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Answer: $4,287
## Explanation To calculate the breakeven premium for a two-year term insurance policy, we need to find the premium amount that makes the present value of expected premiums equal to the present value of expected payouts. **Given:** - Face amount: $2,000,000 - Interest rate: 2% per annum with semiannual compounding - Premiums paid annually at beginning of year - Payouts occur halfway through the year - Mortality rates from the table **Step 1: Calculate effective semiannual rate** Since the interest rate is 2% per annum with semiannual compounding: - Semiannual rate = 2%/2 = 1% = 0.01 **Step 2: Calculate present value of expected payouts** **Year 1 Payout:** - Probability of death in year 1: 0.002092 - Payout occurs at time 0.5 years - Present value factor = (1.01)^(-1) = 0.990099 - Expected payout PV = $2,000,000 × 0.002092 × 0.990099 = $4,141.67 **Year 2 Payout:** - Probability of survival to year 1: 0.95908 (from table) - Probability of death in year 2: 0.002240 - Payout occurs at time 1.5 years - Present value factor = (1.01)^(-3) = 0.970590 - Expected payout PV = $2,000,000 × 0.95908 × 0.002240 × 0.970590 = $4,170.67 **Total PV of expected payouts = $4,141.67 + $4,170.67 = $8,312.34** **Step 3: Calculate present value of $1 premium payments** **Premium at time 0:** $1 **Premium at time 1:** $1 × (Probability of survival to year 1) × Present value factor - Probability of survival to year 1: 0.95908 - Present value factor for time 1: (1.01)^(-2) = 0.980296 - PV of second premium = $1 × 0.95908 × 0.980296 = $0.9400 **Total PV of $1 premium payments = $1 + $0.9400 = $1.9400** **Step 4: Calculate breakeven premium** - Breakeven premium = PV of expected payouts ÷ PV of $1 premium payments - Breakeven premium = $8,312.34 ÷ 1.9400 = $4,284.71 This value is closest to **$4,287** (Option B). The calculation accounts for: - Mortality probabilities in each year - Timing of premium payments (beginning of year) - Timing of death payouts (mid-year) - Semiannual compounding of interest rates - Survival probabilities affecting both premium payments and death probabilities
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The relevant interest rate for insurance contracts is 2% per annum (semiannual compounding applies) and all premiums are paid annually at the beginning of the year. A $2,000,000 term insurance contract is being proposed for a 40-year-old male in average health. Assume that payouts occur halfway throughout the year. Using the mortality rates estimated by the U.S. Social Security Administration, which of the following amounts is closest to the insurance company's breakeven premium for a two-year term?
| Age | Probability of Death within 1 Year | Survival Probability | Life Expectancy |
|---|---|---|---|
| 40 | 0.002092 | 0.95908 | 38.53 |
| 41 | 0.002240 | 0.95708 | 37.61 |
A
$4,246
B
$4,287
C
$4,332
D
$8,482