Modified Reserves: Full Preliminary Term Method

Why Modified Reserves Are Needed

Net level premium reserves assume that expenses are spread evenly over the premium-paying period. In reality, life insurance expenses are heavily front-loaded: first-year commissions can be 50% to 100% of premium, underwriting costs are incurred only at issue, and policy issuance expenses add further to first-year costs. Renewal expenses are much lower, typically 5% to 10% of premium. This mismatch between the level premium reserve assumption and actual expense timing creates what is called "new business strain."

New business strain occurs because the net level premium reserve at the end of year 1 exceeds the asset share (the actual accumulated fund per policy). The insurer must contribute surplus to cover the difference. Modified reserve methods reduce this strain by allowing a lower first-year premium allocation (matching the higher first-year expenses) and a correspondingly higher renewal premium allocation, while still ensuring that the total present value of modified premiums equals the total present value of benefits.

Full Preliminary Term (FPT) Method

The FPT method is the most aggressive standard modification. It treats the first year of any policy as if it were one-year term insurance. The first-year modified premium alpha equals exactly the cost of one-year term coverage: alpha = v * q_x. This means the entire first-year premium (minus the one-year term cost) is implicitly available for expenses, eliminating first-year strain entirely.

The renewal modified premium beta is determined by the equivalence principle applied to years 2 and beyond. Since the first year has been handled as term insurance, the remaining benefits are equivalent to a whole life policy issued at age x+1. Therefore, beta = A_{x+1} / a-ddot_{x+1} = P_{x+1}. The renewal premium under FPT equals the net level premium for a policy issued one year later at the attained age.

FPT Reserves

The FPT reserve at duration t >= 1 has a remarkably simple form: the FPT reserve at duration t equals the net level premium reserve for a whole life policy issued at age x+1, evaluated at duration t-1. In notation, tV^FPT_x = (t-1)V_{x+1}. This follows directly from treating the policy as if it were issued at age x+1 after the first year of term coverage.

At duration 1, the FPT reserve is 0V_{x+1} = 0 (a newly issued policy has zero reserve). Compare this to the net level premium reserve at duration 1, which is positive. The difference between the net level reserve and the FPT reserve at duration 1 represents the first-year strain that the FPT method eliminates. As duration increases, the FPT reserve approaches the net level reserve because the effect of the first-year modification diminishes.

Commissioners Reserve Valuation Method (CRVM)

The CRVM limits the expense allowance in the first year to prevent the FPT method from being too aggressive for high-premium products. Under CRVM, the first-year expense allowance (the difference between the gross premium and the first-year modified premium) is capped. If the FPT allowance exceeds the CRVM limit, the modified premiums are adjusted so that the first-year allowance equals the limit. The CRVM limit is typically defined as the difference between the net level premiums for the actual policy and a 20-payment whole life policy.

For policies where the FPT allowance falls within the CRVM limit, the FPT and CRVM reserves are identical. For policies with very high premiums (such as single-premium or limited-pay policies), the CRVM imposes a tighter constraint, resulting in reserves between the FPT and net level values.

Comparing Modified and Net Level Reserves

Modified reserves are always less than or equal to net level premium reserves at every duration. The maximum difference occurs in the early policy years when the expense allowance has the greatest impact. As the policy ages, the modified reserves converge toward the net level reserves because the first-year modification affects a diminishing fraction of the remaining liability.

The relationship can be summarized as: 0 = tV^FPT <= tV^CRVM <= tV^NLP for all t >= 1 in the first year, and the differences narrow over time. At very long durations, all three reserves converge because the future premium stream and benefit obligations dominate the historical premium allocation pattern.

Common Pitfalls

A frequent mistake is computing the FPT renewal premium using the wrong age. The renewal premium is P_{x+1}, not P_x. This follows from the fundamental FPT construction: years 2 onward are treated as a policy issued at age x+1. Another common error is forgetting the offset in the reserve duration: the FPT reserve at duration t equals the net level reserve at duration t-1 for age x+1, not duration t.

Students also sometimes confuse modified reserves with gross premium reserves. Modified reserves adjust the net premium allocation between first-year and renewal; they do not incorporate actual expense amounts. Gross premium reserves use actual expected expenses and are a separate concept entirely.

Study Tips

Exam LTAM tests your ability to compute FPT and CRVM reserves and compare them to net level reserves. Memorize the key FPT formulas: alpha = v * q_x, beta = P_{x+1}, and tV^FPT = (t-1)V_{x+1}. Practice computing the first-year expense allowance under FPT and checking it against the CRVM limit. Work through problems that ask you to compute reserves under multiple methods for the same policy, as these comparison questions test your understanding of why the methods differ and how the reserves relate to each other.