Question 3 (20 Marks) a) The zero-coupon bonds with par value of $1000 have the

Question

Question 3 (20 Marks) a) The zero-coupon bonds with par value of $1000 have the following term structure Maturity YTM (%) 10% 11% 12% 2 i. What are the implied 1-year forward rates? (5 marks) ii. Assume that the pure expectations hypothesis of the term structure is correct. If market expectations are accurate, what will be the pure yield curve next year? (5 marks) iii. What should be the current price of a 2-year maturity bond with a 12% coupon rate paid annually? If you purchased it at that price, what would your total expected rate of return be over the next year? (5 marks) b) A bond with a coupon rate of 7% makes semi-annual coupon payments on January 15 and July 15 of each year. The Wall Street Journal reports the ask price for the bond on February 15 at 101% of the par value. The par value of the bond is 1000 What is the invoice price of the bond? The coupon period has 182 days (5 marks)

Explanation / Answer

Ans .a.i. Obtain forward rates from the following table:

Maturity (years) YTM Forward rate Price

1 10.0% $909.09 ($1000/1.10)

2 11.0% 12.01% [(1.112/1.10) – 1] $811.62 ($1000/1.112)

3 12.0% 14.03% [(1.123/1.112) – 1] $711.78 ($1000/1.123)

Ans .a.ii. We obtain next year’s prices and yields by discounting each zero’s face value at the forward rates derived in part (a):

Maturity (years) Price (for part iii) YTM

1 $892.08 = ($1000/1.1201) 12.01%

2 $782.93 = ($1000/(1.1201 * 1.1403) 13.02%

This year’s upward sloping yield curve implies, according to the expectations hypothesis, a shift upward in next year’s curve.

Ans.a.iii.

The current price of the bond should be equal to the value of each payment times the present value of $1 to be received at the “maturity” of that payment. The present value can be taken directly from the prices of zero-coupon bonds calculated above.

Current price = ($120 * 0.90909) + ($1120 * 0.81162) = $109.0908 + $909.0144 = $1,018.11

If you purchase a two-year zero-coupon bond now, Next year, the 2-year zero will be a 1 year zero, and will sell at 892.78; likewise, the 3-year zero will be a 2-year zero trading at 782.93. Expected total rate of return:

2-Year (892.78/811.62) – 1 = 10%

3-Year (782.93/711.78) – 1 = 10%

Similarly, the expected prices of zeroes in 1 year can be used to calculate the expected bond value at time: Expected price 1 year from now: = 120*0.89278 + 1120*0.78293 = 984.02

Total expected rate of return = ((120 + 984.02 - 1018.11.68) /1018.11) = 8.44%

Ans.b.

The reported bond price is 101% of par, which equals $1,010.

However, 31 days have passed since the last semiannual coupon was paid, so accrued interest equals $35 × (31/182) = $5.962.

The invoice price is the reported price plus accrued interest = $1015.962

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