Independent solution

How to solve this Premium Amortization Threshold question

Setup

Setup

Scale face amount to 1000. The semiannual coupon is 15 and the semiannual yield is 1%.

C=15,j=0.01,N=40C=15,\qquad j=0.01,\qquad N=40

Model

Model

Compute purchase price and the post-coupon prospective book value.

B0=15a40‾∣j+1,000v40=1,164.1734B_0=15a_{\overline{40}|j}+1{,}000v^{40}=1{,}164.1734
Bk=15a40−k‾∣j+1,000v40−kB_k=15a_{\overline{40-k}|j}+1{,}000v^{40-k}

Compute

Compute

Compare cumulative amortization with the original premium.

B0−B21B0−1,000=0.47537\frac{B_0-B_{21}}{B_0-1{,}000}=0.47537
B0−B22B0−1,000=0.50058\frac{B_0-B_{22}}{B_0-1{,}000}=0.50058

Answer

Answer

The cumulative fraction first exceeds one-half after coupon 22, choice E.

n=22(E)\boxed{n=22\quad\text{(E)}}