Financial perspectives

We have defined the financial terms and composed contracts from them, and every one of those contracts does the same thing: it turns a stream of subject losses into a gross loss. But that gross loss is only half of what a contract produces — the same transformation also fixes what the cedent is left holding. Foundations named these loss perspectives in business terms; here we attach the canonical notation and ask which of them we can actually compute with a single contract in hand.

Perspective	Notation	Definition
Subject	$L_j^{\text{subject}}$	The contract’s input — the cedent’s covered, in-period loss in trial $j$
Gross	$L_j^{\text{gross}} = C(L_j^{\text{subject}})$	What the reinsurer assumes after the contract terms $C$
Cedent retained	$L_j^{\text{ret}} = L_j^{\text{subject}} - L_j^{\text{gross}}$	The part of the subject loss the contract did not absorb — what the cedent keeps

The first two we have computed all chapter long: $C$ is the composition of filter, coverage period, occurrence excess, aggregate excess, and scaling that the dissection built up. The third is new but free — it is the other two rearranged:

$$L_j^{\text{subject}} = L_j^{\text{gross}} + L_j^{\text{ret}}$$

Every trial’s subject loss splits, dollar for dollar, into the part the reinsurer took and the part the cedent kept. This is the clearest single read on what a contract actually does: how much risk moved.

Where is “net”?

Foundations lists a fourth perspective — the reinsurer’s net loss (gross minus its own recoveries). Net needs outward protection — retrocession, ILWs, cat bonds — which only arrives when one contract feeds another. Until then, gross is net. We pick it up under contract chaining; for a single contract, the three perspectives above are the whole story.

What the split buys you

Because the three perspectives partition the same dollars, a metric on one constrains the others — but how it constrains them depends on the metric.

Expected loss splits exactly. EL is an average, and averaging is linear, so the subject’s expected loss is precisely the sum of the ceded and retained expected losses:

$$\text{EL}(L^{\text{subject}}) = \text{EL}(L^{\text{gross}}) + \text{EL}(L^{\text{ret}})$$

For Contract 1 over SunCoast’s 20 trials, that is $43.9\text{M} = 17.4\text{M} + 26.5\text{M}$: the layer absorbs about 40% of SunCoast’s expected Florida-hurricane loss, and SunCoast keeps the rest.

Tail risk does not.

A tail is not a difference

It is tempting to measure the cedent’s residual tail risk as $\text{TVaR}_\alpha(L^{\text{subject}}) - \text{TVaR}_\alpha(L^{\text{gross}})$. Resist it. $\text{TVaR}_\alpha(L^{\text{gross}})$ averages gross over the years when gross is worst; $\text{TVaR}_\alpha(L^{\text{subject}})$ averages over the years when subject is worst. Those are generally different trials, so the subtraction nets two averages taken over mismatched tails.

How wrong it is depends on the data. At $\alpha = 90\%$, Contract 1’s two worst subject years (Trials 9 and 19) happen to be its two worst gross years too, so the subtraction lands on the right answer, $112.3\text{M}$. Drop to $\alpha = 80\%$ and the tails diverge — the fourth-worst subject year is Trial 2, but the fourth-worst gross year is Trial 3 — and the subtraction, $84.6\text{M}$, no longer matches the truth, $85.8\text{M}$. A formula that is right only when the tails coincide is not a formula.

The correct decomposition fixes one tail — the subject’s worst $k$ trials — and adds the gross and retained losses within it. Conditioning on a single fixed set makes expectation linear again, so the split is exact. This is a Co-TVaR (Euler) allocation — the same co-measure used to attribute portfolio capital to contracts:

$$\text{TVaR}_\alpha(L^{\text{subject}}) = \text{Co-TVaR}_\alpha^{\text{subj}}(L^{\text{gross}}) + \text{Co-TVaR}_\alpha^{\text{subj}}(L^{\text{ret}})$$

where $\text{Co-TVaR}_\alpha^{\text{subj}}(\cdot)$ averages its argument over the trials in the subject’s tail. The cedent’s retained tail risk, conditional on its own worst years, is therefore $\text{TVaR}_\alpha(L^{\text{subject}}) - \text{Co-TVaR}_\alpha^{\text{subj}}(L^{\text{gross}})$ — exact, with nothing left over.

A different but equally valid question is “how bad can the retained loss get on its own?” That is simply $\text{TVaR}_\alpha(L^{\text{ret}})$ — the metrics from the toolkit applied to the retained distribution, whose tail is its own worst years. For Contract 1 at $90\%$ that is $124.7\text{M}$, larger than the $112.3\text{M}$ retained slice of the subject tail, because retained peaks in different trials than subject does. Decide which question you are asking before you pick the number.

The generator below reproduces every figure on this page — the per-trial split, the exact EL decomposition, and the Co-TVaR identity at both $90\%$ and $80\%$:

helios_re/perspectives.py Python

▶ Run Reset

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We have dissected one CatXoL, formalized the building blocks, recomposed the canonical contract types, and seen how a single contract partitions risk. The final step is to combine many contracts: portfolio aggregation shows where diversification emerges and how marginal impact becomes the key decision metric.