Yield Duration Statistics: Modified Duration
A semiannual coupon corporate bond with a USD 100 face value currently sells for USD 103.2964. It offers a 5.5% coupon and has six years to maturity. The bond’s modified duration is _closest_ to:
Incorrect.
One possible way to get this answer choice is by not adjusting the modified duration to adjust for the semiannual coupon payment frequency.
Incorrect.
One possible way to get this answer choice is to adjust the Macaulay duration by the interest-rate factor (i.e., 1 + _r_, where _r_ is the annualized yield to maturity). This is incorrect since the bond is a semiannual coupon bond, and the Macaulay duration should be adjusted by the semiannual yield to maturity.
Correct!
First, calculate the bond's Macaulay duration using the following formula:
$$\displaystyle \text{MacDur} = \frac{1+r}{r} - \frac{1+r+[N \times (c-r)]}{c \times \left[ (1+r)^N - 1 \right] + r} - \frac{t}{T} $$
Here, the bond's yield to maturity is unknown. It can be calculated as
$$\displaystyle PV_{0} = \frac{PMT}{(1+r)^{1}} + \frac{PMT}{(1+r)^{2}} + \frac{PMT}{(1+r)^{3}} + ... + \frac{PMT+FV}{(1+r)^{N}}$$
$$\displaystyle 103.2964 = \frac{2.75}{(1+r)^{1}} + \frac{2.75}{(1+r)^{2}} + \frac{2.75}{(1+r)^{3}} + ... + \frac{2.75+100}{(1+r)^{12}}$$
$$\displaystyle r \approx 0.0243$$.
Therefore, the Macaulay duration is
$$\displaystyle \text{MacDur} = \left \{\frac{1+0.0243}{0.0243} - \frac{1 + 0.0243 + [12 \times (0.0275 - 0.0243)]}{0.0275 \times [(1 + 0.0243)^{12} - 1] + 0.0243} \right\} - \frac{0}{180} \approx 10.41328$$.
Since this is a semiannual coupon bond, the annualized value of the Macaulay duration is
$$\displaystyle \frac{10.41328}{2} = 5.20664$$.
Finally, the modified duration is
$$\displaystyle \text{ModDur} = \frac{\text{MacDur} }{(1+r)} = \frac{5.20664}{(1+0.0243)} \approx 5.08$$.
4.97.
5.08.
9.93.
