CS – 15 Risk Premium

for Insurance Product PricingSteve Mildenhall, AON Re

Dave Ingram, Milliman USA

Don Mango, AM Re

Risk Premium for Insurance Product Pricing

Stephen MildenhallCAS/SOA ERM Symposium

Washington DC, July 2003

Why a Risk Premium?

• Need to make a profit• Need to be reasonably confident of making a

profit• Risk Premium is an all encompassing term

– Covers frictional costs

– Covers pure risk (toss of fair coin)

– Compensation for bearing risk under uncertainty

• Philosophical distractions should be resisted

Risk Premium: 2000BC-today

Financial Consequences of policy

Probabilities

State of the world

Policy Payout

L

)Pr()()( LLE

)Pr())(()( LgLE

)(Pr*)()( LLE

)Pr()()( LhLE

All of the above

Risk Premium

• Standard deviation• Variance• Semi-Variance• Percentile/VaR• Tail-VaR• Wang Transform• Esscher Transform• Utility-based

• Micro-view of single risk• SD, Variance,… of what?• Which measure is

appropriate?

Measures of Risk

• Problem: collapse distribution to a number– All moments may not be enough to determine

distribution!

• No consensus methodology• Rothschild-Stiglitz offer four possible definitions

of when X is “more risky” than Y1. X = Y + noise2. Every risk averter prefers Y to X (utility)3. X has more weight in the tails4. Var(X) > Var(Y)

1, 2, and 3 are equivalent and are different from 4

Parameter Risk: don’t delude yourself

• Variance of losses in your model is not the same thing as variance of losses!– Hayne’s Loss Reserving Example (CLRS)

• Leverage, Excess Policies and Jensen’s inequality

– Need to compute the mean correctly

– Risk load should not be used to compensate for miscellaneous actuarial inadequacies

))(E())((E XfXf

Don’t believe a risk load formula that says a new small line is a good thing!

Size: what is a large risk?

• Parameter risk is all that matters…almost• Process risk matters for large risks• Large?

– 100M households in US– $1M loss = 1¢ per household– $100M loss = $1 per household– $1B loss = $10 per household– $10B loss = $100 per household Large

Size: what is a large risk?

• Heterogeneous distribution of wealth

• Demographics– Ultimate risk bearers are individual insureds– Population concentrations correlated to risk

loads

• Frequency of losses, size of market

Don’t believe a risk load formula that does not account for population

demographics

Big Picture: moving beyond individual policy risk

All states of the worldPolicy Payout

L

States of the world relevant for one policy

Projection with loss of information

Multiple states yielding same loss L for one policy

Big Picture: moving beyond individual policy risk

Sim# A B C Total Transf Probs1 77,490 123,643 12,301 213,435 0.1602 58,089 44,276 54,757 157,122 0.1043 78,255 41,085 18,167 137,507 0.0864 8,934 115,909 1,317 126,160 0.0745 45,939 66,417 2,677 115,033 0.0666 37,614 34,151 31,340 103,105 0.0607 5,379 50,342 24,204 79,925 0.0548 16,600 19,034 40,084 75,717 0.0509 36,492 10,658 27,340 74,490 0.04610 53,382 16,521 3,671 73,574 0.04211 6,911 43,635 17,632 68,179 0.03912 42,304 12,079 6,515 60,898 0.03613 4,114 26,544 29,910 60,568 0.03314 31,730 16,976 10,725 59,431 0.03015 15,796 33,017 7,587 56,401 0.02716 19,180 17,466 13,622 50,268 0.02517 6,756 18,021 22,012 46,789 0.02218 3,967 14,584 12,489 31,040 0.01919 9,401 16,660 3,956 30,017 0.01620 11,352 3,810 8,277 23,439 0.011

Mean 28,484 36,241 17,429 82,155Loaded 40,416 53,667 19,733 113,815Load 42% 48% 13% 39%

SD 23,584 31,804 13,564 46,310CV 82.8% 87.8% 77.8% 56.4%

Big Picture: moving beyond individual policy risk

• Rodney Kreps, co-measures

• P/C: Catastrophe (re-)insurance– Cat models explicitly quantify correlation

• Life: Hedging interest rate and investment risk

ii XXg on condition )(E

Three Points to Remember

• Parameter Risk

• Size

• Think Big-Picture

Pricing for Risk

David Ingram

ERM Symposium

Washington DC, July 2003

Pricing for Risk

1. RMTF Survey of current Practices

2. Methods for Setting Risk Marginsa. Charge for Risk Capital

b. Risk Adjusted Hurdle Rates

c. Adjusted Target Calculation

d. Replication

How Do you Price for Risk?

Capital allocation

Risk-adjusted profit target

Stochastic scenario analysis

Assumption PADS

Assumption stress testing

ROI 25% 18% 19% 13% 25%IRR 26% 18% 17% 12% 26%ROE 30% 16% 21% 11% 20%CTS 21% 27% 17% 15% 19%Premium Margin 16% 23% 13% 18% 28%

What is the basis for Risk Adjustment?

Regulatory formula multiple 147 55.26%Internal formula 69 25.94%Economic capital 44 16.54%Other 6 2.26%

Total: 266

3. If you use capital allocations for reflecting risk, how are these allocations determined?

What is the basis for Risk Adjustment?

Analysis of recent experience 66 50.00%Industry standard 36 27.27%Other 5 3.79%Stochastic scenario analysis 25 18.94%

Total: 132

4. If you use assumption PADS, how are these PADS determined?

What is the basis for Risk Adjustment?

Judgment 81 61.83%Formula 44 33.59%Other 6 4.58%

Total: 131

5. If you risk-adjusted profit objective, how is it determined?

What is the basis for Risk Adjustment?

Judgment 137 59.05%Confidence limits 48 20.69%Worst case historical experience 42 18.10%Other 5 2.16%

Total: 232

6. If you use assumption stress testing, how are the parameters determined?

What is the basis for Risk Adjustment?

Percentiles 83 30.51%Mean-variance analysis 44 16.18%Conditional tail expectation (CTE) 44 16.18%Problem scenario analysis 38 13.97%Value at risk 24 8.82%Efficient frontier 23 8.46%Earnings at risk 14 5.15%Other 2 0.74%

Total: 272

7. If you use stochastic scenario analysis, how is the distribution of results analyzed?

Methods for Setting Risk Charge

• Judgment Methods

• Quantitative Methods

Judgment Methods

• Risk Premium based on– Prior products– Market prices– Comfort with particular risks– Relative perceived risk of company products

Quantitative Methods

1. Charge for Risk Capital

2. Risk Adjusted Hurdle Rates

3. Adjusted Target Calculation

4. Replication

Charge for Risk Capital

• Most common quantitative risk adjustment to pricing

• Charge is:– (HR – is) * RCt

• Where HR is Hurdle Rate

• is is the after tax earnings rate on surplus assets

• RCt is the risk capital in year t

Charge for Risk Capital

• Is it actually a charge for risk?– Or just a cost of doing business?

• It is a charge that is proportionate to risk

• If there are other risk charges or adjustments, need to be careful not to double charge for risk

Risk Adjusted Hurdle Rates

• Efficient Frontier Analysis

• Market Analysis

Efficient Frontier Analysis

ProcessA.Brainstorming

B.Modeling

C.Display / Identify Frontier

D.Determine Risk/Reward Trade-off Parameters

Efficient Frontier

Premier III

-

5.00

10.00

15.00

20.00

25.00

30.00

35.00

- 2.00 4.00 6.00 8.00 10.00 12.00 14.00

Risk

Re

turn

Efficient Frontier

Market Analysis

• Study Relationship between Return and – Product Concentration– Income/ ROE volatility

For a group of successful companies.

• Develop Target returns– Based on Products– Based on volatility

Market Analysis

Product Concentration

• Product A – 12%• Product B – 15%• Product C – 10%

ROE Volatility

Target ROE = • Risk-free rate + 3.7

• 22.83% +1.83% ln()

• 7.5% +

Market Analysis

While this is “quantitative”…

Data is so thin that much judgment is needed to develop targets

Study of Insurance Company ROE

ROE Std Dev Ratio

Group I 13.96% 6.71% 48%

Group II 10.52% 11.32% 107%

Group III 10.12% 16.02% 158%

Group IV 4.86% 25.96% 534%

Group V (3.69%) 21.13% NM

Adjusted Target

• Instead of concentrating on 50th Percentile results (or average results)

– In a stochastic pricing model

• Pricing Target adjusted to 60th, 70th or 80th Percentile

Adjusting Target

Monthly Returns

-25%

-20%

-15%

-10%

-5%

0%

5%

10%

15%

20%

100% 95% 90% 85% 80% 75% 70% 65% 60% 55% 50% 45% 40% 35% 30% 25% 20% 15% 10% 5% 0%

50th Percentile0.86%

Average0.72%

80th Percentile(2.35%)

Replication

• Finance – Law of One Price– Two sets of securities that have the same

cashflows under all situations will have the same price

• Replication – if you can replicate the cashflows of an insurance product with marketable securities then market price of securities is the correct price for product

Risk & Return

• Bonds – Volatility of Bond Prices 8.6%– Average Return on Bonds – 5.8% compound

Average, 6.1% Arithmetic Average– Risk/Reward = 139% to 148%

• Stocks – Volatility of Stock Returns 20.5%– Average Return – 10.5%, 12.2%– Risk Reward = 168% to 194%

Insurance Products

• Cannot easily hedge with 100% efficiency

• But can compare…

VA Product vs. Common Stocks

• Insurance Product – VA– $10 B AV– Std Dev = 200, CTE 90=429

Compare to• Common Stock Fund A

– $300 M Fund– Std Dev= 200, CTE 90= 390

• Common Stock Fund B– $330 M Fund– Std Dev= 220, CTE 90= 429

Returns

• Insurance Product – VA– 75 Expected Return

• Common Stock Fund A– 100 Expected Return

• Common Stock Fund B– 110 Expected Return

Recommendations

1. Work on evolving from Judgment to Quantitative

2. Quantitative methods need to be based on Pricing Risk Metric

3. Ultimately should tie to market pricing for risks

Risk Premiums

Don Mango

AM Re

Where Are We Going?

• Commonalities

• Simulation Modeling

• Explicit Valuation

• Aggregate Risk Modeling

• Interaction Effects

Commonalities

• Valuation of Contingent Obligations (“VALCON”)

• Levered investment trusts

• Strong dependencies on economic and capital market conditions

Commonalities

• Long time horizons and held-to-maturity (“HTM”) portfolios

• We sell “long-dated, illiquid, OTC derivatives”

• We have an incomplete, inefficient secondary market

• We retain magnitudes of risk that bankers would never dream of

Commonalities

• IMPLICATIONS:

• This seminar should be the norm, not the exception.

• There may be hybrid products in our future.

• We may not be able to simply borrow capital market techniques.

Simulation Modeling

• Aka “Monte Carlo valuation”

• Financial engineers use it to price long-dated, illiquid, OTC derivatives

• Devil is in the parameters and dependence structure

Simulation Modeling

• IMPLICATIONS:

• We are heading the same direction.

• We need transparency or at least explicitness of assumptions.

Explicit Valuation

• Complete, efficient market affords participants the luxury of not having to think or care or have any opinion of the fundamental value of a product

• Counting on the continued presence of counterparties to limit downside

• Bloomberg gives you “the price”

Explicit Valuation

• Incomplete, inefficient market requires some explicit valuation by its participants

• True, you could be a “delta” off a content provider– 10% below Swiss Re or Met Life

Explicit Valuation

• IMPLICATION: If you want to be a leader, formulate a risk appetite and apply it. – Read Karl Borch, 1961

• What are your desired payoff profiles, and please be specific and use quantities!

Aggregate Risk Modeling

• Valuation develop indicated price based on the impact of the product on your portfolio – a “MARKET OF ONE”

• “One Price” does not mean One Value

• Value is idiosyncratic and in the eye, mind, interpretive filter, and model of the beholder

Aggregate Risk Modeling

• Requires aggregate portfolio risk modeling

• Integration of disparate risks

• A critical goal of our ERM efforts

• Sounds like it might require actuaries of all kinds …

Aggregate Risk Modeling

• IMPLICATIONS:

• Get information content into the indicated prices and (hopefully) the quotes.

• Risk Management is that Content Provider.

Interaction Effects

• Indicated price meets market strategy, premium goals, expense ratios, relationships, history, culture, decision process, …

• Multiple participants selling promises with indistinguishably small probabilities of non-performance

Interaction Effects

• Throw in some “momentum sellers” going delta off the content providers

• Result is an unstable system dynamic = “the insurance market”

• Mutually reinforcing behaviors, for good or bad

Interaction Effects

• IMPLICATIONS: • Theory aside, the attainable risk premium

will rarely be where it “should be.” • Market Price represents somebody’s quote

(usually the LCD – winner’s curse) – no “exogenous” source

• No more isolated strategy development – we have seen the enemy, and it is us.