Long-Term Insurance Coverage

Study guides: Exam FAM, Exam ALTAM, Exam CP-312, Exam CP-321, Exam GH-201C

A Long-Term Insurance Coverage is an insurance coverage written for many years or for life, whose guaranteed terms the insurer cannot change while the contract is in force: life insurance, annuities, long-term health insurance and pension plans. The cash flows run for decades and depend on the survival or health of the insured, so they are valued with interest and mortality, paid for by level premiums, and backed by policy values.

$\text{EPV} = \sum_{k} \text{CF}_k\, v^{k}\,\Pr(\text{CF}_k \text{ is paid})$

$\text{EPV(premiums)} = \text{EPV(benefits)} + \text{EPV(expenses)}$

${}_tV = \text{EPV}_t(\text{outgo}) - \text{EPV}_t(\text{income})$

Coverage	What it pays	Main risks to the provider
Term Life Insurance	Sum insured on death within $n$ years	Mortality, lapse
Whole Life Insurance	Sum insured on death whenever it occurs	Mortality, interest
Endowment Insurance	Sum insured on death within $n$ years or on survival to $n$	Interest
Universal Life Insurance	Death benefit plus an account value credited with interest	Interest, lapse
Life Annuity	Income while the annuitant is alive	Longevity, interest
Disability income insurance	Income while the insured is sick	Morbidity, recovery
Long-term care insurance	Care costs while the insured cannot live independently	Morbidity, longevity
Critical illness insurance	Lump sum on diagnosis of a listed illness	Morbidity
Defined Benefit Plan	Pension set by a formula	Sponsor: investment, longevity
Defined Contribution Plan	Whatever the member's account buys	Member: investment, annuity price

Symbols: $\text{CF}_k$ is a cash flow due at time $k$ , paid only if the life is in the state the contract requires, such as alive, dead or sick. The second block is the equivalence principle, and the third is the policy value, the long-term form of a reserve.
What makes a coverage long-term: The terms are guaranteed, so the insurer cannot reprice when mortality, morbidity or interest rates move against it. A level premium is more than the cost of cover in the early years and less than it later, so the insurer holds a policy value that earns investment income for decades. Interest therefore matters as much as mortality.
Health: Long-term health insurance is modelled with a multiple state model, in which benefits depend on being sick or on falling sick.
Contrast: A short-term coverage is priced one term at a time, so its premium is close to that term's expected claims, and its reserves are for claims already incurred rather than for future guaranteed benefits.

Why a Level Premium Builds a Reserve A fully discrete whole life insurance of $100,000 on $(50)$ has the SULT net premium $1,111.99 at $i = 5\%$ , as in Equivalence Principle. The cost of one year's cover at age $y$ , paid at the start of the year, is $100{,}000\,v\,q_y$ . Compare it with the premium at ages 50, 60, 70 and 80, given $q_{50} = 0.001209$ , $q_{60} = 0.003398$ , $q_{70} = 0.010413$ and $q_{80} = 0.032658$ . At what age does the cost first exceed the premium, given $q_{71} = 0.011670$ and $q_{72} = 0.013081$ ?
Answer
 $\begin{align*} \text{Age 50: } 100{,}000(0.001209)/1.05 &= 115.14 \\ \text{Age 60: } 100{,}000(0.003398)/1.05 &= 323.62 \\ \text{Age 70: } 100{,}000(0.010413)/1.05 &= 991.71 \\ \text{Age 80: } 100{,}000(0.032658)/1.05 &= 3{,}110.29 \end{align*}$ 
The cost is $1,111.43 at age 71 and $1,245.81 at age 72, so it first exceeds the level premium at age 72. Until then, each premium pays more than the year's cover, and the excess is held as the policy value and earns interest. From age 72 on, that policy value pays for the shortfall, which is why a guaranteed level premium needs a reserve that a one-year coverage does not.

Interest Sensitivity of Long-Term Values On the SULT, $\ddot{a}_{65} = 13.5498$ and $A_{65} = 0.35477$ at $5\%$ , and $\ddot{a}_{65} = 14.8746$ and $A_{65} = 0.42790$ at $4\%$ , with $q_{65} = 0.005915$ . Find the change in the EPV of a life annuity-due of $10,000 a year on $(65)$ , in $A_{65}$ , and in the EPV of a one-year term insurance of $100,000 on $(65)$ when the interest rate falls from $5\%$ to $4\%$ .
Answer
 $\begin{align*} \text{Annuity: } 10{,}000(13.5498) &= 135{,}498 \\ 10{,}000(14.8746) &= 148{,}746 \\ \text{Change} &= +9.8\% \\ \text{Whole life: } 0.42790/0.35477 - 1 &= +20.6\% \\ \text{One-year term: } 100{,}000(0.005915)/1.05 &= 563.33 \\ 100{,}000(0.005915)/1.04 &= 568.75 \\ \text{Change} &= +1.0\% \end{align*}$ 
A one-point fall in the interest rate barely moves a one-year cover, but it raises the cost of the long-term benefits by 10% to 21%. An insurer that guaranteed these benefits at $5\%$ bears that cost, which is the interest rate risk of a long-term coverage.

How it appears on the exam

FAM asks candidates to identify the long-term coverages, including life, health, annuities, and DB and DC pension plans, from a description of their benefits, and to contrast them, for example by who bears the investment risk in a pension plan.

Calculations: FAM's sections on present value random variables and on premiums and policy values are the calculation side of the same coverages: EPVs, premiums by the equivalence or portfolio percentile principle, policy values, and the effect of a change in mortality or interest.

ALTAM asks candidates to critique the Markov assumption in multiple state models for long-term benefits. A typical point is that the chance of recovering from sickness depends on how long the life has been sick, which a Markov model ignores.

Exam traps

Long-term health is not group medical: Disability income, long-term care and critical illness are long-term coverages with level premiums. Group medical insurance, renewed and repriced each year, is short-term, so its reserves are chiefly claim reserves.

Who bears the risk: In a DB plan the sponsor bears investment and longevity risk, and in a DC plan the member does. Reversing the two describes the other kind of plan.

Endowment versus pure endowment: An endowment insurance pays on death within the term as well as on survival. A pure endowment pays only on survival, so a question about "the death benefit of an endowment" refers to the endowment insurance.