Price Unequal Premium Patterns for Composite Life Benefits

Pricing Composite Benefits with Unequal Premiums

A life contract can combine several benefits: for example, a payment on death during the term, an additional payment if death occurs in a particular year, and a maturity benefit if the insured survives to the end. Premiums need not be level. A contract might charge one amount at issue and a different amount at the start of each later policy year. The central pricing task is to place all these contingent cash flows on the same present-value basis.

For a hypothetical contract, suppose premiums are payable at the start of each policy year while the insured is alive, and benefits are paid at the end of the year of death or at maturity. Under the equivalence principle, the expected present value (EPV) of premiums equals the EPV of benefits. The method applies to a specified mortality and interest basis; it does not by itself determine whether the basis is appropriate for a particular real-world product.

Definition: A composite life benefit has two or more benefit components, possibly with different payment conditions or amounts. An unequal premium pattern is a schedule whose premium amounts differ by policy year. Under the equivalence principle, the premium scale is chosen so that the EPV of all premiums equals the EPV of all benefits on the stated assumptions.

Set Up the Cash Flows Before Calculating

Use a single time line and distinguish benefit timing from premium timing. For a life aged $x$, the probability of surviving $k$ years is ${}_kp_x$, and the probability of surviving $k-1$ years and then dying during the next year is ${}_{k-1}p_xq_{x+k-1}$. With effective annual interest rate $i$, the discount factor for a payment at time $k$ is $v^k$, where $v=1/(1+i)$.

If $B_k$ is the total amount payable at the end of year $k$ on death in that year, and $M_n$ is payable at time $n$ if the insured survives $n$ years, then the benefit EPV is

\operatorname{EPV}(B) = \sum_{k=1}^{n} v^k\,{}_{k-1}p_xq_{x+k-1}B_k + v^n\,{}_np_xM_n.

If more than one component is payable on the same event, add those component amounts to obtain $B_k$. A survival or maturity benefit is a separate term because it is paid on survival, not on death.

Suppose a premium of $c_kP$ is due at time $k$, provided the insured is then alive, where $P$ is an unknown base premium and $c_k$ is a known schedule multiplier. Its EPV is $v^k\,{}_kp_xc_kP$. Thus the premium EPV is

\operatorname{EPV}(P) = P\sum_{k=0}^{m-1}c_kv^k\,{}_kp_x, \qquad {}_0p_x=1.

The sum is the premium present-value factor for the stated schedule. When the premiums are fixed dollar amounts rather than multiples of one unknown $P$, calculate their EPV directly instead. In either case, include a premium only if its contractual payment condition is met.

Why the Valuation Formula Works

Each payment is a random cash flow: whether it is paid depends on the future lifetime. Its EPV is the sum, over possible payment events, of the payment amount times the probability of the event times the discount factor at its payment time. For a death benefit in year $k$, the event requires survival through $k-1$ years and death in the next year. Its probability is therefore ${}_{k-1}p_xq_{x+k-1}$, and discounting its payment at the end of that year gives the term in the formula.

For a premium due at time $k$, the relevant event is survival to time $k$, with probability ${}_kp_x$. Multiplying its amount by this probability and by $v^k$ gives its expected present value. Adding the expected present values of the individual payments is valid by linearity of expectation; the cash flows do not need to be independent.

List benefits by event and time.
Combine benefit components payable on the same death event, and list any survival payment separately.

Weight each benefit.
Use the probability of death in the specified year or survival to the specified payment date, then discount to time 0.

List premiums at their due dates.
For each premium, apply the condition for payment, such as being alive at the start of that year.

Apply equivalence and solve.
Set premium EPV equal to benefit EPV, solve for the unknown premium, and check the timing and conditions.

Worked Calculations

Worked Example: A Two-Year Contract with an Increasing Premium

A hypothetical two-year contract covers a life aged $x$. The effective annual interest rate is $5\%$, $q_x=0.10$, and $q_{x+1}=0.20$. It pays \$5,000 at the end of year 1 if death occurs in that year, \$8,000 at the end of year 2 if death occurs in that year, and \$3,000 at time 2 if the insured survives. Premiums of $P$ at time 0 and $1.5P$ at time 1 are payable only if the insured is alive when due. Find $P$.

Method. Calculate the expected present value of each benefit event and divide by the present-value factor for the premium schedule. The common mistake to avoid is treating the second premium as certain rather than conditional on survival.

The discount factors are $v=1/1.05=20/21$ and $v^2=400/441$. The probabilities needed are ${}_1p_x=0.90$, death in year 1 $=0.10$, death in year 2 $=0.90(0.20)=0.18$, and survival to time 2 $=0.90(0.80)=0.72$.

The year 1 death benefit has EPV $5{,}000(0.10)v=500(20/21)=\$476.190476$. The year 2 death benefit has EPV $8{,}000(0.18)v^2=1{,}440(400/441)=\$1{,}306.122449$. The maturity benefit has EPV $3{,}000(0.72)v^2=2{,}160(400/441)=\$1{,}959.183673$.

Using the unrounded components, the benefit EPV is $476.190476+1{,}306.122449+1{,}959.183673=\$3{,}741.496598$. The premium EPV is $P+1.5P(0.90)v=P[1+1.5(0.90)(20/21)]=(16/7)P$. Equivalence gives $P=3{,}741.496598(7/16)=\$1{,}636.904762$, or \$1,636.90 to the nearest cent.

Check. The second premium is included with probability 0.90, not with probability 1. Substituting the rounded premium gives a premium EPV of approximately $(16/7)(1{,}636.90)=\$3{,}741.49$, consistent with the benefit EPV to the nearest cent.

Worked Example: Three Benefit Events and a Three-Year Premium Pattern

A hypothetical three-year contract has effective annual interest rate $4\%$, with annual death probabilities $0.10$, $0.20$, and $0.25$ in years 1, 2, and 3, respectively. It pays \$4,000 on death in year 1, \$6,000 on death in year 2, and \$8,000 on death in year 3. It also pays \$2,000 at time 3 if the insured survives. Premiums due at times 0, 1, and 2 are $P$, $1.2P$, and $0.8P$, respectively, if the insured is alive when each is due. Find $P$.

Method. First calculate the probability of each death event and survival to maturity; then form separate benefit and premium EPVs. This avoids confusing the year 3 death probability with the probability of being alive at time 2.

The survival probabilities at the start of the three years are $1$, $0.90$, and $0.90(0.80)=0.72$. Death probabilities by year are $0.10$, $0.90(0.20)=0.18$, and $0.72(0.25)=0.18$. Survival to time 3 is $0.72(0.75)=0.54$. The discount factors are $v=1/1.04=25/26$, $v^2=625/676$, and $v^3=15{,}625/17{,}576$.

The year 1 death benefit EPV is $4{,}000(0.10)v=400(25/26)=\$384.615385$. The year 2 death benefit EPV is $6{,}000(0.18)v^2=1{,}080(625/676)=\$998.520710$. The combined year 3 death and maturity benefits have EPV $[8{,}000(0.18)+2{,}000(0.54)]v^3=(1{,}440+1{,}080)(15{,}625/17{,}576)=\$2{,}240.270824$.

The total benefit EPV, calculated without intermediate rounding, is $\$3{,}623.406919$. The premium factor is $1+1.2(0.90)v+0.8(0.72)v^2=1+1.038461538+0.532544379=2.571005917$. Therefore $P=3{,}623.406919/2.571005917=\$1{,}409.33$ to the nearest cent.

Check. The year 3 premium is weighted by the probability of being alive at time 2, $0.72$; the maturity benefit is weighted instead by survival through all three years, $0.54$. Using these different probabilities reflects their different payment conditions.

Worked Example: A Base Benefit, a Rider, and a Maturity Payment

A hypothetical three-year contract has effective annual interest rate $5\%$, $q_x=0.10$, $q_{x+1}=0.20$, and $q_{x+2}=0.25$. It pays a base benefit of \$4,000 on death in any of the three years, with an additional \$2,000 payable on death in year 3. A \$3,000 maturity benefit is paid at time 3 if the insured survives. Premiums at times 0, 1, and 2 are $P$, $1.25P$, and $1.5P$ if the insured is alive when due. Find $P$.

Method. Add the base benefit and rider only for the event on which both are payable, then include the maturity benefit as a distinct survival event. This avoids applying the rider to deaths in years 1 or 2.

The death probabilities by year are $0.10$, $0.90(0.20)=0.18$, and $0.72(0.25)=0.18$. Survival to time 3 is $0.72(0.75)=0.54$. With $v=20/21$, $v^2=400/441$, and $v^3=8{,}000/9{,}261$, the year 1 death benefit EPV is $4{,}000(0.10)v=\$380.952381$. The year 2 death benefit EPV is $4{,}000(0.18)v^2=\$653.061224$.

In year 3, the death payment is \$6,000, so the death benefit EPV is $6{,}000(0.18)v^3$. The maturity benefit EPV is $3{,}000(0.54)v^3$. Their combined EPV is $[1{,}080+1{,}620](8{,}000/9{,}261)=\$2{,}332.361515$. Thus the total benefit EPV is $\$3{,}366.375120$.

The premium factor is $1+1.25(0.90)v+1.5(0.72)v^2=1+1.071428571+0.979591837=3.051020408$. Therefore $P=3{,}366.375120/3.051020408=\$1{,}103.36$ to the nearest cent.

Check. The rider raises the year 3 death payment to \$6,000, but it does not change the maturity payment. The death and survival events are mutually exclusive, so their separate expected present values can be added.

Common Timing and Modeling Traps

The most consequential error is often not arithmetic but an incorrect payment condition. In the first worked example, using $P+1.5P$ as the premium EPV factor would treat the time 1 premium as certain. That incorrect factor is $2.5$, giving $P=3{,}741.496598/2.5=\$1{,}496.60$, rather than the correct $\$1{,}636.90$. The time 1 premium is payable only if the insured survives year 1, so its EPV contribution is $1.5P(0.90)v$.

Use the right survival probability. A premium at time $k$ payable while alive is weighted by ${}_kp_x$; a death benefit in year $k$ is weighted by ${}_{k-1}p_xq_{x+k-1}$.
Match the discount time to the payment time. A benefit paid at the end of year $k$ is discounted by $v^k$, even though death occurs during that year.
Keep distinct contingencies distinct. Add simultaneous benefit components for the same event, but do not combine a death benefit and a maturity benefit as if both were paid on one outcome.
Do not round intermediate values unnecessarily. Carry full precision through the equivalence equation and round the final premium to cents.

If the contract includes expenses, taxes, or other cash flows, they must be incorporated according to their timing and conditions in the equivalence equation. The calculation here prices the stated benefit and premium cash flows only. This method is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Check Your Understanding

For each hypothetical contract, use the stated mortality and interest assumptions, apply equivalence, and round the final dollar answer to the nearest cent.

A one-year contract has $i=0$ and $q_x=0.20$. It pays \$5,000 on death at the end of the year and \$1,000 on survival at the end of the year. Premiums are $P$ at time 0 and $2P$ at time 1 if alive. Find $P$. (Answer: \$692.31)
A two-year contract has $i=5\%$, $q_x=0.10$, and $q_{x+1}=0.20$. It pays \$3,000 on death in year 1, \$5,000 on death in year 2, and \$2,000 on survival to time 2. Premiums are $P$ at time 0 and $1.5P$ at time 1 if alive. Find $P$. (Answer: \$1,053.57)
A three-year contract has $i=4\%$ and annual death probabilities $0.10$, $0.20$, and $0.25$. It pays \$2,000 on death in any year and \$1,000 on survival to time 3. Premiums are $P$, $1.1P$, and $1.2P$ at times 0, 1, and 2 if alive. Find $P$. (Answer: \$481.78)
A two-year contract has $i=5\%$, $q_x=0.10$, and $q_{x+1}=0.20$. It pays \$4,000 on death in year 1, \$6,000 on death in year 2, and \$1,000 on survival to time 2. A \$500 premium is due at time 0; an unknown premium $Q$ is due at time 1 if alive. Find $Q$. (Answer: \$1,765.87)
A three-year contract has $i=0$ and annual death probabilities $0.10$, $0.20$, and $0.25$. It pays \$1,000 on death in any year and \$2,000 on survival to time 3. Premiums of \$600 at time 0 and \$900 at time 1 are specified; an unknown premium $R$ is due at time 2 if alive. Find $R$. (Answer: \$180.56)

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