Show how mortality tables are used and work out the premium that a policy holder pays.

FRM® Part I · Financial Markets and Products · Insurance Companies and Pension Plans · FRM1.FM.2.b

The idea

A mortality table gives, for each age x, the one-year death probability q(x) or the number of survivors l(x) out of a starting group. From survivor counts, the one-year death probability is the fall in survivors divided by those alive at the start: q = (l(x) − l(x + 1)) ÷ l(x). The chance of being alive at the start of year n is the product of survival probabilities (1 − q) of earlier years, and the chance of dying in year n is that product times that year's q. Premiums are paid at the start of each year only while the insured is alive, and the benefit B at the end of the year of death. The pure premium P equates the present value of expected premiums to that of expected benefits. P = B × Σ(Pr(death in year t) × v^t) ÷ Σ(Pr(alive at start of year t) × v^(t − 1)). Here v is the discount factor 1 ÷ (1 + r) for interest rate r. Expenses and profit are added as a loading. The usual error is treating the premium stream as certain.

Formulas

One-year death probability from a mortality table

q(x) = (l(x) − l(x + 1)) ÷ l(x)

l(x): the number of survivors at age x out of the table's starting group
x: the age at the start of the year

Conditional on being alive at age x, so divide by those alive at the start of the year, not at the end or at birth.

Pure annual premium of a term life policy

P = B × Σ(dₜ × vᵗ) ÷ Σ(aₜ × v^(t − 1))

B: the benefit paid at the end of the year of death
dₜ: the probability, seen from today, of death in year t
aₜ: the probability of being alive at the start of year t
v: the discount factor 1 ÷ (1 + r) at interest rate r
t: the policy year, from 1 to the term

Premiums are paid at the start of each year only while the insured lives. Weighting premiums as if certain, or using the conditional q instead of the probability of death in year t, are the usual errors.

A practice question

From the question bank, with the reason each wrong option is wrong.

A mortality table shows, for each age x, the probability q(x). What does q(x) represent?

The probability that a person aged x survives to the end of the tableWhy not: This is a long-horizon survival probability, not a one-year death probability.
The proportion of the original group of newborns still alive at age xWhy not: This is the survivor fraction l(x)/l(0), not the one-year conditional death probability.
The probability that a newborn dies at exactly age x, measured from birthWhy not: This is an unconditional probability from birth; q(x) is conditional on being alive at age x.
The probability that a person alive at age x dies before age x+1 - correct

q(x) is a one-year conditional death probability: it assumes the person has already reached age x. Survivor counts l(x) are built from successive (1 - q) terms.

A harder one

An insurer sells a one-year term policy with a death benefit of USD 500,000. The one-year death probability is 1.25%, the premium is paid at the start of the year, the benefit is paid at the end of the year, and the discount rate is 4%. The pure premium (before expenses and profit) is closest to:

USD 6,500Why not: Multiplies by (1 + r) instead of discounting, moving the expected benefit forward instead of back.
USD 6,010 - correct
USD 62,500Why not: Slips a decimal place in the probability (treats 1.25% as 12.5%) before discounting is considered.
USD 6,250Why not: Omits discounting; the benefit is paid a year after the premium is received.

Pure premium = benefit x q / (1 + r) = 500,000 x 0.0125 / 1.04 = 6,009.62. The expected benefit at year-end is discounted to the premium date.

The hardest tier

A mortality table shows 91,000 persons alive at age 60 and 86,000 alive at age 65, from a starting group of 100,000 at birth. A pure endowment pays USD 100,000 to a 60-year-old only if the person is alive in five years. Using a 4% annual discount rate, the single premium (ignoring expenses) is closest to:

USD 77,677 - correct
USD 82,193Why not: Ignores the survival probability, assuming the benefit will certainly be paid in five years.
USD 70,686Why not: Divides survivors at 65 by the original 100,000 instead of by the 91,000 alive at age 60.
USD 4,516Why not: Uses the probability of dying within five years instead of surviving; a pure endowment pays on survival.

Probability of surviving from 60 to 65 = 86000/91000 = 0.9451. Single premium = 100,000 x 0.9451 / 1.04^5 = 77,677. The condition is survival, conditional on being alive at 60.

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