Reserve Differences From Premium Basis Changes

Why Isolate a Premium Basis Change?

A prospective reserve depends on the future benefits and expenses valued, the assumptions used to discount and determine decrement probabilities, and the premiums credited in the valuation. Comparing reserves from two bases is informative only when the source of the difference is clear. Here, the benefits, expenses, interest, and decrement assumptions are held fixed; only the valuation premium schedule changes.

The comparison concerns valuation premiums, not a change to the policyholder’s contractual premium. A higher valuation premium represents a larger future premium offset in the prospective reserve, so—other things equal—it produces a lower reserve. The size of that effect depends on when the premiums are due and whether the policy is in a state that permits them to be paid.

Definition: Let $V_k^{(A)}$ and $V_k^{(B)}$ be prospective reserves at the same valuation time and in the same state, calculated on two bases. If every assumption and cash flow is the same except the valuation premium schedule, the reserve difference caused by the premium basis change is $$ \Delta V_k = V_k^{(B)}-V_k^{(A)}. $$ A positive premium difference on basis B generally gives a negative $\Delta V_k$, because future premiums are subtracted in the prospective reserve.

The General Premium-Difference Formula

Use a common timeline for both valuations. At time $k$, suppose future premiums are due at times $j\geq k$, and are payable only if the policy is eligible for a premium at time $j$. Write $P_j^{(A)}$ and $P_j^{(B)}$ for the respective valuation premiums due then. Let $w_{k,j}$ be the probability, conditional on the policy’s state at time $k$, that a premium is payable at time $j$. In particular, $w_{k,k}=1$ when the policy is currently eligible and a premium is due immediately.

Assume the effective annual interest rate is $i$, with $v=1/(1+i)$. The same decrement assumptions determine $w_{k,j}$ on both bases. Let $C_k$ denote the common prospective present value of future benefits and expenses. Since these cash flows and their valuation assumptions are identical, they cancel in the comparison:

V_k^{(r)} = C_k-\sum_{j\geq k}v^{j-k}w_{k,j}P_j^{(r)}, \qquad r\in\{A,B\}.

Subtracting the reserve on basis A from the reserve on basis B gives the central result.

Theorem: If benefits, expenses, interest, decrements, premium timing, and premium eligibility rules are the same on both bases, then $$ \Delta V_k = -\sum_{j\geq k}v^{j-k}w_{k,j}\left(P_j^{(B)}-P_j^{(A)}\right). $$ Thus the reserve difference is the negative of the present value, on the common assumptions, of the difference in future valuation premiums.

The formula is linear in the premium differences: calculate each difference at its payment time, weight it for discounting and eligibility, add the weighted amounts, and reverse the sign to obtain the reserve difference. If the premium differences vary by duration, use each duration’s own difference. If a premium is not payable in a particular state, its weight must reflect that eligibility rule.

Why the Formula Holds

The result follows directly from the prospective reserve definition. On each basis, the reserve equals the expected present value of future benefits and expenses minus the expected present value of future valuation premiums. The first term is identical under the stated conditions. For the second term, the premium due at time $j$ contributes only when the policy is eligible to pay it, an event with probability $w_{k,j}$; discounting that expected payment from $j$ to $k$ multiplies it by $v^{j-k}$.

Write each prospective reserve.
Use the same common benefit-and-expense value $C_k$, and subtract the present value of premiums for that basis.

Subtract basis A from basis B.
The identical $C_k$ terms cancel, leaving only the difference in premium present values.

Apply the reserve sign.
Because premiums are subtracted in a prospective reserve, the premium present-value difference enters with a minus sign.

This argument also explains the scope of the result. If the interest or decrement assumptions differ, the premium weights may differ and the common benefit-and-expense values may not cancel. If benefits or expenses differ, their present-value differences must also be included. Those are not premium-only comparisons.

Worked Calculations

Worked Example: Level Premium Difference at a Term-Endowment Reserve

A hypothetical three-year endowment policy pays $10,000 at the end of the year of death or, if the insured survives, at the end of year 3. At the end of year 1, immediately before the premium due at that time, calculate the reserve difference between two valuation bases. The death probabilities for years 2 and 3, conditional on being alive at the start of each year, are $q_2=0.10$ and $q_3=0.20$. Death benefits are paid at the end of the year of death. Premiums are due at the start of each remaining year while alive. The annual effective interest rate is $i=0.05$. Basis A uses a level annual valuation premium of $1,000; basis B uses $1,200. Expenses are zero.

Method. Calculate the common present value of benefits at time 1, then subtract each basis’s present value of premiums. The reserve difference must also agree with the premium-difference formula.

The discount factors are $v=1/1.05=0.952380952$ and $v^2=0.907029478$. From time 1, the premium is payable immediately, and the next premium is payable at time 2 only if the insured survives year 2. Thus the premium annuity factor is $1+v(1-q_2)=1+0.952380952(0.90)=1.857142857$.

The common benefit value has three mutually exclusive outcomes: death in year 2, death in year 3 after surviving year 2, or survival to maturity. The combined probability of death in year 3 or survival to maturity, conditional on being alive at time 1, is $0.90(0.20)+0.90(0.80)=0.90$. Therefore,

C_1=10{,}000\left(0.10v+0.90v^2\right) =10{,}000(0.095238095+0.816326531) =\$9{,}115.65.

Basis A has a premium present value of $1{,}000(1.857142857)=\$1{,}857.14$, so $V_1^{(A)}=\$9{,}115.65-\$1{,}857.14=\$7{,}258.50$. Basis B has a premium present value of $1{,}200(1.857142857)=\$2{,}228.57$, so $V_1^{(B)}=\$9{,}115.65-\$2{,}228.57=\$6{,}887.07$.

The premium difference is $200 at each payable premium time. Its present value is $200(1+0.90v)=200(1.857142857)=\$371.43$. Hence $\Delta V_1=V_1^{(B)}-V_1^{(A)}=-\$371.43$, which agrees with $\$6{,}887.07-\$7{,}258.50=-\$371.43$. Amounts are rounded to the nearest cent only at the final step.

Common mistake avoided: Counting both future premiums as certain would give a factor $1+v$. The premium at time 2 is payable only if the insured survives year 2, so its correct weight is $0.90v$.

Worked Example: Duration-Varying Premium Differences

A hypothetical three-year endowment pays $5,000 at death during the term or at maturity at the end of year 3. At time 0, premiums are due immediately and at times 1 and 2 if the insured is alive. Conditional death probabilities are $q_1=0.10$, $q_2=0.20$, and $q_3=0.25$; death benefits are paid at the end of the year of death. The annual effective interest rate is $i=0.05$, and expenses are zero. Basis A uses premiums of $700 at each payment time. Basis B uses $820, $780, and $740 at times 0, 1, and 2, respectively. Calculate both reserves and their difference at time 0.

Method. Use the same benefit value for both bases. Weight premiums at times 1 and 2 by survival to those times, then compare the two prospective reserves.

Here $v=0.952380952$, $v^2=0.907029478$, and $v^3=0.863837599$. The probabilities of death in years 1 and 2 are $0.10$ and $0.90(0.20)=0.18$. At the end of year 3, death or survival leads to the same $5,000 payment; the total probability of reaching that payment outcome is $0.90(0.80)=0.72$. The common benefit present value is

C_0=5{,}000\left(0.10v+0.18v^2+0.72v^3\right) =5{,}000(0.095238095+0.163265306+0.621963071) =\$4{,}402.33.

The premium present-value weights at times 0, 1, and 2 are $1$, $0.90v=0.857142857$, and $0.90(0.80)v^2=0.653061224$. Basis A’s premium present value is $700(1+0.857142857+0.653061224)=\$1{,}757.14$, giving $V_0^{(A)}=\$2{,}645.19$.

Basis B’s premium present value is $820+780(0.857142857)+740(0.653061224)=\$1{,}971.84$, giving $V_0^{(B)}=\$2{,}430.50$. The premium differences are $120, $80, and $40, so their weighted present value is $120+80(0.857142857)+40(0.653061224)=\$214.69$. Therefore $\Delta V_0=-\$214.69$, also obtained by subtracting the rounded reserves. All dollar values are rounded to the nearest cent at the final step.

Common mistake avoided: Treating the premium difference as a level $120 at every duration would ignore the actual schedule. Use the three specified differences, each with its own payment-time weight.

Worked Example: A Negative Premium Difference at a Later Duration

At the end of year 2, immediately before the premium due then, a hypothetical five-year term policy has a $20,000 death benefit payable at the end of the year of death. Conditional death probabilities for the remaining years 3, 4, and 5 are $0.10$, $0.20$, and $0.25$. Annual premiums are due at times 2, 3, and 4 while alive; no premium is due at time 5. The annual effective interest rate is $i=0.04$, and expenses are zero. Basis A uses $900 at each due time. Basis B uses $1,100, $950, and $800 at times 2, 3, and 4. Calculate the reserve difference at time 2.

Method. Apply the general formula at the stated valuation time. The premium differences are $+200$, $+50$, and $-100$; the negative difference at time 4 must retain its negative sign.

The discount factors are $v=1/1.04=0.961538462$, $v^2=0.924556213$, and $v^3=0.888996359$. From time 2, the premium payment weights are $1$, $0.90v=0.865384615$, and $0.90(0.80)v^2=0.665680473$. Thus the present value of the premium difference is

200+50(0.865384615)-100(0.665680473) =\$176.70.

For a direct check, the common death-benefit value is $20{,}000(0.10v+0.18v^2+0.18v^3)=\$8{,}451.87$. Basis A’s premium value is $900(1+0.865384615+0.665680473)=\$2{,}277.96$, giving $V_2^{(A)}=\$6{,}173.91$. Basis B’s premium value is $1{,}100+950(0.865384615)+800(0.665680473)=\$2{,}454.66$, giving $V_2^{(B)}=\$5{,}997.21$. Therefore, $\Delta V_2=-\$176.70$, consistent with $V_2^{(B)}-V_2^{(A)}$. Dollar values are rounded to the nearest cent at the final step.

Common mistake avoided: Ignoring the negative $100 premium difference would overstate the present value of the premium increase. A lower premium on basis B at time 4 offsets part of the earlier increases.

Exam Checks and Limits of the Comparison

The formula is a targeted comparison, not a substitute for calculating a reserve when several assumptions change at once. Before using it, confirm that both reserves are measured at the same time and in the same state; that the benefits and expenses valued are identical; and that discounting and decrement assumptions are common. Then list premium differences by payment date and apply the appropriate eligibility probabilities.

A useful sign check is immediate: if basis B has larger premiums at every future payment time and both bases use the same eligibility rules, then each weighted difference is nonnegative, so $\Delta V_k\leq 0$. If all future valuation premiums are identical, the premium-only reserve difference is zero. If the schedule has increases and decreases, the net result depends on the weighted present value, not on an unweighted average of the differences.

A frequent exam error is to use the right premium differences but the wrong time origin. At duration $k$, a premium due immediately has discount factor 1, and a premium one year later has discount factor $v$, multiplied by the probability of being eligible at that later payment time. Another error is to reverse the reserve-difference definition: this tutorial uses basis B minus basis A. Keeping that order visible in the first line of the calculation prevents a sign reversal.

Key takeaway: When only valuation premiums change, subtract the present value of the premium-schedule difference from the reserve difference. Discount each difference from the correct valuation time and weight it by the probability that the premium is payable.

Check Your Understanding

For each problem, calculate the reserve difference for basis B minus basis A. Assume all unmentioned valuation items are common to the two bases.

At a valuation time immediately before a premium, basis B exceeds basis A by $150 immediately and by $100 one year later. The later premium is payable only if the policy survives the year, with probability 0.80. If $i=0.05$, what is the reserve difference? (Answer: $-\$226.19$)
At time 0, a policy has premiums due at times 0, 1, and 2 while in force. The B-minus-A premium differences are $50, $100, and $150. Survival probabilities to times 1 and 2 are 0.90 and 0.72, respectively, and $i=0.04$. What is the reserve difference? (Answer: $-\$236.39$)
At time 3, the B-minus-A premium differences due at times 3 and 4 are $-\$80$ and $+\$120$. The time-4 payment is conditional on survival with probability 0.75, and $i=0.05$. Calculate the reserve difference. (Answer: $-\$5.71$)
A premium difference of $+\$200$ is due at time 0 and $+\$200$ at time 1, but the second premium is payable only if the policy remains in force. At $i=0.04$, the probability of remaining eligible at time 1 is 0.90. Calculate the reserve difference and state its sign. (Answer: $-\$373.08$; negative)

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