At issue, the equivalence principle ensures \(E[L_0] = 0\): expected PV of premiums equals expected PV of benefits. But as time passes, this balance shifts — for level-premium contracts, early premiums are higher than needed (because mortality is low) and later premiums are lower than needed (because mortality is high). The policy reserve is the accumulated surplus the insurer must hold to cover future losses on in-force policies.
The benefit reserve at duration \(t\) on a contract issued to (x), denoted \({}_tV\), is the conditional expectation of the loss-at-duration-\(t\) random variable, given the contract is still in force:
\[ {}_tV = E\big[L_t \mid T(x) > t\big] = (\text{APV of future benefits}) - (\text{APV of future premiums}). \]It is the "prospective" amount the insurer must set aside to honour the remaining policy obligations.
Using \(\bar A_{x+t} = 1 - \delta\,\bar a_{x+t}\):
\[ {}_t\bar V = 1 - \frac{\bar a_{x+t}}{\bar a_x}, \]a particularly clean form: the reserve is the proportion of the original annuity value already "used up."
Note that at duration \(t = n\) the reserve equals 1 (the maturity benefit just before payment).
Whole life on (40), \(A_{40} = 0.16132\), \(\ddot a_{40} = 14.817\), so \(P_{40} = 0.01089\).
At duration 10: \(A_{50} = 0.24905\), \(\ddot a_{50} = 13.267\).
\({}_{10}V_{40} = 0.24905 - 0.01089 \times 13.267 = 0.24905 - 0.14448 = 0.10457.\)
So after 10 years the insurer must hold ₹0.1046 per ₹1 of sum assured — i.e., for a ₹10 lakh policy the reserve is approximately ₹1.05 lakh.
20-year endowment on (40), \(P_{40:\overline{20}|} = 0.02934\). At duration 20 (just before maturity): \(A_{60:\overline 0|} = 1\), \(\ddot a_{60:\overline 0|} = 0\).
\({}_{20}V_{40:\overline{20}|} = 1 - 0.02934 \times 0 = 1.\)
The reserve equals the maturity benefit, as it must — the insurer has fully accumulated the funds to pay the ₹1 endowment.
The reserve can also be computed as the accumulated value of premiums received minus the accumulated value of benefits paid, divided by the survival probability — all from issue date to duration \(t\).
where \(\ddot s_{x:\overline t|} = \ddot a_{x:\overline t|}/{}_tE_x\) is the accumulated annuity-due factor and \({}_{t}k_x = A^1_{x:\overline t|}/{}_tE_x\) is the accumulated cost of insurance. Retrospective = Prospective by equivalence principle.
The reserve at duration \(t+1\) is obtained from \({}_tV\) by:
\[ ({}_tV + \pi_t)(1+i) = q_{x+t}\,(b_{t+1}) + p_{x+t}\,({}_{t+1}V), \]where \(\pi_t\) is the premium received at time \(t\), and \(b_{t+1}\) is the death benefit paid at end of year \(t+1\).
Interpretation: roll forward the reserve plus premium at interest, subtract the expected death benefit, divide by survival probability — gives the reserve held by the survivors.
An ordinary differential equation that is solved numerically when no closed form is available.
Whole-life on (40), \(P_{40} = 0.01089\), \(i = 0.06\), \(q_{40} = 0.0028\), \(b = 1\). Find \({}_1V_{40}\) given \({}_0V_{40} = 0\).
\(({}_0V + P)(1+i) = (0 + 0.01089)(1.06) = 0.01154.\)
\(q_{40}\,b = 0.0028 \times 1 = 0.00280.\)
\({}_1V_{40} = (0.01154 - 0.00280)/(1 - 0.0028) = 0.00874/0.9972 = 0.00877.\)
So after one year the reserve has built up to ₹0.00877 per ₹1 sum assured.
From the rearranged Thiele equation, on a death in year \(t+1\) the insurer pays \(b\) but releases the reserve \({}_{t+1}V\) — net cash outflow at the moment of death is \(b - {}_{t+1}V\), called the net amount at risk (NAR). For whole life with \(b = 1\) and \({}_{t+1}V\) growing over time, the NAR shrinks as the contract ages — meaning mortality risk becomes a smaller and smaller share of the insurer's exposure on each policy.
Insurers incur expenses: initial (commissions, underwriting, policy issue), maintenance (ongoing admin), and settlement (claim handling). The gross premium \(G\) must cover both benefits and expenses.
Typical expense assumption: an initial expense \(E_0\) at time 0, a per-premium fraction \(e_p\) of each premium (commission), and a per-policy maintenance \(E_m\) at the start of each renewal year. Then
\[ G \cdot (1 - e_p) \cdot \ddot a_{x:\overline n|} = A_{x:\overline n|} + E_0 + E_m \cdot \ddot a_{x:\overline n|}. \]Initial expenses are unrecoverable after issue, so the gross-premium reserve is typically lower than the benefit reserve in early durations (sometimes negative — a "deficiency reserve").
On a 20-year endowment with high first-year expense, the insurer initially holds a NEGATIVE gross-premium reserve, because the initial expense has been "lent" to the policy. As renewals arrive the reserve grows positive. Most regulators require the held reserve to be at least zero (or the benefit reserve), so the insurer effectively absorbs the strain.
The benefit reserve assumes level premiums, but initial expenses are very high. Modified reserves redistribute the early-duration "strain" so the held reserve more closely reflects available assets.
The first-year premium funds only a one-year term insurance; the reserve at the end of year 1 is reset to that of a contract issued one year later. Effectively:
\[ {}_1V^{\text{FPT}}_x = 0, \qquad {}_tV^{\text{FPT}}_x = {}_{t-1}V_{x+1} \quad \text{for } t \ge 1. \]Used historically in the U.S. — generous to early-duration solvency.
A "Zillmer-modified" reserve replaces \(P_x\) in the prospective formula by
\[ P^{\text{Z}}_x = P_x + \frac{Z}{\ddot a_{x:\overline n|}}, \]where \(Z\) is the Zillmer expense factor (the unamortised acquisition expense). The reserve becomes
\[ {}_tV^{\text{Z}}_x = {}_tV_x - Z \cdot \frac{\ddot a_{x+t:\overline{n-t}|}}{\ddot a_{x:\overline n|}}. \]Used in many European/UK jurisdictions — explicitly amortises acquisition expense over the premium-paying period.
20-year endowment on (40). Benefit reserve at duration 1: \({}_1V_{40:\overline{20}|} = 0.020\). Under FPT we reset \({}_1V^{\text{FPT}} = 0\) (using the second-year onwards reserves of a 19-year endowment on (41) instead). Saves the insurer 0.020 of reserve at duration 1 — exactly the amount of strain caused by the year-1 commission/expense.
Suppose Zillmer factor \(Z = 0.030\) and \(\ddot a_{40:\overline{20}|} = 11.45\). Then the Zillmer-modified premium is
\(P^{\text{Z}}_{40:\overline{20}|} = 0.02934 + 0.030/11.45 = 0.02934 + 0.00262 = 0.03196\).
The annual premium is bumped up by 0.00262 to amortise the upfront expense, and the held reserve is reduced by \(Z \cdot \ddot a_{x+t:\overline{n-t}|}/\ddot a_{x:\overline n|}\) at every duration.