Pricing a program standalone

The business problem

The post-model report did its job: the underwriter at Helios Re now knows what SunCoast’s book looks like before any contract touches it — a $275.6M expected annual loss, a hurricane tail driven by many-storm years, most of the exposure in Texas and Louisiana rather than Florida. The report ended where its job ended. This story picks up the next question, the one the underwriter is actually paid to answer:

The broker has proposed a structure and a price. Is SunCoast’s renewal adequately priced?

The proposed structure is the three-layer program SunCoast renews every year — Florida hurricane cover, an earthquake aggregate, and a whole-book backstop. The proposed premium is $38M across the three layers. To answer whether that is enough, the underwriter runs the program through the pricing pipeline: transform the subject loss into ceded loss, measure the ceded distribution, and convert those metrics into a technical premium. That price covers expected loss, expenses, and the cost of the capital the program ties up. If the proposed premium clears the technical premium, the program is profitable on its own terms. If it doesn’t, the underwriter negotiates or walks.

In Practice

The “standalone” in standalone pricing is the operative word: we price the program as if it were Helios Re’s only business, quoted — like every standalone result on this site — at the full 100% layer, before participation. We ignore the rest of Helios Re’s book — the Japanese earthquake layer, the European windstorm quota share, every other contract on the portfolio. That is deliberate. Standalone pricing is the baseline; the portfolio roll-up and marginal-pricing stories later in this chapter add the rest of the book back in and show how much the answer moves. As the chapter overview puts it: standalone pricing is marginal pricing with an empty base portfolio.

The pipeline

The standalone program-pricing pipeline is five stages — the same shape as pricing a single contract, with one wrinkle: the “transform” stage is now a whole program, whose net-of-inuring backstop depends on the two layers beneath it.

Every stage is a component from an earlier chapter, pointed at one task:

Stage	What happens	Built in
1. Subject loss	SunCoast’s YELT, validated and profiled	Risk profiling
2. Program	Three layers composed from filter, occurrence excess, aggregate excess, wired by inuring	Programs
3. Ceded distribution	Per-trial gross loss, by layer and for the program	Perspectives
4. Metrics	EL, TVaR, capital on the ceded distribution	Metrics
5. Technical premium	EL + expenses + cost of capital	this page

Prefer to poke at the finished model first? The interactive pricer further down lets you turn every dial — limit, cost of capital, confidence, which layers are in, diversified versus standalone — and watch the premium move. The steps below build up what each dial means.

Step 1: the structure to price

The Programs page built SunCoast’s program in full; here is the shape of it, since the price depends on it:

Layer	Covers	Structure	Proposed premium
C1	Florida hurricane, per occurrence	$30M xs $10M, 1 reinstatement	$20M
C2	CA/AZ earthquake, annual aggregate	$10M xs $15M	$5M
C3	All-US, annual aggregate, net of C1, C2	$50M xs $300M	$13M

C3’s “net of C1 and C2” is the inuring edge — it sees the book only after the first two layers have paid, so the same loss is never recovered twice. Running the program across the 20-trial demo tier gives the ceded loss each layer hands to Helios Re:

C1 · FL hurricane CatXoL C2 · CA/AZ earthquake AggXoL C3 · All-US AggXoL (net of C1, C2)

SunCoast's program ceded to Helios Re, trial by trial. C1 tracks Florida hurricane years; C2 is a steady earthquake band; C3 — the All-US aggregate, net of C1 and C2 — hits its $50M limit in the four heaviest whole-book years (trials 1, 2, 3, 13), pays a token $3.2M in trial 8, and is otherwise dark — including trial 4, where the gross book cleared its attachment but netting C1 and C2 pulled it back under. The layers peak in different trials, which is why the program's capital comes in below the sum of the layers'.

The picture already tells the pricing story in outline. C1 spikes in Florida hurricane years and is capped at $60M by its one reinstatement. C2 is a steady earthquake band, rarely more than its $10M limit. C3 — the backstop, net of C1 and C2 — reaches its $50M limit only in the four heaviest whole-book years (plus a token payment in trial 8), and stays dark in trial 4 even though the gross book cleared its attachment, because netting C1 and C2 pulled the subject back below $300M. The three layers rarely peak in the same trial. Step 4 turns that fact into the program’s capital number.

Step 2: measure the ceded distribution

With the ceded losses in hand, the Toolkit metrics apply unchanged — the ceded distribution is a loss distribution like any other. Expected loss and the tail behave exactly as the loss-set reconciliation in the risk-profiling story warned they would: EL adds across layers, the tail does not.

Distribution	EL	VaR(90%)	TVaR(90%)
C1 — FL hurricane	$17.39M	$60.00M	$60.00M
C2 — CA/AZ earthquake	$5.03M	$10.00M	$10.00M
C3 — All-US backstop	$10.16M	$50.00M	$50.00M
Sum of layers	$32.58M	—	$120.00M
Program	$32.58M	$82.26M	$87.28M

The expected losses add exactly: $17.39M + $5.03M + $10.16M = $32.58M, the program EL, because expectation is linear — trial by trial the program ceded is the sum of the layer ceded, so the means add. The tail behaves differently. The layers’ standalone TVaR(90%)s sum to $120M (each layer reaches its cap in its own worst trials), but the program’s TVaR(90%) is $87.28M — $33M lower. That difference is the same diversification the risk profile found between perils, now operating between layers: they reach their worst trials in different years, so the program’s tail comes in well under the sum of the layers’ tails. This is TVaR’s sub-additivity doing exactly what it promises.

Key Insight

Expected loss is blind to structure — it adds whether you stack layers, split perils, or shuffle the program, because a mean weights every trial equally and never isolates the tail. The tail does not add because of how it is defined. The tail metrics read off the worst trials, and the program’s worst trials are not the same trials as each layer’s worst. Each layer’s TVaR averages its own peak years, while the program’s TVaR averages the years where the total loss is worst — different years, different rows of the data. The two tails select different trials, so they do not combine by addition. Price off EL alone and you would charge the same for three correlated layers as for three independent ones. The tail is what tells them apart.

Step 3: from metrics to premium

The technical premium turns the metrics into a price. The house decomposition has three parts: the expected loss the reinsurer pays out on average, an expense load for running the business, and the cost of the capital the layer ties up:

$$\mathcal{P} = \underbrace{\text{EL}}_{\text{average payout}} + \underbrace{\epsilon \cdot \text{EL}}_{\text{expenses}} + \underbrace{K \cdot r_c}_{\text{cost of capital}}$$

The capital requirement $K$ is the unexpected loss — how far the tail runs beyond the expected level, which is what capital (not premium) has to absorb:

$$K = \text{TVaR}_\alpha - \text{EL}$$

The expected loss is already paid for by the EL term, so capital only has to stand behind the excess.

In Practice

This decomposition is one illustrative framework — a house choice, not the industry’s pricing formula. We use it because each part is a metric the pipeline has already computed, so the price stays a pure function of the ceded distribution. A production rate sheet adds components this page deliberately leaves out: brokerage and ceding commission on the revenue side, premium taxes, loads for parameter and model uncertainty (the modelled EL is an estimate, not a fact), and the underwriter’s rate-adequacy judgment against the market cycle. Clark’s study note is the standard reference for the full decomposition¹; Mata, Fannin, and Verheyen price the loss-sensitive premium features, such as reinstatement premiums and profit commissions, that this page omits². The premium-side features are planned as their own entries in the Financial Modelling term catalog, and the economics of the capital charge as its own reference.

With the house assumptions — $\alpha = 90\%$ confidence, a cost of capital rate $r_c = 10\%$, and an expense load $\epsilon = 5\%$ — each layer prices out like this:

Layer	Expected loss (EL)	TVaR(90%)	Required capital ($K$)	Technical premium	Proposed	Verdict
C1	$17.39M	$60.00M	$42.61M	$22.52M	$20M	underpriced
C2	$5.03M	$10.00M	$4.97M	$5.78M	$5M	underpriced
C3	$10.16M	$50.00M	$39.84M	$14.65M	$13M	underpriced
Sum	$32.58M	—	$87.42M	$42.95M	$38M	—

helios_re/program_pricing.py Python

▶ Run Reset

Priced layer by layer, each layer’s proposed premium sits just below its own technical premium, and the three technical premiums sum to $42.95M. But that sum is not the benchmark for the program: the layer sum prices each layer as if held alone, ignoring the diversification between them, so it overstates what the program should cost. The number to judge the program against is the program’s own technical premium, computed in the next step. The proposed premium falls short of that figure too — unsurprising for a peak-peril Florida hurricane program whose heavy multi-storm tail, flagged in the post-model report, is what drives the capital charge up.

Step 4: the program view — who gets the diversification

The $42.95M sum-of-layers technical premium implicitly assumes each layer is held alone — each carrying its own standalone capital, as if three different reinsurers each took one layer. But the program is placed as one structure. A reinsurer holding all three layers at 100% — the basis every figure in this story is quoted on — would carry the capital of the program distribution, not the sum of the layers’ — and because the layers rarely peak together, that is a much smaller number. Helios Re’s own position is different again: it holds unequal participations (14.5%, 20%, and 10%), and a mixed-weight position is not a uniform scaling of the program distribution. Capital for the book Helios Re actually holds is the portfolio roll-up’s job; the program view here prices the placement itself.

Layer, priced alone Naive sum Program, held together

Capital for SunCoast's program. Each layer priced on its own needs the capital shown; the three sum to $87.4M. But a reinsurer holding all three carries the capital of the program distribution — $54.7M, because the layers rarely hit their worst trials in the same year. The $32.7M gap (37% of the naive sum) is diversification, and it accrues only to whoever holds the whole program.

The three layers’ standalone capital sums to $87.42M. The program — held as one book — needs $54.71M. The $32.72M gap, 37% of the naive sum, is the diversification benefit, and the chart makes its ownership concrete: it accrues only to whoever holds the whole program. Split the layers across three reinsurers and it vanishes; each carries its full standalone capital. Pricing the program as one held book threads the diversified capital through the same premium formula:

	Expected loss (EL)	Required capital ($K$)	Technical premium
Sum of layers (no diversification credit)	$32.58M	$87.42M	$42.95M
Program (held together)	$32.58M	$54.71M	$39.68M

The diversification shaves $3.27M off the technical premium — the capital value of holding the layers together rather than apart. That $3.27M is the reinsurer’s to keep as margin or to compete away in the quote; either way, a system that prices each layer in isolation never sees it, and systematically overcharges for programs whose layers diversify.

Key Insight

Diversification is not an averaging trick that lowers expected loss — EL is identical ($32.58M) in both rows. It lowers the capital required to cover the risk: the tail of the whole is smaller than the sum of the tails, so less capital stands behind it, so the cost-of-capital term shrinks. This is the same mechanism the marginal pricing story scales from three layers in one program to thousands of contracts in one portfolio. Get it right at the scale of a program and the portfolio version is the same idea with more bookkeeping.

Step 5: the verdict

Pricing the program as a single held book — the diversified view from Step 4 — the technical premium is $39.68M against a proposed $38M. The standard pricing ratios say how the proposed premium stacks up:

Ratio	Value	Reading
Rate on line (premium / limit)	42.2%	$38M proposed premium against $90M of placed limit
Loss ratio ($\text{EL} / \mathcal{P}$)	85.7%	Expected loss takes 86 cents of every premium dollar
RAROC	6.9%	Risk-adjusted return on capital — profit after expected loss and expenses, per dollar of capital; below the 10% that capital costs

Two of these levels are artifacts of the demo portfolio. A 42% program rate on line sits far above the roughly 18% (low-attaching) to 3% (remote) schedule a real layered catastrophe program prices at, and an 86% expected loss ratio inverts the market pattern, where premium runs at a multiple of expected loss — both inherited from Helios Re’s arithmetic-convenience premiums.³ The ratios’ mechanics — what each one reads off the distribution and the quote — carry to real placements; their levels here do not.

The RAROC is the one to sit with, because it is where “underpriced” gets its precise meaning. At $38M the program is not a loss-maker on a claims basis — the loss ratio is a healthy 86%, and after expenses the premium still returns a positive amount on capital. But that return is 6.9%, and Helios Re’s cost of capital is 10%. A program that earns 6.9% on capital that costs 10% destroys value with every dollar it ties up, even though it never “loses money” on a claims basis. That is the gap the technical premium closes: at $39.68M the program would earn exactly its cost of capital. The proposed $38M leaves it $1.68M short — close enough to be a live negotiation, not a walk-away, but short.

The verdict is the method’s, not a quote. At 20 trials the program TVaR — and with it the capital charge and the $1.68M gap — rests on the worst two trials, so this walkthrough demonstrates how a desk reaches its answer; a real desk would compute the same figure from a production-scale run before treating the margin as a negotiating position.

Consider

The single 10% cost of capital is a deliberate simplification, and the charge is contested on two fronts. The first is what kind of cost it is: the frictional-cost view (Merton–Perold; Hancock–Huber–Koch) holds that a cedent should be charged only for the deadweight costs of holding capital — double taxation of investment income, agency costs, regulatory drag — not the full opportunity cost. Diversified shareholders are already compensated for systematic risk through market prices. The second is that the capital does not sit idle: held against the treaty it earns an investment yield, so the net charge is closer to $K \times (r_c - y)$, where $y$ is the yield on the held capital — sometimes called the return on allocated capital. Both refinements move the level of $r_c$, not the structure of the pricing. A dedicated treatment of capital economics — frictional versus opportunity cost, investment income, basis risk, the hurdle rate — is planned as its own reference.

Remember

Coming up short on a standalone basis is a baseline, not a decision. The technical premium makes the assumptions explicit — change the cost of capital or the confidence level and the number moves — but it deliberately ignores the rest of Helios Re’s book. The capital this program ties up on its own is more than the incremental capital it ties up as part of an already-diversified portfolio. A US-hurricane program sitting beside Japanese earthquake and European windstorm business adds far less to the book’s required capital than it needs alone — enough to lift that 6.9%, and possibly to clear the hurdle. Whether it actually does is a question this story cannot answer — it needs the portfolio tools the marginal pricing application will build. Standalone pricing tells you where you stand before that conversation, not whether to sign.

Price it yourself

Everything above priced one structure, one way. But the technical premium is a number given a cost of capital rate and a confidence level — both policy choices rather than facts about the risk. It depends equally on the structure: the limit placed, which layers are in the program, and whether the capital is held as a diversified book or carried layer by layer. The explorer below turns every one of those into a dial and reprices the program live. It starts on the as-placed structure — all three layers, the diversified capital basis, the house $\alpha = 90\%$ and $r_c = 10\%$ — so the opening reading is the $39.68M technical premium from Step 4, dashed against the $38M proposal.

Layers in the program

C1 · FL hurricane C2 · CA/AZ earthquake C3 · All-US backstop

C3 limit $50M xs $300M Confidence α 90% drives TVaR Cost of capital 10%

Capital basis

Diversified (program) Standalone (sum of layers)

Reset to as-placed

Underpriced Technical premium $39.68M > proposed $38M — short by 1.68M

Expected loss$32.58M

TVaR(90%)$87.28M

Capital K$54.71M

Technical premium$39.68M

RAROC6.9%

Diversification saving$3.27M

The build-up bar is the technical premium: expected loss, plus a 5% expense load, plus the cost of capital on the program's unexpected loss. The dashed line is the broker's proposed premium — held fixed at the quote for the as-placed structure, so the dials read as fair-price sensitivity, not a re-quote. Turn them and watch the technical premium move: raise the cost of capital and the capital term swells; pull the C3 limit down and both its expected loss and its tail shrink (the \$38M ask does not follow, which is why the verdict can flip — a real broker would re-price a smaller limit); push confidence α further into the tail and TVaR climbs. Switch from diversified to standalone and Capital K jumps from the program's own figure to the sum of the layers' — that jump is the diversification benefit (a capital amount); run through the cost-of-capital charge it becomes the smaller saving on premium in the tiles, which only the holder of the whole program collects. The lower strip shows the program's per-trial ceded loss; the highlighted bars are the worst trials that the TVaR averages.

Three experiments repay the effort. Drop the cost of capital toward 6% and watch the technical premium fall below the $38M line — the deal flips from underpriced to adequate, the most consequential dial on the page. Pull the C3 limit down from $50M and the backstop’s expected loss and tail both shrink, dragging the fair price with them (the proposed $38M stays put, since it is the broker’s quote for the placed structure — so this is a sensitivity, not a re-quote). Flip the capital basis from diversified to standalone and the capital behind the program jumps from $54.71M to $87.42M. That $32.72M is the diversification benefit, worth $3.27M once it flows through the 10% cost-of-capital charge — and it vanishes the moment the layers are held apart.

How sensitive is the answer?

The explorer is for building intuition; to put the answer on the record — rather than wherever you happen to leave the sliders — the sweep below pins the two policy dials, cost of capital and confidence, to a fixed grid and reprices the program at each point:

helios_re/program_sensitivity.py Python

▶ Run Reset

Here the verdict is genuinely assumption-sensitive — and that is the point. At the house 10% cost of capital the program is underpriced by $1.68M. But drop the cost of capital to 6% and the technical premium falls to $37.5M, below the proposed $38M, and the deal clears. The confidence level moves the number far less: across the $\alpha$ sweep the program stays underpriced. So whether this program clears on a standalone basis is decided by the cost-of-capital policy, not by the loss data — a reminder that the technical premium is a framework for making assumptions explicit, not a single objective price.

Watch Out

The $\alpha = 95\%$ row is a trap. With only 20 trials the 95% tail is a single year, so TVaR, VaR, and the worst trial all collapse to the same $92.3M — the resolution floor the risk profile flagged. Even TVaR(90%), which averages the worst two trials, leans on a thin tail. The pricing method is sound at any trial count; the numbers here are only as trustworthy as 20 trials allow. A production book prices peak-peril capital off tens of thousands of trials precisely so the tail metric is not one or two storms away from noise. The demo scale section shows the same quotes deepening on the demo data’s 1,000-trial tier.

At production scale

The pipeline is identical at scale; only the magnitudes change.

Trials and rows. SunCoast’s program ran across 20 trials in milliseconds. A production book runs the same composition across 100,000 to 1,000,000+ trials — and, as the risk profile noted, the row count is the trial count times the occurrences per trial, so a single cedent’s loss set can reach 10⁹ rows. The arithmetic does not change shape: each layer is still a filter plus a clip, the metrics are still a sort and a mean. What changes is that the data stops fitting in memory, and columnar storage, chunked aggregation, and pushing the work into a query engine start to matter.

Evaluation order. The one constraint the program imposes is the inuring edge: C3 cannot be priced until C1 and C2 are evaluated. At scale this is a topological sort over the program’s dependency graph — cheap, because the graph is tiny — wrapped around the genuinely expensive part, which is evaluating each layer across millions of trials. That part stays embarrassingly parallel: partition by trial, evaluate, reduce.

Reproducibility. The chapter’s standing principle — that a financial model is a pure function of its inputs — holds here without modification: the price is a pure function of (subject loss, program structure, pricing parameters). Same inputs, same premium, to the cent. Change the cost of capital and the whole book reprices deterministically — no analyst re-running cells in the right order, no two spreadsheets disagreeing about whose number is right.

Consider

The cleanest way to think about a pricing engine is as a single pure function with three arguments — losses, structure, policy. Everything in this story is that function evaluated once, for one program. The portfolio roll-up is the same function over a base book; marginal pricing is the same function evaluated twice (with and without the new contract) and differenced. Building the function once, cleanly, is what makes the rest of the chapter a matter of calling it differently rather than rewriting it.

What carries forward

Standalone program pricing is five stages — subject loss → program → ceded distribution → metrics → technical premium — and it answers a precise question: is this program adequately priced on its own? For SunCoast’s renewal the answer is no, if narrowly: a $39.68M technical premium against a $38M ask — short by $1.68M at the house cost of capital, earning 6.9% on capital that costs 10%. Along the way it surfaced the number that matters most — the $32.72M of capital diversification that the layers earn by not peaking together, and that only the holder of the whole program collects.

That diversification is the thread out of this story. We found it within one cedent’s program, between three layers. The same mechanism operates between SunCoast’s program and every other program on Helios Re’s book — Japanese earthquake, European windstorm, the lot. Measuring it across the whole portfolio, rather than asserting it, is the portfolio roll-up story; pricing one new contract against that diversified book — the number that decides whether SunCoast’s renewal actually clears — is the marginal pricing application the rest of this Applications chapter builds toward.

Footnotes

1. David R. Clark, “Basics of Reinsurance Pricing”, Casualty Actuarial Society study note, 1996, revised 2014 — the standard study-note treatment of reinsurance pricing in full: loss costs, ceding commissions, expenses, and profit loads, plus the judgment the basic tools require. ↩

2. Ana J. Mata, Brian Fannin, and Mark A. Verheyen, “Pricing Excess of Loss Treaty with Loss Sensitive Features: An Exposure Rating Approach”, Institute and Faculty of Actuaries — when a treaty carries loss-sensitive features, the premium and expenses become random variables that depend on the aggregate loss, and profitability is assessed in expectation across them. ↩

3. Luyang Fu and C. K. Khury, “Optimal Layers for Catastrophe Reinsurance”, Variance 4(2), 2010 — prices a layered catastrophe program from roughly 18% rate on line at the lowest attachment down to about 3% at the most remote, with premium a large multiple of each layer’s expected loss; Kenneth A. Froot, “The Market for Catastrophe Risk: A Clinical Examination”, Journal of Financial Economics 60(2–3), 2001 — documents that premium multiple market-wide. ↩