1. **Problem Statement:** Calculate the net annual premium for a single premium deferred annuity issued to a select life aged 55 with payments starting in 10 years, each payment $50,000 growing at 3% annually, interest rate 5%, using the Standard Select Survival Model. 2. **Step a) Calculation of Net Single Premium:** - The payment at time $t$ (from start) is $$P_t = 50,000 \times 1.03^{t-1}$$ for $t=11, 12, ...$ (since first payment starts at year 10 from issue, i.e. time 10, so payments start at $t=11$). - The annuity is deferred 10 years, so we discount survival probability and interest for 10 years before payments start. - The present value (PV) of the annuity is: $$PV = {}_{10}p_{55} \sum_{k=1}^\infty P_k v^{10+k} {}_{k}p_{65+10}$$ where: - ${}_{10}p_{55}$ is the probability of survival from age 55 to 65 (10 years) under the Standard Select Model. - $v = \frac{1}{1.05}$ is the discount factor per year. - ${}_{k}p_{65}$ is the probability of survival from age 65 for $k$ years. - Because payment grows by 3%, this is a geometric progression, so the sum uses formula for growing annuities. 3. **Step b) Actuarial Present Value (APV) of life annuity immediate paying $1 per year for $n$ years or until death:** - The APV is: $$\text{APV} = \sum_{k=0}^{n-1} v^{k+1} {_{k}p_x}$$ where $v = \frac{1}{1+i}$, ${_{k}p_x}$ is the probability of survival from age $x$ to $x+k$. The sum stops at $n$, reflecting payments for at most $n$ years or death. 4. **Step c) Reason life insurance premiums are paid in advance:** - Premiums are paid at the start of each premium period to ensure the insurer has funds on hand to cover claims as they arise. - Paying in advance reduces the insurer's risk as benefits may be payable immediately upon death. - It aligns with the principle of equivalence and proper cash flow management. Final answers: - a) Calculate deferred annuity PV using survival probabilities and growing annuity formula. Net premium = computed PV. - b) APV formula given. - c) Explained the rationale for advance premium payment.