A premium principle is a rule that assigns a premium \(H(X)\) — a real number — to every loss random variable \(X\). It is a functional on the set of risks; \(H(X)\) represents what the insurer should charge to cover \(X\).
The simplest is the net (pure) premium: \(H(X) = E(X)\). All other principles add a "loading" to account for variability and the insurer's cost of capital.
Charges the expected loss only. Used in life insurance (where mortality predictability is high). Generates zero safety loading, so cannot survive in practice without a separate safety margin.
The expected loss with a proportional safety loading \(\theta\). Most widely used in non-life lines.
Loading is proportional to variance. Penalises volatile risks more heavily.
Loading is proportional to SD; less aggressive than the variance principle on volatile risks. Homogeneous of degree 1.
Derived from exponential utility \(u(w) = -e^{-aw}/a\):
where \(a > 0\) is the risk-aversion parameter. As \(a \to 0\), \(H(X) \to E(X)\) (net premium); as \(a \to \infty\), \(H(X) \to \max X\) (cover-all-extreme premium).
Premium is the \((1-\epsilon)\)-quantile of the loss distribution. \(\epsilon = 0.01\) means the premium covers 99 % of all loss scenarios.
Uses a "tilted" distribution — applicable when MGF exists.
| Principle | Translation invariant | Homogeneous | Sub-additive |
|---|---|---|---|
| Net | Yes | Yes | Yes (= equality) |
| Expected value | Only if θ=0 | Yes | Yes |
| Variance | Yes | No | No |
| Standard deviation | Yes | Yes | Yes |
| Exponential | Yes | No | Yes |
| Percentile | Yes | Yes | Generally no |
A claim \(X\) is ₹10 000 with probability 0.10 and 0 otherwise. So:
\(E(X) = 1{,}000;\;\; E(X^2) = 0.1 \cdot 10000^2 = 10^7;\;\; \text{Var}(X) = 10^7 - 10^6 = 9 \times 10^6;\;\; \sigma_X = 3{,}000\).
Consider \(Y = 2X\) (a policy with double exposure).
In the individual risk model, an insurance portfolio is the collection of \(n\) independent policies. For each policy \(i\), the claim \(X_i\) is a random variable with its own distribution. The aggregate claim is:
\[ S \;=\; X_1 + X_2 + \cdots + X_n. \]Crucially, \(n\) is fixed in advance (one r.v. per policy). Contrast with the collective risk model where \(N\) (number of claims) is random.
A common structure for each policy:
where:
Moments of \(X_i\):
Exact distribution can be computed via convolution but is usually intractable. Two standard approximations:
(a) Normal approximation (when \(n\) is large, individual claim variances are bounded — CLT):
(b) Translated Gamma approximation (when \(S\) is highly skewed). Match the first three moments of a shifted Gamma distribution to those of \(S\).
If every claim size is a fixed amount \(b\) (so \(B_i = b\) constant) and each policy claims with probability \(q\):
and \(\sum I_i \sim\) Binomial(\(n, q\)). Hence \(S = b \cdot \) Binomial(\(n, q\)) — a scaled binomial.
An insurer issues 1 000 one-year term policies, each with face amount ₹1 00 000. Probability of claim is 0.002 per policy (independent).
\(B_i = 100{,}000\) (constant); \(I_i \sim \) Bernoulli(0.002); \(n = 1000\).
\(E(S) = 1000 \cdot 0.002 \cdot 100{,}000 = \text{₹}2{,}00{,}000\).
\(\text{Var}(I_i) = 0.002 \cdot 0.998 = 0.001996;\;\) \(\text{Var}(X_i) = 100000^2 \cdot 0.001996 = 1.996 \times 10^{7}\); \(\text{Var}(S) = 1000 \cdot 1.996 \times 10^{7} = 1.996 \times 10^{10}\); \(\sigma_S = \sqrt{1.996 \times 10^{10}} \approx 141{,}280\).
Probability that aggregate claims exceed ₹3 00 000:
\(P(S > 300000) \approx P\!\left(Z > \dfrac{300000 - 200000}{141280}\right) = P(Z > 0.708) = 0.24\).
The variance is driven by the rare but large ₹1 00 000 payouts, so \(S\) is quite spread out relative to its mean of ₹2 00 000 — a much larger portfolio would be needed for the CLT to give a sharp tail estimate.
Three groups of policies:
| Group | n_i | q_i | E(B_i) | Var(B_i) |
|---|---|---|---|---|
| A | 500 | 0.05 | 1 000 | 250 000 |
| B | 300 | 0.10 | 2 000 | 1 000 000 |
| C | 200 | 0.02 | 5 000 | 4 000 000 |
For Group A: \(E(X_i) = 0.05 \cdot 1000 = 50\). \(\text{Var}(X_i) = 0.05 \cdot 250000 + 0.05 \cdot 0.95 \cdot 1000^2 = 12500 + 47500 = 60000\).
Group B: \(E(X_i) = 200\); \(\text{Var}(X_i) = 0.10 \cdot 1000000 + 0.10 \cdot 0.90 \cdot 4 \times 10^6 = 100000 + 360000 = 460000\).
Group C: \(E(X_i) = 100\); \(\text{Var}(X_i) = 0.02 \cdot 4 \times 10^6 + 0.02 \cdot 0.98 \cdot 25 \times 10^6 = 80000 + 490000 = 570000\).
\(E(S) = 500(50) + 300(200) + 200(100) = 25000 + 60000 + 20000 = \text{₹}1{,}05{,}000\).
\(\text{Var}(S) = 500(60000) + 300(460000) + 200(570000) = 3 \times 10^7 + 1.38 \times 10^8 + 1.14 \times 10^8 = 2.82 \times 10^8\). \(\sigma_S \approx 16{,}793\).
Normal approximation gives a tight bound on aggregate claims.
The insurer wants to choose total premium \(P\) so that the probability of technical ruin (aggregate claims exceeding premium income plus surplus) is small. Using normal approximation:
Solving: \(P \;\ge\; E(S) + z_{1-\epsilon}\,\sigma_S\).
This is exactly the SD principle in disguise, with \(\alpha = z_{1-\epsilon}\).
The insurer holds reserves equal to a high quantile of \(S\) (e.g., 99.5 % VaR under Solvency II). Same calculation gives the required reserve.
Stop-loss reinsurance pays the insurer \(\max(S - d, 0)\). Expected payment is \(E[\max(S - d, 0)]\) — directly computed under normal approximation.
For Example 2 above (\(E(S) = \text{₹}1{,}05{,}000\), \(\sigma_S = \text{₹}16{,}793\)), the premium that limits ruin probability to 5 % is:
\(P = 105000 + 1.645 \cdot 16793 = 105000 + 27{,}625 = \text{₹}1{,}32{,}625\).
For the same portfolio with stop-loss retention \(d = \text{₹}1{,}20{,}000\):
\(E[\max(S - d, 0)] = \sigma_S \phi(k) - (d - E(S))(1 - \Phi(k))\) where \(k = (d - E(S))/\sigma_S = 15000/16793 = 0.893\).
\(\phi(0.893) = 0.268;\; 1 - \Phi(0.893) = 0.186\).
\(E[\max(S - d, 0)] = 16793(0.268) - 15000(0.186) = 4500 - 2790 = \text{₹}1{,}710\).