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?

 Bond YTM Time to Maturity Current Price A 8% 15 \$980 B 8% 15 \$1,000 C 8% 15 \$1,098

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:

 Period Years Forward Rate 1 0.5 1.1% 2 1.0 1.7% 3 1.5 2.2% 4 2.0 2.5%

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 (bond-equivalent yield) spot rates for U.S. Treasury yields provided in the following table, the 6-month forward rate one year from now on a bond-equivalent yield basis is closest to:

 Period Years Spot Rate 1 0.5 1.40% 2 1.0 2.30 % 3 1.5 3.00% 4 2.0 3.50%

A. 4.41%

B. 2.20%

C. 2.30%

Ans: A;

Assume:

xfy represents x-period forward rate y-period from now;

Z x+y represents (x+y)-period spot rate;

Z y represents y-period spot rate.

We have (1+Z x+y)x+y=(1+Zy)y (1+xfy)x

6-month forward rate one year from now in this case is 1 period forward rate 2-period from now.

All spot rates are given on a BEY basis and must be divided by 2 in the calculation:

(1+1f 2)1 (1+0.023/2)2=(1+0.03/2)3

1f 2=0.022038

On a BEY basis, the forward rate is 0.022038*2=4.41%

5. Elaine Wong has purchased an 8%
coupon bond for \$1,034.88 with 3 years to maturity. At what rate must the coupon payments be reinvested to produce a 5% yield-to-maturity rate?

A. 8%

B. 6.5%

C. 5%

Ans: C;

C is correct. Yield-to-maturity measure assumes that the coupon payments can be reinvested at the yield-to-maturity.

In this case, it’s 5%. C is the correct answer.

6. The yield of a 3-year bond issue quoted on an annual-pay basis is 7.84%. The yield-to-maturity on a bond-equivalent basis is closest to:

A. 3.85%

B. 7.69%

C. 7.84%

Ans: B;

(1+bond-equivalent yield/2) 2 =1+annual-pay yield

In this case,

(1+bond-equivalent yield/2) 2 =1+0.0784

Therefore, bond-equivalent yield=7.69%

B is the correct answer.

7. The U.S. Treasury spot rates are provided in the following table:

 Period Years Spot Rate 1 0.5 2.20% 2 1.0 2.50 % 3 1.5 2.70% 4 2.0 3.20%

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, default-free cash flow of \$60,000. She is given the information in the following table:

 Period Years Annual Par Yield to Maturity BEY Theoretical Spot Rate BEY 6-month Forward Rates BEY 1 0.5 2.00% 2.00% 2.00% 2 1.0 2.40 % 2.40% 2.71% 3 1.5 2.70% 2.71% 3.12% 4 2.0 3.20% 3.23% 4.55%

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, default-free cash flows.

Therefore: \$60,000/(1+0.0323/2)4=\$56,276

9. The zero-volatility 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 zero-volatility 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 zero-volatility 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:

 Period Years Spot Rate 1 1 4.000% 2 2 8.167 % 3 3 12.377%

Consider a 3-year, 9% annual coupon corporate bond currently trading at \$89.464. Given the YTM of a 3-year 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 Z-spread, set the present value of the bond’s cash flows equal to today’s market price. Discount each cash flow at the appropriate zero-coupon 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 Z-Spread minus the option cost.

B. OAS is the value of the embedded option.

C. OAS is Z-spread plus the option cost.

Ans: A;

The option-adjusted spread takes the option yield component out of the Z-spread measure. The option-adjusted spread is the spread to the Treasury spot rate curve that the bond would have if it were option-free.

Therefore Z-spread – OAS = option cost in percent. A is the correct answer.

12. The difference between Z-spread 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. mortgage-backed security in a steep upward-sloping yield curve environment

Ans: C;

The difference between the Z-spread 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 Z-spread 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 Z-spread and the nominal spread for a non-Treasury 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 short-term securities, the difference between the nominal spread (which does not account for the shape of the yield curve) and the Z-spread (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 semiannual-pay 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 semi-annual pay=> N=10,

8% coupon and semi-annual 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)/1.10092

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.

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