Worked Comparison of Universal Life Benefit Options

Why Compare Benefit Designs?

Universal life policies can specify a level death benefit or a death benefit equal to the policy account value plus a specified amount. The same premium and charge schedule does not make these designs financially equivalent: the insurer’s net death-benefit exposure can vary by duration, and that changes both prospective reserves and expected profit.

A useful comparison isolates the death-benefit design. In the hypothetical model below, both policies have identical premiums and charges, and account values are fixed by the stated valuation projection. Common premiums are omitted from the comparative profit calculation because they cancel. The model includes no additional expenses, surrender payments, or risk margins. It assumes the account value is funded by the policy account, so the insurer’s net death outgo is the amount payable above that account value.

Definition: Let $B$ be a level death benefit, $S$ a specified amount added to the account value, and $A_j$ the account value at death at the end of policy year $j+1$. The insurer’s net death-benefit amount is $T_j=B-A_j$ under the level-benefit design and $T_j=S$ under the account-value-plus-specified-amount design, provided the level benefit exceeds the account value. A reserve is the present value of future net outgo less future charges, conditional on the policy being in force.

Projection Model and Profit Formula

Use a two-year, discrete-time projection. At the beginning of year 0, a policy is in force and the insurer receives charge $C_0$. During that year, death has conditional probability $q_0$, lapse has conditional probability $l_0$, and survival in force to the beginning of year 1 has probability $p_0=1-q_0-l_0$. At the beginning of year 1, the insurer receives charge $C_1$ if the policy is still in force. During the second year, death has conditional probability $q_1$. A lapse ends coverage with no additional surrender payment in this model. Death benefits are paid at year-end, and the annual effective interest rate is $i$, with $v=1/(1+i)$.

The charges are received at the beginnings of their respective years, while the death outgo is paid at the end of the year of death. The expected present value of profit per policy issued is therefore

\Pi=C_0-v\left[q_0T_0+p_0\left(-C_1+v q_1T_1\right)\right].

Here the first-year death outgo is weighted by $q_0$. The second-year net outgo is weighted by the probability $p_0$ of reaching that year. The term $-C_1$ appears inside the second-year bracket because a charge is income, reducing net outgo. This calculation is an expected profit present value, not a guarantee of profit on each policy.

Theorem: Under the stated timing and assumptions, the prospective reserve at the beginning of year 1 is $V_1=vq_1T_1-C_1$, and the reserve at issue is $V_0=v[q_0T_0+p_0V_1]$. Consequently, $\Pi=C_0-V_0$. These formulas do not apply unchanged if charges, benefits, or lapse payments occur at different times, or if lapses incur a surrender outgo.

The reserve relationship follows directly from conditioning on what happens in the next year. At time 1, death produces discounted net outgo $vT_1$ with probability $q_1$, and the charge $C_1$ is received at that time. Thus $V_1=vq_1T_1-C_1$. At issue, death during the first year produces discounted outgo $vT_0$ with probability $q_0$; otherwise the policy remains in force with probability $p_0$, and its time-1 reserve is discounted one year. A lapse contributes no later payment under the stated assumption, so it has zero continuation liability. This gives $V_0=vq_0T_0+vp_0V_1$, as asserted.

Find net death outgo.
For each benefit design and duration, subtract the account value from a level benefit, or use the specified amount for the account-value-plus-specified-amount design.

Check persistency.
Calculate $p_j=1-q_j-l_j$. Death, lapse, and survival probabilities must sum to 1 for each year.

Calculate reserves and profit.
Work backward from the final projection year, then use $\Pi=C_0-V_0$ to check the issue-date result.

Worked Comparisons

Worked Example: Two-Year Profit and Reserve Comparison

A hypothetical policy has an annual effective interest rate of $i=4\%$. Conditional death probabilities are $q_0=0.0100$ and $q_1=0.0200$; conditional lapse probabilities are $l_0=0.0800$ and $l_1=0.0800$. Charges are $C_0=\$2$ and $C_1=\$1$. Account values at death at the end of years 1 and 2 are $A_0=\$33$ and $A_1=\$36$. Compare a level $\$80$ death benefit with an account-value-plus-$\$45$ benefit. Calculate reserves and expected profits per policy issued.

Method. Calculate the net death amount in each year, then use the prospective reserve recursion. The discount factor is $v=1/1.04=25/26=0.9615384615\ldots$. First-year persistency is $p_0=1-0.0100-0.0800=0.9100$.

Under the level design, the net death amounts are $T_0=80-33=\$47$ and $T_1=80-36=\$44$. Under the account-value-plus-specified-amount design, they are $T_0=T_1=\$45$.

For the account-value-plus-$\$45$ design, the time-1 reserve is $V_1=v(0.0200)(45)-1=0.8653846154-1=-\$0.1346153846$. At issue, the reserve is $V_0=v[0.0100(45)+0.9100(-0.1346153846)]$. The unrounded inner calculation is $0.4500000000-0.1225000000=0.3275000000$, so $V_0=0.3149038462$. Expected profit is $C_0-V_0=2-0.3149038462=\$1.6850961538$.

For the level $\$80$ design, the time-1 reserve is $V_1=v(0.0200)(44)-1=0.8461538462-1=-\$0.1538461538$. At issue, $V_0=v[0.0100(47)+0.9100(-0.1538461538)]$. The unrounded inner calculation is $0.4700000000-0.1400000000=0.3300000000$, so $V_0=0.3173076923$. Expected profit is $2-0.3173076923=\$1.6826923077$.

Rounded to six decimal places, the account-value-plus-$\$45$ design has an issue reserve of $\$0.314904$ and expected profit of $\$1.685096$. The level design has an issue reserve of $\$0.317308$ and expected profit of $\$1.682692$. The profit difference, account-value-plus-specified-amount minus level, is $\$1.6850961538-\$1.6826923077=\$0.0024038461$, or $\$0.002404$ to six decimal places. The corresponding reserve difference is $-\$0.002404$, as required by the common charge $C_0$. This check confirms the reserve and profit calculations are consistent.

Worked Example: A Benefit Design Can Shift Costs Across Years

For another hypothetical policy, let $i=5\%$, $q_0=0.0200$, $l_0=0.1000$, $q_1=0.0300$, and $l_1=0.0700$. Charges are $C_0=\$3$ and $C_1=\$2$. The account values at the two death times are $\$50$ and $\$60$. Compare a level $\$100$ benefit with an account-value-plus-$\$45$ benefit.

Here $v=1/1.05=20/21=0.9523809524\ldots$, and $p_0=1-0.0200-0.1000=0.8800$. The level design has net death amounts $\$50$ and $\$40$; the alternative has net amounts $\$45$ and $\$45$.

For the level design, the year-1 reserve is $V_1=v(0.0300)(40)-2=1.1428571429-2=-\$0.8571428571$. Its issue reserve is $V_0=v[0.0200(50)+0.8800(-0.8571428571)]$. The inner value is $1.0000000000-0.7542857143=0.2457142857$, giving $V_0=\$0.2340136054$ and expected profit $3-0.2340136054=\$2.7659863946$.

For the alternative, $V_1=v(0.0300)(45)-2=1.2857142857-2=-\$0.7142857143$. Its issue reserve is $V_0=v[0.0200(45)+0.8800(-0.7142857143)]$. The inner value is $0.9000000000-0.6285714286=0.2714285714$, giving $V_0=\$0.2585034014$ and expected profit $3-0.2585034014=\$2.7414965986$.

To six decimal places, the account-value-plus-$\$45$ design has an issue reserve $\$0.024490$ higher and expected profit $\$0.024490$ lower than the level design. Its net death amount is lower in year 1 but higher in year 2. In this example, the effect of the later-year exposure more than offsets the first-year reduction. The final check is that the profit difference is the negative of the reserve difference because the initial charge is identical.

Worked Example: Find the Specified Amount for Equal Expected Profit

Use the assumptions in the first worked example, but leave the account-value-plus-specified amount $S$ unknown. Find the value of $S$ that gives the same expected profit as the level $\$80$ design.

Method. Both designs have the same charges, so their expected profits are equal exactly when their issue reserves are equal. The level-design net death amounts are $\$47$ and $\$44$. Equating the reserves and cancelling the common discount factor gives $0.0100S+0.9100v(0.0200)S=0.0100(47)+0.9100v(0.0200)(44)$.

Since $v=25/26$, the coefficient on $S$ is $0.0100+0.9100(25/26)(0.0200)=0.0100+0.0175=0.0275$. The right-hand side is $0.4700+0.0175(44)=0.4700+0.7700=1.2400$. Therefore $S=1.2400/0.0275=\$45.090909\ldots$, or $\$45.09$ to the nearest cent.

As a check, the first example used $S=\$45$, which is $\$0.090909\ldots$ below the break-even amount. Its account-value-plus-specified-amount design consequently had the slightly higher expected profit, by $\$0.0024038461$, as calculated there. The comparison depends on the timing and probabilities of the net death amounts, not just on their average across years.

Common Traps and Interpretation

Using the wrong persistency probability. If $q_0=0.0100$ and $l_0=0.0900$, then $p_0=1-0.0100-0.0900=0.9000$, not $0.9100$. Using $0.9100$ in the continuation term would incorrectly give some policies both a death or lapse exit and continued coverage. In the first worked example, $l_0=0.0800$, so $p_0=0.9100$ is consistent.
Using gross benefit instead of net death outgo. Under a level benefit, the account value is already funded by the account in this model. For example, the insurer’s net amount at risk for an $\$80$ benefit and a $\$33$ account value is $\$47$, not $\$80$.
Reversing reserve and profit signs. Reserves are future net outgo less future charges. A negative reserve means projected future charges exceed projected death outgo under this simplified basis; it is not a positive profit already earned. With identical $C_0$, a higher $V_0$ means lower $\Pi=C_0-V_0$.
Ignoring benefit timing. Equal specified amounts do not imply equal profits when the compared level design has different net amounts at different durations. Weight each amount by the death probability for that year and the probability of persisting to that year.
Changing assumptions between designs. A valid comparison holds interest, mortality, lapse, charges, and timing fixed. If any of these differ, the result no longer isolates the effect of the death-benefit formula.

The comparison is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus: it connects benefit definitions to conditional projection, prospective reserves, and expected profit. In a fuller valuation, account values may depend on premiums, credited interest, and deductions; surrender benefits and other expenses may also matter. Those items must be modeled explicitly rather than silently added to this simplified comparison.

Check Your Understanding

Use the stated timing convention: charges are received at the start of each year, death benefits are paid at year-end, and lapses have no additional payment unless the question says otherwise.

A one-year policy has $i=5\%$, $q=0.0400$, a net death amount of $\$30$, and a start-of-year charge of $\$1.20$. Calculate the reserve at the start of the year. (Answer: $-\$0.0571$)
For a level $\$90$ death benefit, the account value at the death time is $\$52$. What is the insurer’s net death-benefit amount under the stated model? (Answer: $\$38$)
A two-year projection has $i=0\%$, $q_0=0.0200$, $l_0=0.0800$, $q_1=0.0300$, and net death amounts $T_0=T_1=\$25$. Charges are $C_0=\$2$ and $C_1=\$1$. Calculate expected profit per policy issued. (Answer: $\$1.7250$)
Two benefit designs have net death amounts $(\$40,\$30)$ for the level design and $(\$32,\$32)$ for the account-value-plus-specified-amount design. Let $i=5\%$, $q_0=0.0100$, $p_0=0.9500$, and $q_1=0.0200$. With identical charges, calculate the difference in expected profit, alternative minus level. (Answer: $+\$0.0417$)

What You'll Learn