Comparing Gross And Modified Prospective Reserves

Why Compare the Reserves by Component?

A modified reserve can differ from a gross prospective reserve because the valuation basis recognizes future premiums, expenses, or benefits differently. The total difference is useful, but it does not by itself explain why the reserves differ. A component comparison identifies which changes increase the reserve and which reduce it.

The comparison must be made at the same valuation date, for the same contract and in-force condition, with consistent cash-flow timing. The contract’s benefits and premiums are not necessarily changed by a modified reserve basis: the modification is in how future cash flows are valued. Because conventions for modified reserves can vary, use the definition and basis specified in the problem.

Definition: At a stated duration and conditional on the policy being in force, a prospective reserve is the present value of future benefits and expenses less the present value of future premiums, under a specified valuation basis. Write the reserve on basis $b$ as $$ V^{(b)} = B^{(b)} + E^{(b)} - P^{(b)}, $$ where $B^{(b)}$, $E^{(b)}$, and $P^{(b)}$ are the conditional expected present values of future benefits, expenses, and premiums, respectively. A gross prospective reserve uses the gross premium and expense cash flows specified for its gross basis. A modified prospective reserve uses the corresponding modified valuation cash flows specified for its modified basis.

The values $B^{(b)}$, $E^{(b)}$, and $P^{(b)}$ include the effects of the valuation assumptions, such as interest, mortality, and continuation in force. For example, a premium payable only while the policyholder is alive is included in $P^{(b)}$ with its probability of being paid. A benefit payable on death is included with its probability and payment timing. The notation is a bookkeeping identity; it does not assume that every future cash flow is certain.

The Reserve-Difference Bridge

Let $g$ denote the gross basis and $m$ the modified basis. Subtract the gross reserve from the modified reserve. The result separates naturally into three contributions:

V^{(m)}-V^{(g)} = \bigl(B^{(m)}-B^{(g)}\bigr) + \bigl(E^{(m)}-E^{(g)}\bigr) - \bigl(P^{(m)}-P^{(g)}\bigr).

Thus, a larger benefit present value increases the modified reserve, and a larger expense present value also increases it. A larger premium present value decreases the reserve, because more future premium income is available to fund benefits and expenses. Keep the negative sign on the premium contribution visible throughout the calculation.

Key result: Define the benefit contribution as $\Delta B=B^{(m)}-B^{(g)}$, the expense contribution as $\Delta E=E^{(m)}-E^{(g)}$, and the premium contribution as $-\Delta P=-(P^{(m)}-P^{(g)})$. Then $$ V^{(m)}-V^{(g)}=\Delta B+\Delta E-\Delta P. $$ A positive contribution raises the modified reserve relative to the gross reserve; a negative contribution lowers it.

This bridge follows directly from the prospective definition. Substitute $V^{(m)}=B^{(m)}+E^{(m)}-P^{(m)}$ and $V^{(g)}=B^{(g)}+E^{(g)}-P^{(g)}$, then subtract and collect terms. The result requires no independence assumption among cash flows: it uses addition and subtraction of their expected present values. It does require that the two reserves refer to the same valuation date and comparable in-force condition.

Fix the comparison.
State the valuation date, in-force condition, payment timing, and the gross and modified bases being compared.

Calculate the component present values.
Find $B^{(g)}, E^{(g)}, P^{(g)}$ and $B^{(m)}, E^{(m)}, P^{(m)}$, applying survival and discounting consistently.

Calculate both reserves directly.
Use $V=B+E-P$ on each basis. This provides a check on the component values.

Attribute the difference.
Calculate $\Delta B$, $\Delta E$, and $-\Delta P$, then verify that their sum equals $V^{(m)}-V^{(g)}$.

If both bases use the same interest and decrement assumptions, but change only the premium, expense, and benefit cash-flow specifications, the bridge gives a direct component attribution. If assumptions such as mortality or interest also change, they can affect several components at once. To isolate the effect of a particular change, compare counterfactual bases that differ only in that item while holding the remaining assumptions fixed. If several assumptions change together, a component attribution is still possible, but it should not be mistaken for a unique causal allocation among the changed assumptions.

Worked Comparisons

Worked Example: One-Year Reserve With Three Basis Changes

A hypothetical policy is in force at the start of its final coverage year. At that instant, one premium and one renewal expense are due; a death benefit is paid at year-end if death occurs during the year. The annual effective interest rate is $i=4\%$, so $v=1/1.04=0.961538$. The one-year death probability is $q=0.10$. On the gross basis, the benefit is $10,000, the expense is $100, and the premium is $500. On the modified basis, the corresponding valuation amounts are $9,500, $80, and $450. Calculate both reserves and attribute the difference. The policy’s contractual coverage is the same; the stated differences are valuation-basis amounts.

Method. Calculate the expected present value of the year-end death benefit on each basis, then add the expense due immediately and subtract the premium due immediately. The common mistake to avoid is discounting a cash flow due at the valuation instant.

On the gross basis, the benefit present value is

B^{(g)}=0.10(10{,}000)(0.961538)=961.538.

The expense and premium are due immediately, so $E^{(g)}=100$ and $P^{(g)}=500$. Therefore,

V^{(g)}=961.538+100-500=561.538.

On the modified basis,

B^{(m)}=0.10(9{,}500)(0.961538)=913.462, \qquad E^{(m)}=80, \qquad P^{(m)}=450,

V^{(m)}=913.462+80-450=543.462.

The benefit contribution is $913.462-961.538=-48.076$. The expense contribution is $80-100=-20$. The premium contribution is $-(450-500)=+50$, because the modified basis recognizes $50 less future premium. Their sum is $-48.076-20+50=-18.076$, which agrees with $543.462-561.538=-18.076$. Rounded to cents, the gross reserve is $561.54, the modified reserve is $543.46, and the modified reserve is $18.08 lower.

Worked Example: Two-Year Term Coverage With Survival-Contingent Cash Flows

A hypothetical two-year term policy is valued at issue. The annual effective interest rate is $i=1/9=11.1\%$, so $v=0.9$. The conditional death probabilities are $q_1=0.10$ in year 1 and $q_2=0.20$ in year 2; thus $p_1=0.90$. A premium and an expense are payable at the start of each year if the policy is in force. Death benefits are paid at the end of the year of death. The gross basis uses a $5,000 benefit, a $300 premium each year, an initial expense of $50, and a renewal expense of $40. The modified basis uses a $4,800 benefit, a $280 premium each year, an initial expense of $45, and a renewal expense of $30. Calculate and reconcile the reserves.

Method. Enumerate the possible claim years and weight each cash flow by its probability and discount factor. For year 2, the policy must first survive year 1. The common mistake to avoid is omitting that survival factor or discounting a premium due at time 0.

The year 1 death probability is $0.10$. The probability of death in year 2 is $0.90(0.20)=0.18$. Thus, on the gross basis, the benefit present value is

B^{(g)} =(0.10)(5{,}000)(0.9) +(0.90)(0.20)(5{,}000)(0.9^2) =450+729 =1{,}179.

The initial expense is $50 at time 0. The $40 renewal expense is due at time 1 if the policy survives year 1, so its expected present value is $(0.90)(40)(0.9)=32.40$. Hence $E^{(g)}=50+32.40=82.40$. Premiums have the same timing and survival condition: $P^{(g)}=300+(0.90)(300)(0.9)=543$. The gross reserve is

V^{(g)}=1{,}179+82.40-543=718.40.

On the modified basis, the benefit present value is

B^{(m)} =(0.10)(4{,}800)(0.9) +(0.90)(0.20)(4{,}800)(0.9^2) =432+699.84 =1{,}131.84.

The expense present value is $E^{(m)}=45+(0.90)(30)(0.9)=69.30$, and the premium present value is $P^{(m)}=280+(0.90)(280)(0.9)=506.80$. Therefore,

V^{(m)}=1{,}131.84+69.30-506.80=694.34.

The benefit contribution is $1{,}131.84-1{,}179=-47.16$; the expense contribution is $69.30-82.40=-13.10$; and the premium contribution is $-(506.80-543)=+36.20$. The total is $-47.16-13.10+36.20=-24.06$, agreeing with $694.34-718.40=-24.06$. Both reserves and the difference are already exact to cents.

Worked Example: Three-Year Endowment With a Higher Modified Premium

A hypothetical three-year endowment contract is valued at time 0. The annual effective interest rate is $i=0\%$. The probability of death in each year, conditional on being alive at its start, is $q=0.10$, so $p=0.90$. A $10,000 death benefit is paid at the end of the year of death. If the policyholder survives all three years, an $8,000 maturity benefit is paid at time 3 instead. A premium and expense are payable at the start of each year while the policy is in force. The gross basis has a $1,500 annual premium and a $100 annual expense. The modified basis uses a $1,600 annual premium, an $80 annual expense, a $9,500 death benefit, and a $7,600 maturity benefit. Calculate and reconcile the reserves.

Method. Find the probabilities of death in each year and of survival to maturity, then add the corresponding benefits. Since $i=0\%$, there is no discounting. The common mistake to avoid is including both the year 3 death benefit and the maturity benefit for the same policyholder; these are mutually exclusive outcomes.

The probabilities of death in years 1, 2, and 3 are $0.10$, $0.90(0.10)=0.09$, and $0.90^2(0.10)=0.081$. The probability of survival to maturity is $0.90^3=0.729$. Thus,

B^{(g)} =(0.10+0.09+0.081)(10{,}000)+(0.729)(8{,}000) =2{,}710+5{,}832 =8{,}542.

The expected number of premium and expense payments is $1+0.90+0.90^2=2.71$. Therefore, $P^{(g)}=1{,}500(2.71)=4{,}065$ and $E^{(g)}=100(2.71)=271$. The gross reserve is $8{,}542+271-4{,}065=4{,}748$.

On the modified basis, the benefit present value is

B^{(m)} =(0.271)(9{,}500)+(0.729)(7{,}600) =2{,}574.50+5{,}540.40 =8{,}114.90.

The expected number of payments remains 2.71, so $E^{(m)}=80(2.71)=216.80$ and $P^{(m)}=1{,}600(2.71)=4{,}336$. Hence $V^{(m)}=8{,}114.90+216.80-4{,}336=3{,}995.70$.

The benefit contribution is $8{,}114.90-8{,}542=-427.10$; the expense contribution is $216.80-271=-54.20$; and the premium contribution is $-(4{,}336-4{,}065)=-271$. These sum to $-752.30$, exactly the direct difference $3{,}995.70-4{,}748=-752.30$. The modified reserve is lower even though the modified premium is higher: the higher premium present value reduces the prospective reserve.

Common Exam Traps

Reversing the premium sign. If the modified premium present value is larger, its contribution is negative: $-\bigl(P^{(m)}-P^{(g)}\bigr)$. Adding the premium difference instead would reverse the result.
Mixing payment times. A premium or expense due at the valuation instant is not discounted. A year-end benefit is discounted and weighted by the probability of payment.
Changing the population at risk. A renewal cash flow may require survival or continuation to the renewal date. Apply the same in-force condition in both bases unless the problem explicitly changes it.
Calling a bookkeeping bridge a causal explanation. If the mortality or interest assumption changes, benefit, premium, and expense present values may all change. Isolate a specific cause with a comparison that holds other assumptions fixed.

Key takeaway: Compare reserves at the same duration and on clearly stated bases. Calculate each reserve directly, then reconcile the difference as the benefit change plus the expense change minus the premium change. The bridge is complete only when its components sum to the direct reserve difference.

Check Your Understanding

For each problem, use the stated timing and report money to the nearest cent where needed.

A one-year policy is valued at the start of the year. The annual effective interest rate is 4%, the death probability is 0.08, and the benefit is paid at year-end. Expenses and premiums are due immediately. Gross amounts are benefit $20,000, expense $75, and premium $400; modified amounts are benefit $19,000, expense $60, and premium $380. Find both reserves and the three contributions. (Answer: Gross reserve $1,213.46; modified reserve $1,141.54; benefit contribution $-$76.92$, expense contribution $-$15.00$, premium contribution $+$20.00$; difference $-$71.92$.)
The gross present values at a valuation date are $B^{(g)}=2{,}400$, $E^{(g)}=180$, and $P^{(g)}=1{,}900$. The modified values are $B^{(m)}=2{,}520$, $E^{(m)}=150$, and $P^{(m)}=2{,}050$. Find both reserves and attribute the difference. (Answer: Gross reserve $680; modified reserve $620; contributions $+$120$, $-$30$, and $-$150$, respectively; difference $-$60$.)
A gross reserve is $900. Relative to the gross basis, the modified basis increases the benefit present value by $40, increases the expense present value by $25, and decreases the premium present value by $70. Find the modified reserve and the total difference. (Answer: Modified reserve $1,035; difference $+$135$, consisting of $+$40$, $+$25$, and $+$70$.)
A hypothetical three-year term policy is valued at time 0 with $i=0\%$ and annual conditional death probability $q=0.20$. A $1,000 benefit is paid at year-end of death. Premiums of $100 and expenses of $10 are due at the start of each year while the policy is in force. Only the premium changes on the modified basis, to $120 per year. Find both reserves and the premium contribution. (Answer: Gross benefit present value $488; expense present value $24.40; premium present value $244; gross reserve $268.40. Modified premium present value $292.80; modified reserve $219.60; premium contribution $-$48.80$, equal to the reserve difference.)
Two bases use the same cash-flow timing, decrement assumptions, and discounting. Their benefit present values are equal. The modified expense present value is $18 lower, and its premium present value is $30 higher than the gross values. State the reserve difference and explain its sign. (Answer: The modified reserve is $48 lower: the expense contribution is $-$18$, the premium contribution is $-$30$, and the benefit contribution is zero.)

What You'll Learn