Life insurance and annuity values

Why it matters

A life insurer sells a promise: pay a benefit if the insured person dies during the term. The insurer does not know when (or whether) it will pay, but it can compute the expected present value of the payment, which actuaries call the actuarial present value (APV). Do the same for the premiums the customer will pay, set the two equal, and you have a price. This lesson joins the interest rates of module 1 with the life table of the last lesson.

Actuarial present value

For each possible payment time, multiply three things and add them up:

APV = sum over times of  (amount) * (discount factor) * (probability the payment is made)

That is all. The three building blocks, for a life aged x, interest rate i and v = 1/(1+i):

1. Term insurance of 1, paid at the end of the year of death if death happens within n years:

A1_(x:n) = sum for k = 0..n-1 of  v^(k+1) * k_p_x * q_(x+k)

k_p_x * q_(x+k) is the probability of surviving k years and then dying in year k + 1 (the deferred probability k|q_x from the last lesson). The payment comes at the end of that year, hence v^(k+1).

2. Pure endowment of 1, paid at time n only if (x) is alive: n_E_x = v^n * n_p_x. An endowment insurance pays on death within n years or on survival to n, so its APV is A1_(x:n) + n_E_x.

3. Temporary life annuity-due of 1 per year for at most n years while alive (this is how premiums are paid, at the start of each year):

ä_(x:n) = sum for k = 0..n-1 of  v^k * k_p_x

The equivalence principle

The net premium P is the level annual premium that makes the APV of premiums equal the APV of benefits:

P * ä_(x:n) = B * A1_(x:n)     so     P = B * A1_(x:n) / ä_(x:n)

"Net" means no expenses or profit are loaded yet; a gross premium adds them.

Worked example: a 5-year term policy

A person aged 60 buys a 5-year term policy with benefit 100,000, paid at the end of the year of death. Premiums are paid annually in advance. Use the illustrative life table from the last lesson and i = 5%.

APV of benefit = 100,000 * 0.0506876 = 5,068.76
ä_(60:5)       = 4.452300
Net premium    = 5,068.76 / 4.452300 = 1,138.46 per year

Check the reasonableness: about 5.9% of these 60-year-olds die within 5 years (1 - 0.941), so expected claims are about 5,900 undiscounted per policy; discounting brings that to about 5,069, spread over roughly 4.45 expected premium payments.

v = 1 / 1.05
A = sum(v ** (k + 1) * lx[60 + k] / lx[60] * (lx[60 + k] - lx[61 + k]) / lx[60 + k] for k in range(5))
a = sum(v ** k * lx[60 + k] / lx[60] for k in range(5))
print(round(100000 * A / a, 2))  # 1138.46

What real pricing adds

Real premiums add expenses (commission, underwriting, administration), a profit margin, lapses (policyholders who stop paying), and select mortality for newly underwritten lives. Benefits are often paid at the moment of death rather than year-end, handled with continuous versions of these formulas (an A with a bar) or a fractional-age assumption. Exam ALTAM and the SOA's FAM exam cover these extensions; check the current syllabus on the SOA site.

Common mistakes

• Wrong timing of the discount factor. Benefits paid at the end of the year of death use v^(k+1); premiums paid at the start of each year use v^k.
• Using k_p_x alone for a death benefit. The benefit needs survival and then death: k_p_x * q_(x+k).
• Dividing by n instead of the annuity. Premiums are only paid while alive, and they are discounted, so divide by ä_(x:n), not by n.
• Forgetting the premium stops at death. That is why the annuity uses survival probabilities.

Quick recap

• APV = sum of amount x discount x probability.
• Term insurance A1 = sum v^(k+1) k_p_x q_(x+k); pure endowment v^n n_p_x; annuity-due sum v^k k_p_x.
• Net level premium = APV(benefits) / APV(annuity of 1).

Try it

In the Do tab, code these three building blocks and price the 5-year term policy.