Why a Change of Valuation Basis Changes a Reserve
A universal life reserve depends not only on the contract but also on the assumptions used to value its future cash flows. Changing the mortality, expense, interest, or lapse assumptions can therefore change the reserve even when the policy terms and valuation date stay fixed. A sound comparison identifies the cash flows affected by each assumption rather than treating the reserve difference as an unexplained total.
This tutorial focuses on prospective reserve comparisons for policies conditional on being in force at the valuation date. It is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. The calculations use simplified annual models to make the source of each change visible; actual valuation models must reflect the policy’s contract terms, cash-flow timing, and applicable valuation requirements.
A reserve is not the policy account value. The account value is a contract account used to determine amounts under the policy, while a reserve is a valuation measure of future net obligations under specified assumptions. In a universal life calculation, account values and contractual charges can affect the projected benefits and cash flows, but they are not interchangeable with the reserve.
A One-Year Prospective Reserve Model
Consider a policy in force at the start of a year. Assume death and lapse are mutually exclusive exits during the year, and that a policy not exiting remains in force at year-end. Let \(q^D\) be the probability of death, \(q^L\) the probability of lapse, and \(p=1-q^D-q^L\) the probability of persistence. Let \(B\) be the death benefit paid at year-end on death, \(S\) the surrender benefit paid at year-end on lapse, and \(V'\) the reserve at year-end for a policy that persists. Let \(e\) be an expense paid at the start of the year, \(P\) a premium received at the start, and \(i\) the effective annual interest rate.
Here, the probabilities are conditional on the policy being in force at the start of the year. This discrete model does not apply unchanged if exits can occur at different times and benefits depend on the time of exit, if death and lapse are modeled with a different joint structure, or if cash flows occur at other times. Those situations require cash flows and probabilities consistent with their timing and model.
Why the Recursion Holds
The reserve is the expected present value of future benefits and expenses less future premiums, conditional on being in force at the valuation date. Over the next year, exactly one of three outcomes occurs in this model: death, lapse, or persistence. The corresponding year-end amounts are \(B\), \(S\), and \(V'\); their probabilities are \(q^D\), \(q^L\), and \(p\). Thus the expected year-end amount is \(q^D B+q^L S+pV'\). Discounting this expectation by one year and adding the net start-of-year cash flow \(e-P\) gives the formula.
The identity also makes the drivers of a basis change explicit. Mortality changes the weighting of death benefits and persistence reserves; lapse changes the weighting of surrender benefits and persistence reserves; expenses and premiums change start-of-year net cash flow; and interest changes the discount factor. These effects can interact when more than one assumption changes.
Use the same valuation date, policy terms, and starting policy status on both bases.
Identify death benefits, surrender benefits, expenses, premiums, and any reserve carried forward for policies that persist.
Apply each basis’s mortality, lapse, expense, and interest assumptions consistently to all affected future durations.
State which expected cash flows or discount factors changed, and check that the sign of the reserve difference is plausible for those changes.
Worked Comparisons
The examples use hypothetical policies and deliberately simplified cash flows. All monetary amounts are in dollars. Unless an example states otherwise, premiums are zero from the valuation date onward. This keeps attention on the effects of changing the specified assumptions rather than on contract features not included in the model.
Worked Example: Revaluing Mortality and Expenses Over Two Years
A hypothetical policy is valued at time 0 for a two-year period. The death benefit is $10,000, payable at the end of the year of death. There are no lapses, premiums, or interest; expenses are paid at the start of each year, and a policy surviving to the end of year 2 has no further liability. On the original basis, year 1 has expense $20 and death probability 0.01; year 2 has expense $40 and death probability 0.02. On the alternative basis, year 1 has expense $20 and death probability 0.02; year 2 has expense $50 and death probability 0.03. Calculate both reserves and the change.
Method. Work backward from year 2, then use the year 1 recursion. With zero interest and no lapse, the year \(t\) reserve is the start-of-year expense plus the probability-weighted death benefit and continuation reserve.
Original basis, year 2. There is no reserve after year 2. The expected death benefit is \(0.02(10{,}000)=200\). Therefore
Original basis, year 1. The expected death benefit is \(0.01(10{,}000)=100\). The probability of surviving year 1 is \(1-0.01=0.99\), so the expected continuation reserve is \(0.99(240)=237.60\). Thus
Alternative basis, year 2. The expected death benefit is \(0.03(10{,}000)=300\), giving
Alternative basis, year 1. The expected death benefit is \(0.02(10{,}000)=200\). The probability of surviving year 1 is \(1-0.02=0.98\), so the expected continuation reserve is \(0.98(350)=343\). Therefore
The alternative reserve is higher by \(563.00-357.60=\$205.40\). The increase reflects higher death probabilities in both years and a higher year-2 expense; the larger year-2 reserve also affects the year-1 reserve through the probability of survival. Common mistake: using the year-2 reserve of $240 as the original year-1 continuation amount but then making an arithmetic error in \(0.99(240)\). The correct product is $237.60, so the original year-1 reserve is $357.60, not $357.40.
Check. Each year-1 reserve includes its start-of-year expense, expected year-1 death benefit, and survival-weighted year-2 reserve. The calculation uses no discounting, as required by the stated zero interest rate.
Worked Example: Separating Interest and Expense Effects
At the start of a year, a hypothetical policy has death probability \(q^D=0.01\), lapse probability \(q^L=0.04\), death benefit $20,000, surrender benefit $600, and year-end reserve $900 if it persists. The start-of-year expense is $30. There is no premium. Calculate the reserve at an interest rate of 4%. Then recalculate using an interest rate of 2% and expense of $45, keeping all other assumptions fixed.
Method. First form the expected year-end amount. The persistence probability is \(1-0.01-0.04=0.95\). The death, lapse, and persistence contributions are \(0.01(20{,}000)=200\), \(0.04(600)=24\), and \(0.95(900)=855\). Their sum is \(200+24+855=1{,}079\).
On the original basis, the start-of-year expense is added without discounting, and the expected year-end amount is discounted at 4%:
On the alternative basis, the expected year-end amount is unchanged, but it is discounted at 2% and the start-of-year expense is $45:
The reserve increases by \(\$1{,}102.84-\$1{,}067.50=\$35.34\), rounded to cents. To separate the effects, change interest first while retaining the $30 expense: \(30+1{,}079/1.02=\$1{,}087.84\), an increase of $20.34. Then increase the start-of-year expense by $15, bringing the reserve to $1,102.84. Common mistake: discounting the start-of-year expense as if it were paid at year-end. Under the timing stated here, it is included in full at time 0.
Check. A lower interest rate increases the present value of the positive year-end amount, while a larger immediate expense increases the reserve dollar for dollar. The two changes together produce the calculated increase.
Worked Example: Changing the Lapse Assumption
Use the policy cash flows from the preceding example: death probability 0.01, death benefit $20,000, surrender benefit $600, continuation reserve $900, start-of-year expense $30, and interest rate 4%. The original lapse probability is 0.04. Recalculate the reserve if the lapse probability changes to 0.08 and every other assumption remains fixed.
Method. Recalculate both the lapse contribution and the persistence probability. On the original basis the expected year-end amount is \(200+24+855=1{,}079\), so the reserve is \(30+1{,}079/1.04=\$1{,}067.50\).
On the alternative basis, the persistence probability is \(1-0.01-0.08=0.91\). The expected death benefit remains \(0.01(20{,}000)=200\). The expected surrender payment is \(0.08(600)=48\), and the continuation contribution is \(0.91(900)=819\). Therefore the expected year-end amount is \(200+48+819=1{,}067\), and
The reserve decreases by \(\$1{,}055.96-\$1{,}067.50=-\$11.54\). In this example, the surrender payment of $600 is lower than the $900 continuation reserve it replaces, so raising the lapse probability reduces the expected year-end amount. Common mistake: increasing the lapse probability while leaving the persistence probability at 0.95. The probabilities must still sum to 1, so persistence falls to 0.91.
Check. The alternative basis shifts probability from persistence to lapse. The expected year-end amount falls by \(1{,}079-1{,}067=\$12\); discounting that difference by 1.04 gives a reserve decrease of \(12/1.04=\$11.54\), to cents.
Interpreting Assumption Changes
A basis comparison is not complete until the direction of each change has been interpreted in terms of the cash flows. The direction is often intuitive, but it is not universal: a changed probability can shift weight between two outcomes with different values, and a change in one duration’s reserve can feed into an earlier duration’s calculation.
| Changed assumption | Direct effect in the model | Interpretation and caution |
|---|---|---|
| Mortality | Changes the weights on death benefits and continuation reserves | Higher mortality often increases a reserve when death benefits are large, but the net effect depends on the amounts and timing being reweighted. |
| Expense | Changes expense cash flows | A higher immediate expense raises the reserve by its amount in this model. Future expenses must be discounted according to their timing. |
| Interest | Changes the discount factor applied to future net outgo | For positive future net outgo, a lower rate raises present value. The sign need not follow this simple rule if cash flows include material net inflows. |
| Lapse | Shifts weight between surrender benefits and continuation reserves | Higher lapse can raise or lower a reserve. Compare the surrender cash flow with the continuation liability it replaces. |
For a multi-year reserve, recalculate the future reserves consistently on each basis. If year-2 mortality or expenses change, the year-2 reserve changes; that altered reserve then affects the year-1 reserve through the probability of reaching year 2. Changing only the current-year assumptions while retaining a continuation reserve calculated on a different basis may be useful for isolating one effect, but it is not a full alternative-basis valuation.
A practical exam check is to write the timing next to each cash flow, then verify that the probability weights are conditional on the correct in-force status and sum to one across all modeled outcomes. Finally, check whether the result is plausible: if a lapse replaces a large continuation liability with a small surrender payment, for example, a higher lapse rate would generally reduce the expected liability in that period, all else equal.
Check Your Understanding
For each question, state the timing assumptions and apply the reserve formula consistently. Amounts are in dollars.
- A one-year model has no interest, expense $12 at the start of the year, death probability 0.02, death benefit $5,000, lapse probability 0.03, surrender benefit $200, and continuation reserve $400. There is no premium. What is the reserve? (Answer: $498.00)
- A one-year model has interest 5%, expense $25 at the start, death probability 0.01 with benefit $10,000, lapse probability 0.05 with surrender benefit $300, and continuation reserve $700. There is no premium. Calculate the reserve. (Answer: $761.19)
- Keep the data in question 2, but increase the lapse probability to 0.10. Recalculate the reserve, including the revised persistence probability, and state the change from question 2. (Answer: $742.14; decrease of $19.05)
- A two-year model has no lapse, premium, or interest, and a death benefit of $8,000. On the original basis, year 2 has expense $30 and death probability 0.03; year 1 has expense $15 and death probability 0.015. On the alternative basis, year 2 has expense $35 and death probability 0.04; year 1 has expense $20 and death probability 0.02. A survivor at the end of year 2 has no further liability. Find the year-1 reserve on each basis and the change. (Answer: $400.95 originally; $527.90 alternatively; increase of $126.95)
- A policy has a fixed positive continuation reserve and fixed death benefit. In a one-year competing-risk model, explain why the reserve effect of increasing lapse probability cannot be determined from the lapse probability alone. (Answer: Increasing lapse probability shifts weight from persistence to lapse, so the expected year-end amount changes according to the difference between the surrender benefit and the continuation reserve it replaces; the direction depends on their relative values.)