Why Measure Sensitivity?
A participating policy value depends on both contractual benefits and projected discretionary benefits. The valuation may also depend on interest, mortality, lapse, and other assumptions. A sensitivity calculation asks a focused question: how much does the value change when one assumption changes and the remaining assumptions are held fixed? This helps distinguish the effect of an individual assumption from the effect of changing several parts of a valuation at once.
The calculations below use hypothetical policies and simplified cash-flow models. In a full valuation, benefits and probabilities may vary by duration, and discretionary benefits may be projected using a declared or assumed scale. The same one-at-a-time comparison applies, provided that the assumption being varied is clearly identified and the other inputs and valuation rules are held fixed.
Setting Up a One-at-a-Time Comparison
Before calculating, specify the value being measured, the assumption that changes, and the assumptions that remain fixed. For example, changing the mortality assumption while holding the interest rate and bonus scale fixed is different from changing mortality and then recalculating bonuses in response to the new mortality projection. The latter is a combined scenario, not an isolated mortality sensitivity.
For a discrete annual model, a simplified value can be written as
where \(C_t\) is the benefit or other cash flow payable at time \(t\), and \(\mathbb{E}_{\theta}[C_t]\) is its expected amount under the relevant contingencies and assumption \(\theta\). The expression presumes that cash-flow timing and the valuation basis are specified. If benefits depend on the assumption being varied, their expected amounts must be recalculated; it is not enough to change the discount factors or probabilities alone.
Record the baseline value and all assumptions used to obtain it.
Replace only the selected assumption, such as the annual effective interest rate or a mortality probability.
Recalculate every cash flow or probability affected by that input, while keeping the remaining assumptions fixed.
Report the value difference, state its direction, and explain what the comparison does and does not measure.
Interest Sensitivity
When the expected cash flows are fixed and paid at specified future times, a higher effective annual interest rate lowers their present value. For nonnegative expected cash flows, the value is
This formula does not hold unchanged if the cash flows themselves are altered in response to interest—for example, if a bonus projection is recalculated using a new investment-return assumption. In that case, the new projected benefits must also be valued.
For nonnegative cash flows at positive future times, each term in this derivative is nonpositive, so \(dV/di\leq 0\). The derivative is a local rate of change in value per unit change in the interest rate. A finite change between two rates should instead be calculated by valuing at both rates and taking the difference.
Worked Example: Changing the Interest Rate
A hypothetical participating contract has a guaranteed benefit of $100 at the end of year 1 and a projected discretionary benefit of $1,100 at the end of year 2. Treat these amounts as fixed for this comparison; assume no mortality or lapse contingency. Find the value at an effective annual interest rate of 4% and at 5%, and calculate the finite change.
Method. Discount each fixed payment at the stated annual effective rate. The only changed assumption is \(i\); the projected benefits and payment times are held fixed.
At 4%,
At 5%,
The finite change is
Rounded to cents, the values are $1,113.17 at 4% and $1,092.97 at 5%; the change is \(-$20.20\). The lower value at the higher rate is consistent with discounting fixed positive payments more heavily.
Common mistake avoided. Rounded discount factors should not be used to produce intermediate amounts at a precision they cannot support. The calculations above use the stated rates and full precision before rounding the final dollar results.
Mortality Sensitivity
A mortality change affects expected benefits when payment depends on whether death or survival occurs. In a one-year model with death probability \(q\), survival probability \(1-q\), and end-of-year benefits \(B_D\) on death and \(B_S\) on survival, the present value at effective annual rate \(i\) is
This assumes death and survival are the only outcomes during the year, benefits are paid at year-end, and the stated benefits are unchanged when \(q\) changes. If benefit amounts or other assumptions are linked to mortality, those effects must be included separately.
Worked Example: Isolating a Mortality Change
A hypothetical one-year participating policy pays $500 at year-end if the insured dies and $240 at year-end if the insured survives. The survival benefit consists of a $200 guaranteed amount and a fixed $40 projected bonus. Use an effective annual interest rate of 5%. Calculate the value when \(q=0.02\), then when \(q=0.03\), keeping the benefits and interest rate fixed.
Method. Weight each end-of-year benefit by its probability, then discount the expected payment for one year.
At \(q=0.02\), the survival probability is \(1-0.02=0.98\). The expected end-of-year benefit is
Thus \(V(0.02)=245.2/1.05=233.5238095238\), or $233.52 to cents.
At \(q=0.03\), the survival probability is \(0.97\). The expected end-of-year benefit is
Thus \(V(0.03)=247.8/1.05=236.0000000000\), or $236.00 to cents. The mortality sensitivity for this change is \(236.00-233.5238095238=2.4761904762\), which rounds to an increase of $2.48.
As a check, increasing the death probability transfers probability from the $240 survival benefit to the $500 death benefit. The expected benefit rises by \(0.01(500-240)=2.60\); discounting that difference gives \(2.60/1.05=2.4761904762\), matching the valuation comparison.
Common mistake avoided. Do not increase the death probability without reducing the survival probability by the same amount. The two probabilities must still sum to one in this model.
Lapse and Bonus Sensitivity
Lapse sensitivity also depends on how the outcomes are defined. In the following one-year competing-outcome model, let \(q\) be the probability of death, \(l\) the probability of lapse, and \(1-q-l\) the probability of remaining in force to the end of the year. These are mutually exclusive unconditional probabilities, so \(q+l\leq 1\). If the end-of-year payments are \(B_D\) on death, \(B_L\) on lapse, and \(B_S\) on survival, then
This expression assumes all three payments occur at year-end and that the stated lapse probability is an unconditional competing probability. If a lapse rate is instead specified conditionally on being alive at a particular time, the probabilities must first be converted consistently.
Worked Example: Isolating a Lapse Change
A hypothetical policy pays $500 at year-end on death, $80 at year-end on lapse, and $240 at year-end if the policy remains in force. Use an effective annual interest rate of 5%, death probability \(q=0.02\), and baseline lapse probability \(l=0.08\). Revalue with \(l=0.10\), holding all other assumptions fixed.
Method. Treat death, lapse, and continued insurance as mutually exclusive outcomes. First find the remaining in-force probability as \(1-q-l\).
At \(l=0.08\), the in-force probability is \(1-0.02-0.08=0.90\). The expected payment is
Therefore \(V(0.02,0.08)=232.4/1.05=221.3333333333\), or $221.33 to cents.
At \(l=0.10\), the in-force probability is \(1-0.02-0.10=0.88\). The expected payment is
Therefore \(V(0.02,0.10)=229.2/1.05=218.2857142857\), or $218.29 to cents. The change is \(218.2857142857-221.3333333333=-3.0476190476\), an decrease of $3.05 to cents.
The result can be checked directly: the 0.02 increase in lapse probability replaces the same amount of in-force probability. The expected payment falls by \(0.02(240-80)=3.20\), so its present value falls by \(3.20/1.05=3.0476190476\).
Common mistake avoided. A lapse increase does not simply add lapse claims while leaving survival probability unchanged. In this competing-outcome model, the in-force probability must fall when lapse probability rises.
Worked Example: Sensitivity to a Projected Bonus
Use the same hypothetical one-year outcome probabilities and cash flows as the lapse example, except that the $240 survival benefit consists of a $200 guaranteed benefit plus a $40 projected bonus. Find the value if the projected bonus increases to $50. Hold the interest rate at 5%, \(q=0.02\), and \(l=0.08\).
Method. Vary only the bonus amount. The probability of receiving the survival benefit remains \(1-0.02-0.08=0.90\), and the death and lapse benefits remain $500 and $80.
With a $40 bonus, the baseline expected payment is \(0.02(500)+0.08(80)+0.90(240)=232.4\), so the baseline value is \(232.4/1.05=221.3333333333\), or $221.33.
With a $50 bonus, the survival benefit is \(200+50=250\). The expected payment becomes
The revised value is \(241.4/1.05=229.9047619048\), or $229.90 to cents. The change is \(229.9047619048-221.3333333333=8.5714285715\), an increase of $8.57 to cents.
As a check, the bonus rises by $10 only on survival, which has probability 0.90. The value increase is therefore \(0.90(10)/1.05=8.5714285714\), the same apart from rounding.
Common mistake avoided. The bonus is not payable on death or lapse in this scenario. Applying the $10 increase to every policy would overstate its sensitivity.
Interpreting Sensitivities Carefully
A one-at-a-time sensitivity is a controlled comparison, not a claim that assumptions are independent in practice. For example, an actual bonus projection may be revised when investment or mortality assumptions change. Holding the bonus fixed while testing interest gives an isolated interest effect under the stated setup; it does not capture that possible interaction. A full scenario comparison can be useful for combined effects, but it answers a different question.
A derivative is useful for a small local change in a smooth valuation model. A finite difference is usually clearer for a specified change, such as a one-percentage-point increase in a rate or a particular adjustment to a bonus scale. If a guaranteed minimum, option, or bonus rule causes a change in the form or amount of benefits, the value may not be differentiable at the point considered. In that case, calculate values on both sides of the change rather than relying on a local derivative.
The following checks help keep a sensitivity result interpretable:
- State whether the result is a change in dollars, a percentage change, or a derivative per unit change in an assumption.
- Use the same valuation date, payment timing, and benefit definitions in both valuations.
- Recalculate every probability or cash flow affected by the changed assumption, but do not silently change other inputs.
- Check that mutually exclusive probabilities remain valid and sum to one where the model requires it.
- Explain whether the direction of change is reasonable for the cash flows in the scenario.
Check Your Understanding
For each problem, vary only the assumption specified and keep the other stated inputs fixed. Monetary answers are rounded to cents.
- A hypothetical one-year policy pays a guaranteed $300 plus a $25 bonus at year-end if the policy survives. Survival probability is 0.94 and the effective annual interest rate is 4%. Find the value. (Answer: $293.75)
- A one-year policy pays $700 on death and $180 on survival, both at year-end. The death probability is 0.04 and the effective annual interest rate is 5%. Find the expected present value. (Answer: $191.24)
- A one-year policy pays $600 on death, $90 on lapse, and $250 on survival, at year-end. At an effective annual interest rate of 4%, \(q=0.03\), and \(l=0.07\). Find the value after lapse probability increases to 0.09, with death probability fixed. (Answer: $236.63; change: \(-$3.08\))
- A projected terminal bonus is paid only if the policy survives. Survival probability is 0.85, and the effective annual interest rate is 3%. If the bonus increases by $12, find the increase in value. (Answer: $9.90)
- In a one-year competing-outcome model, \(q=0.02\), \(l=0.08\), and survival probability is \(1-q-l\). Benefits at year-end are $400 on death, $100 on lapse, and $300 on survival. At 5% interest, find the value change when lapse probability rises by 0.01, with death probability and benefits fixed. (Answer: \(-$1.90\))