1. **Problem statement:** A beneficiary aged 40 has two payment options upon death benefit payment: (i) A lump sum of 10,000. (ii) An annual payment of $c$ at the beginning of each year, guaranteed for 10 years and continuing as long as the beneficiary is alive. The two options are actuarially equivalent. Given: $i=0.04$, $A_{40}=0.3$, $A_{50}=0.35$, $A_{40:10}^1=0.09$. Calculate the value of $c$. 2. **Key formulas and definitions:** - The lump sum payment is 10,000. - The annual payment is an annuity-due for 10 years plus a life annuity from year 10 onward. - The actuarial present value (APV) of option (ii) is: $$\text{APV} = c \times \left( \ddot{a}_{40:10} + v^{10} \times A_{50} \right)$$ where $\ddot{a}_{40:10}$ is the present value of an annuity-due for 10 years at age 40, $v=\frac{1}{1+i}$, and $A_{50}$ is the present value of a whole life insurance at age 50. 3. **Calculate $v$:** $$v = \frac{1}{1+0.04} = 0.9615385$$ 4. **Calculate $\ddot{a}_{40:10}$:** We use the relation: $$A_{40:10}^1 = v \ddot{a}_{40:10} - A_{40:10}$$ Given $A_{40:10}^1 = 0.09$ and $A_{40:10} = A_{40} - v^{10} A_{50}$. Calculate $A_{40:10}$: $$A_{40:10} = A_{40} - v^{10} A_{50} = 0.3 - (0.9615385)^{10} \times 0.35$$ Calculate $v^{10}$: $$v^{10} = (0.9615385)^{10} \approx 0.6651$$ So: $$A_{40:10} = 0.3 - 0.6651 \times 0.35 = 0.3 - 0.2328 = 0.0672$$ 5. **Calculate $\ddot{a}_{40:10}$:** Rearranging the formula: $$A_{40:10}^1 = v \ddot{a}_{40:10} - A_{40:10} \Rightarrow v \ddot{a}_{40:10} = A_{40:10}^1 + A_{40:10}$$ $$\ddot{a}_{40:10} = \frac{A_{40:10}^1 + A_{40:10}}{v} = \frac{0.09 + 0.0672}{0.9615385} = \frac{0.1572}{0.9615385} \approx 0.1635$$ 6. **Set up the actuarial equivalence equation:** $$10,000 = c \times \left( \ddot{a}_{40:10} + v^{10} A_{50} \right) = c \times (0.1635 + 0.6651 \times 0.35)$$ Calculate the term in parentheses: $$0.1635 + 0.6651 \times 0.35 = 0.1635 + 0.2328 = 0.3963$$ 7. **Solve for $c$:** $$c = \frac{10,000}{0.3963} \approx 25233.5$$ **Final answer:** $$\boxed{c \approx 25233.5}$$ This means the annual payment $c$ at the beginning of each year is approximately 25233.5 to be actuarially equivalent to the lump sum of 10,000.