Independent solution

How to solve this Annuity Exchange under Changing Rates question

Setup

Setup

Value the original perpetuity under the two interest-rate regimes: 10% for the first ten years and 8% thereafter. The first ten payments and the deferred tail must be separated.

PVperp=15000(a¨10‾∣ 0.10+11.101010.08)=179457.87PV_{\rm perp}=15000\left(\ddot a_{\overline{10}|\,0.10}+\frac{1}{1.10^{10}}\frac1{0.08}\right)=179457.87

Model

Model

Value the replacement twenty-five-payment annuity in the same two segments, using annuity-due factors because payments begin immediately in each segment.

179457.87=X(a¨10‾∣ 0.10+1.10−10a¨15‾∣ 0.08)179457.87=X\left(\ddot a_{\overline{10}|\,0.10}+1.10^{-10}\ddot a_{\overline{15}|\,0.08}\right)

Compute

Compute

The original stream is worth 179,457.87. Dividing by the replacement annuity factor gives X = 17,384.

X=17384X=17384

Answer

Answer

The calculation gives 17,384 for annuity exchange under changing rates, matching published choice B.

X=17384(B)\boxed{X=17384\quad\text{(B)}}