Mortality Level Shocks for Protection Contracts

Why Mortality Shocks Need a Cash-Flow Revaluation

A mortality increase raises the probability of a death claim during a given period, which tends to increase the present value of protection benefits. It also reduces the probability that a policy remains in force to pay later premiums. The net effect on a contract’s present-value result therefore depends on both sides of the cash flow, not only on the change in claim probability.

This tutorial considers hypothetical term-assurance and whole-life contracts under an immediate mortality level shock. The shock changes future mortality assumptions at the valuation date; it does not change the contractual benefit or reprice premiums already specified in the policy. All examples use annual periods, a constant effective annual interest rate, no expenses, no lapses, and premiums payable annually in advance while the insured is alive. Death benefits are payable at the end of the year of death. These timing assumptions matter: different benefit or premium payment times require different present-value formulas.

Definition: If the original annual mortality probability in year $k$ is $q_k$, a proportional mortality level shock of size $s$ sets $q'_k=(1+s)q_k$, with $p'_k=1-q'_k$. The shocked rates must satisfy $0\le q'_k\le1$. A positive $s$ is a mortality increase; a negative $s$ is a mortality decrease. The shock is applied to the rates used for future projections, while contractual premiums remain fixed unless the problem explicitly says otherwise.

For a term contract, a mortality shock changes not just the probability of death in the first year, but also the probability of surviving to each later year. Write ${}_kp'$ for the probability, under the shocked assumptions, of surviving $k$ years from issue. Under the stated timing assumptions, the present value of benefits is the sum of the discounted death benefits, each weighted by its shocked probability of payment. Premiums are weighted by the shocked probability that the insured is alive when each premium falls due.

\operatorname{EPV}(B_n)=B\sum_{k=0}^{n-1}v^{k+1}{}_kp'\,q'_k, \qquad \operatorname{EPV}(P_n)=P\sum_{k=0}^{n-1}v^k{}_kp'.

Here $B$ is the benefit, $P$ is the fixed annual premium, and ${}_0p'=1$. The premium term at $k=0$ is payable immediately at issue, so it is not discounted. For insurer cash flows, define the present-value loss as benefit outgo minus premium income. A positive result is a loss on this basis; a negative result is a net gain. This is a present-value comparison under the stated assumptions, not a complete solvency measure.

Worked Term-Assurance Revaluation

Worked Example: A Mortality Increase on a Two-Year Term Contract

A hypothetical two-year term policy pays $1,000 at the end of the year of death. The original annual mortality probability is $q=0.10$ in each year, and the effective annual interest rate is $i=0.05$. The annual premium is a level net premium, payable at issue and at time 1 if the insured is alive. An immediate mortality shock raises each annual probability to $q'=0.15$. Find the original net premium, the stressed benefit and premium present values, the present-value loss under stress, and the reserve at time 1 immediately after the second premium has been paid if the policyholder is alive.

Method. First value the original two-year benefits and premiums to determine the contractual net premium. Then revalue both cash-flow streams using stressed survival and mortality. Finally, calculate the reserve at the stated instant, after identifying whether any premiums remain due.

The discount factor is $v=1/1.05=20/21$. Under the original mortality assumption, the present value of benefits per $1,000 of cover is $0.10v+0.90(0.10)v^2=0.0952381+0.0816327=0.1768707$. Thus the benefit present value is $1000(0.1768707)=\$176.87$.

The original premium annuity present value is $1+0.90v=1+0.8571429=1.8571429$. The level net premium is therefore $P=176.8707/1.8571429=\$95.2381$, or $95.24 to the nearest cent.

Under the shock, the first-year death probability is $0.15$, and the probability of surviving the first year and dying in the second is $0.85(0.15)=0.1275$. The stressed benefit present value is $1000[0.15v+0.1275v^2]=1000[0.1428571+0.1156463]=\$258.50$.

The first premium is payable at issue. The second is payable at time 1 only if the insured survives the first year, with stressed probability $0.85$. Thus the stressed premium present value is $95.2381[1+0.85v]=95.2381(1.8095238)=\$172.34$. The present-value loss is $258.5034-172.3356=\$86.17$, rounded to the nearest cent. The mortality increase raises benefit outgo and reduces the expected value of premium income.

At time 1, immediately after the second premium has been paid, one year of cover remains and no future premium remains due. Conditional on being alive, the reserve is just the present value at time 1 of the remaining death benefit: $V_1=1000(0.15)v=\$142.86$. It would be incorrect to subtract another premium at this reserve date. Immediately before the time-1 premium, the valuation would instead include that premium as a future receipt; the reserve would then be $150v-95.2381=\$47.62$.

Check. The stressed probabilities sum to one across the two death years and survival through year 2: $0.15+0.1275+0.85^2=1$. The premium present value includes two possible payments, with the second conditional on first-year survival. The reserve timing agrees with the statement that the time-1 premium has already been paid.

Whole-Life Cash Flows and the Premium-Stop Effect

For whole life with constant annual mortality probability $q$, let $p=1-q$. Under annual premiums payable in advance while alive and a benefit payable at the end of the year of death, the benefit present value is a geometric series. The probability of dying in year $k+1$ is $p^kq$, and the benefit is discounted $k+1$ years. Premiums are payable at times $0,1,2,\ldots$ while the insured survives to those times.

\operatorname{EPV}(B)=B\sum_{k=0}^{\infty}(pv)^k qv =\frac{Bqv}{1-pv}, \qquad \operatorname{EPV}(P)=P\sum_{k=0}^{\infty}(pv)^k =\frac{P}{1-pv}.

The geometric-series formulas require $pv<1$, which holds when $i>0$ or when $q>0$ under the usual nonnegative interest assumption. For a net annual premium, equating the two present values gives $P=Bqv$. Under a shock, however, a premium set on an earlier basis is generally held fixed; it is the probabilities weighting future premium payments that change.

Why the whole-life formulas hold: In year $k+1$, the probability of a claim is $p^kq$, so its discounted expected amount is $Bv^{k+1}p^kq=Bqv(pv)^k$. Summing these terms gives $Bqv/(1-pv)$. A premium at time $k$ is paid with probability $p^k$ and discounted by $v^k$, so its expected present value is $P(pv)^k$. Summing gives $P/(1-pv)$. The same survival-discount factor $1/(1-pv)$ appears in both expressions, which is why the net premium simplifies to $Bqv$.

Worked Example: Whole-Life Revaluation with a Fixed Gross Premium

A hypothetical whole-life policy pays $1,000 at the end of the year of death. The original annual mortality probability is $q=0.10$, and $i=0.05$. The contractual gross annual premium is $188.63, payable annually in advance while the insured is alive; it is not recalculated after the shock. An immediate shock raises mortality to $q'=0.12$. Calculate the stressed benefit present value, premium present value, and present-value loss. Also compare the result with the original-basis present-value loss.

Method. Apply the whole-life geometric-series formulas separately to benefits and premiums, using the mortality basis relevant to each valuation. The premium amount remains $188.63 under both bases.

For the stressed basis, $p'=0.88$ and $v=20/21$. Therefore $q'v=0.12(20/21)=4/35$, and $1-p'v=1-0.88(20/21)=17/105$. The stressed benefit present value is $1000(4/35)/(17/105)=1000(12/17)=\$705.88$.

The stressed premium annuity present value is $1/(1-p'v)=105/17=6.1764706$. Hence the premium present value is $188.63(105/17)=\$1,165.07$. The stressed present-value loss is benefit outgo minus premium income: $705.8824-1165.0676=-\$459.19$, rounded to the nearest cent. The negative sign means that, on this simplified present-value basis, premium income exceeds benefit outgo.

On the original basis, $p=0.90$, $qv=0.10(20/21)=2/21$, and $1-pv=1-0.90(20/21)=1/7$. The original benefit present value is $1000(2/21)/(1/7)=\$666.67$. The original premium annuity present value is $7$, so premium income has present value $188.63(7)=\$1,320.41$. The original-basis loss is $666.6667-1320.41=-\$653.74$. The shock changes the loss by $-459.1853-(-653.7433)=\$194.56$, an improvement in this measure.

The example illustrates why the sign of the net effect cannot be inferred from the benefit side alone. Stressed benefit outgo rises, but higher mortality also shortens the expected stream of premium payments. Here the specified gross premium is high enough that the reduction in premium income present value is important, yet premium income still exceeds benefit outgo. The figures describe this hypothetical contract only.

Mortality Decreases and Reserve Timing

A mortality decrease reverses the direct effect on death claims but does not necessarily reverse every change in a contract’s net present-value result. For protection insurance, lower mortality generally reduces the expected present value of death benefits. It also increases the expected duration of premium income. The reserve at a particular date must still be calculated using the stated premium timing and the shocked future assumptions.

Worked Example: A Mortality Decrease on Term Assurance

A hypothetical two-year term policy pays $1,000 at the end of the year of death. The original annual mortality probability is $0.10$ in each year, the effective annual interest rate is $0.05$, and the net annual premium is the original-basis premium. An immediate shock reduces mortality to $q'=0.08$ in each year. Find the stressed benefit and premium present values, the present-value loss, and the reserve at time 1 immediately after the premium at that time has been paid, conditional on survival.

Method. Use the same original premium as in the preceding two-year term calculation, $P=\$95.2381$, and substitute the shocked rates in both the benefit and premium present values.

With $v=20/21$, the probability of dying in year 1 is $0.08$, and the probability of surviving year 1 and dying in year 2 is $0.92(0.08)=0.0736$. The stressed benefit present value is $1000[0.08v+0.0736v^2]=1000[0.0761905+0.0667574]=\$142.95$.

The premium present value is $95.2381[1+0.92v]=95.2381(1.8761905)=\$178.69$. Thus the present-value loss is $142.9479-178.6859=-\$35.74$, rounded to the nearest cent. Lower mortality reduces expected claims and increases the probability of receiving the second premium.

At time 1, immediately after the premium at that time has been paid, one year of cover remains and there is no later premium payment. The reserve conditional on survival is $V_1=1000(0.08)v=\$76.19$. This is lower than the corresponding original-basis reserve $1000(0.10)v=\$95.24$, as expected for this one-year remaining protection benefit. The calculation uses a reserve date after—not before—the time-1 premium.

Check. Under the shock, the probability of death in the second year is $0.0736$, not $0.08$, because the insured must first survive the first year. The probability of survival through both years is $0.92^2=0.8464$, and $0.08+0.0736+0.8464=1$.

Interpreting the Results and Avoiding Common Traps

A mortality level shock is a change in the assumptions used to value future contingent payments. It is not automatically a change in contractual terms. For a protection contract, the standard direction is clear for the benefit stream: higher mortality increases expected death claims, while lower mortality decreases them, all else equal. The direction of the total present-value result also depends on premiums, expenses if included, and any other cash flows. In particular, a whole-life contract can have substantial future premium income that is itself sensitive to survival.

A useful calculation sequence is to apply the shock to each annual rate, derive the corresponding survival probabilities, value benefits and premiums separately, and only then combine them using a clearly stated sign convention. Do not apply a shock to a present value after it has been calculated on the original basis. The shock changes the probabilities at each duration, so both the probability of a claim and the probability of reaching later premium dates must be reconsidered.

Using the first-year shocked rate for every duration without checking the model. This is appropriate only if the problem specifies a constant rate, or if the shock is a proportional level change applied to each duration-specific rate.
Forgetting survival before a later claim. In a two-year model, the second-year death probability is $p'_0q'_1$, not simply $q'_1$.
Discounting an immediate premium. A premium payable at time 0 has discount factor 1. Later premiums are discounted and weighted by the probability of being alive when due.
Subtracting a premium that has already been paid in a reserve. State whether the reserve is immediately before or immediately after the premium date. A payment already made is not a future premium receipt.
Assuming a mortality increase must worsen the net result. It raises the expected value of protection benefits, but it can also reduce the present value of future premium income. Compare both streams before deciding the net direction.

Key takeaway: Revalue protection benefits and survival-contingent premiums separately under the shocked mortality assumptions. Then combine them using explicit cash-flow timing and a stated sign convention. The reserve at a specified date depends just as critically on whether a premium is still due.

Check Your Understanding

For each question, use the stated timing assumptions and identify whether the requested amount is benefit outgo, premium income, or a net loss.

A one-year term policy pays $2,500 at the end of the year of death. The interest rate is $i=0.04$. An immediate mortality shock increases $q$ from $0.08$ to $0.10$. By how much does the benefit present value increase? (Answer: $48.08.)
A two-year term policy pays $1,000 at the end of the year of death. Interest is zero. Mortality is $0.05$ in each year originally and $0.06$ in each year after a shock. What is the increase in benefit present value? (Answer: $18.90.)
A whole-life policy pays $1,000 at the end of the year of death. Under a constant shocked mortality probability $q'=0.10$ and $i=0.04$, calculate the benefit present value. (Answer: $714.29.)
A two-year term policy has $B=\$2,000$, $i=0.05$, and constant shocked mortality $q'=0.12$. Benefits are paid at the end of the year of death. Calculate the benefit present value. (Answer: $429.71.)
A one-year term policy has a $5,000 benefit, $i=0.03$, and shocked mortality $q'=0.07$. At time 1, immediately after the only premium has been paid and conditional on the insured being alive, what is the prospective reserve? (Answer: $339.81.)

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