Claims Allocation in Reinsurance

Calculating Shares for Proportional Reinsurance

As a brief recap, proportional reinsurance involves the cedent and reinsurer sharing premiums and losses in an agreed-upon proportion. The calculation method, however, differs significantly between Quota Share and Surplus Share treaties.

Quota Share Calculation

This is the most straightforward calculation. The reinsurer takes a fixed percentage of all premiums and pays the same fixed percentage of all losses for a given book of business.

Let's say an insurer has a 40% quota share treaty. This means the insurer cedes 40% of its business to the reinsurer and retains 60%.

• Insurer's Retention: 60%
• Reinsurer's Share: 40%

If a loss of $200,000 occurs on a policy covered by this treaty, the split is simple:

• Insurer pays: $0.60\times \$200,000=\$120,000$
• Reinsurer pays: $0.40\times \$200,000=\$80,000$

The video "Treaty Reinsurance" provides another clear, numerical example of this.

Surplus Share Calculation

Surplus share calculations are more involved because the proportion is not fixed across the entire portfolio; it is calculated individually for each risk. The key is to first determine the insurer's retained line and then calculate the ceding percentage for any policy that exceeds this line.

An insurer defines its retained line (e.g., $100,000), which is the maximum amount of risk it wishes to keep for itself on any single policy. The reinsurer then covers a "surplus" amount above this line, up to a specified limit (e.g., 4 "lines," or 4 x $100,000 = $400,000).

For any given policy, the percentage of risk retained by the insurer is:

The reinsurer's share is simply $1-\text{Percent Retained}$. This percentage is then applied to any loss that occurs.

The following reading provides an excellent, detailed example of this calculation for several different policies.

Let's break down Risk 3 from that example:

• Insured Value: $250,000
• Retained Line: $100,000
• Insurer's Retained Portion: $100,000 (the maximum it will keep)
• Reinsured Portion: $150,000 (the surplus)
• 1st Surplus Percent (Reinsurer's Share): $\frac{\$150,000}{\$250,000}=60\%$.

Therefore, if a $50,000 loss occurs on this policy (Risk 3):

• Insurer pays: $(1-0.60)\times \$50,000=0.40\times \$50,000=\$20,000$
• Reinsurer pays: $0.60\times \$50,000=\$30,000$

It is crucial to remember, as the text notes, that this is not excess insurance. The reinsurer pays 60% of the entire loss from the first dollar, not just the amount of loss exceeding the retained line. The line is only used to establish the percentage share.

Calculating Shares for Non-Proportional Reinsurance

In non-proportional (or Excess of Loss, XL) reinsurance, the calculation is driven by a monetary threshold (the retention), not a percentage. The reinsurer only pays if a loss exceeds this retention.

Excess of Loss (Per Risk) Calculation

The formula for splitting a single loss $X$ with a retention level $d$ and a reinsurance cover limit of $L$ is:

• Insurer pays: $\min (X,d)$
• Reinsurer pays: $\min (\max (0,X-d),L)$ or $\min ((X-d{)}_{+},L)$
• Unreinsured amount (paid by insurer): $\max (0,X-d-L)$

Let's consider a reinsurance treaty described as $400,000 xs $100,000. This means:

• Retention (d): $100,000
• Reinsurance Cover (L): $400,000
• Reinsurance layer: The reinsurer covers losses in the layer from $100,000 up to $500,000.

Now, let's calculate the split for three different loss amounts:

1. Loss of $80,000:

   • Insurer pays: $\min (\$80,000,\$100,000)=\$80,000$
   • Reinsurer pays: $\max (0,\$80,000-\$100,000)=\$0$

2. Loss of $350,000:

   • Insurer pays: $\min (\$350,000,\$100,000)=\$100,000$
   • Reinsurer pays: $\min (\$350,000-\$100,000,\$400,000)=\min (\$250,000,\$400,000)=\$250,000$

3. Loss of $700,000:

   • Insurer pays: $100,000 (retention) + ($700,000 - $100,000 - $400,000) (unreinsured amount) = $100,000 + $200,000 = $300,000.
   • Reinsurer pays: $\min (\$700,000-\$100,000,\$400,000)=\min (\$600,000,\$400,000)=\$400,000$ (the full limit).

The "Treaty Reinsurance" video walks through a similar example.

Stop-Loss (Aggregate XL) Calculation

The calculation for a Stop-Loss treaty is mathematically identical to an Excess of Loss treaty, but the input is different. Instead of applying to a single loss $X$, it applies to the aggregate loss $S={\sum }_{i=1}^N{X}_i$ for a portfolio over a specific period (usually one year).

If the aggregate retention is $M$ and the cover limit is $L$, then for a total annual loss of $S$:

• Insurer pays: $\min (S,M)$ + any amount over the cover limit.
• Reinsurer pays: $\min (\max (0,S-M),L)$

To understand this in context, it's helpful to distinguish it from "specific" stop-loss, which is just another name for per-risk Excess of Loss. The following video explains both concepts clearly.

For example, consider an insurer with an aggregate stop-loss treaty of $1M xs $5M.

• Aggregate Retention (M): $5,000,000
• Cover Limit (L): $1,000,000

If the total claims for the year amount to $6.2 million:

• The total loss $S$ is $6.2M.
• This exceeds the retention $M$ of $5M.
• The excess is $S-M=\$1.2M$.
• The reinsurer pays $\min (\$1.2M,\$1M)=\$1,000,000$.
• The insurer pays the first $5M (its retention) plus the amount above the reinsurance cover ($0.2M), for a total of $5.2 million.

Advanced Calculation Scenarios

In practice, and on exams, you will encounter more complex scenarios. Two common ones are the layering of different reinsurance treaties and the handling of loss adjustment expenses.

Inuring Reinsurance

Sometimes, one reinsurance treaty applies to the risk that remains after another treaty has already been applied. This is called inuring reinsurance. A common case is a Surplus Share treaty that inures to the benefit of a Per-Risk XL treaty.

To calculate the shares, you must apply the treaties in order:

1. First, calculate the insurer's net retained risk after the proportional (Surplus Share) treaty.
2. Then, apply the non-proportional (Excess of Loss) treaty to that net loss.

The Basics of Reinsurance Pricing PDF has an excellent worked example of this.

Handling Loss Adjustment Expenses (ALAE)

Allocated Loss Adjustment Expenses (ALAE) are the costs directly associated with settling a specific claim, such as legal fees. Reinsurance treaties must specify how ALAE is handled. There are two common methods:

1. Pro-rata with loss: The ALAE is split between the insurer and reinsurer in the same proportion as the loss itself.
2. ALAE as Part-of-Loss: The ALAE is added to the loss amount before applying the retention and limits. The sum (Loss + ALAE) is treated as the total loss amount.

The following reading provides clear examples that show the difference in calculation.

Let's re-examine Example 1 from the text:

• Loss = $640,000; ALAE = $320,000
• Treaty = $600,000 xs $400,000

Case 1: Pro-rata Treatment

• Loss in layer: $\max (0,\$640,000-\$400,000)=\$240,000$ goes to the reinsurer. The insurer retains $400,000.
• Reinsurer's share of loss is $\frac{\$240,000}{\$640,000}=37.5\%$.
• Reinsurer pays ALAE: $0.375\times \$320,000=\$120,000$.
• Total paid by reinsurer: $\$240,000+\$120,000=\$360,000$.

Case 2: ALAE as Part-of-Loss

• Total amount: $\$640,000+\$320,000=\$960,000$.
• Amount in treaty layer: $\min (\max (0,\$960,000-\$400,000),\$600,000)=\min (\$560,000,\$600,000)=\$560,000$.
• Total paid by reinsurer: $560,000.

As you can see, the method for handling ALAE can have a significant financial impact.

Conclusion

In this lesson, we have moved from theory to practice, focusing on the essential calculations for dividing claims under the most common reinsurance treaties.

Key Takeaways:

• Quota Share: Claims are split by a simple, fixed percentage.
• Surplus Share: The sharing percentage is calculated for each risk based on its value relative to the insurer's retained line, and this percentage is then applied to any loss.
• Excess of Loss / Per Risk: The insurer pays up to a monetary retention $d$, and the reinsurer pays the excess, $(X-d{)}_{+}$, up to a specified limit.
• Stop-Loss / Aggregate: The calculation is the same as Excess of Loss, but it applies to the total annual claims $S$ rather than a single claim $X$.
• Complexities: You must be prepared to handle layered treaties (inuring reinsurance) and different methods for allocating loss adjustment expenses (pro-rata vs. part-of-loss).

Preview of the Next Lesson

You are now equipped to determine the financial impact of reinsurance on individual or aggregate losses. In the next lesson, we will broaden our perspective to the collective risk model. You will learn to state and apply formulas for the moments (mean, variance) of the aggregate claims for an entire portfolio, a fundamental building block for modeling an insurer's overall financial position. This will set the stage for later analyzing how reinsurance affects this aggregate claims distribution and, ultimately, the insurer's probability of ruin.