Actuarial Notes

Insurance Coverage

Study guide: Exam FAM

An insurance coverage is the protection an insurance policy provides: the insured event, the benefit the insurer pays when it occurs, the period over which the promise runs, and the limits, exclusions and conditions that bound it. Every coverage is either an indemnity — it reimburses a measured loss — or a fixed benefit — it pays a stated amount when the event happens.

0 \le Y = g(X) \le X

Y = b \cdot \mathbf{1}\{\text{insured event occurs}\}

  • X is the loss the event causes and g the payment function set by the deductible, coinsurance and limits (Coverage Modifications); indemnity never pays more than the loss. A fixed benefit b — a life insurance sum insured, a monthly disability income — is paid whatever the loss, which is why it needs insurable interest and underwriting of the amount rather than loss adjustment.
  • Short-term coverages (Short-Term Insurance Coverage) — property, liability, auto, group health — run a year or less and are repriced at renewal, so the work is in frequency, severity and outstanding claims.
  • Long-term coverages — life insurance (whole life, term, endowment), annuities, long-term health and disability cover, and defined benefit and defined contribution pension plans — fix premiums and benefits for many years, so interest and survival dominate and the insurer must hold reserves.
  • The same event can be covered either way: a hospital stay can trigger an expense-reimbursing indemnity and a fixed daily cash benefit at once.

Indemnity Versus Fixed Benefit for One Illness A policyholder incurs \$12{,}000 of medical expenses. Her medical expense coverage has a \$1{,}000 deductible and pays 80\% above it; she also holds a critical illness policy paying a fixed \$25{,}000 on diagnosis. What does each coverage pay?

Answer

\begin{align*}
Y_{\text{medical}} &= 0.80\,(12{,}000 - 1{,}000) \\
&= \$8{,}800 \\
Y_{\text{critical illness}} &= \$25{,}000
\end{align*}

The indemnity leaves her \$3{,}200 of the expense and could never pay more than \$12{,}000; the fixed benefit pays \$25{,}000 regardless of what the illness cost.

How Coverage Design Moves the Cost For a life aged x, q_x = 0.010, q_{x+1} = 0.012, q_{x+2} = 0.015 and i = 5\%. Compare the expected present value of a 3-year term insurance and a 3-year endowment insurance, each with sum insured \$100{,}000 payable at the end of the year of death (and, for the endowment, on survival to x+3).

Answer

\begin{align*}
\text{Term} &= 100{,}000\big[v q_x + v^2 p_x q_{x+1} + v^3\, {}_2p_x\, q_{x+2}\big] \\
&= 952.38 + 1{,}077.55 + 1{,}267.41 \\
&= \$3{,}297.34
\end{align*}
\begin{align*}
{}_3p_x &= (0.990)(0.988)(0.985) \\
&= 0.963448 \\
\text{Pure endowment} &= 100{,}000\, v^3 (0.963448) \\
&= \$83{,}226.28 \\
\text{Endowment} &= 3{,}297.34 + 83{,}226.28 \\
&= \$86{,}523.62
\end{align*}

The death cover is the same in both; adding the survival benefit makes the endowment cost 26 times the term, because it pays with near certainty — it is mostly a savings contract.

How it appears on the exam

  • FAM (understand the key features of insurance coverages and retirement financial security programs): identify which long-term coverage — whole life, term, endowment, annuity, health, DB or DC pension — fits a described need, or what a given contract pays under a stated scenario.
  • The same section opens with insurable interest, so a question may combine the two: who can buy the coverage, and what it pays.
  • The calculation side is in the rest of FAM: the present value random variables, premiums and policy values for these coverages, and coverage modifications for the short-term ones.

Exam traps

  • Treating a fixed benefit as an indemnity. A life or critical illness benefit is not reduced to the "loss" the event caused; capping it at expenses is wrong.
  • Mixing up term and endowment. Term pays only on death within the term; endowment also pays on survival to the end, so an answer that gives the endowment a zero survival payout (or the term one) has swapped them.
  • DB versus DC risk-bearer. In a defined benefit plan the sponsor bears investment and longevity risk; in a defined contribution plan the member does. Reversing them is the usual written-answer slip.