Bodoff’s Percentile Layer of Capital-Method

Capital Allocation by Percentile Layer by Bodoff

Abstract

Bodoff’s paper¹ describes a new approach to capital allocation; the catalyst for this new approach is a new formulation of the meaning of holding Value at Risk (VaR) capital. This new formulation expresses the firm’s total capital as the sum of many granular pieces of capital, or “percentile layers of capital.” As a result, one must allocate capital separately on each layer and perform the capital allocation across all layers. The resulting capital allocation procedure, “capital allocation by percentile layer,” exhibits several salient features. First, it allocates capital to all losses, rather than allocating capital only to extreme losses in the tail of the distribution. Second, despite allocating capital to this broad range of loss events, the proposed procedure does not allocate in proportion to average loss; rather, it allocates disproportionate capital to severe losses. Third, it allocates capital by relying neither upon esoteric parameters nor upon elusive risk preferences. Ultimately, on the practical plane, capital allocation by percentile layer produces allocations that are different from many other methods. Concomitantly, on the theoretical plane, capital allocation by percentile layer leads to new continuous formulas for risk load and utility.

Capital, premiums and losses

• Bodoff’s paper is an important contribution to capital allocation and actuarial science

• It’s key insight is that layers of capital respond to a range of loss events and not just tail events; it is not appropriate to focus solely on default states when allocating capital

• Bodoff takes capital to mean total claims paying ability, comprised of equity and premium

• Bodoff allocates capital by considering loss outcomes and assumes the same allocation for expected loss, margin, premium, and equity

• Bodoff blurs the distinction between events and outcomes

  • Allocates to identifiable events (wind-only loss, etc.) rather than to outcomes

  • In examples, outcomes amounts selected to distinguish events

  • In the Lee diagram, events are on the horizontal axis and outcomes on the vertical axis

Assumptions and model framework

• Two independent lines \(X_1\) and \(X_2\), usually wind and quake

• Total \(X = X_1 + X_2\)

• Lines independent

• \(F\) and \(S\) represent the distribution and survival function of \(X\) and \(q\) its lower quantile function

• Capital requirement set at (lower) \(p=0.99\)-VaR capital \(:=a\)

Three strawman allocation methods

1. Conditional VaR: coVaR, method allocates using \[a=\mathsf{E}[X\mid X=a] = \mathsf{E}[X_1\mid X=a] + \mathsf{E}[X_1\mid X=a]\]

2. Alternative conditional VaR: alt coVaR, method allocates using \[a = a\,\mathsf{E}\left[\frac{X_1}{X}\mid X\ge a \right] + a\,\mathsf{E}\left[\frac{X_1}{X}\mid X\ge a \right]\]

3. Naive conditional TVaR: naive coTVaR, method allocates \(a\) proportional to \(\mathsf{E}[X_1\mid X \ge a]\) and \(\mathsf{E}[X_2\mid X \ge a]\)

Critique of strawman allocations

Bodoff’s principal criticism of these methods is that they all ignore the possibility of outcomes \(<a\)

1. coVaR allocates based proportion of losses by line on the events \(\{X=a\}\) of exact size \(a\)
   • Result ignores other events near \(X=a\) and all events \(X<a\), which seems unreasonable
   • The allocation is not numerically stable: in simulation output, \(\{X=a\}\) is often only a single event
2. alt coVaR allocates based proportion of losses by line on the events \(\{X \ge a\}\)
   • Still ignores all events \(<a\)
   • alt coVaR relies on the relationship \[\begin{align*} a &= a\, \left(\mathsf{E}\left[\frac{X_1}{X}\mid X\ge a\right] + a\mathsf{E}\left[\frac{X_2}{X}\mid X\ge a\right] \right) \\ &= a\,\alpha_1(a) + a\,\alpha_2(a)\end{align*}\]

Critique and extension of straw-person allocations

3. naive coTVaR resorts to a pro rata kludge because \(\mathsf{E}[X\mid X \ge x]\ge x\) and is usually \(>x\)
   • Pro rata adjustments signal the lack of a rigorous rational and should generally be avoided
   • Note: TVaR would usually condition on \(X>a\) rather than \(X\ge a\)
4. Alternative conditional TVaR: coTVaR, method (not considered by Bodoff but introduced by Mango, Venter, Kreps, Major)
   • Solve \(a=\mathsf{TVaR}(p^*)\) for \(p^*\le p\) (we shall see below we really need to use expected shortfall, not TVaR)
   • Determine \(a^*=q(p^*)\), the \(p^*\)-VaR
   • Allocate using \(a=\mathsf{E}[X\mid X\ge a^*] =\mathsf{E}[X_1\mid X\ge a^*] + \mathsf{E}[X_2\mid X\ge a^*]\)

General comments

• All methods are presented as an actuarial allocation exercise without an economic motivation
• Methods do not consider premium: additional assumptions needed to derive a premium from an asset or capital allocation, e.g., a target return on allocated capital
• Methods provide an allocation of premium plus equity

Percentile layer allocation: definition

• To address the criticism that methods 1-4 all ignore events causing losses below the level of capital, whereas capital is certainly used to pay such losses, Bodoff introduces the percentile layer of capital, plc, allocation method
• plc allocates capital in the same proportion as losses for each layer
• Loss payments under equal priority are contractually stipulated
• In a dollar-wide all-or-nothing cover, attaching with probability \(s=1-p\) at \(x=q(p)\) (\(=p\)-\(\mathsf{VaR}\)), line \(i\) receives a proportion \(\alpha_i(x):=\mathsf{E}\left[\dfrac{X_i}{X}\mid X > x\right]\) of assets, conditional on a loss
• Unconditional expected loss recoveries equal \(\alpha_i(x)S(x)\), part of total layer losses \(S(x)\)
• Allocating each layer of capital between 0 and \(a\) in the same way gives the percentile layer of capital plc allocation \[a_i:=\int_0^a \alpha_i(x)\,dx = \int_0^a \mathsf{E}\left[ \frac{X_i}{X}\mid X >x \right]\,dx\]
• By construction, \(\sum_i a_i=a\)

Percentile layer allocation: discussion

• plc allocation can be understood better by decomposing \[\begin{align*} a &= \int_0^a 1\, dx \\ &= \int_0^a \alpha_1(x) + \alpha_2(x)\, dx\\ &= \int_0^a \alpha_1(x)S(x) + \alpha_1(x)F(x)\, dx + \int_0^a \alpha_2(x)S(x) + \alpha_2(x)F(x)\, dx \\ &= \left(\mathsf{E}[X_1(a)] + \int_0^a \alpha_1(x)F(x)\, dx\right) + \left(\mathsf{E}[X_2(a)] + \int_0^a \alpha_2(x)F(x)\, dx\right) \end{align*}\]
• plc splits unfunded assets (i.e., assets in excess of expected losses) in the same proportion as losses in each asset layer, using \(\alpha_i(x)\)
• plc says nothing about how to split the allocated unfunded capital \(\int_0^a \alpha_2(x)F(x)\, dx\) into premium margin and equity
  • Not surprising: no pricing assumptions made
  • Natural allocation introduces a pricing distortion to compute an allocation of premium, and hence margin
  • Extension to pricing considered below

Capital Allocation by Percentile Layer by Bodoff

Summary of allocations considered by Bodoff

no	Method Name	Allocation of assets \(a\) to line \(1\)
1.	pct EX	\(\mathsf{E}[X_1] / \mathsf{E}[X]\)
2.	coVaR	\(\mathsf{E}[X_1\mid X=a]\)
3.	adj VaR	\(a\,\mathsf{E}\left[\dfrac{X_1}{X}\mid X\ge a \right]\)
4.	naive coTVaR	\(a\,\dfrac{\mathsf{E}[X_1\mid X \ge a]}{\mathsf{E}[X\mid X \ge a]}\)
5.	coTVaR	\(\mathsf{E}[X_1\mid X > a^*]\), where \(a=\mathsf{TVaR}(p^*)\)
6.	plc	\(\displaystyle\int_0^a \alpha_i(x)\,dx\), where \(\alpha_i(x):=\mathsf{E}\left[\dfrac{X_i}{X}\mid X > x\right]\)

Bodoff’s Examples

Introduces four thought experiments

1. Thought experiment number 1: wind and quake, wind losses 0 or 99, quake 0 or 100, 0.2 probability of a wind loss and 0.01 probability of a quake loss

2. Thought experiment number 2: wind and quake, wind 0 or 50, quake 0 or 100, same probabilities

3. Thought experiment number 3: wind and quake, wind 0 or 5, quake 0 or 100, same probabilities

4. Thought experiment number 4: Bernoulli / exponential compound distribution

Thought experiment number 1

• Wind \(X_1\) and earthquake \(X_2\)
  • \(X_1=99\) with probability 0.2 and \(X_1=0\) otherwise
  • \(X_2=100\) with probability 0.05 and \(X_2=0\) otherwise
• Total \(X=X_1+X_2\)
• Perils independent
• Four possible events \(\omega\), leading to the following loss outcomes \(X(\omega)\)

Assumptions for Bodoff Example 1

Event, \(\omega\)	\(X_1\)	\(X_2\)	\(X\)	\(\mathsf{Pr}(\omega)\)	\(F\)	\(S\)
No loss	0	0	0	0.76	0.76	0.24
Wind	99	0	99	0.19	0.95	0.05
Quake	0	100	100	0.04	0.99	0.01
Both	99	100	199	0.01	1.00	0.00

Bodoff Example 1 Results

Table 1: shows expected unlimited loss by line. and \tty{sa TVaR

Issues With Discrete Distributions

TVaR vs. expected shortfall

• The failure of the coTVaR method to allocate the correct capital is caused by the difference between TVaR (not coherent) and expected shortfall (average of worst \(1-p\) outcomes, coherent)
• \(p\)-\(\mathsf{TVaR}\) is defined as \(\mathsf{E}[X \mid X > q(p)]\), where \(q\) is the quantile (VaR) function of \(X\)
• Recall \(p^*\) solves \(p^*\)-\(\mathsf{TVaR}(X)=q(p)=100.0\) when \(p=0.99\), which does not have a solution because TVaR only takes the values 103.333, 119.8, 199
• ES is a continuous, increasing function with range \(\mathsf{E}[X]\) to \(\sup(X)\) and so it is guaranteed there is a solution to \(p^*\)-\(\mathsf{E}S(X)=q(p)\)
• Solution is \(p^*=0.752\), and then
  • \(q(p^*)=0.0\)
  • Expected shortfall \(\dfrac{1}{1-p^*}\displaystyle\int_{1-p^*}^1 q(s)ds = q(p) = 100.0\) as expected (note ES\(=\mathsf{E}[X]/(1-p^*)\) in this case)
  • But \(\mathsf{E}[X \mid X > q(p^*)]=103.333\)
• A co ES allocation would re-scale the coTVaR allocation shown

Survival and Allocation Function for Bodoff Example 1

Top row: survival functions, bottom row: \(\alpha_i(x)\) allocation functions. Left side shows full range of \(0\le x\le 200\) and right side highlights the functions around the loss points, \(96\le x \le 103\).

Expected Shortfall vs Tail Value at Risk

For a discrete distribution ES and TVaR have different behaviors. TVaR is a jump function. ES is a continuous, increasing function taking all values between the mean and maximum value of \(X\). Graph illustrates the functions for Bodoff Example 1.

Bodoff Examples 1-3 Results

Table 2: Example 2 illustrates that can produce an answer that is different from expected losses. is not computed for Example 3; it illustrates fungibility of pooled capital. suffers the same issues in Examples 2 and 3 as it does in Example 1.

Bodoff Example 4

Table 3: This table recreates the exhibit in Section 9.1 of Bodoff’s paper. There are three lines labelled , , and . It shows the percent allocation of capital to each line across different methods. Breakeven percentile, interpreted as the percentile equal to expected losses, equals \(p=0.831588\). Bodoff’s calculation used 10,000 simulations. The table shown here uses FFTs to obtain a close-to exact answer. The exponential distribution is borderline thick tailed, and so is quite hard to work with for both simulation methods and FFT methods.

Bodoff Summary

• Allocates all capital like loss; does not distinguish expected loss, margin and equity
• Does not get to a price
• Event-centric approach allocates to events, but really allocating to peril=lines
• Premium not mentioned until Section 7 (of 10)
• Uses basic formula \(P=vL + da\) (eq. 8.2)

CAS Exam 9 Sample Question

Spring 2018 Question 15

An insurer has exposure to two independent perils, wind and earthquake:

• Wind has a 15% chance of a $5 million loss, and an 85% chance of no loss.
• Earthquake has a 1 % chance of a $15 million loss, and a 99% chance of no loss.

Using the capital allocation by percentile layer methodology with a 99.5% VaR capital requirement, determine how much capital should be allocated to each peril.

CAS Spring 2018 Question 15: Answer

Table 4: The last row gives the percentile layer capital.

CAS Part 9 2018 Q 16 (4.5 Points)

The co-measures table below displays simulated values of the aggregate loss distribution for a company in descending order. The component loss sources underlying the aggregate loss are displayed for each scenario.

Sorted Scenarios	Aggregate Underwriting Loss	Line A Underwriting Loss	Line B Underwriting Loss
100	1000	700	300
99	500	400	100
98	200	0	200
97	100	0	100
96	0	0	0
95	0	0	0
94	0	0	0
93	0	0	0
92	0	0	0
91	0	0	0
1	0	0	0

• Ignore other risks for the company.
• The target return on capital is 15%.

Answer the following questions.

1. (3 points) Calculate the 98% VaR risk capital allocated to Line A in proportion to the 98% VaR, the 98% Co-TVAR, and the allocation by Percentile Layer method.
2. (0.75 point) Explain the characteristics of each methodology above that gives rise to the different results.
3. (0.75 point) Calculate premium net of expenses for Line A using the 98% VaR risk capital allocated using the Percentile Layer method.

CAS Part 9 2018 Q 16 (4.5 Points) Solution

1. (3 points) Calculate the 98% VaR risk capital allocated to Line A in proportion to the 98% VaR, the 98% Co-TVAR, and the allocation by Percentile Layer method.

Value at Risk

• (Lower) 98% VaR, smallest value \(x\) so the probability of a loss \(\le x\) is \(\ge 0.98\) is the 98th observation: 200 in total, 0 or A and 100 for B, hence 0
• (Upper) 98% VaR, smallest value \(x\) so the probability of a loss \(\le x\) is \(> 0.98\) is the 99th observation: 500 in total, 400 or A and 200 for B, would give a different answer

Tail Value at Risk

• 98% coTVaR Bodoff uses \(\ge \mathsf{VaR}\) definition: in total events 200, 500, and 1000 of which line A is 0, 400, and 700.
  • Thus TVaR in total is 17/3 x 100 and for A is 11/3 x 100.
  • A’s Pro rata share is (11/3) / (17/3) = 11/17 part of 200 equals 129.41
• Standard usage for TVaR is \(> \mathsf{VaR}\): in total events 500 and 1000 of which line A is 400 and 700.
  • Thus TVaR in total is 15/2 x 100 and for A is 11/2 x 100.
  • A’s Pro rata share is (11/2) / (15/2) = 11/15 part of 200 equals 146.6

CAS Part 9 2018 Q 16 (4.5 Points) Solution

1. (3 points) Calculate the 98% VaR risk capital allocated to Line A in proportion to the 98% VaR, the 98% Co-TVAR, and the allocation by Percentile Layer method.

plc: take 98% VaR to be 200

• Layer 100 x 0
  • Four events, of which A has a loss in two, with shares 4/5 and 7/10
  • Conditional average share is (4/5 + 7/10) / 4 = 15/40 (1/4 since all events equally likely); notice this computes \(\alpha_1\) in the layer
  • Share times layer width of 100 = 1500 / 40; notice this is the integral of \(\alpha_1\)
• Layer 100 x 100:
  • Three events, of which A has a loss in two, with same shares 4/5 and 7/10
  • Conditional average share is (4/5 + 7/10) / 3 = 15/30 (1/3 since all events equally likely)
  • Share times width of 100 = 1500 / 30
• Total = 1500 x (1/40 + 1/30) = 7/12 x 1500 = 87.50

CAS Part 9 2018 Q 16 (4.5 Points) Solution

2. (0.75 point) Explain the characteristics of each methodology above that gives rise to the different results.

• coVaR only looks at one point on the loss distribution

• coTVaR only looks at large losses, at or above the VaR threshold

• plc considers all loss outcomes, recognizing that capital pays for losses in solvent and default states

3. (0.75 point) Calculate premium net of expenses for Line A using the 98% VaR risk capital allocated using the Percentile Layer method.

• EL without regard for default is 1100 / 100 = 11

• ROE target is 0.15, set v = 1/1.15 and d = 1 - v

• Capital a = 87.50 from part 1

• Premium = v x EL + d x a = (11 + 0.15 x 87.50) / 1.15 = 20.978

• EL with default and equal priority

  • Not considered in sample answers; only appropriate for whole portfolio, not subportfolios
  • EL = 87.50 x (7/10 + 4/5) / 100 = 1.3125

Pricing for Bodoff Example 4

Portfolio specification

• Bodoff Example 4 is based on a three line portfolio

• Each line has a Bernoulli 0/1 frequency and exponential severity

Frequency and severity assumptions for Example 4.

Line	Pr Claim	Avg Severity
a	0.25	4
b	0.05	20
c	0.01	100

• All lines have unlimited expectation 1.0

• Other statistics shown right

Table 5: Audit statistics for Bodoff example 4. The distributions become more skewed. has stand-alone VaR of 0 at the 0.99 regulatory probability threshold. The portfolio is unrealistically skewed for a normal insurance portfolio. Pricing will mirror catastrophe reinsurance pricing.

Pricing for Bodoff’s Example 4

Pricing assumptions

• Bodoff does not consider pricing per se, his allocation can be considered as \(P_i+Q_i\), with no opinion on the split between margin and equity
• Making additional assumptions we can compare the plc capital allocation with other methods
• Assume total roe = 0.1 at 0.99-VaR capital standard
• For example 4 these assumptions produce
  • assets 52
  • expected losses (limited by available assets) 2.31, unlimited loss 3
  • premium 6.83
  • loss ratio 0.339
  • premium leverage 0.151 to 1
• Calibrating other distortion and traditional methods to these assumptions produces the table right
  • agg corresponds to the PIRC approach and bod to Bodoff’s methods
  • Only additive methods shown
  • method ordered by allocation to line a the least skewed; c is the most skewed

Table 6: Allocation of premium plus equity to each line across different pricing methods. All methods except percentile layer capital calibrated to the same total premium and capital level. Distortions that price tail loss will allocate the most to line , the most volatile. More bowed distortions will allocate most to . The three lines have the same expected loss (last row). is covariance method; is conditional VaR.

Pricing for Bodoff’s Example 4

Table 7: Implied premium by method sorted by premium for . All methods produce the same total premium by calibration. Very considerable differnces evident across the methods.

Table 8: Implied loss ratios by method sorted by .

Table 1: shows expected unlimited loss by line. and \tty{sa TVaR Top row: survival functions, bottom row: \(\alpha_i(x)\) allocation functions. Left side shows full range of \(0\le x\le 200\) and right side highlights the functions around the loss points, \(96\le x \le 103\). For a discrete distribution ES and TVaR have different behaviors. TVaR is a jump function. ES is a continuous, increasing function taking all values between the mean and maximum value of \(X\). Graph illustrates the functions for Bodoff Example 1. Table 2: Example 2 illustrates that can produce an answer that is different from expected losses. is not computed for Example 3; it illustrates fungibility of pooled capital. suffers the same issues in Examples 2 and 3 as it does in Example 1. Table 3: This table recreates the exhibit in Section 9.1 of Bodoff’s paper. There are three lines labelled , , and . It shows the percent allocation of capital to each line across different methods. Breakeven percentile, interpreted as the percentile equal to expected losses, equals \(p=0.831588\). Bodoff’s calculation used 10,000 simulations. The table shown here uses FFTs to obtain a close-to exact answer. The exponential distribution is borderline thick tailed, and so is quite hard to work with for both simulation methods and FFT methods. Table 4: The last row gives the percentile layer capital. Table 5: Audit statistics for Bodoff example 4. The distributions become more skewed. has stand-alone VaR of 0 at the 0.99 regulatory probability threshold. The portfolio is unrealistically skewed for a normal insurance portfolio. Pricing will mirror catastrophe reinsurance pricing. Table 6: Allocation of premium plus equity to each line across different pricing methods. All methods except percentile layer capital calibrated to the same total premium and capital level. Distortions that price tail loss will allocate the most to line , the most volatile. More bowed distortions will allocate most to . The three lines have the same expected loss (last row). is covariance method; is conditional VaR. Table 7: Implied premium by method sorted by premium for . All methods produce the same total premium by calibration. Very considerable differnces evident across the methods. Table 8: Implied loss ratios by method sorted by .

Footnotes

1. Capital Allocation by Percentile Layer, by Neil Bodoff (Variance 3(1), 13–30, 2007)