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Problems <strong>and</strong> <strong>Solutions</strong><br />
We take a long position of $100,000,000 in the 5.5%/12-year bond. The<br />
hedge ratio HR is equal <strong>to</strong><br />
HR = 0.06738<br />
= 1.10818<br />
0.06081<br />
Then we have <strong>to</strong> take a short position of x in the 5%/10-year bond, where x<br />
is given by<br />
6 CHAPTER 6—Problems<br />
x = HR × $100,000,000 = $110,818,000<br />
Exercise 6.1 We consider a 20-year zero-coupon bond with a 6% YTM <strong>and</strong> $100 face value.<br />
Compounding frequency is assumed <strong>to</strong> be annual.<br />
1. Compute its price, modified duration, $duration, convexity <strong>and</strong> $convexity?<br />
2. On the same graph, draw the price change of the bond when YTM goes from<br />
1% <strong>to</strong> 11%<br />
(a) by using the exact pricing formula;<br />
(b) by using the one-order Taylor estimation;<br />
(c) by using the second-order Taylor estimation.<br />
Solution 6.1 1. The price P of the zero-coupon bond is simply<br />
$100<br />
P =<br />
= $31.18<br />
(1 + 6%) 20<br />
Its modified duration is equal <strong>to</strong> 20/(1 + 6%) = 18.87<br />
Its $duration, denoted by $Dur, is equal <strong>to</strong><br />
$Dur=−18.87 × 31.18 =−588.31<br />
Its convexity, denoted by RC, is equal <strong>to</strong><br />
RC = 20 × 21 ×<br />
= 373.80<br />
100<br />
(1 + 6%) 22<br />
Its $convexity, denoted by $Conv, is equal <strong>to</strong><br />
$Conv = 373.80 × 31.18<br />
= 11,655.20<br />
2. Using the one-order Taylor expansion, the new price of the bond is given by<br />
the following formula:<br />
New Price = 31.18 + $Dur×(New YTM − 6%)<br />
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