1. **Problem statement:** Harold, aged 60, buys a life annuity paying 1000 annually starting at age 61. The death probability at age 60 is increased by 0.1, i.e., $q_{60}^* = q_{60} + 0.1$. Given $N_{60} = 4650$, $N_{61} = 3950$, $N_{62} = 3350$, and interest rate $i = 0.07$, find the net single premium. 2. **Key formulas and definitions:** - Probability of death at age 60 from table: $q_{60} = \frac{N_{60} - N_{61}}{N_{60}} = \frac{4650 - 3950}{4650} = \frac{700}{4650}$. - Adjusted death probability: $q_{60}^* = q_{60} + 0.1$. - Probability of survival at age 60: $p_{60}^* = 1 - q_{60}^*$. - Interest factor: $v = \frac{1}{1+i} = \frac{1}{1.07}$. - Net single premium (NSP) for life annuity is the expected present value of payments: $$ NSP = 1000 \times \sum_{k=1}^\infty v^k \, {}_k p_{60}^* $$ where ${}_k p_{60}^*$ is the probability Harold survives $k$ years from age 60 under the adjusted mortality. 3. **Calculate adjusted probabilities:** - Calculate $q_{60}$: $$ q_{60} = \frac{700}{4650} \approx 0.1505 $$ - Adjusted $q_{60}^*$: $$ q_{60}^* = 0.1505 + 0.1 = 0.2505 $$ - Adjusted survival probability at age 60: $$ p_{60}^* = 1 - 0.2505 = 0.7495 $$ 4. **Calculate survival probabilities for subsequent years:** - For $k=1$ (age 61 to 62), use original mortality since adjustment applies only at age 60: $$ q_{61} = \frac{N_{61} - N_{62}}{N_{61}} = \frac{3950 - 3350}{3950} = \frac{600}{3950} \approx 0.1519 $$ - Survival probability at age 61: $$ p_{61} = 1 - q_{61} = 1 - 0.1519 = 0.8481 $$ 5. **Calculate ${}_k p_{60}^*$ for $k=1,2$:** - ${}_1 p_{60}^* = p_{60}^* = 0.7495$ - ${}_2 p_{60}^* = p_{60}^* \times p_{61} = 0.7495 \times 0.8481 = 0.6357$ 6. **Calculate present value factors:** - $v = \frac{1}{1.07} = 0.9346$ - $v^1 = 0.9346$ - $v^2 = 0.9346^2 = 0.8735$ 7. **Calculate expected present value of payments for first two years:** $$ 1000 \times (v^1 \times {}_1 p_{60}^* + v^2 \times {}_2 p_{60}^*) = 1000 \times (0.9346 \times 0.7495 + 0.8735 \times 0.6357) $$ $$ = 1000 \times (0.7003 + 0.5555) = 1000 \times 1.2558 = 1255.8 $$ 8. **Approximate further years:** Assuming mortality stabilizes to $q_{61}$ for subsequent years, the annuity can be approximated as a 2-year annuity plus a perpetuity from year 3 onward. 9. **Calculate perpetuity from year 3 onward:** - Survival from year 2 to 3: $p_{61} = 0.8481$ - Present value factor from year 3 onward: $$ \text{Perpetuity} = \frac{v^3 \times {}_3 p_{60}^*}{1 - v \times p_{61}} $$ - Calculate $v^3 = v^2 \times v = 0.8735 \times 0.9346 = 0.8163$ - Calculate ${}_3 p_{60}^* = {}_2 p_{60}^* \times p_{61} = 0.6357 \times 0.8481 = 0.5393$ - Denominator: $$ 1 - v \times p_{61} = 1 - 0.9346 \times 0.8481 = 1 - 0.7927 = 0.2073 $$ - Perpetuity value: $$ \frac{0.8163 \times 0.5393}{0.2073} = \frac{0.4401}{0.2073} = 2.123 $$ - Present value of perpetuity payments: $$ 1000 \times 2.123 = 2123 $$ 10. **Total net single premium:** $$ NSP = 1255.8 + 2123 = 3378.8 $$ **Final answer:** The net single premium for the annuity is approximately **3379**.