A catastrophe bond's secondary spread identifies roughly its spread divided by expected loss. Safarveisi establishes that limit through a short expansion of the survival probability. Anyone reproducing the paper should put most of their effort into the cash-flow convention. The Riemannian optimizer needs much less attention.
Eighteen quarterly quotes, one bond
The paper starts with a physical loss model and a two-parameter family of pricing measures. A compound Poisson process generates losses, with generalized Pareto severities. Physical intensity is 0.02 per year; the GPD has shape 2.2 and scale 2.6×10^7 USD. Each pricing measure multiplies event frequency by 1+λ′ and applies a Wang distortion, a severity tilt controlled by κ. The observed Treasury curve supplies discounting. That leaves the interest-rate premium outside the selection problem.
Financial entropy is the selection criterion. Under exponential utility, it equals minus the KL divergence from the physical measure. Used alone, it selects the physical measure and produces a zero premium. Safarveisi brings market prices into the problem through two formulations. The pooled formulation maximizes entropy less squared pricing errors weighted by 1/(2α); the date-wise formulation matches each quarter's price exactly, then chooses the measure nearest to physical.
The sample contains 18 quarter-end indicated spreads for Golden State Re II 2014-1. Sponsored by the California State Compensation Insurance Fund, the bond covers California earthquakes. Its indications span September 2014 to December 2018. They come from Lane Financial annual reviews and represent quarter-end average secondary-market indications. Issuance was at 2.20%; subsequent indications range from 1.65% to 4.56%.
The bond being priced is hypothetical. Its aggregate trigger, D = 2.483×10^8 USD, matches the reference bond's 0.49% annual attachment probability. The paper's pooled estimates are λ′ = 2.549 and κ = 1.037, giving a risk-neutral trigger intensity 9.20 times physical. Across individual dates, that multiple ranges from 6.58 to 10.45, apart from a spike to 18.26 at end-2015.
What can the spreads identify?
Almost all the price dependence on (λ′, κ) passes through ℓ: the risk-neutral rate of individual events large enough to breach D. This is exact for a per-occurrence trigger. With an aggregate trigger, the remaining terms are of order (λ̂t)². At the pooled optimum, λ̂T is 0.302. Across the 18 dates, the largest price gap between per-occurrence and aggregate triggers is 0.0459 per 100.
Ignoring that structure changes the fitted parameters dramatically. Of 100 random starts for unregularized least squares, 97 converge to λ′ ≈ 24.78 and κ ≈ −0.697. The resulting multiple is 8.70, with RMSE 1.418. But λ̂T reaches 2.19 there, outside the approximation's useful range. The paper attributes this fit to accumulated sub-threshold losses.
Entropy pulls the solution back to the pooled estimate. KL falls to 0.3275 from 5.55 at the naive solution, while the multiple changes only from 8.70 to 9.20. Meanwhile, λ′ drops from 24.78 to 2.549, roughly a tenth of its previous value.
Safarveisi explicitly assigns the frequency-severity split to the selection rule. The abstract calls the 53.5% to 63.7% frequency share "a property of the selection principle and the trigger level rather than of the market data". In the same passage, the defence is that entropy resolves the ambiguity. It does so by selecting the KL-nearest point along a curve whose points the market cannot distinguish.
The trigger level changes that choice. At D = 6×10^7, the mean frequency share increases to 71.6%. For a per-occurrence trigger, every date-wise solution follows a single path indexed by ℓ; the aggregate trigger approximately shares this property. This assumes a unique minimiser, which the paper checks numerically. Quarter-to-quarter co-movement between λ′ and κ therefore conveys nothing beyond the multiple.
Coupon choice cuts the multiple from 9.20 to 4.30
Two decisions have the largest effect on the multiple: the conversion of a quoted spread into a price, and recovery on trigger. In the base case, the coupon equals the spread plus the risk-free par yield for the remaining term. The price must then equal par. Li et al.'s convention instead makes the coupon equal to the spread. That change lowers the pooled multiple from 9.20 to 4.30. It also leaves the exact-fit problem feasible on only 14 of 17 dates.
Changing the discount curve has little effect. CIR discounting produces 9.15, compared with 9.20 using the Treasury curve.
The base case admits a simple first-order interpretation: par requires ℓ(1−ζ) ≈ s, with recovery on trigger set at ζ = 0.5. The model's expected loss is 0.245%, close to the bond's reported 0.25%. At issuance, the date-wise multiple is 8.87, compared with a spread-to-expected-loss ratio of 8.81. At end-2015, those figures are 18.26 and about 18.6.
Recovery sensitivity follows the same arithmetic. Setting ζ = 0.25 produces 6.13; setting ζ = 0.75 produces 18.45. These are close to scaling 9.20 by 0.5/0.75 and 0.5/0.25.
The optimizer itself behaves well. Algorithm 1 takes 27 iterations to converge and matches an independent quasi-Newton solution within 1.7×10^−5. The date-wise solver matches a one-dimensional search within 9.7×10^−7.
Implementation choices
The paper documents the following decisions, each of which matters to someone implementing the valuation:
- Regularization. A noise target of 0.25 per 100 represents half a typical bid-ask range. No fit meets it: the best RMSE is 1.418. The paper therefore sets α = 83.2 using the in-sample noise estimate, targeting a sum of squared errors of 40.71. Varying that α from 0.1x to 10x moves the multiple between 4.66 and 9.96. At 10x, RMSE reaches 3.448.
- The end-2015 indication. The pooled residual is 6.25 per 100 for this quote; every other residual is at most 0.80. Removing it along with the maturity-date quote reduces RMSE from 1.504 to 0.335. The author interprets it as a year-end effect in the averaged indications.
- Loss model basis. The severity parameters are point estimates drawn from US earthquake losses over 1906 to 2005. The bond instead covers a California workers' compensation modelled index. The paper acknowledges this basis difference as a limitation. A 20% change in shape or intensity produces multiples from 7.66 to 11.57.
- Grid and trigger. Coupons follow a stylised quarterly grid, with maturity 17/4. Because the final quote falls at maturity, it supplies no information. The usable sample is therefore 17 informative quotes.
The result is a valuation. The bond trades at par on issuance, against an actuarial value of 107.876 using physical survival and market discounting. That makes the premium 7.876 per 100. Under the pooled measure, the price is 99.694. Every date has a positive premium, ranging from 0.357 to 12.620. Any liquidity or year-end component is absorbed into it as well; a single bond cannot separate those components.
We did not run this.
Treasury yields are readily available, and the paper prints the loss parameters and trigger terms. The missing input for us is the Lane Financial series of quarter-end indications.
Safarveisi leaves for later the result that could make the frequency-severity split tradeable. Bonds exposed to the same peril at different attachment points sample the risk-neutral loss distribution at different levels. That cross-section could identify what this bond's 17 informative quotes cannot.