasury spot curve. 

Ans: C; Instead of measuring the spread to YTM, the zerovolatility spread measures the spread to Treasury spot rates necessary to produce a spot rate curve that correctly prices a risky bond. Therefore B is incorrect. The zerovolatility spread is the equal amount that we must add to each rate on the Treasury spot yield curve in order to make the present value of the risky bond’s cash flow equal to its market price. Therefore A is incorrect.


10. The U.S. Treasury spot rates are provided in the following table:
Consider a 3year, 9% annual coupon corporate bond currently trading at $89.464. Given the YTM of a 3year Treasury is 12%, the Z spread of the corporate bond is closest to: A. 1.50%. B. 1.67%. C. 1.76%. 

Ans: B; The Z spread is the equal amount that we must add to each rate on the Treasury spot yield curve in order to make the present value of the risky bond’s cash flow equal to its market price. To compute the Zspread, set the present value of the bond’s cash flows equal to today’s market price. Discount each cash flow at the appropriate zerocoupon bond spot rate plus a fixed spread named ZS. 89.464 = _{}+_{} +_{} Solve for ZS. Note that ZS can be found by replacing Choice A, B and C into the equation to see which is the correct answer. ZS=1.67%


11. Which of the following statement is correct about the option adjusted spread ( OAS ): A. OAS is ZSpread minus the option cost. B. OAS is the value of the embedded option. C. OAS is Zspread plus the option cost. 

Ans: A; The optionadjusted spread takes the option yield component out of the Zspread measure. The optionadjusted spread is the spread to the Treasury spot rate curve that the bond would have if it were optionfree. Therefore Zspread – OAS = option cost in percent. A is the correct answer.


12. The difference between Zspread and nominal spread will most likely be the most significant for a: A. Treasury security with short maturity in a flat yield curve environment B. zero coupon Treasury security. C. mortgagebacked security in a steep upwardsloping yield curve environment 

Ans: C; The difference between the Zspread and the nominal spread is greater for issues in which the principal is repaid over time rather than only at maturity. Therefore B is incorrect. In addition, the difference between the Zspread and the nominal spread is greater in a steep yield curve environment. Therefore, B is incorrect and C is the correct answer.


13. All else being the same, the difference between the Zspread and the nominal spread for a nonTreasury security will be greater when: A. maturity of the security is longer. B. yield curve is flatter.
C. security
has a bullet maturity rather than an amortizing structure.


Ans: A; A is correct because for shortterm securities, the difference between the nominal spread (which does not account for the shape of the yield curve) and the Zspread (the spread over the entire theoretical spot rate curve) is small. This difference grows with the maturity of the security and as the slope of the yield curve increases. 

14. A semiannualpay bond is callable in five years at $106. The bond has an 8% coupon and 15 years to maturity. If the bond is currently trading at $98 today, the yield to call is closest to: A. 8.22% B. 8.49%. C. 9.48%. 

Ans: C; Use the calculator to calculate yield to call: Time to call is 5 years and semiannual pay=> N=10, 8% coupon and semiannual pay=> PMT=4, The call price is $106 => FV=106, PV=98 CPT > 1/Y=4.7386 4.7386*2=9.48 Therefore the yield to call is 9.48%. C is the correct answer.


15. A 10% annual coupon bond with 3 years to maturity is currently trading at $1,010. The bond is callable in one year at a call price of $1,008 and in two years at a call price of $1,005. The bond’s yield to worst most likely occurs when the bond is: A. held until maturity in 3 years. B. called in year 1. C. called in year 2. 

Ans: A; The yield to worst for a callable bond is the lowest of the yields to call for each possible call date and the yield to maturity. The yield to call if the bond is called in one year is 10.45%, because 1,005=(100+1,010)/1.1045 The yield to call if the bond is called in two years is 10.09% , because 1,005=100/1.1009+(100+1,008)/1.1009^{2} The yield to maturity of the bond is 9.80%, because 1,005=100/1.0980+100/1.0980^{ 2}+(100+1,000)/1.0980^{ 2} The yield to worst is the lowest of these and occurs when the bond is held until maturity. Therefore A is the correct answer.
