1. [12pts] Consider the following bond (assume: credit risk free, no embedded op
ID: 2783261 • Letter: 1
Question
1. [12pts] Consider the following bond (assume: credit risk free, no embedded options, pays interest semiannually): Coupon 9% YTM 8% Term (yrs) 5 Par 100.00 Compute the following (a) The price value of a basis point; (b) The exact Macaulay duration; (c) The exact modified duration; (d) An approximate value for the modified duration, obtained by changing the yield by ±20bp (i.e., use numerical approximation to compute the first derivative). How does it compare to the exact value obtained in part (c)? (e) The exact convexity measure; (f) An approximate value of the convexity measure (i.e., use numerical approximation to compute the second derivative), obtained by changing the yield by ±20bp. How does it compare to the exact value obtained in part (e)? 2. [8pts] For the bond in question 1, (a) [1pt] Calculate the actual, exact price change for a 100bp increase in interest rates; (b) [2pt] Using duration only, estimate the price of the bond for a 100bp increase in rates; (c) [2pt] Using duration and convexity, estimate the price change of the bond for a 100bp increase in rates; (d) [1pt] Comment on the accuracy of the approximations in (b) and (c) and explain why one is better than the other; (e) [2pt] Without an actual calculation, indicate (“explain”) whether the duration of the bond would be higher or lower if the YTM were 10%, rather than 8%. Hint: recall that the price-yield curve is convex for risk- and option-free bonds. 3. [5pts] The federal housing GSEs (Fannie and Freddie) hold mortgages on balance sheet, financed by issuing debt. Their `”net duration’’ is said to be zero, while they are “short convexity.” a) Explain what these two phrases mean. b) Sketch the (book) value of their equity as a function of yield. c) How would this diagram look if their net portfolio had positive convexity? d) Given the market’s perception that the Federal government would step in to make bondholders whole in the event of bankruptcy, why would the GSEs want negative convexity in their net portfolios? e) Why might your answer in d) not apply to a fully private, much smaller firm that is in the exact same business as the GSEs?
Explanation / Answer
1a.
Change in price = 103.9927023 - 103.99271 = - 0.0000077 = Price value of a basis point
2. Macalauy duration is time weighted average of cash flows:
Thus macaulay duration = 4.26
c. Modified duration = macaulay duration /(1+YTM)
= 4.26/(1+0.08)
= 3.94
d. Changing yield by 20 basis points each side will give following bond prices:
Price of bonds = 103.1774, 103.2083
Change in bond price = 0.0308643
Modified duration approximate = PV(-) - PV(+)/2*change in yield*PVo
=0.0308643/(2*0.4*103.9927)
=3.71
It is slightly lesser than what was obtained in (c), this is because it is an approximation on a straightline instead of on the curve.
Chegg allows only 4 parts of a question to be answered in one post. You may post the remaining question separately.
Coupon 9% Yield 8% Time 1 2 3 4 5 Bond cash flows 9 9 9 9 109 Discount factor 0.9259 0.8573 0.7938 0.7350 0.6806 Price of the bond now 103.99271 Suppose YTM is increased by 1 basis point i.e. 0.01 percent, we need to calculate bond price thenRelated Questions
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