Suppose that $q_{61}=0.2, q_{61}=0.25, q_{62}=0.4, q_{63}=0.5$ and the interest rate is a constant $5%$ for the first $2$ years, $7%$ after that.A $5$-year life annuity on $(60)$ provides for payments of $100(1+k)$ at time $k$, where $k=0,1,2,3,4$.
(a) Find the present value.
(b) Suppose that instead of being a straight life annuity, the first three annuity payments are guaranteed regardless of whether $(60)$ is alive or not. Find the present value.
Solution for (a)
Cashflow vector $c_k:100,200,300,400,500$
Discount: $v(0)=1 v(1)=\frac{20}{21}, v(2)=\frac{400}{441}, v(3)=\frac{400}{441}(1.07)^{-1}, v(4)=\frac{400}{441}(1.07)^{-2}$
$_0P_{60}=1, _1P_{60}=0.8, _2P_{60}=0.6, _3P_{60}=0.36, _4P_{60}=0.18$
P.V=$100+(200)(\frac{20}{21})(0.8)+(300)(\frac{400}{441})(0.6)+(400)(\frac{400}{441}(1.07)^{-1})(0.36)+(500)(\frac{400}{441}(1.07)^{-2})(0.18)=609.01$
Solution for (b) is a mess.
Why would (b) be any more difficult? You simply change ${}_1 p_{60} = {}_2 p_{60} = 1$, but leave the other survival probabilities unchanged, then recalculate. This gives you the correct present value because you can think of ${}_k p_{60}$ as not the $k$-term survival probability of $(60)$, but rather, the probability of payment in the $k^{\rm th}$ period. Then it becomes clear what you need to do.