Risk Lighthouse, LLC by Dr. Shaun Wang October 5, 2012 - - PDF document

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Risk Lighthouse, LLC by Dr. Shaun Wang October 5, 2012 - - PDF document

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09/25/2012 Capital Allocation: A Benchmark Approach Risk Lighthouse, LLC by Dr. Shaun Wang October 5, 2012 Acknowledgement: Support from Tokio Marine Technologies LLC www.risklighthouse.com 2 1 09/25/2012 Part 1. Review of Capital

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09/25/2012 1

Capital Allocation: A Benchmark Approach

Risk Lighthouse, LLC

by Dr. Shaun Wang October 5, 2012

Acknowledgement: Support from Tokio Marine Technologies LLC

www.risklighthouse.com 2

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09/25/2012 2

Part 1. Review of Capital Allocation Methods

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Part 1. Review of Capital Allocation Methods

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Portfolio Theory of Capital Allocation

  • Most current capital allocation methods are

variations of the Markowitz Portfolio Theory, based on portfolio Value-at-Risk and marginal contributions. Diversification benefit is a key driver that impacts allocated capital. It is hard to select correlation parameters among lines of business.

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Limitations of the Portfolio Theory of Capital Allocation 1) Allocation results are highly unstable. 2) Adding a new risk can significantly alter allocated capital for existing risks. 3) Allocation is highly sensitive to the correlation parameters used. 4) Allocated capital to a risk can exceed its policy limit.

Part 1. Review of Capital Allocation Methods

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Insurance Market Cycles Last for Several Years

  • Unlike a stock market which is characterized

by random walks, insurance market cycles are played out in slow-motion and last for multiple years.

  • Capital allocations need to reflect the

through-the-cycle profitability.

  • In insurance, customer relation is an

important factor in long-term profitability.

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09/25/2012 4

  • 5%

0% 5% 10% 15% 20% 25%

71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10E 11F

Part 1. Review of Capital Allocation Methods

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Net Written Premium Growth Rate

(Percent)

1975-78 1984-87 2000-03 Shaded areas denote “hard market” periods Sources: Insurance Information Institute, A.M. Best, ISO Net Written Premiums fell 0.7% in 2007 (first decline since 1943) by 2.0% in 2008, and 4.2% in 2009, the first 3-year decline since 1930-33. NWP was up 0.9% in 2010 with forecast growth of 1.4% in 2011

Part 1. Review of Capital Allocation Methods

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A Case Example: Wildly Different Capital Allocation Methods

Gary Venter, Feb 2002 Actuarial Review

In 2001, the CAS Call For Papers to analyze a hypothetical insurer, recommend a reinsurance program, allocate capital, etc.

Philbrick & Painter * Bohra & Weist ** % of Surplus Allocated % of Surplus Allocated Relative Ratio Workers Comp 41% 11% 3.73 Auto Liab 26% 29% 0.90 HO/CMP Prop 11% 51% 0.22 Auto Phys Dmg 1% 1% 1.00 GL/CMP Liab 21% 8% 2.63 Total 100% 100% * From Swiss Re ** From Munich-American Re

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09/25/2012 5

Part 2. An Alternative Method: Risk Margins and Benchmark Approach

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Part 2. Risk Margins and Benchmark Approach

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The Concept of Risk Margins

  • Explicit Risk Margin is now required by some

financial reporting proposals.

  • It recognizes risk and uncertainty in the

amount and timing of future payments needed to satisfy insurance liabilities.

  • It reflects market-based price an insurer

would rationally pay to be relieved of the insurance liabilities.

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Average Risk Margin

  • Insurance markets vary widely across

products and market segments. We define Average Risk Margin as an aggregated average, or central value, over a portfolio of insurance contracts for a fixed time period.

  • At any specific time, the prevailing market risk

margin may differ from the Average Risk Margin.

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Benchmark Capital Method

  • The basic idea: allocated capital is calculated

as the ratio of Average Risk Margin and Target Excess Return.

  • We estimate Average Risk Margin and Target

Excess Return using aggregate industry statutory report data.

𝐵𝑤𝑓𝑠𝑏𝑕𝑓 𝑆𝑗𝑡𝑙 𝑁𝑏𝑠𝑕𝑗𝑜 𝑈𝑏𝑠𝑕𝑓𝑢 𝐹𝑦𝑑𝑓𝑡𝑡 𝑆𝑓𝑢𝑣𝑠𝑜 = Allocated Capital

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Benchmark Capital Method

Average Risk Margin Allocated Capital

Target Excess Return

(over risk-free rate)

𝐵𝑤𝑓𝑠𝑏𝑕𝑓 𝑆𝑗𝑡𝑙 𝑁𝑏𝑠𝑕𝑗𝑜 𝑈𝑏𝑠𝑕𝑓𝑢 𝐹𝑦𝑑𝑓𝑡𝑡 𝑆𝑓𝑢𝑣𝑠𝑜 = Allocated Capital

Part 3. Risk Margins Using Wang Transform

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09/25/2012 8 Part 3. Risk Margins Using Wang Transform

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Theory of Market Price of Risk

  • Fund performance (also called Sharpe Ratio):

 = { E[R] r } /[R]

  • Capital Asset Pricing Model:

i = Corr(RM, Ri)  M

  • Black-Scholes-Merton model for options

Call Option = Underlying Asset

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Financial & Insurance Pricing

Mapping between 1. Loss Curve – physical measure – S(x) = 1- F(x) 2. Pricing Curve – risk-neutral measure – S*(x) = 1- F*(x)

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Wang Transform

  • Maps a loss curve to a price curve:

F*(x) ) = = [–1(F(x)) )  ] E.g. 0. 0.97 = [–1(0.99) )  0.4 0.45]

  is the standard normal distribution   extends the Sharpe Ratio concept

  • Recovers CAPM and Black-Scholes-Merton

formula for (log)normally distributed risks

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Risk Margin Using Wang Transform

  • Let F(x) denote the loss distribution
  • Apply Wang transform to derive a risk-

adjusted distribution F*(x) ) = = 𝒖𝟕[–1(F(x)) )  ]

  • We get risk margin from the transformed

distribution 𝑆𝑗𝑡𝑙 𝑁𝑏𝑠𝑕𝑗𝑜 = 𝐹∗ 𝑀𝑝𝑡𝑡 − 𝐹(𝑀𝑝𝑡𝑡)

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Estimated “lambda” values from CAT bond transactions: Effects of 2005 Katrina

Peril Zone Before 2005 Katrina After 2005 Katrina U.S. Wind 0.48 0.77 Europe Wind 0.41 0.53 Japan Earthquake 0.50 0.50

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Part 3. Risk Margins Using Wang Transform

Estimated “lambda” values from CAT bond transactions: Effect of 2001 Japan Earthquake

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Part 3. Risk Margins Using Wang Transform

Peril Before 2011 After 2011 U.S. Earthquake 0.54 0.55 Japan Earthquake 0.50 0.64

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Part 4. Proposed Benchmark Capital Allocation

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Example of Coefficient of Variation of Net Loss Ratios (AY 1987-2004)

  • Apply Wang transform

to stylized risk ratio distribution for a line of business

  • Use benchmark price

to back out required capital charge

Bi-Model Distribution

  • 0.005

0.005 0.01 0.015 0.02 0.025 0.03

0% 100% 200% 300% 400% 500% Loss Ratio Probability Density

Density

Part 4. Proposed Benchmark Capital Allocation

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Applications in Calculating Capital Charges

Part 4. Proposed Benchmark Capital Allocation

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Use Wang transform to derive Capital Charge Factors for ground-up risks

Sharpe Ratio Target Excess Return Over Risk- free Rate

0.3 10%

UW Year Payout Annualized Annual Capital Line of Business Volatility Duration Volatility Charge Factor PPA Liab 4.0% 2.3 2.6% 0.08 Prem/Ops Small 11.3% 3 6.5% 0.20 Prem/Ops Large 26.4% 6 10.8% 0.32 Comml Auto NonFleet 6.9% 3.8 3.5% 0.11 Comml Auto Fleet 37.1% 3.8 19.0% 0.57 Worker Comp Small 12.6% 10 4.0% 0.12 Worker Comp Large 28% 11.3 8.2% 0.25

Part 4. Proposed Benchmark Capital Allocation

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Apply Wang transform to derive relativity (excess vs. ground-up) in capital charge factors

150 xs 100 250xs250 500xs500 1M xs 1M 3M xs 2M 5M xs 5M Pers Auto Liab 1.67 Comm Auto Liab NonFleet 1.67 Comm Auto Liab Fleet 1.2 1.45 1.67 2 2.8 3.5 Prems/Op Small 1.2 1.45 1.67 2 2.8 3.5

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Adjust for Payment Duration “D”

  • Let D denote the duration of payment pattern

for a line of business.

  • The market price of risk for an Accident Year

𝜇𝐵𝑍 can be adjusted for duration to derive an 1-year parameter: 𝜇1 =

𝜇𝐵𝑍 𝐸 .

  • This gives a middle ground of the two

extremes: MunichRe vs. SwissRe methods.

Part 4. Proposed Benchmark Capital Allocation

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Thank you!

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Contact

  • Dr. Shaun Wang, FCAS, MAAA

One Atlanta Plaza, Suite 2160 950 E. Paces Ferry Road NE Atlanta, GA 30326-1384 Phone: 678-732-9112 shaun.wang@risklighthouse.com www.risklighthouse.com