Explanation:
The correct answer is B.
Bootstrapping is the process of carving out spot rates from the market prices of a set of coupon-paying bonds. The spot rates are determined in a consequential manner: after obtaining the first spot rate, say, for six months, we use that to obtain the spot rate for one year. Then we can use the six-month and one-year spot rates to obtain the one and a half-year spot rate, and so on.
We know that the price of a bond is the discounted value of all of its future cashflows.
For the six month T-bond, therefore:
101.5` = \frac{(100 + 5/2)}{\left(1 + \frac{Z_{0.5}}{2}\right)^1}
$`$101.5`\left[\left(1 + \frac{Z_{0.5}}{2}\right)^1\right] = \`$102.5`Thus, the six-month spot rate is 1.98% with semiannual compounding.
For the one-year T-bond:
102.6` = \frac{(5.25/2)}{\left(1 + \frac{0.0198}{2}\right)^1} + \frac{(100 + 5.25/2)}{\left(1 + \frac{Z_{1.0}}{2}\right)^2}
$`$102.6` = \frac{2.625}{1.0099} + \frac{102.625}{\left(1 + \frac{Z_{1.0}}{2}\right)^2}102.6` = 2.599 + \frac{102.625}{\left(1 + \frac{Z_{1.0}}{2}\right)^2}
$`$100.00`1 = \frac{102.625}{\left(1 + \frac{Z_{1.0}}{2}\right)^2}Thus, the one-year spot rate is 2.61% with semiannual compounding.
Q.4825 Four U.S. T-bonds currently on the market have the following characteristics:
Price per $100 par value | Coupon (paid semiannually) | Maturity (yrs.) | Semiannual period |
|---|---|---|---|
$101.50 | 5.0% | 0.5 | 1 |
$102.60 | 5.25% | 1.0 | 2 |
$103.15 | 5.75% | 1.5 | 3 |
$103.95 | 6.20% | 2.0 | 4 |
Determine the one-year spot rate via the bootstrapping method, assuming semiannual compounding.
A
1.98%
B
2.61%
C
1.31%
D
2.25%
No comments yet.