Let's analyze the problem step by step. --- ## Given Data: | \( x \) | \( q[x] \) | \( q[x+1] \) | \( q[x+2] \) | \( q[x+3] \) | \( x+3 \) | |---------|------------|--------------|--------------|--------------|-----------| | 71 | .10 | .16 | .22 | .28 | 74 | | 72 | .15 | .21 | .27 | .33 | 75 | | 73 | .20 | .26 | .32 | .39 | 76 | | 74 | .25 | .31 | .37 | .44 | 77 | | 75 | .30 | .36 | .42 | .49 | 78 | | 76 | .35 | .41 | .47 | .54 | 79 | | 77 | .40 | .46 | .52 | .58 | 80 | --- ## Question Summary: - Client age: **73** - Insurance paid: **$200,000** - End of year: **Before age 77** - Goal: Find the **expected present value** of the additional coverage, considering an 8% annual effective interest rate. --- ## Step 1: Determine the probability of dying within each year The probability that the client aged 73 will die in year \( x \) (from age \( x \) to \( x+1 \)): \[ p_{x} = q[x] \] And the probability of surviving through \( x \) years: \[ S_{x} = \prod_{k=71}^{x} (1 - q[k]) \] --- ## Step 2: Find the probability of dying in each year before age 77 From the table, the probabilities: | Age | \( q[x] \) | |-------|------------| | 73 | .20 | | 74 | .25 | | 75 | .30 | | 76 | .35 | --- ## Step 3: Calculate survival probabilities up to each age - **Survival from age 73 to 74:** \[ S_{73} = 1 \] \[ S_{74} = (1 - .20) = .80 \] - **Survival from age 73 to 75:** \[ S_{75} = S_{74} \times (1 - .25) = .80 \times .75 = .60 \] - **Survival from age 73 to 76:** \[ S_{76} = S_{75} \times (1 - .30) = .60 \times .70 = .42 \] - **Survival from age 73 to 77:** \[ S_{77} = S_{76} \times (1 - .35) = .42 \times .65 = .273 \] --- ## Step 4: Probabilities of dying in each year \[ \begin{aligned} p_{73} &= .20 \\ p_{74} &= S_{74} \times q[74] = .80 \times .25 = .20 \\ p_{75} &= S_{75} \times q[75] = .60 \times .30 = .18 \\ p_{76} &= S_{76} \times q[76] = .42 \times .35 = .147 \end{aligned} \] --- ## Step 5: Calculate the present value of the death benefit in each year The death benefit is paid at the end of the year, so the present value (PV) of the \$200,000 death benefit paid in year \( t \): \[ PV_t = \frac{200,000}{(1 + i)^t} \] where \( i = 8\% \) or .08. The relevant \( t \) (number of years from age 73): | Year | \( t \) | Probability \( p_{x+t} \) | PV factor \( (1 + .08)^t \) | Present Value of death benefit | |--------|--------|--------------------------|----------------------------|------------------------------| | 73 | | .20 | 1 | \( 200,000 / 1 = 200,000 \) | | 74 | 1 | .20 | 1.08 | \( 200,000 / 1.08 \approx 185,185 \) | | 75 | 2 | .18 | 1.1664 | \( 200,000 / 1.1664 \approx 171,467 \) | | 76 | 3 | .147 | 1.2597 | \( 200,000 / 1.2597 \approx 158,720 \) | --- ## Step 6: Calculate the expected present value (EPV) \[ EPV = \sum_{t=}^{3} p_{x+t} \times PV_t \] \[ EPV = .20 \times 200,000 + .20 \times 185,185 + .18 \times 171,467 + .147 \times 158,720 \] Calculations: \[ \begin{aligned} &.20 \times 200,000 = 40,000 \\ &.20 \times 185,185 \approx 37,037 \\ &.18 \times 171,467 \approx 30,864 \\ &.147 \times 158,720 \approx 23,329 \end{aligned} \] Total: \[ EPV \approx 40,000 + 37,037 + 30,864 + 23,329 = \boxed{\$131,230} \] --- ## **Final Answer:** ### The expected present value of the additional coverage is approximately **\$131,230**.