A $2$-year term insurance policy on $(60)$ provides for a death benefit of $100$ payable at the end of the year of death. This is purchased by a single premium. If $(60)$ lives to age $62$, the single premium is returned without interest. Given $q_{60}=0.1, q_{61}=0.15$, and interest is a constant $10%$, find the single premium.
Present value of death benefit
$A_{60}(100)=\sum b_k v(k+1) q_x$ $_k$$p_x=100(1.10)^{-2}$ X $0.15$ X $0.9=11.1570$
Present value of endowment
$a_{60}(P)=P(0.9)(0.85)(1.10)^{-2}$
Premium pattern
$a_{60}(P)=P$
$a_{60}(P)=A_{60}(100)+a_{60}(P)$
Hence P= 30.33
But the answer is $55.06$
The problem is simple enough that you can find the actuarial present value as follows: with probability $q_{60} = 0.1$, $(60)$ dies in the first year, and the benefit paid is $100v$, where $v = (1+i)^{-1}$ is the annual present value discount factor. With probability $p_{60} q_{61} = (0.9)(0.15)$, $(60)$ dies in the second year and the benefit paid is $100v^2$. The only remaining possibility is if $(60)$ survives the term of the policy, in which case the premium is returned without interest, so the present value of this is $Pv^2$, which occurs with probability $p_{60} p_{61} = (0.9)(0.85)$. Therefore, our equation of value is $$P = (0.1)(100v) + (0.9)(0.15)(100v^2) + (0.9)(0.85)(Pv^2).$$ Solving for $P$, we obtain $$P = \frac{100v(20+27v)}{200-153v^2}$$ and substituting $v = 0.90909$, we get $P = 4900/89 \approx 55.0562$ as claimed.