Gross Premium Recursions with Acquisition and Renewal Expenses

Why Expense Timing Changes a Reserve Recursion

A gross premium covers more than expected benefits: it must also support the expenses attached to issuing and maintaining the policy. Those expenses do not necessarily occur at the same times. An acquisition expense may be paid once at issue, while renewal expenses may be paid each time a later premium is collected. Because a reserve is a balance of future cash flows, moving an expense to the wrong duration changes both the reserve recursion and any gross premium calculated from it.

This tutorial uses a specific annual timing convention. State it before applying a recursion: reserves are measured immediately before the premium and expense due at that policy duration. The premium is received and the expense is paid at the start of the year; a death benefit is paid at the end of the year. A reserve at the end of the year for a survivor is measured immediately before the next scheduled premium and expense.

Definition: Let $V_k$ be the gross premium reserve immediately before the premium and expense due at duration $k$, conditional on the policy being in force. Let $P$ be the level annual gross premium, $e_k$ the expense paid at that time, $i$ the effective annual interest rate, $q_k$ the probability of death during the next year, and $p_k=1-q_k$ the probability of survival. If death during the year produces an end-of-year benefit $B_{k+1}$, the one-year gross reserve recursion is

(1+i)(V_k+P-e_k)=q_kB_{k+1}+p_kV_{k+1}.

This form assumes annual premium and expense payments at the start of the year, an end-of-year death benefit, and a reserve $V_{k+1}$ for survivors just before the next payment. It does not directly apply without adjustment if, for example, the benefit is paid immediately on death, premiums are paid monthly, or an expense is paid at a different time. In those cases, the cash flows must be placed at their actual times.

For a policy with an acquisition expense $a$ paid at issue and a fixed renewal expense $r$ paid at later annual premium dates, set $e_0=a$ and $e_k=r$ for $k\geq 1$, for as long as the policy is in force and the expense is due. The expenses enter separately because their timing differs, even if their dollar amounts happen to be equal.

Deriving the One-Year Balance

At duration $k$, the insurer holds the reserve $V_k$, receives the premium $P$, and pays the expense $e_k$. The funds available for the coming year are therefore $V_k+P-e_k$, and they accumulate to $(1+i)(V_k+P-e_k)$ by year-end. A death occurs with probability $q_k$, requiring the benefit $B_{k+1}$. With probability $p_k$, the policy survives and the insurer must hold $V_{k+1}$, just before the next payment. Equating the accumulated funds to the probability-weighted year-end requirement gives the stated recursion.

Solving for the current reserve gives a useful backward-calculation form:

V_k=v\left(q_kB_{k+1}+p_kV_{k+1}\right)-P+e_k, \qquad v=\frac{1}{1+i}.

The expense appears with a plus sign in this expression because a larger expense reduces the funds available from the premium and therefore increases the reserve required before that payment. The premium appears with a minus sign because receiving a larger premium reduces the reserve requirement. At the end of a finite term, if no benefit or further premium obligation remains for a surviving policy, the terminal reserve is $V_n=0$. Recursion then proceeds backward from duration $n-1$.

Fix the reserve date.
Here $V_k$ is immediately before the duration-$k$ premium and expense, not after them.

Assign expenses to durations.
Use the acquisition expense at issue and renewal expenses only at the later dates when they are payable.

Set the boundary reserve.
For a finite contract with no remaining obligations after maturity, start with $V_n=0$.

Apply the recursion backward.
Use the benefit, survival probabilities, premium, and expense that apply at each duration.

Worked Gross Premium Recursions

Worked Example: A Level Gross Premium with Issue and Renewal Expenses

A hypothetical two-year term policy pays \$100 at the end of the year of death. At the start of each year, a premium is due while the policyholder is alive. The annual death probability is $q=0.5$, the annual interest rate is $i=0$, the acquisition expense at issue is \$10, and the renewal expense at time 1 is \$4. Find the level gross premium $P$ by the equivalence principle, and check the resulting reserves. Premiums stop at the end of year 2.

Method. Equate the expected present value of premiums to that of benefits and expenses. With $i=0$, there is no discounting. The probability of surviving one year is $p=0.5$, so the expected present value of premiums is $P(1+0.5)$. The benefit is paid in year 1 with probability $0.5$, or in year 2 with probability $0.5(0.5)=0.25$. Thus its expected value is $100(0.5)+100(0.25)=75$.

The issue expense costs \$10 with certainty. The renewal expense costs \$4 only if the policyholder survives to time 1, so its expected value is $4(0.5)=2$. The premium equation is

1.5P=75+10+2=87, \qquad P=58.

Reserve check at duration 1. Here $V_2=0$, $e_1=4$, and $B_2=100$. Applying the recursion gives

$$ V_1=0.5(100+0)-58+4=-4. $$

Reserve check at issue. With $e_0=10$, the recursion gives

V_0=0.5\left(0.5(100)+0.5(-4)\right)-58+10 =0.5(48)-48=0.

The initial reserve is zero, as expected from equivalence pricing. The negative duration-1 reserve is not an arithmetic error: at that point, the remaining premium is large relative to the expected remaining benefit and renewal expense. The check confirms the premium and the timing of both expenses. A common mistake is to charge the \$10 acquisition expense again at duration 1; that would incorrectly raise $V_1$ by \$6.

Worked Example: Backward Reserves with a Specified Premium

A hypothetical three-year term policy pays \$200 at the end of the year of death. At every duration, $q_k=0.25$ and $p_k=0.75$; the effective annual interest rate is $4\%$. The annual premium is \$50. The acquisition expense is \$10 at issue, and the renewal expense is \$5 at durations 1 and 2. Find $V_2$, $V_1$, and $V_0$, given $V_3=0$.

Method. Work backward using $v=1/1.04$. At duration 2, the expected year-end requirement is $0.25(200)+0.75(0)=50$. Therefore,

V_2=\frac{50}{1.04}-50+5 =48.076923\ldots-45 =3.076923\ldots.

At duration 1, the year-end requirement is $0.25(200)+0.75V_2$. Substituting the unrounded value of $V_2$,

V_1=\frac{50+0.75(3.076923\ldots)}{1.04}-50+5 =\frac{52.307692\ldots}{1.04}-45 =5.295858\ldots.

At issue, use the acquisition expense of \$10, rather than the \$5 renewal expense:

V_0=\frac{50+0.75(5.295858\ldots)}{1.04}-50+10 =11.896\ldots.

Rounded to the nearest cent, $V_2=\$3.08$, $V_1=\$5.30$, and $V_0=\$11.90$. The calculation keeps full precision until the final rounding. These reserves are consistent with the specified \$50 premium; that premium was not stated to satisfy equivalence pricing, so $V_0$ need not be zero. A common mistake is to use the issue expense at every duration, which would overstate the reserves at durations 1 and 2 by \$5 each.

Worked Example: Solving a Premium and Diagnosing an Expense-Timing Error

A hypothetical two-year term policy pays \$1,000 at the end of the year of death. At each year, the death probability is $0.2$, and the survival probability is $0.8$. The effective annual interest rate is $5\%$. The insurer pays an \$80 acquisition expense at issue and a \$20 renewal expense at time 1 if the policyholder survives. Find the equivalence gross premium.

Method. Set $v=1/1.05=20/21$. Premiums are paid at issue and at time 1 if the policyholder survives, so their expected present value is $P(1+0.8v)$. The expected present value of benefits is

1000(0.2v)+1000(0.8)(0.2)v^2 =200v+160v^2 =\frac{148000}{441}.

The expected present value of expenses is the \$80 issue expense plus the renewal expense weighted for survival and discounted one year:

80+0.8v(20)=80+\frac{320}{21}.

The equivalence equation is therefore

P\left(1+\frac{16}{21}\right) =\frac{148000}{441}+80+\frac{320}{21} =\frac{190000}{441}.

Since $1+16/21=37/21=777/441$,

P=\frac{190000}{777}=244.530\ldots.

The gross premium is \$244.53. For a timing check, incorrectly swapping the expenses would give an expense value of $20+0.8v(80)=80.95238\ldots$, rather than the correct $95.23810\ldots$. That error would produce a premium of $183700/777=\$236.42$, which is too low. The issue expense is certain at time 0; only the renewal expense is conditional on surviving and discounted to time 1.

Interpreting and Checking the Recursion

The recursion is a one-year cash-flow balance, so its signs and timing offer practical checks. Increasing $e_k$, while holding the other inputs fixed, increases $V_k$ dollar for dollar in the solved form. Increasing $P$ decreases $V_k$ dollar for dollar. If the expense is due only when the policy is in force at a later date, its expected value in a premium-equivalence equation must reflect that probability of being in force and the appropriate discount factor.

For a level premium with one acquisition expense $a$ at issue and a fixed renewal expense $r$ at each premium date $k=1,\ldots,n-1$, the equivalence equation can be written using the survival probabilities and discount factors directly. Suppose premiums are due at times $0,\ldots,n-1$ while alive, and benefits $B_{k+1}$ are paid at the end of the year of death. Then

P\sum_{k=0}^{n-1}v^k\,{}_kp_x = a+r\sum_{k=1}^{n-1}v^k\,{}_kp_x +\sum_{k=0}^{n-1}v^{k+1}\,{}_kp_xq_{x+k}B_{k+1}.

Here ${}_0p_x=1$. The renewal-expense sum starts at $k=1$, while the premium sum includes time 0. The benefit term uses the probability of surviving to the start of year $k+1$ and then dying during that year. This expression assumes the stated annual timing and no other expenses or benefits; changing the payment timing requires changing the corresponding terms.

Key takeaway: First identify when each premium, expense, benefit, and reserve is measured. With reserves immediately before each scheduled payment, the recursion is $V_k=v(q_kB_{k+1}+p_kV_{k+1})-P+e_k$. Use the acquisition expense at issue and renewal expenses only at the later durations when they are payable.

A negative gross reserve can occur when future premiums, after allowing for expenses, exceed the expected value of remaining benefits and other obligations. It should prompt a timing and sign check, but it is not by itself proof of an error. The most reliable check is to substitute the calculated reserve back into the one-year balance and verify that accumulated funds equal the probability-weighted year-end requirements.

Check Your Understanding

Use the reserve convention and payment timing stated in each question. Round money to the nearest cent when needed.

A one-year policy has $q=0.10$, $p=0.90$, $i=5\%$, an end-of-year death benefit of \$500, a premium of \$70, and a renewal expense of \$6 at the start of the year. The terminal reserve is zero. Find the reserve immediately before the payment. (Answer: $-\$16.38$)
A one-year term policy has a \$600 end-of-year death benefit, $q=0.10$, $i=5\%$, and a \$12 acquisition expense at issue. There is no renewal expense and only one premium, due at issue. Find the equivalence gross premium. (Answer: \$69.14)
A two-year term policy has $q=0.20$, $p=0.80$ each year, $i=0$, and a \$100 end-of-year death benefit. A \$8 acquisition expense is paid at issue and a \$3 renewal expense at time 1 if the policyholder survives. Find the level equivalence gross premium. (Answer: \$25.78)
A two-year policy has $q=0.10$, $p=0.90$, and $i=4\%$ in each year. It pays \$1,000 at the end of the year of death. The premium is \$120 at each scheduled date while alive; the acquisition expense is \$15 at issue and the renewal expense is \$8 at time 1. Given $V_2=0$, find $V_1$ and $V_0$. (Answer: $V_1=-\$15.85$, $V_0=-\$22.56$)

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