Deriving Premiums from Expected Benefit Values

Why Premiums Depend on State Probabilities

A long-term care contract can pay benefits only while an insured meets its care-state trigger, while premiums may be due only in selected states. The expected present value of benefits therefore depends on the probability of being in an eligible care state at each payment time. The premium value depends separately on the probability that a premium is payable at each premium date. Equating these two values gives the equivalence-principle net premium.

This is a pricing calculation under specified assumptions, not a prediction of an individual insured’s payment history. It is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. The model below excludes expenses, profit margins, and other loadings: it considers benefits and premiums only.

Definition: Under the equivalence principle, the level net premium is set so that the expected present value of premiums equals the expected present value of benefits, using the same interest and state-transition assumptions for both. A net premium here means a premium for benefits only, with no allowance for expenses or profit.

Set Up the Benefit and Premium Values

Suppose payments are made at specified times $t$, and the care model has states $s$. Let $\pi_s(t)$ be the probability that the insured is in state $s$ at time $t$. A state may represent active and eligible for a benefit, a particular care level, or a state in which no payment is made. Let $b_s(t)$ be the benefit paid at time $t$ if the insured is in state $s$.

For premiums, let $c_s(k)$ be 1 if a premium is payable at time $k$ in state $s$, and 0 otherwise. If the level premium is $P$, the premium at that date is $P c_s(k)$. For example, if premiums are waived after entry into care, $c_s(k)=1$ in the active state and 0 in care states.

Assume an effective annual interest rate $i$, with discount factor $v=1/(1+i)$. The payment dates and state probabilities must refer to the same instants: a benefit at the end of year 2 is weighted by the probability of being benefit-eligible at time 2, not by a probability at some other time.

\operatorname{EPV}(\text{benefits}) = \sum_t v^t \sum_s \pi_s(t)b_s(t), \qquad \operatorname{EPV}(\text{premiums}) = P\sum_k v^k \sum_s \pi_s(k)c_s(k).

The sums include only the specified benefit and premium payment dates. A benefit that is conditional on being in care contributes its amount multiplied by the probability of occupying an eligible care state at its payment time. If more than one state qualifies, add the state-specific contributions. A premium payable only while active is multiplied by the active-state probability at its due time.

Result: If the expected present value of the unit premiums is positive, the equivalence-principle level net premium is

P = \frac{\operatorname{EPV}(\text{benefits})} {\operatorname{EPV}(\text{unit premiums})} = \frac{\displaystyle\sum_t v^t \sum_s \pi_s(t)b_s(t)} {\displaystyle\sum_k v^k \sum_s \pi_s(k)c_s(k)}.

Why the Ratio Gives the Premium

The ratio follows from writing the present value of the actual premiums as the level amount $P$ times a unit-premium present value. To justify taking expectations, let $I_{s,t}$ equal 1 if the insured is in state $s$ at time $t$, and 0 otherwise. The benefit at time $t$ is $\sum_s I_{s,t}b_s(t)$. Since $E[I_{s,t}]=\pi_s(t)$, linearity of expectation gives the expected benefit at that time as $\sum_s\pi_s(t)b_s(t)$. Discounting and summing over dates gives the benefit expression above. The same argument applied to premium eligibility gives the unit-premium expression. Equating these expected values and dividing by the unit-premium value yields the ratio.

This argument uses expected values of payments, not the probability that a particular policyholder pays every premium or receives every benefit. It does not require the benefit and premium events to be independent: linearity of expectation allows the benefit and premium values to be calculated separately.

List benefit dates and amounts.
For each date, identify the states that trigger payment and the amount paid in each state.

Calculate the benefit value.
Multiply each state-specific amount by its occupancy probability and the discount factor for that payment date, then add the results.

List premium dates and eligibility.
For a unit premium, use 1 in each state where a premium is due and 0 in each state where it is waived.

Calculate and divide.
Discount and add the unit premiums, then divide the benefit value by that unit-premium value. Check that the result has the correct currency units.

Worked Premium Calculations

Worked Example: Premiums Waived After Care Entry

A hypothetical contract pays \$9,000 at the ends of years 1 and 2 if the insured is in a qualifying care state at the respective payment date. The care-state probabilities are 0.10 at time 1 and 0.18 at time 2. Premiums are due at time 0 and time 1 only if the insured is active; the active-state probabilities at those dates are 1.00 and 0.75. Find the level net premium at an effective annual interest rate of 4%.

Method. Calculate the expected present value of the benefits, then divide by the expected present value of unit premiums. Benefits are paid at the ends of years 1 and 2; premiums are due at times 0 and 1.

First, $v=1/1.04=0.96153846$, and $v^2=1/1.04^2=0.92455621$. The expected benefit amounts before discounting are $9{,}000(0.10)=\$900$ at time 1 and $9{,}000(0.18)=\$1{,}620$ at time 2. Therefore,

\operatorname{EPV}(\text{benefits}) = 900(0.96153846)+1{,}620(0.92455621) = 865.38461+1{,}497.78106 = \$2{,}363.16567.

The time-0 premium is certain, so its unit-premium value is 1. The time-1 premium is payable only if active, with probability 0.75. Thus,

\operatorname{EPV}(\text{unit premiums}) = 1+0.75(0.96153846) = 1.72115385.

The level net premium is

P=\frac{2{,}363.16567}{1.72115385} =\$1{,}373.01 \quad\text{(rounded to the nearest cent).}

Check. Multiplying the unit-premium value by the rounded premium gives approximately $1.72115385(1{,}373.01)=\$2{,}363.16$, agreeing with the benefit value to the nearest cent. The common mistake this method avoids is a misplaced decimal: the premium is about \$1,373, not \$137. The full benefit value, in dollars, must be divided by the unit-premium value; that denominator is not a percentage or a number of thousands.

Worked Example: Premiums Due While Active at Three Dates

A hypothetical policy pays \$6,000 at the end of each of years 1 and 2 if the insured is in care at that date. The care-state probabilities are 0.12 at time 1 and 0.20 at time 2. Level premiums are due at times 0, 1, and 2, but only while active. The active-state probabilities at those premium dates are 1.00, 0.82, and 0.68. Find the net premium at an effective annual rate of 5%.

Method. Keep the benefit-date probabilities separate from the premium-date probabilities. They describe different payment conditions and enter different parts of the ratio.

Here, $v=1/1.05=0.95238095$ and $v^2=1/1.05^2=0.90702948$. The expected benefit amounts before discounting are $6{,}000(0.12)=\$720$ at time 1 and $6{,}000(0.20)=\$1{,}200$ at time 2. Hence,

\operatorname{EPV}(\text{benefits}) = 720(0.95238095)+1{,}200(0.90702948) = 685.71428+1{,}088.43538 = \$1{,}774.14966.

For a unit premium, the time-0 contribution is 1. The time-1 and time-2 contributions use the active-state probabilities on their respective dates:

\operatorname{EPV}(\text{unit premiums}) = 1+0.82(0.95238095)+0.68(0.90702948) = 1+0.78095238+0.61678005 = 2.39773243.

Therefore,

P=\frac{1{,}774.14966}{2.39773243} =\$739.93 \quad\text{(rounded to the nearest cent).}

Check. The premium times the unit-premium value is approximately $739.93(2.39773243)=\$1{,}774.15$, matching the benefit value after rounding. Using care probabilities in the premium denominator, or active probabilities in the benefit numerator, would apply the wrong payment condition.

Worked Example: Several Benefit Dates with Waiver in Care

A hypothetical contract pays \$5,000 at the ends of years 1, 2, and 3 if the insured is in care. The care-state probabilities at those dates are 0.08, 0.15, and 0.20. Premiums are due at times 0, 1, and 2 while active, and are waived in care. The active-state probabilities at the premium dates are 1.00, 0.88, and 0.70. Find the net premium at an effective annual interest rate of 4%.

Method. Discount the three expected benefits at times 1 through 3. Separately, discount unit premiums at times 0 through 2, weighted by the probability of being active on each premium date.

The discount factors are $v=1/1.04=0.96153846$, $v^2=1/1.04^2=0.92455621$, and $v^3=1/1.04^3=0.88899636$. The expected benefits before discounting are $\$400$, $\$750$, and $\$1{,}000$. Consequently,

\begin{aligned} \operatorname{EPV}(\text{benefits}) &=400(0.96153846)+750(0.92455621)+1{,}000(0.88899636)\\ &=384.61538+693.41716+888.99636\\ &=\$1{,}967.02890. \end{aligned}

The time-0 unit premium is certain. The later premiums are weighted by the active probabilities at times 1 and 2:

\begin{aligned} \operatorname{EPV}(\text{unit premiums}) &=1+0.88(0.96153846)+0.70(0.92455621)\\ &=1+0.84615385+0.64718935\\ &=2.49334320. \end{aligned}

Thus,

P=\frac{1{,}967.02890}{2.49334320} =\$788.91 \quad\text{(rounded to the nearest cent).}

Check. The unit-premium value multiplied by the rounded premium is approximately $2.49334320(788.91)=\$1{,}967.02$, which agrees with the benefit value to the nearest cent. The time-3 care probability belongs in the benefit value because there is a benefit at time 3; it does not belong in the premium value because there is no premium due at time 3.

Common Errors and Model Checks

An equivalence-principle calculation is only as sound as its payment rules. Before substituting numbers, identify the date and state condition for every cash flow. In particular, do not assume that a premium is payable merely because the policy is in force: a contract may waive premiums after care entry. Conversely, a benefit may be due at a date when a premium is not.

Using the wrong probability. Weight a care benefit by the probability of being in an eligible care state at the benefit date. Weight a premium by the probability of being in a premium-eligible state at its due date.
Discounting from the wrong time. A payment at time 0 has discount factor 1. A payment at the end of year $t$ has factor $v^t$. Do not discount a time-0 premium by one year.
Mixing up benefit and premium dates. A benefit at time 3 does not create a premium at time 3 unless the contract says one is due then.
Changing the unit of the benefit amount. If amounts are stated in dollars, calculate the numerator in dollars and report the resulting premium in dollars. Do not inadvertently divide by 10 or 1,000.
Assuming unrelated events must be independent. The numerator and denominator use their own state probabilities. Their expected values can be calculated separately by linearity; independence is not required for this equation.

The state probabilities must describe a coherent model, but the states not listed as eligible need not have zero probability. For example, the remaining probability may represent death, recovery, or another state that triggers neither payment. Include such states explicitly in the sum if they have nonzero payments; otherwise, their payment contributions are zero.

Key takeaway: Build the benefit value and the unit-premium value as separate discounted sums. In each sum, use the probability of the state that makes that particular payment due, at that payment’s date. The level net premium is their ratio.

Check Your Understanding

For each problem, use the stated payment dates and state probabilities. Round monetary answers to the nearest cent.

A hypothetical policy pays \$4,000 at the end of year 1 if the insured is in care, with care probability 0.15. One premium is due at time 0 with certainty. At an effective annual rate of 5%, find the level net premium. (Answer: \$571.43)
A contract pays \$3,000 at the ends of years 1 and 2 if the insured is in care, with care probabilities 0.10 and 0.20. Premiums are due at times 0 and 1 while active, with active probabilities 1.00 and 0.80. At $i=5\%$, find the level net premium. (Answer: \$471.04)
A policy pays \$1,000 at the ends of years 1 and 2 if the insured is in care, with care probabilities 0.20 and 0.30. Premiums are due at times 0, 1, and 2 while active, with active probabilities 1.00, 0.90, and 0.70. At $i=10\%$, find the level net premium. (Answer: \$179.31)
A hypothetical benefit pays \$2,000 at the end of year 1 for care level A and \$5,000 for care level B. At time 1, the respective state probabilities are 0.10 and 0.04. At the end of year 2, the same benefits apply, with respective state probabilities 0.12 and 0.06. Premiums are due at times 0 and 1 while active, with active probabilities 1.00 and 0.85. At $i=5\%$, find the level net premium. (Answer: \$481.20)

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