Going Spectral

Why and how to implement spectral pricing.

Abstract

Going Spectral: Why and how to implement spectral pricing challenges the constant cost of capital (aka risk-adjusted return on capital or RAROC) assumption commonly used insurance pricing—an assumption the weighted average cost of capital calculation reveals as invalid. The presentation demonstrates the advantages of using spectral risk measure (SRM) pricing rules, illustrating how SRMs not only generalize traditional methods like CoXTVaR but also address their limitations. Instead of prescribing a single solution, SRM methods offer a range of results corresponding to different risk appetites. Capital allocation is interpreted as sourcing capital from investors with the lowest return requirements across the risk spectrum.

There is also a shorter verion of this presentation.

Setup Code

Running the examples requires loading some Python libraries. See blog post version of this presentation for the code.

1 Listen to Management

Management’s Complaint

Too many actuarial pricing models

• over-weight tail catastrophe risk, and
• under-weight earnings-hit volatility risk.

This behavior drives portfolio steering and reinsurance purchase decisions materially at odds with management’s risk preferences.

Diagnosis

Management’s complaint is entirely explained by an assumption that the cost of capital is constant.

The assumption is often implicit and hidden.

Prescription

Stop assuming the cost of capital is constant!

There are other good alternatives … enter Spectral Risk Measures (SRMs).

2 ToyCo Model

Example: Scenario Losses and Portfolio Statistics

Table 1: Assumptions for ToyCo two-unit portfolio across 10 equally likely scenarios.

	X1	X2 net	X2 ceded	X2	total
0	36.0	0.0	0.0	0.0	36.0
1	40.0	0.0	0.0	0.0	40.0
2	28.0	0.0	0.0	0.0	28.0
3	22.0	0.0	0.0	0.0	22.0
4	33.0	7.0	0.0	7.0	40.0
5	32.0	8.0	0.0	8.0	40.0
6	31.0	9.0	0.0	9.0	40.0
7	45.0	10.0	0.0	10.0	55.0
8	25.0	40.0	0.0	40.0	65.0
9	25.0	40.0	35.0	75.0	100.0
EX	31.700	11.400	3.500	14.900	46.600
CV	0.215	1.299	3.0	1.545	0.455

• Unit X1 is non-cat
• Unit X2 is cat exposed, shown split into net and ceded to 35 xs 40 cover

Example: Pricing

Keep these numbers in mind!

• Market pricing for the total portfolio earns a 15% return on a fully capitalized basis

• Premium $P$, capital $Q$, total assets $a=P+Q=max(L)=100$, and return $r=0.15$ are related by $$P=\mathsf{E}[L]+r(a-P)⟹P=v\mathsf{E}[L]+dmax(L)$$ where $v=1/(1+r)$ and $d=r/(1+r)$ are the discount rate and factor

• Facts imply $P=53.565$, loss ratio 87%, and premium to surplus ratio of 1.15:1

3 The “Industry Standard Approach”

Portfolio Pricing

In order to make insurance a trade at all, the common premium must be sufficient to compensate the common losses, to pay the expense of management, and to afford such a profit as might have been drawn from an equal capital employed in any common trade.”

 Adam Smith, Book 1, Ch X, Part I, 5th Edition, 1789

Portfolio Pricing

Adam Smith’s pricing rule

• Portfolio pricing rule

• Common loss = expected loss

• Dollar cost of capital (CoC), reflects

  • different forms of capital: equity, debt, reinsurance;
  • each with different cost rates.

• Premium is a technical price and excludes expenses, investment income

Determining the Cost of Capital

Constant CoC assumption

• Constant cost of capital (CCoC) is a standard assumption, ignoring alternatives

  • Vary across lines of business (too hard)
  • Vary across layers of capital (debt, equity, reins.)

• CCoC of capital $r$ is called target return on capital, WACC, opportunity cost of capital, average CoC

• CCoC assumption
  Dollar CoC = (average CoC) × (amount of capital)

• CCoC Premium = expected loss + (avg CoC) x (amount of capital)

CCoC Critique

Capital cost and capital use both vary by layer

• Different costs manifest in WACC calculation!
• Published daily in credit yield curves (Treasury $<$ AAA $<\cdots <$ Junk yields at fixed duration)
• Capital allocation methods assume all capital has the same cost
• Average CoC $\times$ amount of capital
  • $=$ (Avg cost of capital across layers) $\times$ (Avg use of capital across layers)
  • $\ne$ Average(cost of capital by layer $\times$ use of capital by layer)
• Compare $\mathsf{E}[XY]\ne \mathsf{E}[X]\mathsf{E}[Y]$ unless $X$ and $Y$ are uncorrelated

• Cost and use are correlated because higher layers are bigger and cheaper (per unit of capital)

• Cat exposed layers use a lot of cheap capital $⟹$ CCoC will overstate cost of cat risk and be too tail centric

CCoC Pricing Rule Formula

“The [actuarial analyst] must understand symbols and speak in words.” John Maynard Keynes

For premium $P$, expected loss $\mathrm{EL}$, capital $Q$, assets $a$, and cost of capital $r$

• $P=\mathrm{EL}+rQ$

• $\phantom{P}=\mathrm{EL}+r(a-P)$

• $\phantom{P}={\displaystyle \frac{1}{1+r}\mathrm{EL}+{\displaystyle \frac{r}{1+r}a}}$

• $\phantom{P}=v\mathrm{EL}+dmax(\mathrm{loss})$

using $v$ and $d$ for risk discount factor and rate of discount, $v+d=1$, $d=rv$

Price of binary loss of $1$ with probability $s$ is $g(s)=vs+d$

CCoC Pricing Rule Formula

CCoC pricing rule has strange formulation

for a 15% target return, $v=1/1.15=0.87$ and $d=0.13$

• Rule is additive for independent losses!
• Rule does not “discount” all but max loss because coverage is for a given loss and larger losses
• $g(s)=vs+d>s$ for all $s$ shows a positive margin

• Interpretation: re-weighting of scenarios or probabilities?
  • Outcome x (Adjustment x Probability) not (Outcome x Adjustment) x Probability
  • 0.87 x EL: weight all scenario (probabilities) by factor of 0.87
  • Increase worst possible outcome probability to 0.13

CCoC Pricing Rule Formula

• Left plot shows CCoC risk (probability) adjustment factor distortion function relative to base at 1 (dashed line), equals $g^{\prime }(s)$
• All outcome probabilities except the largest (“100%-percentile”) discounted by 0.87
• Largest outcome probability increased to 0.13 (red star)

• Right plot shows ToyCo total loss outcome as a quantile plot
• Low (good) loss outcomes shown on left
• High (bad) loss outcomes shown on right
• $s\leftrightarrow p=1-s$

Obvious Question

What about using other distortions?

The all-or-nothing nature of CCoC seems unsatisfactory and is numerically unstable

• What other options are available?

• How are different choices interpreted?

• Do business decisions vary materially by distortion?

Alternatives to CCoC Assumption

Re-weight using risk-adjusted probabilities

Imagine spreadsheet of equally likely scenarios. Want to re-weight with risk-adjusted probabilities. What properties must rational adjusted probabilities possess?

1. Non-negative

2. Sum to 1

3. Weight should increase with increasing loss

All bad outcomes that occur at a lower losses also occur for any larger loss

Spectral distortion functions are a systematic way to derive weights satisfying 1-3

4 The Spectral World: Why and How

Spectral Pricing

• SRM pricing uses a distortion function to add a risk load

• Distortion functions make bad outcomes more likely and good ones less, resulting in a positive loading

• Distortions express a risk appetite, Mango called them a “care/don’t care curve”

• Many existing methods, including CCoC and CoXTVaR, are special cases of SRMs

• Portfolio SRM premium has a natural allocation to individual units

• Different distortions can produce same total portfolio pricing but have materially different natural allocations to units, reflecting distinct risk appetites

• Different allocations, in turn, drive materially different business decisions

Spectral Pricing

• A distortion function $g:[0,1]\to [0,1]$ maps a probability to a larger probability, used to fatten the tail

  • Increasing
  • Concave (decreasing derivative)

• $g(s)$ can be interpreted as the (ask) price to write a binary risk paying 1 with probability $s$ and 0 otherwise; c.f. $g(s)=d+sv$ for CCoC

• $S(x)=Pr(X>x)$, the survival function of a random variable $X$ on sample space $\mathrm{\Omega }$

• Recall expected Loss cost $\mathsf{E}[X]={\displaystyle {\int }_0^{\mathrm{\infty }}xf(x)dx={\displaystyle {\int }_0^{\mathrm{\infty }}S(x)dx}}$

Spectral Pricing

• $g(S(x))>S(x)$ is the risk-adjusted survival function

• Spectral pricing rule associated with a distortion $g$ is given by $$\rho (X)={\int }_0^{\mathrm{\infty }}g(S(x))dx$$ It is intrepreted as a price, technical premium, risk-adjusted loss cost, or risk measure

• Integration by parts trick gives an alternative expression $$\rho (X)={\int }_0^{\mathrm{\infty }}x{g}^{\prime }(S(x))f(x)dx=\mathsf{E}[X{g}^{\prime }(S(X))]$$ which makes the spectral risk adjustment $g^{\prime }(S(X))$ explicit

• $g^{\prime }(s)\ge 0$ measures risk attitude to losses with an exceedance probability $s$ across the spectrum of losses as $s$ varies

Distortion Functions and Insurance Statistics

Figure 1: A distortion in orange compared to expected loss in blue (left). Relation to meaningful insurance statistics for each layer of loss (right).

Spectral Pricing Rules Have Nice Properties

All risk measures with the following four properties are SRM rules

1. Monotone: Uniformly higher risk implies higher price

2. Sub-additive: diversification decreases price

3. Comonotonic additive: no credit when no diversification; if out-comes imply same event order, then prices add

4. Law invariant: Price depends only on the distribution

Since layer losses are comonotonic (they are increasing functions of total loss), SRM pricing adds-up by layer

SRM Pricing Adds Up Pricing by Layer

• Low layers, below horizontal line at $x$ (left) have high EL and premium but low capital, driving low margin to loss but high return and high leverage

• High layers, above the line, have low EL and premium but high capital, driving high margin to loss but low return and low leverage

SRM Pricing Adds Up Pricing by Layer: ToyCo Example $x=60$

• ToyCo losses (left and center)
• Proportional hazard distortion calibrated to 15% overall return at $a=100$, $g(s)=s^{0.72}$

SRM Pricing has a Natural Allocation to Sub-units

• If $X=\sum _i{X}_i$, define the natural allocation to unit $i$ as $$\mathsf{NA}(X_i)=\mathsf{E}[X_i{g}^{\prime }(S(X))]$$

• Example: TVaR corresponds to $g(s)=min(1,s/(1-p))$
  • $\rho (X)={\mathsf{TVaR}}_p(X)$
  • $\mathsf{NA}(X_i)={\mathsf{CoTVaR}}_p(X_i)$

• The natural allocation pricing has nice properties
  • It is natural because it involves no additional assumptions
  • It adds-up because the sum of natural allocations is the original SRM price
  • It equals marginal volume pricing when marginal pricing is well defined

Risk-Adjusted Probabilities Reflecting “Volatility Aversion”

Volatility aversion…

worries about

• Earnings miss
• Plan miss
• Bonus miss

…and is concerned with outcomes near the mean

A volatility averse SRM applies maximal weight, consistent with (1)-(3) to a scenario at exceedance probability around 50%

Corresponding risk-adjusted probabilities

• Result: Tail Value at Risk at $p$ around 0.27; graph is $g^{\prime }$
• TVaR pricing: ignore best $\approx 27\mathrm{\%}$ of outcomes and average the rest
• Comparison with usual XTVaR approach using $p\approx 0.99$ presented in Appendix

SRMs to Reflect a Range of Risk Appetites

Five parametric families of distortion functions

• CCoC $\to$ PH (Proportional hazard) $\to$ Wang $\to$ dual $\to$ TVaR
  • Express progressively less tail-centric appetite
  • Five different one-parameter families of risk-adjusted probabilities, see Section 7.3 for formulas
  • Each easily parameterized to desired pricing
• Graph shows weight adjustments $g^{\prime }(s)$ for comparably calibrated distortions

• Dual distortion popular in applications: bounded, but weights all scenarios

CCoC Portfolio Pricing with XTVaR Capital Standard

XTVaR is a special case of SRM pricing

• CCoC implementation with XTVaR capital: $$P(X)=\mathsf{E}[X]+r{\mathsf{XTVaR}}_p(X)=(1-r)\mathsf{E}[X]+r{\mathsf{TVaR}}_p(X)$$

• Corresponding distortion is $$g(s)=(1-r)s+rmin(1,s/(1-p))$$
  • Weight $1-r$ applied to all events: risk neutral part
  • Weight $r$ applied to $p$-tail events: extremely risk averse
  • Example of a bi-TVaR, an average of two TVaRs, since $\mathsf{E}[X]={\mathsf{TVaR}}_0(X)$

• Easy to check corresponding $\rho (X)=(1-r)\mathsf{E}[X]+r{\mathsf{TVaR}}_p(X)$

CCoC Portfolio Pricing with XTVaR Capital Standard

XTVaR natural allocation is CoXTVaR

• Corresponding natural allocation is simply CoXTVaR pricing $$\mathsf{NA}(X_i)=(1-r)\mathsf{E}[X_i]+r\mathsf{CoTVaR}(X_i)=\mathsf{E}[X_i]+r\mathsf{CoXTVaR}(X_i)$$

• Shows SRM approach generalizes existing methods

5 ToyCo Numerical Example

Example: Scenario Losses and Portfolio Statistics (recap)

Table 2: Assumptions for ToyCo two-unit portfolio across 10 equally likely scenarios.

	X1	X2 net	X2 ceded	X2	total
0	36.0	0.0	0.0	0.0	36.0
1	40.0	0.0	0.0	0.0	40.0
2	28.0	0.0	0.0	0.0	28.0
3	22.0	0.0	0.0	0.0	22.0
4	33.0	7.0	0.0	7.0	40.0
5	32.0	8.0	0.0	8.0	40.0
6	31.0	9.0	0.0	9.0	40.0
7	45.0	10.0	0.0	10.0	55.0
8	25.0	40.0	0.0	40.0	65.0
9	25.0	40.0	35.0	75.0	100.0
EX	31.700	11.400	3.500	14.900	46.600
CV	0.215	1.299	3.0	1.545	0.455

• Unit X1 is non-cat
• Unit X2 is cat exposed, shown split into net and ceded to 35 xs 40 cover

Example: Cat Pricing Across a Range of Risk Appetites

Mechanics

	p	S	gS	q	dx
loss
22.0	0.100	0.900	0.975	0.025	22.0
28.0	0.100	0.800	0.923	0.051	6.0
36.0	0.100	0.700	0.853	0.070	8.0
40.0	0.400	0.300	0.434	0.420	4.0
55.0	0.100	0.200	0.299	0.134	15.0
65.0	0.100	0.100	0.155	0.145	10.0
100.0	0.100	0.0	0.0	0.155	35.0
Total	46.600	46.600	53.565	53.565	100.0

Columns

• p : probability of each total loss value
• S : the survival function
• gS: distortion applied to $S$ $g(s)=1-(1-s{)}^{1.59515}$
• q : backward differences $dgS$ of gS, fill value 1 at the top
• dx: backward differences of loss fill value 0

Total row calculation with $x$ denoting loss value

• p : $\sum _{x}x{p}_x$, expected loss
• S : ${\int }_0^{100}S(x)dx$, expected loss
• gS: ${\int }_0^{100}g(S(x))dx$, premium (risk adjusted expected loss)
• q : $\sum _{x}x{q}_x$, premium
• dx: $\sum _{x}dx=a$

Example: Cat Pricing Across a Range of Risk Appetites

Natural allocation pricing and loss ratios implied by dual distortion

Table 3: Expected loss L, premium P, and loss ratio LR by unit

	L	P	LR
unit
X1	31.700	32.310	0.981
X2	14.900	21.256	0.701
X2 ceded	3.500	5.415	0.646
X2 net	11.400	15.841	0.720
total	46.600	53.565	0.870

• Gross pricing at 87% loss ratio calibrated to 15% return with assets $a=100$ sufficient to pay all claims with no-default
• Total $P=53.565=vEL+dmax(L)=46.6/1.15+(0.15/1.15)\times 100$
• Loss ratio for X2 ceded loss represents model minimum acceptable ceded loss ratio

Example: Cat Pricing Across a Range of Risk Appetites

Pricing and loss ratios implied by dual distortion (details)

Table 4: Expected loss, premium, margin, capital, assets, loss ratio, leverage PQ, and cost of capital by unit.

	L	P	M	Q	a	LR	PQ	COC
unit
X1	31.700	32.310	0.610	13.826	46.136	0.981	2.337	0.044
X2	14.900	21.256	6.356	32.609	53.864	0.701	0.652	0.195
X2 ceded	3.500	5.415	1.915	13.125	18.539	0.646	0.413	0.146
X2 net	11.400	15.841	4.441	19.484	35.325	0.720	0.813	0.228
total	46.600	53.565	6.965	46.435	100.0	0.870	1.154	0.150

• Displays natural allocation of capital and associated average cost by unit; reflects lower capital cost for tail cat risk (see PIR Ch. 14.3.8)
• Very low cost of capital for X1 reflects its value as a hedge; negative tail correlation

Example: Cat Pricing Across a Range of Risk Appetites

Implied loss ratios by unit across distortion risk appetites

Table 5: Model loss ratios by distortion

unit	X1	X2 net	X2 ceded	total
distortion
ccoc	102.8%	75.3%	46.0%	87.0%
ph	101.7%	72.5%	52.5%	87.0%
wang	100.1%	72.1%	57.5%	87.0%
dual	98.1%	72.0%	64.6%	87.0%
tvar	95.7%	72.9%	72.9%	87.0%

• All risk appetites calibrated to same total loss ratio
• Distortions shown from tail-centric to volatility-centric
• Ceded loss ratios show decreasing value of tail-reinsurance as risk appetite becomes more volatility driven
• Conversely X1 loss ratio increases as tail-hedge becomes more valuable

Example: Cat Pricing Across a Range of Risk Appetites

CoC by capital tranche: equity vs. reinsurance

Table 6: Model indicated CoC for equity and reinsurance capital.

	Reins	Equity	Capital
distortion
ccoc	15.0%	15.0%	15.0%
ph	11.2%	21.0%	15.0%
wang	8.9%	25.0%	15.0%
dual	6.5%	30.0%	15.0%
tvar	4.3%	34.9%	15.0%

• Gross calibrated to 15% average return, determined by market dynamics
• Purchase reinsurance when implied CoC at or below indicated return
• Reflects lower value ascribed to reinsurance by volatility-sensitive management

Example: Cat Pricing Across a Range of Risk Appetites

CoC can be computed by unit but hard to interpret and not useful

Table 7: Natural allocation CoC by distortion

unit	X1	X2 net	X2 ceded	total
distortion
ccoc	15.0%	15.0%	15.0%	15.0%
ph	-8.9%	18.9%	18.0%	15.0%
wang	-0.3%	22.4%	18.3%	15.0%
dual	4.4%	22.8%	14.6%	15.0%
tvar	10.0%	22.0%	10.1%	15.0%

• CCoC distortion results in constant CoC but perverse negative allocation to X1
• CoC hard to interpret without CCoC assumption; better to allocate total margin directly
• Interpret margin as the CFO’s “cost to enter the theme park” and expose all capital

6 Determining the Distortion

What is the “Market $g$”?

Thesis

• Different investor market participants have different distortions

• Insurer optimal capital structure uses the efficiently

• Market $g$ emerges from participant’s pricing rules

Supporting market observations

• Equity investors: 15%+ target return, volatility averse concerned you “make earnings”
  $\leftrightarrow$ CCoC distortion

• Alternative cat investors: accept mid-single digits for uncorrelated bond returns
  $\leftrightarrow$ TVaR distortion

• Active brokering in market expends great effort to match risk with capital

Calibrate a Set of Distortions to Market Pricing

• Distortions calibrated to common market pricing benchmark
• 15% return with full capitalization $a=100$
• For PH lower parameter is more expensive; for others a higher parameter
• These parameters are moderate, realistic dual parameter ranges $2.0-3.0$
• TVaR $p=0.271$ means average worst $1-0.271=0.729$ proportion of outcomes
• Distortions have one parameter and are easy to parameterize in Excel using Solver

Table 8: Distortion parameters at 15 percent target pricing.

	P	COC	param	error
method
ccoc	53.565	0.150	0.150	0.0
ph	53.565	0.150	0.720	3.2978e-10
wang	53.565	0.150	0.343	1.2525e-08
dual	53.565	0.150	1.595	-3.3927e-07
tvar	53.565	0.150	0.271	7.6102e-06

What is the “Market $g$”?

Figure 2: Five distortions calibrated to 15 percent return pricing (left). The minimum of CCoC and TVaR pricing (right).

Distortions correspond with investor’s risk tolerances. Figure is consistent with thesis that cat (TVaR) and equity (CCoC) are two most important source of capital.

ToyCo Tranching between CCoC Equity and TVaR Debt Investors

• Tranche with debt (or cat bond) attaching at $p=0.9$ quantile of loss

• Turns out to be sub-optimal

Premium, Return and Loss Ratio by Tranche by Market Agent

metric	Premium			ROE			Loss ratio
layer	Equity	Debt	Total	Equity	Debt	Total	Equity	Debt	Total
distortion
ccoc	45.957	7.609	53.565	0.150	0.150	0.150	0.938	0.460	0.870
tvar	48.762	4.803	53.565	0.349	0.043	0.150	0.884	0.729	0.870
min_g	45.658	4.803	50.461	0.132	0.043	0.078	0.944	0.729	0.923

• Rows: pricing from each agent and minimum across agents (min_g)
• Columns: equity and debt correspond to capital tranches
• Key takeaway: min_g price for the equity layer is not placeable
• min_g of 45.658 lower than equity agent’s price 45.957 and debt’s 48.762
• Neither equity nor debt agent’s distortion is uniformly lower over full range of the equity tranche $⟹$ sub-optimal tranching

Optimal Capital Structure

• Optimal spit occurs at $p=0.74$ because the CCoC and TVaR distortions cross at $1-p$, see Figure 2
• Pricing by tranche by break point, with implied placeable premium and its split between tranches

metric	Premium			ROE			Loss ratio
layer	Equity	Debt	Total	Equity	Debt	Total	Equity	Debt	Total
distortion
ccoc	42.913	10.652	53.565	0.150	0.150	0.150	0.958	0.516	0.870
tvar	46.018	7.548	53.565	0.547	0.055	0.150	0.893	0.729	0.870
min_g	42.913	7.548	50.461	0.150	0.055	0.078	0.958	0.729	0.923

• The table shows both layers are now placeable

Placeable Premium vs. Debt Attachment Point

• Pricing for equity and debt tranches, and implied total placeable premium, compare to premium using the minimum distortion by size of equity layer
• See next slide for graphic

Placeable Premium vs. Debt Attachment Point

• Orange line shows total cost of capital varying with different debt/equity splits
• Material costs from sub-optimal tranching of capital
• Potential cost savings explain active broker rôle in reinsurance markets

7 Appendix

Actuarial Chestnuts — 1 of 2

Margin or return?

• Distinguish capital from equity

• Insureds concerned with margin to premium

• Investors concerned with return to capital

• Cat risk has cheap capital but expensive premium

• Leverage reconciles the two viewpoints

aka premium is expensive for whom?

Actuarial Chestnuts — 2 of 2

Capital vs equity

• Equity: book value of owner’s residual interest

• Capital: net assets subordinate to policyholder claims

Five Parametric Families of Distortion Functions

The “usual suspect” distortions

• CCoC: $g(s)=d+vs$ for $s>0$ and $g(0)=0$ where
  $d=1/(1+r)$, $v=1-d$ are discount rates

• PH proportional hazard: $g(s)=s^{\alpha }$, $0\le \alpha \le 1$

• Wang: $g(s)=\mathrm{\Phi }({\mathrm{\Phi }}^{-1}(s)+\lambda )$

• Dual: $g(s)=1-(1-s{)}^{\beta }$, $\beta \ge 1$

• TVaR: $g(s)=min(1,s/(1-p))$

• Higher $r$, $\lambda$, $\beta$ and $p$ and lower $\alpha$ correspond to higher prices

Algorithm for (Linear) Natural Allocation

1. Compute unit average loss grouped by total loss & sum group probabilities
2. Sort by ascending total loss (all values now distinct)
3. Compute survival function S
4. Apply distortion function g(S)
5. Difference step 4 to compute risk adjusted probabilities Q
6. Compute sum-products by unit and in total with respect to Q to obtain SRM pricing and natural allocation pricing by unit

Step 1 replaces $X_i$ with the conditional expectation $\mathsf{E}[X_i\mid X]$, a random variable defined by $\mathsf{E}[X_i\mid X](\omega )=\mathsf{E}[X_i\mid X=X(\omega )]$

See PIR Algorithms 11.1.1 p.271 and 15.1.1, p.397 for more detail

See Why SRMs presentation for calculation details

Python Source Code for imported Python functions

path: tranching_problem_code.py
lang: python

Further Reading

Python source for presentation (RMarkdown): use Code drop down at top of page

Mildenhall and Major (2022) for the theory of spectral risk measures, natural allocation, and implementation details

Actuarial Standards Board (2019) Modeling Standard of Practice for US Actuaries

References

Actuarial Standards Board. (2019). Modeling. Actuarial Standard of Practice No. 56, (56).

Mildenhall, S. J., & Major, J. A. (2022). Pricing insurance risk: Theory and practice. John Wiley & Sons, Inc.