Modified Reserves And Their Valuation Bases

Why the Valuation Basis Matters

A reserve is a value assigned to future cash flows at a specified time and under specified assumptions. The same policy can therefore have different reserves when the assumptions used to value it differ. The key is to state the valuation basis explicitly: otherwise, a comparison between reserves may confuse a change in assumptions with a change in the contract itself.

This tutorial defines a modified reserve by comparison with a stated basis and distinguishes it from a gross prospective reserve. It is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: A valuation basis is the set of assumptions and conventions used to value future cash flows. It can include the benefit and premium amounts, expenses, mortality or other transition probabilities, interest or discounting, and the timing of cash flows. A modified reserve is a reserve calculated using a valuation basis in which at least one specified assumption about future cash flows, their probabilities, or their discounting differs from the assumptions used for the comparison reserve.

The comparison must be explicit. A reserve may be called modified relative to a particular reference basis, but that label alone does not say which assumption changed. In particular, a modification can be a change in discounting even when the benefit, premium, expense, and mortality assumptions are all unchanged.

A modified valuation assumption does not automatically change the policyholder’s contractual rights or the insurer’s actual expected cash flows. For example, a valuation may use a different mortality assumption while the policy benefit and contractual premium remain fixed. State which amounts are contractual and which assumptions have been altered for valuation.

Gross Prospective Reserves and Modified Bases

A prospective reserve at time $t$ is the expected present value of future outgo less the expected present value of future income, conditional on the policy being in the stated condition at time $t$. A gross prospective reserve includes future gross premiums and future expenses in that calculation. “Gross” distinguishes the premium basis from a net-premium calculation; it does not mean that the reserve is a gross benefit amount.

{}_tV^{\mathrm{gross}} = \operatorname{EPV}(\text{future benefits}+\text{future expenses}) - \operatorname{EPV}(\text{future gross premiums}).

All terms in this expression must use one stated valuation basis. If a basis is modified, recalculate the relevant expected present values on that modified basis. Do not combine, for example, a benefit value using one mortality assumption with a premium value using another unless the problem expressly defines such a basis.

Feature	Gross prospective reserve	Modified reserve
What is being described?	A prospective calculation including future gross premiums and expenses	A reserve calculated on a basis altered relative to a stated comparison basis
Does the label identify the assumptions?	No; the valuation basis still has to be stated	Only if the assumptions changed are identified
Can discounting differ?	Yes, according to its stated basis	Yes; a discounting change alone can define a modified basis
Must the contractual policy change?	No	No; the modification may be to valuation assumptions only

One-Year Prospective Formula and Its Justification

Consider a policy in force at time 0 for one year. Let ${}_1q_x$ be the probability of death during the year for a life aged $x$. A benefit $B$ is paid at the end of the year if death occurs. A gross premium $P$ and an expense $e$ are both paid immediately at time 0, whether or not death occurs during the year. Assume an effective annual interest rate $i$, with $v=1/(1+i)$, and no other cash flows.

{}_0V^{\mathrm{gross}} = {}_1q_x Bv + e - P.

Write the benefit present value.
The benefit has present value $Bv$ if death occurs and zero otherwise. The expected present value is therefore ${}_1q_x Bv$.

Include time-0 expenses and premiums.
Because both are paid immediately, their present values are $e$ and $P$. The expense is outgo and the premium is income.

Subtract income from outgo.
The prospective reserve is ${}_1q_x Bv+e-P$. If a valuation assumption changes, replace the affected term with its value under the modified assumption.

This derivation also explains the reserve’s sign convention: future benefits and expenses increase the reserve, while future premiums decrease it. The formula applies only with the stated timing and cash-flow pattern. If the premium or expense is payable only on survival, or if the benefit is paid at a different time, its expected present value must reflect that condition or timing.

Worked Example: Changing Discounting Only

A hypothetical one-year term policy covers a life aged $x$. The death benefit of $10,000 is paid at the end of the year of death. The probability of death during the year is ${}_1q_x=0.10$. A gross premium of $950 and an expense of $80 are paid at time 0. Calculate the reserve on a basis with $i=10\%$, then calculate it on a modified basis with $i=0\%$, keeping all other assumptions unchanged.

Method. Use the one-year prospective formula. Only the discounting assumption changes; neither the policy cash flows nor the mortality assumption changes.

On the original basis, $v=1/1.10=0.9090909$. The expected present value of the benefit is ${}_1q_x Bv=0.10(10{,}000)(0.9090909)=\$909.0909$. The time-0 expense is $80 and the time-0 premium is $950. Thus ${}_0V^{\mathrm{gross}}=\$909.0909+\$80-\$950=\$39.0909$, or $39.09.

On the modified basis, $i=0$, so $v=1$. The expected present value of the benefit is $0.10(10{,}000)(1)=\$1{,}000$. The expense and premium are unchanged, giving ${}_0V^{\mathrm{mod}}=\$1{,}000+\$80-\$950=\$130.00$.

Check and trap avoided. The modified reserve is $90.91 higher because the expected benefit is no longer discounted. The premium and expense occur at time 0, so changing the interest assumption does not alter their present values. It would be incorrect to leave the benefit at $909.09 after changing the discount rate.

Modifying Mortality or Expenses

A modified basis may instead change the probability of a future event or an expense assumption. The calculation method is the same: identify which expected present value terms depend on the changed assumption, recalculate those terms, and keep unchanged terms consistent with the stated comparison.

Worked Example: Changing the Mortality Assumption

A hypothetical one-year policy pays $12,000 at the end of the year if the insured life, aged $x$, dies during the year. A gross premium of $600 and an expense of $100 are paid at time 0. The effective annual interest rate is $i=4\%$. Calculate the gross prospective reserve using ${}_1q_x=0.05$, then calculate a modified reserve using ${}_1q_x=0.08$. Keep the benefit, premium, expense, and interest assumption fixed.

Method. Recalculate the probability-weighted benefit value. The present value factor is $v=1/1.04=0.9615385$.

On the original basis, the expected present value of the benefit is $0.05(12{,}000)(0.9615385)=\$576.9231$. Therefore, ${}_0V^{\mathrm{gross}}=\$576.9231+\$100-\$600=\$76.9231$, or $76.92.

On the modified basis, the expected present value of the benefit is $0.08(12{,}000)(0.9615385)=\$923.0769$. The other terms are unchanged, so ${}_0V^{\mathrm{mod}}=\$923.0769+\$100-\$600=\$423.0769$, or $423.08.

Check and trap avoided. The increase is $\$423.0769-\$76.9231=\$346.1538$. This also equals the increase in the benefit value: $(0.08-0.05)(12{,}000)(0.9615385)=\$346.1538$. The modified mortality assumption changes the valuation probability, not the contractual benefit or premium.

Worked Example: Changing a Future Expense Assumption

A hypothetical two-year term policy pays $10,000 at the end of the year of death. At time 0, a gross premium of $1,000 and an expense of $100 are paid. If the policyholder survives the first year, a further premium of $1,000 and an expense are paid at time 1. Death probabilities are 0.10 in each year conditional on being alive at the start of that year. The effective annual interest rate is $i=0\%$. Under the original basis, the time-1 expense is $100; under the modified basis, it is $160. Find the reserve at time 0 on each basis.

Method. Sum expected cash flows at their payment times. With $i=0\%$, all discount factors are 1. The time-1 premium and expense are payable only if the policy is in force at time 1, which has probability $1-0.10=0.90$.

The expected benefit is $1,000 from deaths in year 1, plus $0.90(0.10)(\$10{,}000)=\$900$ from deaths in year 2. Total expected benefits are $\$1{,}000+\$900=\$1{,}900$.

Expected premiums are $\$1{,}000+0.90(\$1{,}000)=\$1{,}900$. On the original expense basis, expected expenses are $\$100+0.90(\$100)=\$190$. Hence ${}_0V^{\mathrm{gross}}=\$1{,}900+\$190-\$1{,}900=\$190.00$.

On the modified basis, expected expenses are $\$100+0.90(\$160)=\$244$. Benefits and premiums are unchanged, so ${}_0V^{\mathrm{mod}}=\$1{,}900+\$244-\$1{,}900=\$244.00$.

Check and trap avoided. The modified reserve exceeds the original reserve by $54, equal to the increase in expected time-1 expense, $0.90(\$160-\$100)=\$54$. Charging the time-1 expense with probability 1 would be wrong because it is paid only if the policy survives to that time.

Common Exam Traps and Practical Checks

A reserve calculation is easier to audit when the basis, state, and cash-flow timing are written before the arithmetic. In particular, keep these distinctions clear:

Changing a basis is not necessarily changing the contract. A valuation mortality or interest assumption can change while contractual benefits and premiums stay fixed.
Discounting is part of the valuation basis. Even when all benefit amounts and event probabilities remain the same, a changed interest assumption can produce a modified reserve.
Apply probabilities to contingent cash flows. A payment made only if the policy survives to a future date must be weighted by the probability of survival to that date.
Do not mix bases silently. If mortality, expenses, or discounting are modified, identify which terms use the modified assumption and which remain unchanged.
Respect timing. A time-0 premium or expense is not discounted; an end-of-year benefit uses the appropriate end-of-year discount factor.
Do not assume a modified reserve is higher. The direction depends on the change. Higher expected benefits or expenses tend to increase a reserve, while higher future premiums tend to decrease it; a discount-rate change has an effect determined by the timing and balance of future outgo and income.

Key takeaway: A modified reserve is defined relative to a stated comparison basis. State the changed assumption, preserve the contract cash flows unless they are explicitly changed, and value all future benefits, expenses, and premiums consistently on the chosen basis.

Check Your Understanding

For each problem, identify the applicable basis and use the stated payment timing. Answers are rounded to the nearest cent.

A one-year policy pays $15,000 at year-end if death occurs. For a life aged $x$, ${}_1q_x=0.04$. A premium of $700 and expense of $80 are paid at time 0. At $i=5\%$, what is the gross prospective reserve? (Answer: $-\$48.57$)
For the policy in Question 1, recalculate the reserve on a modified basis with $i=0\%$, leaving mortality and all cash flows unchanged. (Answer: $-\$20.00$)
A two-year policy pays $5,000 at the end of the year of death. The conditional death probability is 0.20 in each year, $i=0\%$, and premiums of $600 and expenses of $50 are paid at times 0 and 1 while the policy is in force. What is the time-0 gross prospective reserve? (Answer: $810.00)
A one-year policy pays $20,000 at year-end on death, with $i=0\%$, a time-0 premium of $900, and a time-0 expense of $100. Calculate the original reserve at ${}_1q_x=0.03$ and the modified reserve at ${}_1q_x=0.05$. (Answer: Original $-\$200.00$; modified $200.00)
A one-year policy has a $25,000 year-end death benefit, time-0 premium of $1,200, and time-0 expense of $100. The original basis has ${}_1q_x=0.04$ and $i=5\%$; the modified basis has ${}_1q_x=0.05$ and $i=4\%$. Calculate the reserve on each basis. (Answer: Original $-\$147.62$; modified $101.92)

What You'll Learn