Let's solve the problem **step by step**. ### **Given:** - **Fully discrete 10-pay 20-year insurance** of 10,000 on (30) - **Mortality:** Illustrative Life Table - **Interest rate, i = 5%** - **Premiums:** Equivalence Principle - **Calculate reserves at:** t = 1, t = 10, t = 18 --- ## **Step 1: Notation** - \( A_{x:n} \) = Present value of $1 payable at end of year of death, for n years, for life aged x - \( \ddot{a}_{x:n} \) = Present value of $1 per year for n years, for life aged x - \( P \) = Annual premium (to be calculated) - \( V_t \) = Reserve at time t --- ## **Step 2: Calculate the Premium (\( P \))** The premium is calculated by the **Equivalence Principle**: \[ P \times \ddot{a}_{30:10} = 10,000 \times A_{30:20} \] \[ P = \frac{10,000 \times A_{30:20}}{\ddot{a}_{30:10}} \] --- ## **Step 3: Reserves Formula** For a fully discrete whole life insurance (or n-year term with limited premiums), reserve at time \( t \): \[ V_t = 10,000 \times A_{30+t:20-t} - P \times \ddot{a}_{30+t:\min(10-t, 20-t)} \] - For \( t < 10 \): Premiums due for \( 10-t \) more years - For \( t \geq 10 \): No more premiums due So: - For \( t < 10 \): \[ V_t = 10,000 \times A_{30+t:20-t} - P \times \ddot{a}_{30+t:10-t} \] - For \( t \geq 10 \): \[ V_t = 10,000 \times A_{30+t:20-t} \] --- ## **Step 4: Find Values from the Illustrative Life Table** Assume you have access to the **Illustrative Life Table** (common values): - \( i = 5\% \to v = 1/1.05 = 0.95238 \) - \( A_{30:20} \approx 0.220 \) - \( \ddot{a}_{30:10} \approx 7.72 \) - \( A_{31:19} \approx 0.229 \) - \( \ddot{a}_{31:9} \approx 6.97 \) - \( A_{40:10} \approx 0.330 \) - \( A_{48:2} \approx 0.505 \) (for t=18) - \( \ddot{a}_{48:2} \approx 1.87 \) *(These values are for illustration; **use your table for exact calculations**!)* --- ## **Step 5: Calculate Premium \( P \)** \[ P = \frac{10,000 \times 0.220}{7.72} \approx \frac{2,200}{7.72} \approx 284.97 \] --- ## **Step 6: Calculate Reserves** ### **(a) t = 1** \[ V_1 = 10,000 \times A_{31:19} - P \times \ddot{a}_{31:9} \] \[ V_1 = 10,000 \times 0.229 - 284.97 \times 6.97 \] \[ = 2,290 - 1,986.26 = \boxed{303.74} \] --- ### **(b) t = 10** Premiums stop at t=10. Reserve is just the present value of the remaining benefit: \[ V_{10} = 10,000 \times A_{40:10} \] \[ = 10,000 \times 0.330 = \boxed{3,300} \] --- ### **(c) t = 18** Only 2 years remain, and no premiums are left. \[ V_{18} = 10,000 \times A_{48:2} \] \[ = 10,000 \times 0.505 = \boxed{5,050} \] --- ## **Step 7: Boxed Final Answers** ### **Final Answers:** \[ \boxed{ \begin{align*} \text{Reserve at } t=1: & \quad 303.74 \\ \text{Reserve at } t=10: & \quad 3,300 \\ \text{Reserve at } t=18: & \quad 5,050 \\ \end{align*} } \] **Note:** You must use the precise values from your Illustrative Life Table for exact answers, but the steps and method above will always apply! If you provide the table, I can use the *exact* values for you.