Purpose and Valuation Setup
The full preliminary term (FPT) method modifies the valuation premium used to calculate reserves. It treats the first policy year as a separate one-year term insurance and, after that year, values the contract as if it had been issued one year later. Consequently, the first-year premium is based on the original issue age, while renewal premiums and subsequent reserves use the shifted issue age.
The method below applies to annual discrete premiums and benefits paid at the end of the year of death. Assume a fixed sum insured \(B\), annual effective interest rate \(i\), and mortality probabilities on the valuation basis. Premiums are payable in advance while the insured is in force. Unless stated otherwise, expenses and lapses are excluded, and the same benefits are used in the modified valuation as in the contract. The FPT valuation premiums need not equal the premiums actually charged.
For an \(n\)-year term insurance, write \(A_{y:\overline{m}|}\) for the actuarial present value of a unit death benefit payable at the end of the year of death during the next \(m\) years, for a life aged \(y\). Write \(\ddot{a}_{y:\overline{m}|}\) for the present value of premiums of one per year in advance, payable while in force for at most \(m\) years. The level net premium at age \(y\) for an \(m\)-year term insurance with benefit \(B\) is \(B A_{y:\overline{m}|}/\ddot{a}_{y:\overline{m}|}\). Whole-life quantities are denoted \(A_y\) and \(\ddot{a}_y\).
For a term insurance initially issued at age \(x\) for \(n\) years, the FPT first-year premium is \(B A_{x:\overline{1}|}/\ddot{a}_{x:\overline{1}|}\). Since the premium is payable at issue, \(\ddot{a}_{x:\overline{1}|}=1\), and \(A_{x:\overline{1}|}=v q_x\), where \(v=1/(1+i)\). The renewal premium is \(P_{x+1:\overline{n-1}|}=B A_{x+1:\overline{n-1}|}/\ddot{a}_{x+1:\overline{n-1}|}\). For whole life, it is \(P_{x+1}=B A_{x+1}/\ddot{a}_{x+1}\).
How to Value a Reserve After the First Year
At duration \(k\geq 1\), the reserve is calculated for the benefits still payable and the renewal premiums still due, conditional on the policy being in force. The premium used is the fixed renewal premium calculated at age \(x+1\), not a newly calculated net premium at age \(x+k\). For an \(n\)-year term contract, with the reserve taken immediately before any renewal premium due at duration \(k\), the formula is:
At duration \(k=1\), the remaining benefits and premiums are exactly those of the shifted-age contract at issue. The renewal premium was calculated by the equivalence principle for that contract, so the reserve at duration one is zero as well. For \(k>1\), the reserve reflects the shifted-age premium still being used, even though the mortality and remaining term have changed. For whole life, replace the term quantities by whole-life quantities: at duration \(k\geq1\), the prospective value of future benefits is \(B A_{x+k}\), and the premium value is \(P_{x+1}\ddot{a}_{x+k}\).
Worked Example: A Three-Year Term Contract
A hypothetical three-year term policy has a benefit of $10,000, paid at the end of the year of death. Assume \(i=10\%\), annual premiums in advance, and \(q_x=0.10\), \(q_{x+1}=0.10\), and \(q_{x+2}=0.30\). Calculate the FPT first-year premium, the renewal premium, and the reserve at durations one and two.
Method. Calculate the first-year net premium for one-year term insurance at age \(x\). Then calculate one level net premium for the remaining two-year term at age \(x+1\). Use that renewal premium unchanged when valuing the duration-two reserve.
First-year premium. Here \(v=1/1.10=0.9090909\). The one-year death-benefit value is \(10{,}000(0.10)v=$909.0909\), and only one premium is due at issue. Thus the first-year FPT premium is $909.09, rounded to cents.
Renewal premium. At age \(x+1\), the death-benefit value for the next two years is \(10{,}000[0.10v+(0.90)(0.30)v^2]\). The two terms inside the brackets are \(0.0909091\) and \(0.2231405\), giving a total of \(0.3140496\). The benefit value is therefore $3,140.4959.
The premium annuity value at age \(x+1\) is \(1+(0.90)v=1.8181818\). The renewal premium is \(3{,}140.4959/1.8181818=$1{,}727.2727\), or $1,727.27 to cents.
Duration one. The duration-one benefit value is $3,140.4959, as calculated for the shifted-age two-year contract. The value of premiums due, using the same basis, is \(1.8181818($1{,}727.2727)=$3{,}140.4959\). The reserve is therefore $3,140.4959−$3,140.4959=$0.00.
Duration two. One year of coverage remains. The benefit value is \(10{,}000(0.30)v=$2{,}727.2727\). One renewal premium is due at duration two, so its value is $1,727.2727. The reserve is $2,727.2727−$1,727.2727=$1,000.00.
Check. The first-year premium covers only the first-year term valuation. The duration-two reserve uses the renewal premium established for the two-year shifted-age contract; it does not reset the premium to the one-year net premium at age \(x+2\).
Why the First-Year Reserve Is Zero
The zero reserve at issue follows directly from the one-year net premium equivalence. The present value of the first-year death benefit is \(Bq_xv\), while the first-year FPT premium is \(Bq_xv\) paid at issue. Immediately after receipt of that premium, the prospective value of benefits less the value of future premiums is therefore zero:
The same reasoning explains why the duration-one reserve is zero under the stated timing convention. Conditional on being in force at duration one, the remaining contract is the shifted-age contract at its issue. By the net premium equivalence principle, the present value of its future benefits equals the present value of the renewal premiums. This zero at duration one is distinct from later reserves, which value the remaining benefits using the same fixed shifted-age premium.
Use the one-year term net premium at age \(x\) as the first-year valuation premium.
Use the remaining term, or whole-life benefit, as if issued at age \(x+1\).
At each duration, subtract the value of future renewal premiums from the value of future benefits, using the original shifted-age renewal premium.
Worked Example: A Whole-Life Contract
A hypothetical whole-life policy has a benefit of $12,000. Assume \(i=10\%\), annual premiums in advance, \(q_x=0.05\), \(q_{x+1}=0.10\), and \(q_y=0.20\) for every age \(y\geq x+2\). Calculate the first-year premium, the shifted-age whole-life premium, and the duration-two reserve.
First-year premium. With \(v=10/11\), the first-year premium is \(12{,}000(0.05)(10/11)=$6000/11=$545.4545\), or $545.45 to cents.
Shifted-age values. At age \(x+2\), mortality is constant at \(q=0.20\), so \(p=0.80\). The whole-life insurance and annuity values satisfy \(A_{x+2}=0.20v+0.80vA_{x+2}\) and \(\ddot{a}_{x+2}=1+0.80v\ddot{a}_{x+2}\). Since \(0.80v=8/11\), these give \(A_{x+2}=(2/11)/(3/11)=2/3\) and \(\ddot{a}_{x+2}=1/(3/11)=11/3\).
At age \(x+1\), \(p_{x+1}=0.90\). Thus \(A_{x+1}=0.10(10/11)+0.90(10/11)(2/3)=7/11\). The annuity value is \(\ddot{a}_{x+1}=1+0.90(10/11)(11/3)=4\). The shifted-age renewal premium is therefore \(12{,}000(7/11)/4=$84{,}000/44=$1{,}909.0909\), or $1,909.09 to cents.
Duration-two reserve. At duration two, mortality follows the constant \(q=0.20\) assumption. The benefit value is \(12{,}000(2/3)=$8{,}000\). The value of future premiums is \($1{,}909.0909)(11/3)=$7{,}000\). The reserve is therefore $8,000−$7,000=$1,000.00.
Check. At duration one, the prospective benefit value is \(12{,}000(7/11)\), while the premium value is \($1{,}909.0909)(4)\); both equal $7,636.3636. The duration-one reserve is zero, consistent with the shifted-age equivalence calculation.
Using the Shifted Premium Correctly
A frequent error is to use the original level net premium at every duration. That is not the FPT valuation basis: after the preliminary year, the renewal premium is determined once at age \(x+1\). It remains the premium used in subsequent prospective reserves, even when a new net premium at a later attained age would be different.
Worked Example: Avoiding the Original-Premium Trap
Use the three-year term data from the first example: \(B=$10{,}000\), \(i=10\%\), and annual mortality probabilities \(0.10, 0.10, 0.30\) by policy year. At duration two, compare the FPT reserve with the result from mistakenly using the original three-year level net premium.
Correct FPT calculation. The age-\(x+1\) renewal premium is $1,727.2727, as calculated earlier. The duration-two benefit value is $2,727.2727 and one renewal premium of $1,727.2727 is due. Thus the correct reserve is $2,727.2727−$1,727.2727=$1,000.00.
Calculate the original level premium to identify the error. Its three-year benefit value is \(10{,}000[0.10v+0.09v^2+0.243v^3]\). The terms in brackets are \(0.0909091\), \(0.0743802\), and \(0.1825695\), summing to \(0.3478588\); the benefit value is $3,478.5875. The premium annuity value is \(1+0.90v+0.81v^2=2.4876033\). Hence the original net premium is approximately \(3{,}478.5875/2.4876033=$1{,}398.37\).
Wrong calculation and correction. Using the original premium at duration two gives \(2{,}727.2727-1{,}398.37\approx$1{,}328.90\). This is not an FPT reserve because $1,398.37 is not the renewal premium on the FPT basis. Use the shifted-age premium instead, giving the correct reserve of $1,000.00.
Practical Checks and Scope
Before finalizing an FPT calculation, confirm the timing convention, the benefit pattern, and the premium basis. The formulas above assume annual premiums in advance and death benefits at the end of the year of death. A different payment timing or benefit timing changes the actuarial present values and must be reflected in both the premium and reserve calculations.
- Check the shift. The renewal premium is based on age \(x+1\), not the original age \(x\), and not a newly selected age for each reserve duration.
- Check the remaining term. An \(n\)-year term policy has an \(n-1\)-year renewal-premium calculation after the preliminary year.
- Check what is due at the valuation time. In the examples, a reserve at duration \(k\) is taken before the renewal premium due at that duration; its annuity value includes that payment.
- Check the valuation basis. Mortality, interest, benefits, and premium timing used in the net premium and reserve must be consistent. Expenses or other contract features require an appropriately specified valuation model.
FPT is a modified reserve basis: it deliberately changes the valuation premium pattern to recognize a preliminary first year and a shifted-age renewal basis. It should not be confused with a reserve calculated using the original level net premium, or with a method that independently recalculates the premium at every duration. This distinction is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.
Check Your Understanding
For each problem, state the premium or reserve on the FPT basis and retain unrounded values until the final answer.
- A two-year term policy has a $25,000 benefit, \(i=5\%\), and \(q_x=0.04\). What is the FPT first-year premium, and what is the reserve at issue immediately after that premium is received? (Answer: first-year premium $952.38; reserve $0.00.)
- A three-year term policy has a $1,000 benefit, \(i=0\%\), and mortality probabilities \(q_{x+1}=0.10\) and \(q_{x+2}=0.20\). Calculate the shifted-age renewal premium for the two remaining years and the duration-two reserve. (Answer: renewal premium $147.37; duration-two reserve $52.63.)
- A whole-life policy has benefit \(B\), \(i=5\%\), and \(q_y=0.20\) at every age \(y\geq x+1\). Express the shifted-age renewal premium as a multiple of \(B\), and state the reserve at duration two. (Answer: renewal premium \(4B/21\), approximately \(0.1905B\); duration-two reserve \(0\).)
- A four-year term policy has a $5,000 benefit, \(i=0\%\), and mortality probabilities at ages \(x+1,x+2,x+3\) of \(0.20,0.10,0.40\), respectively. Calculate the renewal premium based on age \(x+1\), then the duration-two reserve. (Answer: renewal premium $1,126.98; duration-two reserve $158.73.)
- A term policy has three years remaining at the shifted issue age. Its renewal premium is $800. At duration two, the remaining one-year death probability is \(0.25\), \(i=0\%\), and the benefit is $4,000. Calculate the reserve immediately before the premium due at duration two. (Answer: $200.00.)