Question

Calculate the minimum premium an insurance company should charge for $1 million 2-year term life insurance policy issued to a woman age 50. Assume that the premium is paid at the beginning of each year and that the interest rate is zero. Note that: the probability of death within one year at age 50 = 0.003182, and the probability of death within one year at age 51 = 0.003473.

Q: What is the value of X if the interest rate is 3% per annum?

Q: What is the value of X if the interest rate is 4% per annum compounded semiannually and the payout occurs halfway through the year or happens at time 18 months (moving to second year)?

Answer 1

Insured Amount = $1,000,000

Term of contract = 2 years

Probability of death within one year at age 50 = 0.003182

Probability of death within one year at age 51 = 0.003473

Solution

a)If the interest rate is 3% per annum

The expected payout for 1 year term = Probability of death within one year at age 50 * Insured Amount

The expected payout for 1 year term = 0.003182 * $1,000,000

The expected payout for 1 year term = $3,182

Assuming the payout takes place in 6 months,

Present value of expected payout = The expected payout for 1 year term / (1 + interest rate)¹

Present value of expected payout = $3,182 / (1 + (3% / 2))¹

Present value of expected payout = $3134.98

Probability of death in the 2^nd year = (1 - Probability of death within one year at age 50) * Probability of death within one year at age 51

Probability of death in the 2^nd year = (1 - 0.003182) *  0.003473

Probability of death in the 2^nd year = 0.003462

The expected payout for 2^nd year = Probability of death in the 2^nd year * Insured Amount

The expected payout for 2^nd year = 0.003462 * $1,000,000

The expected payout for 2^nd year = $3,462

Assuming the payout takes place in 18 months,

Present value of expected payout  = The expected payout for 1 year term / (1 + interest rate)¹

Present value of expected payout = $3,462 / (1 + (3% / 2))³

Present value of expected payout = $3,310.77

The total present value of the payout = Present value of expected payout in year 1 + Present value of expected payout in year 2

The total present value of the payout = $3134.98 + $3,310.77

The total present value of the payout = $6445.75

The first premium payment is received immediately.

However the probability of the second premium payment being made at the beginning of the 2^nd year is the probability of not dying the in first year

Probability of not dying the in first year = (1 - Probability of death within one year at age 50)

Probability of not dying the in first year = (1 - 0.003182)

Probability of not dying the in first year = 0.996818

Let X be the breakeven premium

The present value of the premium payments = X + (0.996818X / (1 + (3% / 2))²

The present value of the premium payments = X + 0.967573X

The present value of the premium payments = 1.967573X

The total present value of the payout = The present value of the premium payments

$6445.75 = 1.967573X

X = $3275.99

The breakeven annual premium = $3275.99

b) If the interest rate is 4% per annum

The expected payout for 1 year term = Probability of death within one year at age 50 * Insured Amount

The expected payout for 1 year term = 0.003182 * $1,000,000

The expected payout for 1 year term = $3,182

Assuming the payout takes place in 6 months,

Present value of expected payout = The expected payout for 1 year term / (1 + interest rate)¹

Present value of expected payout = $3,182 / (1 + (4% / 2))¹

Present value of expected payout = $3119.61

Probability of death in the 2^nd year = (1 - Probability of death within one year at age 50) * Probability of death within one year at age 51

Probability of death in the 2^nd year = (1 - 0.003182) *  0.003473

Probability of death in the 2^nd year = 0.003462

The expected payout for 2^nd year = Probability of death in the 2^nd year * Insured Amount

The expected payout for 2^nd year = 0.003462 * $1,000,000

The expected payout for 2^nd year = $3,462

Assuming the payout takes place in 18 months,

Present value of expected payout  = The expected payout for 1 year term / (1 + interest rate)¹

Present value of expected payout = $3,462 / (1 + (4% / 2))³

Present value of expected payout = $3,262.32

The total present value of the payout = Present value of expected payout in year 1 + Present value of expected payout in year 2

The total present value of the payout = $3119.61 + $3,262.32

The total present value of the payout = $6,381.93

The first premium payment is received immediately.

However the probability of the second premium payment being made at the beginning of the 2^nd year is the probability of not dying the in first year

Probability of not dying the in first year = (1 - Probability of death within one year at age 50)

Probability of not dying the in first year = (1 - 0.003182)

Probability of not dying the in first year = 0.996818

Let X be the breakeven premium

The present value of the premium payments = X + (0.996818X / (1 + (4% / 2))²

The present value of the premium payments = X + 0.958110X

The present value of the premium payments = 1.958110X

The total present value of the payout = The present value of the premium payments

$6,381.93 = 1.958110X

X = $3259.23

The breakeven annual premium = $3259.23