Reinsurance and Risk Portfolio Management

The insurer is not himself insured: a hurricane, epidemic, or mass bankruptcy can wipe out the capital of the largest company. The solution is reinsurance, the “insurance for the insurer.” Part of the risks is transferred to the reinsurer (Munich Re, Swiss Re, Hannover Re), who specializes in taking on catastrophic losses. This is a key instrument for managing the risk portfolio—without it, the modern insurance industry is unimaginable. The global reinsurance market: ~$700 billion in premiums in 2023.

Types of Reinsurance

1. Proportional (Quota Share, QS). The reinsurer takes a fixed share q of each loss and receives a share q of the insurance premium. The cedent (direct insurer) retains (1 − q)·X.

Example: q = 30%. Loss of 1 million → reinsurer pays 300 thousand, cedent — 700 thousand. Premiums are distributed in the same proportion.

Advantages: simplicity, stability of relationships. Drawback: the cedent also pays for "small" losses, which he could afford to bear.

2. Excess of Loss (XL). The reinsurer pays only for losses above the retention d, up to the limit u:

• Reinsurer’s payment: min(max(X − d, 0), u − d).
• Cedent’s retention: min(X, d) + max(X − u, 0).

Example: d = 5 million, u = 50 million. Loss of 30 million → reinsurer 25 million, cedent 5 million. Loss of 70 million → reinsurer 45 million, cedent 5 + 20 = 25 million.

Advantage: protection from catastrophic losses. Drawback: more expensive per unit of insurance sum (tail premium).

3. Stop Loss. The reinsurer pays aggregate portfolio losses S exceeding level L: payment = max(S − L, 0).

Protects from accumulation: epidemics, hurricanes, mass bankruptcies, COVID-2020. Has been actively used in crop, travel, and event insurance.

4. Catastrophe XL (CatXL). A special XL for natural disasters. Activates on an “occurrence event” (a single hurricane, earthquake). Limits of hundreds of millions or billions of USD.

Optimal Reinsurance

Task: choose a reinsurance structure R(X) ∈ [0, X] (the portion of loss transferred to the reinsurer), minimizing the chosen risk measure of the net losses of the cedent (X − R(X) + π(R), where π is the reinsurance premium) under a budget constraint.

Arrow’s Theorem (1963). For a risk-neutral reinsurer (π(R) = (1 + θ)·E[R]) and risk-averse cedent (minimizing variance or CVaR), the optimal structure is stop-loss: R(X) = max(X − d, 0). Geometrically—“cutting off the tail” of the loss distribution.

Cai et al. (2008): for CVaR-optimization, stop-loss with a specific d* is also optimal. The proof uses variational calculus on the form of R.

Cat Bonds (Catastrophe Bonds)

An alternative risk transfer mechanism. The cedent issues a bond. Investors receive a coupon (LIBOR/SOFR + spread of 5–15%). If, during the life of the bond, a trigger event occurs (hurricane of strength ≥ 4, earthquake with M ≥ 7, parametric index), the principal is partly or fully lost and used to cover the cedent’s losses.

Triggers:

• Indemnity: the actual loss of the insurer.
• Industry index: industry-wide loss (PCS index).
• Parametric: value of a natural parameter (magnitude, wind speed at point).
• Modeled loss: loss as determined by a predefined model.

Cat Bonds market: $40 billion outstanding in 2023. Issuers: USAA, Allstate, Florida Citizens, Caltrans. Used after Katrina (2005), Sandy (2012), Maria (2017), Mexican earthquake (2017), COVID (World Bank Pandemic Bond).

Numerical Example: Effect of XL-Reinsurance

Portfolio with aggregate loss S distribution S ~ Lognormal(μ = 14, σ = 0.8) (mean ≈ 1.65 million RUB, median e^14 ≈ 1.20 million).

E[S] = e^{14 + 0.32} = 1.65 million. Var[S] = (e^{0.64} − 1)·e^{28 + 0.64} = 0.896·2.43·10^12 ≈ 2.18·10^12. σ(S) ≈ 1.48 million. VaR_{0.99}(S) (lognormal): exp(μ + σ·Φ^{−1}(0.99)) = exp(14 + 0.8·2.326) ≈ 7.61 million. CVaR_{0.99}(S) ≈ 9.52 million.

XL-reinsurance with d = 2 million, u = 5 million (premium π = 0.4 million). Net loss of cedent: Y = min(S, 2) + max(S − 5, 0) + π.

Simulation (Python, 100,000 iterations):

• E[Y] = E[min(S, 2)] + E[max(S − 5, 0)] + 0.4 ≈ 1.31 + 0.27 + 0.4 = 1.98 million (higher than E[S] by 0.33 million — the cost of reinsurance).
• σ(Y) ≈ 0.85 million (sharply lower than σ(S) = 1.48 million).
• VaR_{0.99}(Y) ≈ 2 + (max(S − 5, 0))_{0.99} ≈ 4.6 million (vs. 7.61 million without reinsurance).
• CVaR_{0.99}(Y) ≈ 5.5 million (vs. 9.52 million).

Conclusion: reinsurance increases expected costs by 20%, but decreases CVaR_{0.99} by 42%. A trade-off between profit and stability.

Real Applications

• Munich Re, Swiss Re, Hannover Re. The top 3 reinsurers in the world, each with $50–60 billion in premiums. They reinsure risks from natural disasters, aviation, marine insurance, cyber.
• Lloyd’s of London. Unique syndicate market: 70+ syndicates specializing in complex risks (art, maritime piracy, concert cancellation). Premium ≈ £45 billion/year.
• Florida Hurricane Catastrophe Fund. State “reinsurer” of Florida for hurricanes. After Andrew (1992) and Wilma (2005) was restructured, uses Cat Bonds.
• World Bank Pandemic Bond (2017). $320 million to cover pandemics in developing countries. Was triggered in March 2020 — payout of $195 million (but heavily criticized for overly strict triggers).

Task. An insurer with a risk portfolio, S ~ Lognormal(μ = 14, σ = 0.8). XL-reinsurance: priority d = 2 million, limit u = 5 million, premium π = 0.4 million. (a) By simulation (100,000 iterations) calculate E[cedent’s loss], σ, VaR_{0.99}, CVaR_{0.99} with and without reinsurance. (b) Plot histograms of both distributions on one graph. (c) Find the optimal priority d* to minimize CVaR_{0.99} under the constraint π ≤ 0.5 million (assuming π depends linearly on the trigger probability). (d) Compare XL and stop-loss for the same budget—which is more effective?