The equivalence principle sets premium so that the expected present value of premiums received equals the expected present value of benefits paid:
\[ E[\text{PV of premiums}] = E[\text{PV of benefits}]. \]This gives the net (benefit) premium — the price that makes the insurer's expected profit zero, before expenses.
Let \(L\) be the present value of insurer's loss on the contract:
\[ L = (\text{PV of benefits}) - (\text{PV of premium income}). \]Equivalence principle: choose premium \(P\) so that \(E[L] = 0\). The premium is then unique.
Benefit 1 paid at moment of death (\(T = T(x)\)); premiums paid continuously at annual rate \(\bar P(\bar A_x)\) while (x) is alive. Then
\[ L = v^T - \bar P(\bar A_x) \cdot \bar a_{\overline T|}. \]Setting \(E[L] = 0\) and using \(\bar a_x = E[\bar a_{\overline T|}]\),
\[ \bar P(\bar A_x) = \frac{\bar A_x}{\bar a_x} = \frac{\delta\,\bar A_x}{1 - \bar A_x}. \]Substituting \(\bar a_{\overline T|} = (1 - v^T)/\delta\) into \(L\),
\[ L = \left(1 + \frac{\bar P}{\delta}\right) v^T - \frac{\bar P}{\delta}. \] \[ \mathrm{Var}[L] = \left(1 + \frac{\bar P}{\delta}\right)^{\!2} \big[{}^{2}\!\bar A_x - (\bar A_x)^2\big]. \]The variance is driven entirely by the variance of \(v^T\), inflated by the premium-to-discount factor.
\(\mu = 0.04\), \(\delta = 0.06\). From Unit 2: \(\bar A_x = 0.40\), \({}^{2}\!\bar A_x = 0.25\), \(\bar a_x = 10\). Compute \(\bar P(\bar A_x)\) and \(\mathrm{SD}[L]\).
\(\bar P = 0.40 / 10 = 0.04\).
Loss factor: \((1 + 0.04/0.06) = (1 + 2/3) = 5/3\). Variance of \(v^T = 0.25 - 0.16 = 0.09\).
\(\mathrm{Var}[L] = (5/3)^2 \times 0.09 = 25/9 \times 0.09 = 0.25\); \(\mathrm{SD}[L] = 0.50\).
The standard deviation of the per-policy loss is 0.50 on a 1-unit benefit — enormous compared to the premium of 0.04, which is why insurers need large portfolios for risk pooling.
Benefit 1 paid at end of year of death; premium \(P_x\) paid at start of each year of life.
\[ L = v^{K+1} - P_x \cdot \ddot a_{\overline{K+1}|}. \] \[ P_x = \frac{A_x}{\ddot a_x} = \frac{d\,A_x}{1 - A_x}. \]At age 40: \(A_{40} = 0.16132\), \(\ddot a_{40} = 14.817\), \(i = 0.06\) (so \(d = 0.05660\)).
\(P_{40} = 0.16132 / 14.817 = 0.01089\).
So for every ₹1 of sum assured the annual net premium is ₹0.01089 — i.e. a ₹10 lakh policy costs ₹10,890 per year as net premium (before expenses).
The benefit period is whole life; the premium-payment period is only \(h\) years. So the same \(A_x\) is divided by a shorter annuity — premium per year is correspondingly larger.
The "1" sits over \(n\), meaning the contingency is survival to time \(n\).
At age 40, 20-year term: \(A^1_{40:\overline{20}|} = 0.024\), \(\ddot a_{40:\overline{20}|} = 11.45\).
20-year endowment: \(A_{40:\overline{20}|} = 0.336\).
\(P^1_{40:\overline{20}|} = 0.024/11.45 = 0.00210\).
\(P_{40:\overline{20}|} = 0.336/11.45 = 0.02934\).
The endowment is roughly 14× more expensive than the term — the survival payout dominates.
Whole life on (40), premiums payable for 20 years. From the table: \(A_{40} = 0.16132\), \(\ddot a_{40:\overline{20}|} = 11.45\).
\({}_{20}P_{40} = 0.16132 / 11.45 = 0.01409\).
Compared to lifetime-pay (\(P_{40} = 0.01089\) from Example 2 above), the 20-pay version costs about 29% more per year — but stops after 20 years, so total premiums paid are typically less.
"Semi-continuous" means the benefit is paid continuously at the moment of death, while premiums are paid discretely at the start of each year. The premium is
\[ P(\bar A_x) = \frac{\bar A_x}{\ddot a_x}. \]Under UDD this simplifies to \(P(\bar A_x) = (i/\delta)\,P_x\).
The equivalence-principle premium leaves \(E[L] = 0\), but the variance is large. For a portfolio of \(N\) iid policies, the total loss \(S = L_1 + L_2 + \cdots + L_N\) has \(E[S] = 0\) and \(\mathrm{Var}[S] = N\,\mathrm{Var}[L]\). For the insurer to be ruined with at most probability \(\alpha\) we charge a higher premium so that \(P[S > 0] \le \alpha\).
Using normal approximation,
\[ P[\,L \le \ell\,] \approx \alpha \quad\Leftrightarrow\quad \ell = E[L] + z_{\alpha}\,\sqrt{\mathrm{Var}[L]}. \]Set \(E[L] < 0\) by a margin of \(-z_{\alpha}\,\sqrt{\mathrm{Var}[L]/N}\). For each individual policy the required premium thus exceeds the equivalence-principle premium.
Continuing the constant-force example: equivalence \(\bar P = 0.04\), \(\mathrm{SD}[L] = 0.50\). A single insurer covering only 1 policy who wants 95% confidence of solvency must charge a much higher premium than 0.04 — solve \(0 = E[L] + 1.645 \times \mathrm{SD}[L]\). The equivalence-principle premium is totally inadequate for a one-policy book.
For \(N = 1000\) similar policies: \(\mathrm{SD}[S] = \sqrt{1000} \times 0.50 = 15.8\). The 95% percentile margin per policy is \(1.645 \times 15.8 / 1000 = 0.026\) — small but non-zero. Total premium needed: 0.04 + 0.026 = 0.066 per policy. This is the "law of large numbers" in action.
For \(N\) iid policies, the per-policy margin scales as \(1/\sqrt N\):
| \(N\) | Margin per policy |
|---|---|
| 1 | \(1.645 \times 0.50 = 0.823\) |
| 100 | \(1.645 \times 0.05 = 0.082\) |
| 1{,}000 | \(1.645 \times 0.0158 = 0.026\) |
| 10{,}000 | \(1.645 \times 0.005 = 0.008\) |
By 10,000 policies the safety margin per policy drops to 0.008 — only 20% of the equivalence-principle premium. This is exactly why insurers grow their book.
An apportionable (refundable) premium policy returns the unused fraction of the last premium upon death. Under UDD,
\[ P_x^{\{m\}} = \frac{A_x}{\ddot a_x^{\{m\}}}, \]with the apportionable annuity \(\ddot a_x^{\{m\}} \approx \ddot a_x - 1/(2m)\). For \(m=1\) (yearly), this equals roughly \(\ddot a_x - 0.5\); for monthly (\(m=12\)) it equals \(\ddot a_x - 1/24\).