Reserves for Policies with Multiple Benefit Components

Why Separate Benefit Components?

A contract can pay different amounts depending on how and when coverage ends. It might pay one benefit if death occurs during a policy year, another if the insured survives to a specified date, and a maturity amount if the insured survives to the end of the term. A reserve for such a contract is not a choice between benefit formulas: it is the value of all benefits payable under the contract, less the value of future premiums, assessed at the valuation date.

The useful method is to value each contingent payment separately. For a death benefit, use the probability of death in the relevant future policy year. For a survival or maturity benefit, use the probability of surviving to its payment date. Add the resulting present values, then subtract the present value of future premiums. This component-by-component approach makes payment timing and the event that triggers each amount explicit.

Definition: A prospective reserve at duration $k$, conditional on the insured being alive and the policy being in force at that time, is the expected present value of future benefits minus the expected present value of future premiums. The valuation date and whether a premium due on that date has already been paid must be specified.

The calculations below use a discrete annual model. The effective annual interest rate is $i$, so a payment one year after the valuation date has discount factor $v=1/(1+i)$. For an insured aged $x+k$, ${}_tp_{x+k}$ is the probability of surviving $t$ years from the valuation date, and ${}_tq_{x+k}$ is the probability of dying within those $t$ years. In particular, $q_{x+k}$ is the probability of death during the next year and $p_{x+k}=1-q_{x+k}$.

Valuing Each Component

Suppose the remaining term is $n$ years. Let $B_j$ be a benefit payable at the end of future year $j$ if death occurs during that year. The probability of reaching the start of that year and dying during it is ${}_{j-1}p_{x+k}q_{x+k+j-1}$. Thus the expected present value of the death benefits is the sum of the present values for the separate years.

\operatorname{EPV}(\text{death benefits}) = \sum_{j=1}^{n} v^j\,{}_{j-1}p_{x+k}\,q_{x+k+j-1}\,B_j

If a distinct maturity amount $M$ is payable at the end of year $n$ only if the insured survives to that date, its expected present value is $v^n\,{}_np_{x+k}M$. A survival benefit payable at an earlier time $r$ has value $v^r\,{}_rp_{x+k}C_r$, where $C_r$ is the amount payable on survival to that time. These terms are added to the death-benefit values when the contract promises them.

Premiums are handled in the same way: discount each future premium and multiply by the probability it is payable. For example, a premium $P_s$ due at the beginning of future year $s+1$, at time $s$ after valuation, is payable only if the insured survives to that date. Its expected present value is $v^s\,{}_sp_{x+k}P_s$. A premium due immediately at the valuation date has $s=0$; it is included if the reserve is measured just before it is paid and excluded if the reserve is measured just after it is paid.

Result: For a policy with multiple benefits, add the expected present values of all distinct benefit payments, then subtract the expected present values of all future premiums. Payments triggered by mutually exclusive events are valued with their respective probabilities; do not assign the same payment to more than one event.

Why the Component Formula Works

The justification is the linearity of expectation. For each possible payment, represent its present value by its amount times a discount factor times an indicator that the payment event occurs. The expectation of that indicator is the probability of the event. The expected present value of that payment is therefore its discounted amount multiplied by that probability.

E\!\left[\sum_j Z_j\right] = \sum_j E[Z_j], \qquad E\!\left[v^j B_j\,\mathbf{1}_{\{\text{death in year }j\}}\right] = v^j B_j\,P(\text{death in year }j)

Here $Z_j$ denotes the present value of an individual cash flow, and the indicator $\mathbf{1}$ equals 1 when its stated event occurs and 0 otherwise. The events do not need to be independent for the sum of expected values to equal the expected value of the sum. They do need to be described correctly: for example, a death benefit and a maturity benefit may be payable under different outcomes, while two separate survival benefits can both be payable if the contract promises both.

Worked Example: Death and Maturity Benefits in the Final Year

A hypothetical three-year contract is valued at time 2, just before the premium due at that time is paid. The insured is alive. The premium due immediately is $1,500. During the final year, the probability of death is $q=0.08$; survival probability is $p=0.92$. The effective annual interest rate is 5%. The contract pays $14,000 at the end of the year on death during that year, or a separate $6,000 maturity benefit at year-end on survival. Calculate the prospective reserve.

Method. Value the death payment and survival payment as separate components. The two outcomes are mutually exclusive, so each amount is multiplied by the probability of its own payment event.

The one-year discount factor is $v=1/1.05=0.95238095\ldots$. The death-benefit component is

v q(14{,}000) = \frac{0.08(14{,}000)}{1.05} = \frac{1{,}120}{1.05} = \$1{,}066.6666\ldots

The maturity-benefit component is

v p(6{,}000) = \frac{0.92(6{,}000)}{1.05} = \frac{5{,}520}{1.05} = \$5{,}257.1428\ldots

The total expected present value of benefits is $1{,}066.6666\ldots+5{,}257.1428\ldots=\$6{,}323.8095\ldots$. Because the reserve is measured immediately before the premium is paid, subtract the premium due now:

V = 6{,}323.8095\ldots-1{,}500 = \$4{,}823.8095\ldots

The reserve is $4,823.81, rounded to the nearest cent. As a check, the benefit components total $6,000 before discounting, weighted by their outcome probabilities: $0.08(14{,}000)+0.92(6{,}000)=\$6{,}640$. Discounting gives $6{,}640/1.05=\$6{,}323.81$, consistent with the component calculation. A common mistake avoided here is adding the two benefit amounts as though both were paid on the same outcome.

Worked Example: Two Future Years with Multiple Benefits and Premiums

A hypothetical policy is valued at time 1, immediately before a premium is due. There are two years remaining. In the first future year, $q_{x+1}=0.10$ and $p_{x+1}=0.90$; in the second, $q_{x+2}=0.20$ and $p_{x+2}=0.80$. The effective annual interest rate is 4%. The policy pays $10,000 on death at the end of the first future year, $15,000 on death at the end of the second future year, and a separate $6,000 maturity benefit at the end of the second year on survival. Premiums of $2,000 are due at time 1 and time 2 if the insured is alive. Find the reserve at time 1 before the premium is paid.

Method. First value each benefit conditional on being alive at time 1. Then value the premium at time 1 immediately and the premium at time 2 only if the insured survives the first future year.

Here $v=1/1.04=0.96153846\ldots$ and $v^2=1/1.04^2=0.92455621\ldots$. The first-year death benefit has value

0.10(10{,}000)v = \frac{1{,}000}{1.04} = \$961.5384\ldots

To receive the second-year death benefit, the insured must first survive year 1 and then die in year 2. Its value is

(0.90)(0.20)(15{,}000)v^2 = \frac{2{,}700}{1.0816} = \$2{,}496.3017\ldots

The maturity benefit requires survival through both years, so its value is

(0.90)(0.80)(6{,}000)v^2 = \frac{4{,}320}{1.0816} = \$3{,}994.0828\ldots

Thus the expected present value of benefits is $961.5384\ldots+2{,}496.3017\ldots+3{,}994.0828\ldots=\$7{,}451.9230\ldots$. The premium at time 1 is payable immediately, so its value is $2,000. The time-2 premium is payable only if the insured survives year 1:

\operatorname{EPV}(\text{premiums}) = 2{,}000+(0.90)(2{,}000)v = 2{,}000+\frac{1{,}800}{1.04} = \$3{,}730.7692\ldots

Subtracting the unrounded premium value from the unrounded benefit value gives

V = 7{,}451.9230\ldots-3{,}730.7692\ldots = \$3{,}721.1538\ldots

The reserve is $3,721.15, rounded to the nearest cent. The second-year probabilities are conditional on survival through the first year, which is why both second-year benefits include the factor $0.90$. The second premium also includes that survival factor, but is discounted for one year, not two, because it is paid at time 2.

Worked Example: Determine a Level Premium, Then Find a Reserve

A hypothetical two-year contract pays $8,000 at the end of year 1 on death in that year, $12,000 at the end of year 2 on death in that year, and $3,000 at the end of year 2 on survival to maturity. The annual death probabilities are $q_x=0.15$ and $q_{x+1}=0.10$, so $p_x=0.85$ and $p_{x+1}=0.90$. The effective annual interest rate is 4%. A level premium $P$ is payable at time 0 and time 1 if the policy is in force. Determine the net premium by the equivalence principle, then find the reserve at time 1 immediately before the second premium is paid.

Method. At issue, set the expected present value of benefits equal to that of premiums to determine $P$. For the time-1 reserve, recalculate future values conditional on survival to time 1 and include the premium due immediately.

At issue, $v=25/26$ and $v^2=625/676$. The first-year death benefit has expected present value

(0.15)(8{,}000)v = 1{,}200\left(\frac{25}{26}\right) = \$1{,}153.8461\ldots

The second-year death benefit has expected present value $(0.85)(0.10)(12{,}000)v^2=637{,}500/676=\$943.0473\ldots$. The maturity benefit has expected present value $(0.85)(0.90)(3{,}000)v^2=1{,}434{,}375/676=\$2{,}121.8565\ldots$. Together, the benefits have expected present value

\frac{780{,}000+637{,}500+1{,}434{,}375}{676} = \frac{2{,}851{,}875}{676} = \$4{,}218.75

The time-0 premium is certain, while the time-1 premium is payable only if the insured survives year 1. Therefore,

\operatorname{EPV}(\text{premiums}) = P\left(1+0.85v\right) = P\left(1+\frac{85}{104}\right) = P\left(\frac{189}{104}\right)

Equating the two expected present values gives

P = 4{,}218.75\left(\frac{104}{189}\right) = \$2{,}321.4285\ldots

The level net premium is $2,321.43, rounded to the nearest cent. At time 1, conditional on being alive, the expected present value of the remaining benefits is

\left[(0.10)(12{,}000)+(0.90)(3{,}000)\right]v = \frac{1{,}200+2{,}700}{1.04} = \$3{,}750

The reserve is measured just before the second premium is paid, so that premium is included as a future payment due immediately:

{}_1V_x = 3{,}750-2{,}321.4285\ldots = \$1{,}428.5714\ldots

The time-1 reserve is $1,428.57, rounded to the nearest cent. As a timing check, immediately after the premium is paid, the reserve would be $3,750 because no premiums remain. The difference between the before- and after-premium values is exactly the premium due.

Common Exam Trap: Forgetting the Conditioning

A prospective reserve is conditional on the insured having survived to the valuation date. The probabilities used for future events therefore start at that date. They are not unconditional probabilities measured from policy issue. Applying an extra probability of surviving to the valuation date understates the reserve.

For example, suppose an unconditional calculation from issue would multiply a correctly calculated time-1 reserve of $3,721.1538 by a probability of 0.80 of surviving to time 1. The incorrect result would be $0.80(3{,}721.1538)=\$2{,}976.92$. That is a survival-weighted amount from an earlier perspective, not the reserve conditional on being alive at time 1. The correct time-1 reserve remains $3,721.15. Check that every survival probability in a prospective reserve is measured from the valuation date, while still including survival between that date and a future payment date where needed.

Payment timing is another frequent source of error. A premium due at the valuation date is included if the reserve is measured just before it is paid, but not if the reserve is measured just after. A benefit due at the end of a year is discounted for that year even when its payment probability is conditioned on survival to the start of the year. State the valuation instant before writing the cash-flow terms.

Key takeaway: List each benefit by amount, payment date, and triggering event. Calculate its conditional probability and present value from the valuation date, do the same for each future premium, and subtract total premium value from total benefit value. Keep the timing of any payment due at the valuation date explicit.

Check Your Understanding

For each problem, identify the relevant payment events and timing before calculating.

A hypothetical one-year contract is valued immediately before a $500 premium is due. The effective annual interest rate is 4%; the probability of death during the year is 0.25. The contract pays $9,000 on death at year-end or $3,000 on survival at year-end. Find the reserve. (Answer: $3,826.92)
At time 1, immediately after any premium due then has been paid, a two-year contract has $q_{x+1}=0.10$, $p_{x+1}=0.90$, $q_{x+2}=0.20$, and $p_{x+2}=0.80$. At 5% effective annual interest, it pays $7,000 on death in the first future year, $11,000 on death in the second future year, and $5,000 on survival to the end of the second year. A $1,000 premium is due at time 2 if the insured is alive. Find the reserve at time 1. (Answer: $4,870.75)
At the start of a final policy year, immediately after the premium has been paid, the death probability is 0.12. At 3% effective annual interest, the policy pays $20,000 on death during the year or $4,000 on survival to year-end. Find the reserve. (Answer: $5,747.57)
A two-year policy has $q_x=0.15$, $p_x=0.85$, $q_{x+1}=0.10$, and $p_{x+1}=0.90$. At 4% effective annual interest, it pays $8,000 on death in year 1, $12,000 on death in year 2, and $3,000 on survival to maturity. A level premium is payable at times 0 and 1 if the insured is alive. Find the reserve at time 1 immediately before the premium is paid. (Answer: $1,428.57)
A hypothetical two-year contract has $q_x=0.20$, $p_x=0.80$, $q_{x+1}=0.25$, and $p_{x+1}=0.75$. At 5% effective annual interest, it pays $10,000 on death in year 1, $16,000 on death in year 2, and $4,000 on survival to maturity. A level premium is payable at time 0 and time 1 if the insured is alive. Find the net premium and the reserve at time 1 immediately before the second premium. (Answer: net premium $3,963.96; reserve $2,702.70)

What You'll Learn