Deriving Full Preliminary Term Reserves

Purpose and valuation setup

Full preliminary term (FPT) is a modified reserve method that changes the valuation premium pattern. The first-year premium is calculated for one-year term insurance; premiums from the second policy year onward are calculated as a whole-life net premium at the attained age at the end of the first year. The resulting reserve is zero at the end of year one, immediately before the first renewal premium, and thereafter follows the whole-life reserve pattern for the later premium basis.

The timing convention matters. In this tutorial, premiums are annual and payable at the start of each policy year, provided the insured is alive. A death benefit is paid at the end of the policy year of death. Reserves at integer durations are measured immediately before the premium due at that duration. Interest is at an effective annual rate $i$, and $v=1/(1+i)$. There are no expenses, lapses, or other cash flows. Benefits and premiums are valued on the same mortality and interest basis.

Definition: Under the full preliminary term method for a whole-life benefit $B$ issued at age $x$, the first premium is the net single-year term premium for the first policy year. Each premium thereafter is the whole-life net premium for a life aged $x+1$, applied to the same benefit $B$. At duration $k$, the reserve is measured just before the premium due at time $k$.

For precision, the discrete whole-life insurance and life annuity-due values at age $y$ are

A_y=\sum_{j=0}^{\infty}v^{j+1}\,{}_jp_y q_{y+j}, \qquad \ddot{a}_y=\sum_{j=0}^{\infty}v^j\,{}_jp_y.

Here ${}_jp_y$ is the probability of surviving $j$ years from age $y$, and $q_{y+j}$ is the probability of death during the following year, conditional on reaching age $y+j$. The first-year term insurance value per dollar of benefit is $v q_x$: death during that year is paid at its end. The term premium is due at time 0, so its present value per dollar is the premium itself.

Deriving the FPT premium and reserve pattern

Write $P_0^{\mathrm{FPT}}$ for the first-year premium per dollar of benefit, and $P^*_{x+1}$ for each later annual premium per dollar. Equivalence for the one-year term contract gives the first premium. Equivalence for whole life at age $x+1$ gives the renewal premium:

P_0^{\mathrm{FPT}}=v q_x, \qquad P^*_{x+1}=\frac{A_{x+1}}{\ddot{a}_{x+1}}.

For benefit amount $B$, multiply both premiums by $B$. In particular, the first premium is not the whole-life premium at age $x$: it pays for first-year term coverage only.

At duration $k\geq 1$, the future benefit value is $B A_{x+k}$. Because the reserve is measured before the premium due at time $k$, the future premium stream includes payments at times $k,k+1,\ldots$, with value $B P^*_{x+1}\ddot{a}_{x+k}$. Thus

{}_kV_x^{\mathrm{FPT}} = B A_{x+k}-B P^*_{x+1}\ddot{a}_{x+k}, \qquad k\geq 1.

At $k=1$, the premium was set by whole-life equivalence at age $x+1$, so

{}_1V_x^{\mathrm{FPT}} = B\left(A_{x+1}-\frac{A_{x+1}}{\ddot{a}_{x+1}}\ddot{a}_{x+1}\right)=0.

This is the central feature of the method: the end-of-first-year reserve, before the renewal premium, is zero. For later durations the expression is exactly the prospective net premium reserve for whole life valued from age $x+1$, at duration $k-1$. The first-year term valuation and later whole-life valuation therefore join at the first anniversary without inserting or omitting a premium.

Value first-year coverage.
Use the end-of-year death benefit and the one premium at issue to obtain $P_0^{\mathrm{FPT}}=v q_x$ per dollar of benefit.

Set the renewal premium.
At time 1, value whole-life benefits and premiums for a life aged $x+1$, giving $P^*_{x+1}=A_{x+1}/\ddot{a}_{x+1}$.

Calculate each reserve on its stated date.
Just before a premium at duration $k$, include that premium in the future annuity-due value. Multiply the per-dollar reserve by $B$.

Worked applications

Worked Example: Deriving the Premiums and Two Later Reserves

A hypothetical whole-life policy has unit death benefit. The effective annual interest rate is $i=10\%$, so $v=10/11$. The conditional death probabilities are $q_x=0.05$, $q_{x+1}=0.10$, $q_{x+2}=0.20$, and $q_{x+3}=1$. Find the first-year premium, the renewal premium, and the reserves at durations 1, 2, and 3. Premiums are due at the start of each year; benefits are paid at the end of the year of death. Reserves are just before premiums.

Method. Value first-year term insurance, then value whole life from age $x+1$. For the later reserves, use the prospective formula at the attained age. This avoids using the issue-age whole-life premium in the first year.

The first-year premium is

P_0^{\mathrm{FPT}}=v q_x=\frac{10}{11}(0.05)=0.045455

The whole-life values at age $x+1$ are

A_{x+1} =\frac{10}{11}(0.10) +\left(\frac{10}{11}\right)^2(0.90)(0.20) +\left(\frac{10}{11}\right)^3(0.90)(0.80)(1) =\frac{1039}{1331},

\ddot{a}_{x+1} =1+\frac{10}{11}(0.90) +\left(\frac{10}{11}\right)^2(0.90)(0.80) =\frac{292}{121}.

The three insurance terms correspond to death in the first, second, and third future years. The annuity terms correspond to premiums at time 0, year 1 if alive, and year 2 if alive. Therefore,

P^*_{x+1} =\frac{1039/1331}{292/121} =\frac{1039}{3212} =0.323474.

At duration 1, the reserve is zero by equivalence at age $x+1$:

{}_1V_x^{\mathrm{FPT}}=A_{x+1}-P^*_{x+1}\ddot{a}_{x+1}=0.

At duration 2, the attained age is $x+2$. Death in the next year has probability $0.20$; otherwise death in the following year is certain. Thus

A_{x+2} =\frac{10}{11}(0.20) +\left(\frac{10}{11}\right)^2(0.80) =\frac{102}{121}, \qquad \ddot{a}_{x+2}=1+\frac{10}{11}(0.80)=\frac{19}{11}.

The duration-2 reserve, just before its premium, is

{}_2V_x^{\mathrm{FPT}} =\frac{102}{121}-\frac{1039}{3212}\frac{19}{11} =\frac{10043}{35332} =0.284247.

At duration 3, death is certain in the next year, so $A_{x+3}=v$ and $\ddot{a}_{x+3}=1$. Hence

{}_3V_x^{\mathrm{FPT}} =v-P^*_{x+1} =\frac{10}{11}-\frac{1039}{3212} =0.585616.

This last calculation uses $A_{x+3}=v$, not 1, because the certain death benefit is paid at the end of the year. As a timing check at duration 1, the reserve recursion is

(1.1)\left({}_1V_x^{\mathrm{FPT}}+P^*_{x+1}\right) =0.1+0.9\,{}_2V_x^{\mathrm{FPT}}.

Both sides equal approximately $0.355822$, confirming the duration-1 to duration-2 cash-flow timing. Reserve values are rounded to six decimal places.

Worked Example: Scaling the FPT Pattern to a Dollar Benefit

Use the mortality and interest assumptions from the preceding calculation, but suppose the hypothetical policy benefit is \$50,000. Find the first-year premium, the later annual premium, and the reserves at durations 1 and 2. Give money to the nearest cent.

Method. The per-dollar premiums and reserves scale linearly with the benefit. The first-year premium is $0.045454545\ldots$ per dollar, the renewal premium is $1039/3212$ per dollar, and the duration-2 reserve is $10043/35332$ per dollar.

Multiplying by \$50,000 gives

P_0^{\mathrm{FPT}}=50{,}000\left(\frac{1}{22}\right) =\$2{,}272.73,

P^*_{x+1}=50{,}000\left(\frac{1039}{3212}\right) =\$16{,}173.72.

At duration 1, before the first renewal premium, the reserve is \$0. At duration 2, just before the next premium, it is

{}_2V_x^{\mathrm{FPT}} =50{,}000\left(\frac{10043}{35332}\right) =\$14{,}212.33.

The later premium is much larger than the first-year premium because it funds whole-life coverage rather than only one year of term coverage. The reserve remains zero at duration 1 because the reserve is measured before the renewal premium and that premium is exactly the net whole-life premium at age $x+1$. All dollar amounts are rounded to the nearest cent.

Worked Example: Distinguishing Before-Premium and After-Premium Reserves

A hypothetical unit-benefit policy has $i=0$. Its mortality assumptions are $q_x=0.12$, $q_{x+1}=0.25$, $q_{x+2}=0.50$, and $q_{x+3}=1$. Find the first premium and renewal premium, the reserve at duration 1 immediately before the renewal premium, the reserve immediately after it is received, and the reserve at duration 2 immediately before the next premium.

Method. Keep the two duration-1 valuation times separate. The before-premium prospective reserve includes the premium due at time 1; the after-premium reserve excludes the premium just received.

With $v=1$, the whole-life values at age $x+1$ are

A_{x+1}=0.25+(0.75)(0.50)+(0.75)(0.50)(1)=1, \qquad \ddot{a}_{x+1}=1+0.75+(0.75)(0.50)=2.125=\frac{17}{8}.

Thus the first premium is $P_0^{\mathrm{FPT}}=q_x=0.12$, and the renewal premium is

P^*_{x+1}=\frac{A_{x+1}}{\ddot{a}_{x+1}}=\frac{8}{17}.

Immediately before the renewal premium, the reserve is

{}_1V_x^{\mathrm{FPT}}=A_{x+1}-P^*_{x+1}\ddot{a}_{x+1}=0.

Immediately after receiving that premium, the premium is no longer a future inflow. The remaining premium stream begins one year later, so its value uses $\ddot{a}_{x+1}-1=9/8$:

{}_1V_x^{\mathrm{FPT,after}} =A_{x+1}-P^*_{x+1}(\ddot{a}_{x+1}-1) =1-\frac{8}{17}\frac{9}{8} =\frac{8}{17}.

At duration 2, $A_{x+2}=0.50+(0.50)(1)=1$ and $\ddot{a}_{x+2}=1+0.50=1.5$. Therefore, just before the duration-2 premium,

{}_2V_x^{\mathrm{FPT}} =1-\frac{8}{17}(1.5) =\frac{5}{17}.

A useful check is the one-year balance from immediately after the renewal premium to duration 2: $8/17=0.25+(0.75)(5/17)$. Both sides agree. The after-premium reserve at duration 1 is not the zero before-premium reserve; it is a different valuation instant.

Common timing and basis errors

A frequent error is to calculate the duration-1 reserve after the renewal premium while using the before-premium formula $A_{x+1}-P^*_{x+1}\ddot{a}_{x+1}$. That formula includes the premium due at time 1 in the annuity-due value. Applying it after the premium has been received counts that premium twice: once as already received and once as a future inflow. For a post-premium reserve, remove the time-1 payment from the annuity value, using $\ddot{a}_{x+1}-1$, as in the third example.

Another tempting but incorrect calculation is to set the first-year premium equal to the whole-life net premium at age $x$. FPT deliberately uses a one-year term premium for that year. In the first example, using $A_x/\ddot{a}_x$ would not price the first-year term coverage by itself and would break the specified FPT premium pattern.

Check the benefit date. A benefit paid at the end of the year is discounted by $v$, including a certain benefit payable one year from the valuation date.
Check the premium date. A reserve immediately before a premium includes that premium among future premiums; a reserve immediately after it does not.
Check the age used. The renewal premium is based on age $x+1$, and a duration-$k$ reserve uses attained age $x+k$.
Check the scope. The simple formulas here assume annual premiums and benefits, no expenses, and a consistent interest and mortality basis. Other cash flows or payment timings require corresponding changes to the prospective values.

FPT is useful because it isolates first-year acquisition funding from the later whole-life premium valuation pattern. Its defining zero reserve at duration 1 is not a general statement that an insurer has no obligations at that date; it follows from the chosen net valuation basis and the before-renewal-premium timing. This distinction is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each question, assume annual premiums are payable at the start of each year, death benefits at the end of the year of death, no expenses, and reserves immediately before premiums unless stated otherwise.

A unit-benefit policy has $i=0$ and $q_x=0.07$. What is the FPT first-year premium per dollar, and what is the first-year premium for a \$20,000 benefit? (Answer: $0.07$ per dollar; \$1,400.00.)
At age $x+1$, suppose $i=0$, $q_{x+1}=0.20$, $q_{x+2}=0.40$, and $q_{x+3}=1$. Find the FPT renewal premium per dollar. (Answer: $25/57=0.438596$ per dollar.)
At age $x+1$, suppose $i=0$, $q_{x+1}=0.25$, and $q_{x+2}=1$. Find the renewal premium per dollar and the duration-2 reserve per dollar. (Answer: renewal premium $4/7$; duration-2 reserve $3/7$.)
If $i=10\%$ and $q_x=0.08$, find the FPT first-year premium per dollar. (Answer: $0.08(10/11)=0.072727$ per dollar.)
At age $x+1$, suppose $i=0$, $q_{x+1}=0.20$, and $q_{x+2}=1$. Find the whole-life renewal premium per dollar and the reserve immediately after that premium is received at duration 1. (Answer: $5/9$ per dollar for each.)

What You'll Learn