Going Spectral

A shorter version of Going Spectral.

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

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

• Adam Smith - 1789

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 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

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: ToyCo Example $x=60$

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

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
  • 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

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

Implied loss ratios by unit across distortion risk appetites

Table 4: 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 5: 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

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.

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

• Debt attaching at 90th percentile or 65
• 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

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