Sensitivity to Credited Rates and Charges

Why Sensitivity Matters

A universal life account projection depends on assumptions about credited interest and policy deductions. A small difference in a credited rate can compound over several years; a higher expense or mortality charge reduces the amount left to earn interest. Those effects also influence how soon an account may be unable to meet a scheduled deduction. Sensitivity analysis makes these relationships visible by changing one assumption at a time while holding the others fixed.

The calculations below use hypothetical policies and simplified annual timing. Actual policy provisions may specify monthly deductions, different charge bases, minimum credited rates, premium timing, or a grace period. A projection that omits those provisions is an illustration of sensitivity, not a determination of an actual policy’s coverage status. This treatment is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: Let $A_{t-1}$ be the account value at the start of policy year $t$, $P_t$ the premium added at the start of that year, $E_t$ the expense deduction, $M_t$ the mortality charge, and $j_t$ the annual credited rate. Under the convention that the premium is added first, both deductions are taken next, and interest is credited on the remaining amount at year-end, $$ A_t=(A_{t-1}+P_t-E_t-M_t)(1+j_t). $$ This recurrence assumes the amount before interest is nonnegative and that there are no other account transactions during the year. If deductions occur at different times, or charges are defined differently by the contract, the recurrence must be adjusted.

How a Change in Credited Interest Affects the Account

The credited rate applies to the amount remaining after the model’s premium and deductions. Therefore, its effect is not generally the rate change multiplied by the opening account alone. The relevant base is the post-transaction amount that earns interest. When rates differ over several years, the resulting account values diverge further because each year’s credited interest is applied to a different accumulated balance.

Worked Example: Comparing Three Credited Rates

A hypothetical policy starts with $A_0=\$5{,}000$. At the start of each of the next three years, the owner pays a premium of $\$700$, and the insurer deducts $\$100$ for expenses. Assume no mortality charge, no other transactions, and a constant annual credited rate within each scenario. Compare year-three account values at 2%, 4%, and 6%. All premiums and deductions occur at the start of each year, before interest is credited.

Method. The net addition before interest is $700-100=\$600$ each year, so $A_t=(A_{t-1}+600)(1+j)$. For the 2% scenario:

\begin{aligned} A_1&=(5{,}000+600)(1.02)=5{,}712,\\ A_2&=(5{,}712+600)(1.02)=6{,}312(1.02)=6{,}438.24,\\ A_3&=(6{,}438.24+600)(1.02)=7{,}038.24(1.02)=7{,}179.0048. \end{aligned}

For the 4% scenario:

\begin{aligned} A_1&=(5{,}000+600)(1.04)=5{,}824,\\ A_2&=(5{,}824+600)(1.04)=6{,}424(1.04)=6{,}680.96,\\ A_3&=(6{,}680.96+600)(1.04)=7{,}280.96(1.04)=7{,}572.1984. \end{aligned}

For the 6% scenario:

\begin{aligned} A_1&=(5{,}000+600)(1.06)=5{,}936,\\ A_2&=(5{,}936+600)(1.06)=6{,}536(1.06)=6{,}928.16,\\ A_3&=(6{,}928.16+600)(1.06)=7{,}528.16(1.06)=7{,}979.8496. \end{aligned}

Rounded to the nearest cent only at the end, the year-three values are $\$7{,}179.00$, $\$7{,}572.20$, and $\$7{,}979.85$, respectively. The higher rate produces a higher projected value in this example because the cash flows and timing are otherwise identical. Common mistake avoided: applying interest to the opening account alone, or failing to include the premium and deduction before calculating interest, uses the wrong interest-bearing balance.

How Deductions Affect the Projection

A deduction reduces the amount that remains in the account and, under the stated timing convention, also reduces the amount on which interest is credited. The direct effect of an additional deduction of $c$ in year $t$, with the other year-$t$ inputs fixed, is a reduction of $c(1+j_t)$ in $A_t$. In later years the difference can persist or change: for example, a mortality charge defined using the net amount at risk may itself change when the account value changes.

For a simplified mortality-charge calculation, suppose the death benefit is $B$, the opening account value is $A_{t-1}$, and the annual charge rate is $q_t$ dollars per $\$1{,}000$ of net amount at risk. If the model defines net amount at risk as $B-A_{t-1}$ and this is positive, then:

M_t=q_t\frac{B-A_{t-1}}{1{,}000}.

This is a specified simplified charge basis, not a universal rule for every policy. It does not apply unchanged if a contract defines the charge using a different amount, uses a different death-benefit option, or specifies different timing.

Worked Example: Separating Mortality and Expense Sensitivity

A hypothetical policy has opening account value $\$10{,}000$, annual premium $\$1{,}200$, annual expense deduction $\$120$, and death benefit $\$50{,}000$. The mortality charge is assessed at the start of the year at $q=\$2.50$ per $\$1{,}000$ of opening net amount at risk. The annual credited rate is 4%. Repeat the projection with $q=\$3.75$, leaving all other assumptions unchanged. For each scenario, project two years. Assume the same premium, expense, and charge basis in both years.

Method and original rate. In year one, the net amount at risk is $50{,}000-10{,}000=\$40{,}000$, so the mortality charge is $2.50(40{,}000/1{,}000)=\$100$. Thus:

A_1=(10{,}000+1{,}200-120-100)(1.04) =10{,}980(1.04)=11{,}419.20.

At the start of year two, the net amount at risk is $50{,}000-11{,}419.20=\$38{,}580.80$, giving a charge of $2.50(38{,}580.80/1{,}000)=\$96.452$. The unrounded calculation is:

A_2=(11{,}419.20+1{,}200-120-96.452)(1.04) =12{,}402.748(1.04)=12{,}898.85792.

Higher mortality rate. In year one, the charge is $3.75(40{,}000/1{,}000)=\$150$, so:

A_1=(10{,}000+1{,}200-120-150)(1.04) =10{,}930(1.04)=11{,}367.20.

At the start of year two, the net amount at risk is $50{,}000-11{,}367.20=\$38{,}632.80$. The charge is $3.75(38{,}632.80/1{,}000)=\$144.873$. Therefore:

A_2=(11{,}367.20+1{,}200-120-144.873)(1.04) =12{,}302.327(1.04)=12{,}794.42008.

Rounded to the nearest cent at the end, the original-rate year-two account is $\$12{,}898.86$, versus $\$12{,}794.42$ at the higher mortality rate, a difference of $\$104.44$. The higher charge lowers the account in year one and also changes the next year’s net amount at risk. Common mistake avoided: using the initial $\$40{,}000$ net amount at risk for year two ignores the changed opening account.

Why the Sensitivity Relationships Hold

The one-year effects follow directly from the recurrence. Let $C_t=E_t+M_t$, and hold the start-of-year account, premium, and deductions fixed. If the credited rate changes from $j$ to $j+\Delta j$, then:

\begin{aligned} A_t(j+\Delta j)-A_t(j) &=(A_{t-1}+P_t-C_t)(1+j+\Delta j)\\ &\quad -(A_{t-1}+P_t-C_t)(1+j)\\ &=(A_{t-1}+P_t-C_t)\Delta j. \end{aligned}

Thus, when the pre-interest balance is positive, increasing the credited rate increases that year’s ending account value. If instead the credited rate is fixed and an additional deduction $c$ is taken before interest, the ending account changes by:

(A_{t-1}+P_t-C_t-c)(1+j) -(A_{t-1}+P_t-C_t)(1+j)=-c(1+j).

These comparisons hold with the other inputs fixed. They do not alone establish the exact multi-year impact when charges depend on account value, when rates or charges change in response to policy conditions, or when an account becomes insufficient to pay a deduction. In those cases, reproject the sequence year by year using the appropriate charge rules.

Account Values and the Timing of Coverage Risk

A projected account can be described as being at risk of failing to cover a scheduled deduction when its value immediately before that deduction is less than the amount due. This is a projection threshold, not necessarily the date coverage terminates. Actual contract provisions may provide a grace period or other conditions, so an annual simplified model cannot determine those outcomes unless they are explicitly included.

Worked Example: Credited Rate and the Year Charges Become Unaffordable

Consider a hypothetical policy with opening account value $\$2{,}500$, no future premiums, and a fixed annual deduction of $\$660$. The deduction is taken at the start of each year, followed by interest crediting. Compare annual credited rates of 3% and 7%. In this simplified illustration, the account is at risk at the start of a year if it is less than $\$660$ before that year’s deduction.

At 3%:

\begin{aligned} A_1&=(2{,}500-660)(1.03)=1{,}895.20,\\ A_2&=(1{,}895.20-660)(1.03)=1{,}272.256,\\ A_3&=(1{,}272.256-660)(1.03)=630.62368. \end{aligned}

At the start of year four, $\$630.62368<\$660$, so this projection cannot fund the scheduled deduction in year four.

At 7%:

\begin{aligned} A_1&=(2{,}500-660)(1.07)=1{,}968.80,\\ A_2&=(1{,}968.80-660)(1.07)=1{,}400.416,\\ A_3&=(1{,}400.416-660)(1.07)=792.24512,\\ A_4&=(792.24512-660)(1.07)=141.5022784. \end{aligned}

The year-four deduction is affordable in the 7% scenario, but the resulting $\$141.5022784$ is below the $\$660$ due at the start of year five. Rounded to the nearest cent only at the end, the projected values after year three are $\$630.62$ at 3% and $\$792.25$ at 7%; the first unaffordable deduction occurs in year four and year five, respectively. This comparison shows how a higher credited rate can defer, rather than eliminate, the point at which charges exceed the account.

Common mistake avoided: checking affordability after interest is credited for the year would give the account interest it cannot earn under the stated timing convention. The test must use the account immediately before the deduction.

A Practical Sensitivity Workflow

Use a controlled comparison so that each result can be traced to one changed assumption. The following sequence is useful for a projected account table:

Fix the timing and charge basis.
State when premiums, expenses, mortality charges, and interest are applied. Specify how each charge is calculated.

Establish a baseline projection.
Record each opening account, transaction, credited rate, and ending account using the stated recurrence.

Change one assumption.
For example, change only the credited rate or only the mortality-charge rate. This isolates the direct comparison.

Recalculate all dependent quantities.
If the charge depends on the opening account or net amount at risk, recompute it in every year after the account changes.

Check affordability at the correct time.
Compare the account immediately before each scheduled deduction with the amount due; do not infer actual termination without the relevant contract provisions.

A useful consistency check is to verify that each year’s ending account equals the post-premium, post-deduction balance multiplied by the factor for the stated credited rate. Keep full precision during a multi-year projection and round only the reported results. This avoids accumulating rounding differences and helps distinguish a genuine sensitivity effect from an arithmetic or timing error.

Key takeaway: Under a fixed timing convention, higher credited interest raises the projected account when the pre-interest balance is positive, while higher deductions lower it. The timing of deductions and any account-dependent charge rules determine how those changes carry forward and when a scheduled charge may become unaffordable.

Check Your Understanding

Use the annual timing convention stated in each problem, show intermediate calculations, and round money to the nearest cent only at the end.

A hypothetical policy starts with $\$8{,}000$. At the start of the year it receives a $\$500$ premium and has deductions of $\$80$ for expenses and $\$120$ for mortality. The annual credited rate is 5%. Find the year-end account value. (Answer: $\$8{,}715.00$)
A policy has opening account value $\$6{,}000$, a start-of-year premium of $\$400$, and total start-of-year deductions of $\$150$. Compare the one-year account values at credited rates of 3% and 5%, and find their difference. (Answer: 3%: $\$6{,}437.50$; 5%: $\$6{,}562.50$; difference: $\$125.00$)
With all other inputs fixed, an expense deduction rises from $\$90$ to $\$130$ immediately before interest is credited at 4%. By how much does the year-end account decrease? (Answer: $\$41.60$)
A hypothetical account starts at $\$1{,}800$, has no further premiums, and incurs a fixed $\$500$ deduction at the start of each year before interest at 2%. Find the account after each affordable deduction and identify the first year in which the deduction is unaffordable. (Answer: after years one, two, and three the values are $\$1{,}326.00$, $\$842.52$, and $\$349.37$; the year-four deduction is unaffordable.)
A policy has opening account $\$12{,}000$, death benefit $\$40{,}000$, premium $\$1{,}000$, expense deduction $\$100$, and mortality rate $\$3$ per $\$1{,}000$ of opening net amount at risk. With a 4% annual credited rate and charges applied before interest, calculate the mortality charge and year-end account. (Answer: mortality charge $\$84.00$; year-end account $\$13{,}328.64$)

What You'll Learn