Modified Prospective Reserve Equivalence

What the Equivalence Says

A prospective reserve is a value at a particular time, conditional on the policy being in force then. The word modified specifies that the valuation uses a modified basis: the interest, decrement, premium, or expense assumptions designated for that reserve. The basis determines how future cash flows are discounted and how their probabilities are assessed; it does not change the contract’s actual past cash flows.

The equivalence in this tutorial is a valuation identity. A modified prospective reserve is the expected present value, on the modified basis, of future benefits and expenses less future premiums. This sounds direct, but errors often arise from mixing bases or from overlooking whether a payment is due immediately or at the end of a period.

Definition: At time $t$, conditional on the policy being in force in state $r$, the modified prospective reserve, denoted $V_{t,r}^{M}$, is the modified-basis expected present value of all future benefits and expenses minus the modified-basis expected present value of all future premiums. Cash flows due at time $t$ are included if the valuation instant is defined as immediately before those cash flows; they are excluded if it is immediately after them.

Use a single, explicitly stated timing convention throughout. In the examples, premiums and expenses described as due at a specified time are valued at that time, and benefits described as paid at year-end are valued at the end of that year. A cash flow at the valuation instant has discount factor 1.

V_{t,r}^{M} = \operatorname{EPV}_{M}(\text{future benefits}\mid r\text{ at }t) + \operatorname{EPV}_{M}(\text{future expenses}\mid r\text{ at }t) - \operatorname{EPV}_{M}(\text{future premiums}\mid r\text{ at }t).

The superscript $M$ is a reminder that all three present values use the same modified basis. For example, if the basis specifies an effective annual interest rate $i^M$, the discount factor for a payment one year later is $v^M=1/(1+i^M)$. Probabilities of survival or transition must also be those specified by that basis. A negative reserve is possible under this definition: it means the modified-basis value of future premiums exceeds that of future benefits and expenses.

Deriving the Prospective Equivalence

The justification follows from the definition of present value and conditional expectation. Consider the random future cash flows of a policy in state $r$ at time $t$. From the insurer’s perspective, benefits and expenses are positive outgo, while premiums are negative outgo. Discount each future cash flow to time $t$, add them along the policy’s future path, and then take the conditional expectation using the modified basis.

Form the future net outgo.
For each possible future path, add benefits and expenses and subtract premiums. Include only cash flows after the stated valuation instant, together with any cash flow due immediately if valuation is just before payment.

Discount each cash flow.
Discount a payment at future time $s$ by the modified-basis accumulation from $t$ to $s$. Under a constant effective annual rate $i^M$, a payment $j$ years later has discount factor $(v^M)^j$.

Take the conditional expectation.
Average the discounted future net outgo over possible outcomes, conditional on being in state $r$ at time $t$, with probabilities from the modified basis.

Separate the three cash-flow categories.
Linearity of expectation makes the expected present value of future net outgo equal to expected benefits plus expected expenses minus expected premiums. That is the prospective equivalence.

For a discrete-time single-life contract, suppose future death benefits $B_j$ are paid at the end of year $j$, premiums $P_j$ and expenses $E_j$ are due at the start of year $j$, and the valuation is immediately before the start-of-year cash flows. If $n$ years remain, the modified-basis expression is

V_k^M = \sum_{j=0}^{n-1}(v^M)^j\,{}_jp_{x+k}^{M}(E_{k+j}-P_{k+j}) + \sum_{j=1}^{n}(v^M)^j\,{}_{j-1}p_{x+k}^{M}\,q_{x+k+j-1}^{M} B_{k+j}.

Here ${}_jp_{x+k}^{M}$ is the modified-basis probability of surviving $j$ years from age $x+k$, and $q_{x+k+j-1}^{M}$ is the modified-basis probability of death during the following year, conditional on reaching its start. The first sum weights start-of-year amounts by the probability the insured is alive to pay or incur them. In the second sum, the insured must survive to the start of the year and then die during it. This expression assumes annual timing, premiums and expenses only while alive, and no maturity benefit. Add any maturity benefit or other state-contingent cash flow at its specified time with its corresponding probability.

For a multi-state contract, the same reasoning applies, but cash flows and probabilities depend on the current state and the transitions between states. A transition benefit is weighted by the probability of the transition; a state payment is weighted by the probability of occupying that state at the payment time. The formula does not apply unchanged if it omits a possible cash flow, uses a probability inconsistent with the modified basis, or discounts from the wrong valuation instant.

Worked Applications

Worked Example: One Remaining Year With an Immediate Premium

A hypothetical one-year term policy is valued at time 1, immediately before a premium and an expense due at that instant. On the modified basis, the effective annual interest rate is $10\%$, and the probability of death during the next year is $0.10$; hence the probability of survival is $0.90$. The premium due now is $100, the expense due now is $20, and a $1,000 death benefit is paid at time 2 if death occurs. If the insured survives, a $10 maintenance expense is paid at time 2. Find the modified prospective reserve.

Method. Value each future cash flow from time 1, applying the modified probability to cash flows contingent on the year’s outcome. The discount factor for time 2 is $v^M=1/1.10=0.9090909\ldots$.

At time 1, the expense is an outgo of $20 and the premium is an income of $100. Their net contribution to the reserve is $20-100=-\$80$.

At time 2, expected outgo is $0.10(\$1{,}000)+0.90(\$10)=\$100+\$9=\$109$. Its present value at time 1 is $\$109/1.10=\$99.0909\ldots$.

V_{1}^{M} =20-100+\frac{0.10(1{,}000)+0.90(10)}{1.10} =-80+99.0909\ldots =\$19.09.

The reserve is $19.09, rounded to the nearest cent. As a check, the time-1 immediate net inflow of $80 offsets most, but not all, of the $99.09 present value of the next year’s expected outgo. A common mistake is to discount the $20 expense and $100 premium as if they were due at time 2; they are due at the valuation instant and therefore have discount factor 1.

Worked Example: Two-Year Modified-Basis Term Reserve

A hypothetical two-year term policy is valued at time 0, immediately before any cash flow due then. On the modified basis, $i^M=5\%$. The first-year death probability is $0.10$, and the second-year death probability, conditional on surviving the first year, is $0.20$. A $500 benefit is paid at the end of the year of death. A premium of $80 and an expense of $10 are due at the start of each year in which the insured is alive. Calculate the reserve at time 0.

Method. Calculate the present values of claims, expenses, and premiums separately. Use $v^M=1/1.05=20/21$. The probability of reaching the second year is $0.90$.

The expected present value of first-year death claims is $v^M(0.10)(\$500)=\$1{,}000/21=\$47.6190\ldots$. The expected present value of second-year claims is $(v^M)^2(0.90)(0.20)(\$500)=\$12{,}000/147=\$81.6327\ldots$. Therefore, total claim value is $\$19{,}000/147=\$129.2517\ldots$.

Expenses are due at the start of each year, while the insured is alive. Their present value is $\$10+(0.90)(\$10)(20/21)=\$10+\$60/7=\$18.5714\ldots$. Premiums have the same timing and survival weighting, so their present value is $\$80+(0.90)(\$80)(20/21)=\$80+\$480/7=\$148.5714\ldots$.

V_0^M = \frac{19{,}000}{147} +\frac{130}{7} -\frac{1{,}040}{7} = -\frac{110}{147} = -\$0.75.

The reserve is -$0.75, rounded to the nearest cent. The unrounded value is $-\$0.748299\ldots$. This is consistent: on this modified basis, expected premiums slightly exceed expected claims and expenses. The calculation avoids a common mistake of treating the second-year probability as $0.20$ from time 0 without also multiplying by the $0.90$ probability of surviving year 1.

Worked Example: A Reserve Conditional on a Care State

A hypothetical long-term care contract is valued at time 1, immediately before an expense due then. The insured is in the disabled-care state. On the modified basis, $i^M=10\%$; over the next year, the probability of remaining in care is $0.70$, and the probability of recovery is $0.30$. There are no future premiums. An expense of $6 is due now. At time 2, a $120 benefit and a $10 expense are paid if the insured remains in care; a $40 recovery benefit and a $10 expense are paid if the insured recovers. Find the modified reserve.

Method. Condition on the state at time 1, then value each possible time-2 outcome using its modified transition probability. The time-2 discount factor is $1/1.10$.

The time-1 expense contributes $6. If the insured remains in care, time-2 outgo is $\$120+\$10=\$130$, with probability $0.70$. If recovery occurs, time-2 outgo is $\$40+\$10=\$50$, with probability $0.30$. Expected time-2 outgo is $0.70(\$130)+0.30(\$50)=\$91+\$15=\$106$.

V_{1,\mathrm{care}}^M = 6+\frac{0.70(130)+0.30(50)}{1.10} = 6+\frac{106}{1.10} = \$102.36.

The reserve is $102.36, rounded to the nearest cent. The state condition matters: this is a reserve given that the insured is already in care, not a reserve at issue averaged over all possible states. The calculation also shows why recovery benefits cannot be omitted merely because recovery ends the care state.

Basis Consistency and Common Traps

Modified prospective equivalence is useful as both a definition and a verification method. Whatever procedure produced a reserve, its value should agree with the expected present value of the future benefits and expenses less future premiums under the stated modified basis and timing convention. This makes the equivalence a check on signs, probabilities, and payment dates.

Mixing valuation bases. Discounting at a modified interest rate while using decrement probabilities from a different basis does not produce a modified-basis prospective value.
Reversing the sign of premiums. From the insurer’s perspective, premiums reduce future net outgo. The expression is benefits plus expenses minus premiums.
Using the wrong conditioning event. A reserve in a particular state is conditional on being in that state at the valuation time. Do not average across states unless the question asks for an unconditional value.
Misplacing cash flows in time. A premium due immediately is not discounted one year. A benefit paid at year-end is discounted for the full year.
Confusing a reserve with an actual cash balance. The reserve is a conditional expected present value under stated assumptions. It is not a guarantee that the future realized cash flows will equal the reserve.

Key takeaway: To establish modified prospective reserve equivalence, state the valuation instant and modified basis, list all future benefits, expenses, and premiums with their timing and conditions, and take their conditional expected present values. The reserve is outgo less premium income.

Check Your Understanding

For each problem, use the stated modified basis and the cash-flow timing given.

A one-year policy is valued immediately before a $70 premium and a $12 expense. On the modified basis, $i^M=4\%$ and the death probability is $0.08$. A $800 benefit is paid at year-end on death, with no other future cash flows. Find the reserve. (Answer: $3.54)
A two-year term policy is valued at time 0 before cash flows. On the modified basis, $i^M=10\%$, the first-year death probability is $0.20$, and the second-year death probability conditional on surviving the first year is $0.25$. A $400 benefit is paid at the end of the year of death. A $60 premium and a $5 expense are due at the start of each year the insured is alive. Find the reserve to the nearest cent. (Answer: $43.84)
A policy is valued at time 0 before immediate cash flows. The insured is active. On the modified basis, $i^M=5\%$; over the next year, transition probabilities are $0.90$ to remain active and $0.10$ to enter care. A $30 premium and $4 expense are due now. At year-end, active status incurs a $5 expense; care status incurs a $200 benefit and a $15 expense. Find the reserve to the nearest cent. (Answer: -$1.24)
A contract is valued at time 1 while the insured is in the disabled-care state. On the modified basis, $i^M=8\%$; over the next year, the probability of remaining in care is $0.60$, and the probability of recovery is $0.40$. An expense of $2 is due now. At time 2, outgo is $96 if the insured remains in care and $36 if recovery occurs. There are no future premiums. Find the reserve to the nearest cent. (Answer: $68.67)

What You'll Learn