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 — everything the cedent’s book presents in trial $j$, before any term applies
Covered	$L_j^{\text{cov}}$	The covered, in-period slice left after the contract’s filter and coverage period
Gross	$L_j^{\text{gross}} = C(L_j^{\text{subject}})$	What the reinsurer assumes after the full composition $C$
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

Subject and covered differ whenever the filter is not the identity, and conflating them mislabels real dollars: on Trial 9, SunCoast presents a $272.1M subject, and Contract 1’s Florida-hurricane filter keeps $216.8M of it — the covered loss. Neither quantity is new: $C$ is the composition the dissection built up, its first two terms are the filter and the period, and the covered loss is their output — the point in the pipeline three terms short of the gross. Subtracting the gross from the subject gives the cedent’s retained loss, and the coverage boundary splits it in two:

$$L_j^{\text{subject}} = \underbrace{\left(L_j^{\text{subject}} - L_j^{\text{cov}}\right)}_{\text{never covered}} + \underbrace{\left(L_j^{\text{cov}} - L_j^{\text{gross}}\right)}_{\text{covered, kept}} + \underbrace{L_j^{\text{gross}}}_{\text{ceded}}$$

Every trial’s subject loss splits, dollar for dollar, into three parts: what the contract never covered, what it covered but did not absorb, and what the reinsurer took. The first two together are the cedent’s retained loss $L_j^{\text{ret}}$. The never-covered part carries no information about the contract — the filter and period drop it by construction, whatever the amounts. What the transforming terms — attachment, limit, aggregate cap — actually did is recorded entirely in how the covered loss divides between kept and ceded. The next section therefore measures the covered loss $L^{\text{cov}}$ — for Contract 1, SunCoast’s Florida-hurricane loss — and its kept part, written $L^{\text{ret,cov}} = L^{\text{cov}} - L^{\text{gross}}$. That split is the clearest single read on what a contract actually does: how much of the risk it covers actually moved.

Where is “net”?

Foundations also lists 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 four perspectives above are the whole story.

What the split buys you

Because the perspectives divide 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 covered stream’s expected loss is precisely the sum of the ceded and kept expected losses:

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

For Contract 1 across SunCoast’s 20-trial demo tier, that is $43.9\text{M} = 17.4\text{M} + 26.5\text{M}$: the layer absorbs about 40% of SunCoast’s expected covered (Florida-hurricane) loss, and SunCoast keeps the rest — along with everything the filter never covered.

Tail risk does not.

A tail is not a difference

It is tempting to measure the cedent’s residual tail risk as $\text{TVaR}_\alpha(L^{\text{cov}}) - \text{TVaR}_\alpha(L^{\text{gross}})$. Resist it. $\text{TVaR}_\alpha(L^{\text{gross}})$ averages gross over the years when gross is worst; $\text{TVaR}_\alpha(L^{\text{cov}})$ averages over the years when the covered loss 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 covered 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 covered 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 covered stream’s worst $k$ trials — and adds the ceded and kept 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:

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

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

A different but equally valid question is “how bad can the kept loss get on its own?” That is $\text{TVaR}_\alpha(L^{\text{ret,cov}})$ — the metrics from the toolkit applied to the kept distribution, whose tail is its own worst years. For Contract 1 at $90\%$ that is $124.7\text{M}$, larger than the $112.3\text{M}$ kept slice of the covered tail, because the kept loss peaks in different trials than the covered loss does. Each number has a job. The Co-TVaR slice attributes an agreed total: it says how much of the covered tail the contract removed and how much it left, and the parts reconcile exactly — the right frame for judging the contract’s tail relief, and the same frame the portfolio chapter uses to allocate capital across contracts. The standalone figure sizes the retained position itself: the cedent’s balance sheet suffers the kept loss’s own worst years whether or not the covered total was extreme in them, so capital and risk appetite for what stays behind read from $\text{TVaR}_\alpha(L^{\text{ret,cov}})$. Attribution takes the conditional number; holding the residual takes the standalone one.

The generator below reproduces every figure on this page — the per-trial covered 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.