Benefit Modifications In Reserve Bases

Why Benefit Assumptions Change a Reserve

A reserve is a valuation, not a restatement of the contract. The contract specifies what the insurer owes, but a reserve basis may value those obligations using benefits that differ from the contractual amounts. For example, a hypothetical valuation basis might recognize a lower death benefit or a lower amount payable on survival to maturity. The reserve then reflects the valuation benefits, not the contractual benefits.

This tutorial considers benefit changes while holding the other valuation assumptions fixed. In particular, interest, mortality, and the future premium stream are the same when comparing the modified-benefit reserve with the reserve using contractual benefits. This isolates the effect of changing the benefit assumption; if premiums or other assumptions also change, their effects must be valued separately.

Definition: A contractual benefit is the amount payable under the policy terms. A valuation benefit is the amount used to value that payment in a reserve basis. A modified benefit reserve uses valuation benefits that differ from the contractual benefits, with the other reserve-basis assumptions specified separately.

Prospective Valuation with Modified Benefits

Use the following discrete-time model. A policy is in force at integer times, death benefits are paid at the end of the year of death, and a survival benefit, if any, is paid at the end of the final year. Annual premiums are paid at integer times while the insured is alive. The reserve $V_t$ is measured immediately after any premium due at time $t$ has been paid. Thus, future premiums are included only when they are due after time $t$, and only if the insured survives to the payment date.

Let $i$ be the effective annual interest rate and $v=1/(1+i)$. Given survival to time $t$, write ${}_jp_t$ for the probability of surviving a further $j$ years, and $q_{t+j-1}$ for the probability of death during the year ending at time $t+j$, conditional on being alive at its start. Let $B_{t+j}$ be the contractual death benefit for death in that year, and $\widetilde B_{t+j}$ its valuation amount. Let $S_n$ and $\widetilde S_n$ be the contractual and valuation survival benefits at the end of year $n$. Finally, let $P_{t+j}$ be a future premium payable at time $t+j$ if the insured is then alive.

Prospective reserve: Under this model, the reserve at time $t$ is the present value of future valuation benefits minus the present value of future premiums, conditional on being alive at $t$. For a contract ending at time $n$,

$$ \widetilde V_t = \sum_{j=1}^{n-t} v^j\,{}_{j-1}p_t\,q_{t+j-1}\widetilde B_{t+j} + v^{n-t}\,{}_{n-t}p_t\,\widetilde S_n - \sum_{j=1}^{n-t-1} v^j\,{}_jp_t P_{t+j}. $$
The survival-benefit term is omitted if there is no such benefit. A premium due at time $n$ is also omitted here because the contract has ended at that time.

The death-benefit probability in year $t+j$ is ${}_{j-1}p_tq_{t+j-1}$: the insured must survive the preceding $j-1$ years and then die in the next year. A survival benefit at maturity requires survival for all $n-t$ years. A future premium is payable only on survival to its due date, which explains the factor ${}_jp_t$ in the premium term.

Why the Reserve Difference Is a Benefit Present Value

Suppose the modified-benefit reserve and the contractual-benefit reserve use the same interest, mortality, and premium assumptions. Subtracting the two prospective reserves cancels the common future premium present value. What remains is the present value of the difference between the valuation and contractual benefits.

Write each reserve.
Each is the present value of its own benefit stream minus the same present value of future premiums.

Subtract the contractual-benefit reserve.
The common premium present value cancels, leaving the difference between the two benefit present values.

Interpret the sign.
If the valuation benefits are lower in every relevant outcome, the modified reserve is lower. The size of the change depends on each benefit’s probability and payment time.

\widetilde V_t-V_t = \sum_{j=1}^{n-t}v^j\,{}_{j-1}p_tq_{t+j-1} \bigl(\widetilde B_{t+j}-B_{t+j}\bigr) + v^{n-t}\,{}_{n-t}p_t \bigl(\widetilde S_n-S_n\bigr).

This identity is a direct consequence of prospective valuation and linearity of present values. It applies only when the premium stream and all other assumptions are common to both reserves. If the modified basis also changes premiums, that premium difference does not cancel.

Worked Example: Lower Death Benefits with a Future Premium

A hypothetical two-year policy is valued immediately after the premium at time 0. The effective annual interest rate is $10\%$, and the probability of death in each year, conditional on being alive at its start, is $0.10$. The contractual death benefit is \$10,000 in either year; the valuation death benefit is \$8,000 in either year. There is no survival benefit. A premium of \$500 is due at time 1 if the insured survives to that time. Calculate the modified reserve and compare it with the reserve using contractual benefits.

Method. Apply the prospective reserve formula at time 0. Here $v=1/1.10=10/11$, the probability of death in year 1 is $0.10$, and the probability of death in year 2 is $0.90(0.10)=0.09$. The time-1 premium is payable with probability $0.90$.

The present value of the modified death benefits is

(0.10)(8{,}000)v+(0.09)(8{,}000)v^2 = \frac{800}{1.10}+\frac{720}{1.21} = 727.2727+595.0413 = \$1{,}322.3140.

The future premium present value is

(0.90)(500)v = 450\left(\frac{10}{11}\right) = \frac{4{,}500}{11} = \$409.0909.

Therefore, the modified reserve is

\widetilde V_0 = 1{,}322.3140-409.0909 = \$913.2231 \approx \$913.22.

Using the contractual benefit instead, its present value is

(0.10)(10{,}000)v+(0.09)(10{,}000)v^2 = \frac{1{,}000}{1.10}+\frac{900}{1.21} = 909.0909+743.8017 = \$1{,}652.8926.

Thus the contractual-benefit reserve is $1{,}652.8926-409.0909=\$1{,}243.8017$, or \$1,243.80 to the nearest cent. The difference is $\$913.2231-\$1{,}243.8017=-\$330.5786$, which rounds to $-\$330.58$. As a check, reducing each death benefit by \$2,000 reduces the benefit present value by $(0.10)(2{,}000)v+(0.09)(2{,}000)v^2=\$330.5785$, with the small difference in the last decimal due to intermediate rounding. The reserve difference is negative, as expected for lower valuation benefits.

Worked Example: A Modified Maturity Benefit

A hypothetical two-year policy has a $10\%$ effective annual interest rate and independent annual mortality probabilities of $0.10$ conditional on survival to the start of each year. Death benefits are \$5,000 at the end of either year under both the contract and the valuation basis. A survival benefit of \$12,000 is payable at time 2 under the contract, but the valuation basis uses \$10,000. A \$400 premium is due at time 1 if the insured survives. Calculate the modified reserve at time 0 immediately after the initial premium.

Method. The probability of death in year 1 is $0.10$, death in year 2 is $0.90(0.10)=0.09$, and survival to maturity is $0.90^2=0.81$. Also, $v=10/11$ and $v^2=100/121$.

The modified death-benefit present value is

(0.10)(5{,}000)v+(0.09)(5{,}000)v^2 = \frac{500}{1.10}+\frac{450}{1.21} = 454.5455+371.9008 = \$826.4463.

The modified survival-benefit present value is

(0.81)(10{,}000)v^2 = \frac{8{,}100}{1.21} = \$6{,}694.2149.

The premium present value is $(0.90)(400)v=360(10/11)=\$327.2727$. Consequently,

\widetilde V_0 = 826.4463+6{,}694.2149-327.2727 = \$7{,}193.3885 \approx \$7{,}193.39.

The contractual survival benefit exceeds the valuation benefit by \$2,000. The reserve difference is therefore $\widetilde V_0-V_0=-(0.81)(2{,}000)v^2=-\$1{,}338.8430$, or $-\$1{,}338.84$. The contractual-benefit reserve is thus $\$7{,}193.3885+\$1{,}338.8430=\$8{,}532.2315$, which rounds to \$8,532.23. This check isolates the effect of the maturity benefit: the death benefits and future premium are identical on both bases.

Worked Example: Separating Death and Survival Benefit Changes

At time 1, immediately after the premium due then, a hypothetical policy has two years remaining. The effective annual interest rate is $5\%$. For the first remaining year, the death probability is $0.20$; for the second, it is $0.10$, conditional on survival to that year. The contractual death benefit is \$8,000 in either year, while the valuation death benefit is \$7,000. A survival benefit of \$17,000 is payable at the end of the second remaining year under the contract, but the valuation amount is \$15,000. A \$600 premium is due at time 2 if the insured survives the first remaining year. Calculate the modified reserve and its difference from the contractual-benefit reserve.

Method. Use $v=1/1.05=20/21$, so $v^2=400/441$. The probabilities of death in the two remaining years are $0.20$ and $(0.80)(0.10)=0.08$. The probability of survival to maturity is $(0.80)(0.90)=0.72$.

The modified death-benefit present values are

(0.20)(7{,}000)v = 1{,}400\left(\frac{20}{21}\right) = \$1{,}333.3333, $$ $$ (0.08)(7{,}000)v^2 = 560\left(\frac{400}{441}\right) = \$507.9365.

The modified survival-benefit present value is $(0.72)(15{,}000)v^2=10{,}800(400/441)=\$9{,}795.9184$. The time-2 premium present value is $(0.80)(600)v=480(20/21)=\$457.1429$. Therefore,

\widetilde V_1 = 1{,}333.3333+507.9365+9{,}795.9184-457.1429 = \$11{,}180.0453 \approx \$11{,}180.05.

For the reserve difference, the death-benefit reduction is \$1,000 in each year and the maturity-benefit reduction is \$2,000:

\widetilde V_1-V_1 = -(0.20)(1{,}000)v -(0.08)(1{,}000)v^2 -(0.72)(2{,}000)v^2 = -\frac{692{,}000}{441} = -\$1{,}569.1610.

The contractual-benefit reserve is $\$11{,}180.0453+\$1{,}569.1610=\$12{,}749.2063$, or \$12,749.21. The difference is negative because every valuation benefit is lower. This calculation also checks the timing: the year-two death and maturity payments are both discounted for two years, but their probabilities differ because they are mutually exclusive outcomes.

Common Exam Trap: Substituting the Contract Benefit

A frequent error is to calculate a reserve using the contractual benefit even though the question specifies a modified valuation benefit. In the first example, using \$10,000 in place of the \$8,000 valuation death benefit gives a reserve of \$1,243.80. That is the contractual-benefit reserve, not the modified reserve of \$913.22. The correction is to use the valuation benefit in each outcome when calculating the modified reserve, then use the reserve-difference identity as a check.

A second error is to reverse the difference. If the requested difference is “modified minus contractual,” write $\widetilde V_t-V_t$, not $V_t-\widetilde V_t$. When valuation benefits are lower and all other assumptions are common, this difference should be non-positive. A positive result signals either a sign reversal or a benefit assumption that has not been applied as stated.

Key takeaway: Value each benefit using its amount on the specified reserve basis, its probability of payment, and its payment time. When premiums and all other assumptions are held fixed, the change in reserve is exactly the present value of the change in benefits.

Check Your Understanding

For each question, compare reserves on the same interest and mortality basis. Unless stated otherwise, report reserve differences as modified minus contractual.

A one-year policy has an effective annual interest rate of $4\%$ and a death probability of $0.05$. There is no future premium or survival benefit. The contractual death benefit is \$20,000 and the valuation death benefit is \$18,000. Find the modified reserve and its difference from the contractual-benefit reserve. (Answer: modified reserve \$865.38; difference $-\$96.15$)
For a two-year policy, $i=10\%$, and each annual death probability is $0.10$. The valuation death benefit is \$1,000 lower than the contractual benefit in each year. Find the reserve difference at time 0 immediately after the initial premium. (Answer: $-\$165.29$)
A survival benefit payable in two years is reduced by \$3,000 on the valuation basis. The annual survival probability is $0.95$, and $i=5\%$. Find the reserve difference attributable to this change. (Answer: $-\$2{,}455.78$)
Only the second-year death benefit changes: it is \$5,000 lower on the valuation basis. The first-year survival probability is $0.90$, the second-year death probability conditional on survival is $0.20$, and $i=10\%$. Find the reserve difference at time 0 attributable to the change. (Answer: $-\$743.80$)

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