Percentage-of-Premium Expense Loadings

Why a Percentage Loading Changes the Premium Equation

A premium can be reduced by expenses before the remaining amount is available to fund benefits. When the expense is a percentage of each premium, the expense amount is not a fixed dollar charge: it rises or falls with the premium itself. The gross premium must therefore be solved from an equation in which both the premium income and the expense depend on the unknown premium.

This distinction matters for policies with several premiums. If premiums are payable only while the insured is alive, the expected present value of expenses follows the same survival and discount pattern as the premiums. A percentage expense rate can then be incorporated directly into the premium-value factor. The method below is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: A percentage-of-premium expense loading at rate $e$ means that the expense associated with a premium of amount $P$ is $eP$. The amount remaining after that expense is $(1-e)P$. In this tutorial, assume the same rate $e$ applies to every premium, the expense is incurred when the premium is received, and $0\le e<1$.

Building the Gross Premium Equation

Consider a discrete policy with level annual gross premiums $P$, payable in advance at times $0,1,\ldots,n-1$ provided the insured is alive. Let $B$ be the expected present value of the policy benefits. Define $C$ as the expected present value of one unit of premium at each scheduled premium time, including the condition that the insured is alive:

C=\sum_{k=0}^{n-1} v^k\,{}_kp_x.

Here $v=1/(1+i)$, and ${}_kp_x$ is the probability that a life aged $x$ survives $k$ years. At time 0, ${}_0p_x=1$, so the first premium is included without discount or a survival reduction. If premiums continue only while the insured survives, each later premium is discounted and weighted by the probability it is payable.

The expected present value of all gross premiums is $PC$. Each premium produces an expense equal to $e$ times that premium, so the expected present value of expenses is $ePC$. Under the equivalence principle, gross premiums fund both benefits and expenses:

$$ PC=B+ePC. $$

Collect the terms involving $P$ on one side. This gives the central equation and its solution:

P(1-e)C=B \qquad\Longrightarrow\qquad P=\frac{B}{(1-e)C}.

The equation also explains why the expense rate cannot be applied as an addition to a net premium without solving for the gross amount. The expenses are a fraction of $P$, not a fraction of the amount available after expenses. If $e=1$, no premium is left for benefits; with positive benefit value, there is no finite solution. A rate greater than 1 would imply negative amounts available for benefits and is not a meaningful loading in this model.

A Direct Derivation from the Cash Flows

The formula follows by valuing each premium and its associated expense at the same time. For premium $k$, the expected present value of the premium is $Pv^k{}_kp_x$. The expected present value of its expense is $ePv^k{}_kp_x$. Subtracting the expense from the premium leaves $(1-e)Pv^k{}_kp_x$. Summing across all scheduled premium times gives:

\sum_{k=0}^{n-1}(1-e)Pv^k{}_kp_x =(1-e)P\sum_{k=0}^{n-1}v^k{}_kp_x =(1-e)PC.

That available premium value must equal $B$, the expected present value of benefits. This derivation relies on the expense being proportional to each premium and incurred at the same time. If expenses have a different timing or are based on something other than the premium amount, they cannot automatically be represented by $ePC$.

Worked Example: A Two-Year Term Policy

A hypothetical two-year term policy pays $2,000 at the end of the year of death. Annual premiums are payable in advance at times 0 and 1 while the insured is alive. Assume an effective annual interest rate $i=0.05$, a conditional probability of death $q=0.10$ in each year, and an expense rate $e=0.20$ on each premium. Find the level gross premium, to the nearest cent.

Method. Calculate the benefit present value $B$, then calculate the survival-weighted premium factor $C$, and solve $P=B/[(1-e)C]$. The constant annual mortality assumption gives death probabilities of 0.10 in year 1 and $0.90(0.10)=0.09$ in year 2.

The discount factor is $v=1/1.05=20/21$. Therefore:

B=2{,}000\left(0.10v+0.90(0.10)v^2\right) =2{,}000\left(\frac{2}{21}+\frac{36}{441}\right) =2{,}000\left(\frac{78}{441}\right) =\frac{52{,}000}{147} \approx 353.7415.

The premium at time 0 is certain, and the premium at time 1 is payable with probability 0.90. Thus:

C=1+0.90v =1+\frac{18}{21} =\frac{13}{7}.

The proportion of each premium remaining after expenses is $1-e=0.80$. Substitution gives:

P=\frac{52{,}000/147}{0.80(13/7)} =\frac{5{,}000}{21} \approx 238.0952.

Rounded to the nearest cent, the gross premium is $238.10. As a check, its available premium present value before rounding is $(5{,}000/21)(0.80)(13/7)=52{,}000/147$, exactly the benefit present value. This avoids the common mistake of discounting the time-0 premium or treating the time-1 premium as certain.

Using the Equation with More Than Two Premiums

The same equation applies if there are more scheduled premiums. The premium factor $C$ accounts for survival to each payment time; it is not generally the present value of a certain annuity. The benefit value $B$ must likewise reflect the benefit timing and the probability of each benefit payment.

For example, if a term benefit $b$ is paid at the end of the year of death during an $n$-year term, and the conditional probability of death in year $k+1$ is $q_{x+k}$, then:

B=b\sum_{k=0}^{n-1}v^{k+1}{}_kp_xq_{x+k}, \qquad C=\sum_{k=0}^{n-1}v^k{}_kp_x.

The benefit is discounted to the end of the year of death, while a premium is paid at the beginning of a year. That one-year difference in timing is important. Under the assumptions above, the gross premium is then $B/[(1-e)C]$.

Worked Example: Three Premiums with Constant Mortality

A hypothetical three-year term policy pays $5,000 at the end of the year of death. Premiums are payable at times 0, 1, and 2 while the insured is alive. Assume $i=0$, a conditional probability of death $q=0.20$ in each year, and a percentage expense rate $e=0.10$ on every premium. Find the level gross premium, to the nearest cent.

Method. With $i=0$, all discount factors equal 1. First determine the probabilities of death in the three years. They are $0.20$, $0.80(0.20)=0.16$, and $0.80^2(0.20)=0.128$. Their sum is 0.488.

The benefit present value is therefore:

$$ B=5{,}000(0.20+0.16+0.128) =5{,}000(0.488) =2{,}440. $$

The premiums are payable at times 0, 1, and 2 with respective probabilities $1$, $0.80$, and $0.80^2=0.64$. Hence:

$$ C=1+0.80+0.64=2.44. $$

Each premium contributes 90% of its amount after expenses. The gross premium is:

P=\frac{2{,}440}{0.90(2.44)} =\frac{2{,}440}{2.196} =\frac{10{,}000}{9} \approx 1{,}111.1111.

Rounded to the nearest cent, $P=$1,111.11$. The available premium value before rounding is $0.90(10{,}000/9)(2.44)=2{,}440$, which equals the expected present value of benefits. A common error is to count all three premiums as certain: only the time-0 premium is certain, and later premiums depend on survival.

Recovering an Expense Rate from a Gross Premium

The equation can also be rearranged when the gross premium is known and the percentage expense rate is the unknown. If $P$, $B$, and $C$ are given under the same timing assumptions, then:

e=1-\frac{B}{PC}.

This is useful for checking whether a stated premium is consistent with a proposed expense rate. The value $B/(PC)$ is the proportion of the total premium present value needed for benefits; the remaining proportion is available for expenses. The resulting rate is meaningful under this model only if it lies between 0 and 1.

Worked Example: Finding the Implied Expense Percentage

A hypothetical two-year term policy pays $900 at the end of the year of death. Annual premiums of $200 are payable in advance at times 0 and 1 while the insured is alive. Assume $i=0$ and $q=0.10$ in each year. Find the common percentage expense rate implied by this gross premium.

Method. Calculate the benefit value and the present value factor for the premiums, then use $e=1-B/(PC)$. The death probabilities are 0.10 in year 1 and $0.90(0.10)=0.09$ in year 2, so:

$$ B=900(0.10+0.09)=171. $$

The premium at time 0 is certain and the premium at time 1 is payable with probability 0.90. Since $i=0$:

C=1+0.90=1.90, \qquad PC=200(1.90)=380.

The amount available for expenses in present-value terms is $380-171=209$. Therefore:

e=\frac{209}{380} =0.55.

The implied expense rate is 55%. As a check, expenses have present value $0.55(380)=209$, and benefits plus expenses have value $171+209=380$, equal to the gross premium present value.

Common Mistakes and a Reliable Check

The most important algebra trap is to calculate the net premium and then add the expense percentage to it. Suppose the net premium is $500 and expenses are 20% of the gross premium. Adding 20% of $500 gives a proposed premium of $600. But 20% of $600 is $120, leaving only $480 for benefits. The correct equation is $P(1-0.20)=500$, so $P=500/0.80=$625$; the expense is then $125 and the remaining $500 funds benefits.

Other checks help catch mistakes before rounding:

Match the expense basis. For an expense defined as a proportion of each premium, use $ePC$. Do not treat it as a fixed dollar amount per policy.
Use the correct payment times. A premium in advance is discounted to its payment time; the time-0 premium has discount factor 1.
Condition premiums on survival. If a future premium is due only while the insured is alive, weight it by the probability of being alive at that time.
Check the balance. At the solved premium, verify that $(1-e)PC=B$, or equivalently $PC=B+ePC$.
Round only at the end. Keep unrounded values of $B$, $C$, and $P$ through the calculation, then report money to the nearest cent.

The percentage-loading formula is simple once its assumptions are clear: the same share is removed from every premium, at the time that premium is received. The survival and interest calculations determine $C$ and $B$; the loading changes the amount of each premium available for benefits by the factor $1-e$.

Check Your Understanding

For each problem, assume expenses are incurred at the time each premium is received and are a common percentage of every premium. Retain full precision until the final answer.

A hypothetical two-year term policy pays $1,200 at the end of the year of death. Premiums are payable at times 0 and 1 while the insured is alive. Assume $i=0$, $q=0.25$ each year, and $e=0.20$. Find the level gross premium to the nearest cent. (Answer: $375.00)
A hypothetical three-year term policy pays $2,000 at the end of the year of death. Premiums are payable at times 0, 1, and 2 while the insured is alive. Assume $i=0$, $q=0.10$ each year, and $e=0.10$. Find the level gross premium to the nearest cent. (Answer: $222.22)
A hypothetical two-year term policy pays $2,500 at the end of the year of death. Premiums are payable at times 0 and 1 while the insured is alive. Assume $i=0.05$, $q=0.20$ each year, and $e=0.12$. Find the level gross premium to the nearest cent. (Answer: $541.13)
A hypothetical two-year term policy pays $900 at the end of the year of death. Premiums of $300 are payable at times 0 and 1 while the insured is alive. Assume $i=0$ and $q=0.10$ each year. Find the implied common percentage expense rate. (Answer: 70%)
A hypothetical three-year term policy pays $4,000 at the end of the year of death. Conditional probabilities of death are 0.10 in year 1, 0.20 in year 2, and 0.25 in year 3. Premiums are payable at times 0, 1, and 2 while the insured is alive. Assume $i=0$ and $e=0.05$. Find the level gross premium to the nearest cent. (Answer: $739.25)

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