Independent solution

How to solve this Duration and Convexity question

Setup

Setup

List the ten equal end-of-year mortgage payments and discount them at 6%.

DM=∑t=110tP(1.06)−t∑t=110P(1.06)−tD_M=\frac{\sum_{t=1}^{10}tP(1.06)^{-t}}{\sum_{t=1}^{10}P(1.06)^{-t}}

Model

Model

The common payment cancels from the time-weighted present-value ratio.

DM=(Ia)10‾∣ 0.06a10‾∣ 0.06D_M=\frac{(Ia)_{\overline{10}|\,0.06}}{a_{\overline{10}|\,0.06}}

Compute

Compute

The increasing-annuity numerator divided by the ordinary-annuity denominator is 5.02201.

DM=36.962417.36009=5.02201D_M=\frac{36.96241}{7.36009}=5.02201

Answer

Answer

The mortgage Macaulay duration is 5.02 years, corresponding to choice B.

DM≈5.02 years(B)\boxed{D_M\approx5.02\text{ years}\quad\text{(B)}}