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CS – 15 Risk Premium for Insurance Product Pricing. Steve Mildenhall, AON Re Dave Ingram, Milliman USA Don Mango, AM Re. Risk Premium for Insurance Product Pricing. Stephen Mildenhall CAS/SOA ERM Symposium Washington DC, July 2003. Why a Risk Premium?. Need to make a profit

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Cs 15 risk premium for insurance product pricing l.jpg

CS – 15 Risk Premium for Insurance Product Pricing

Steve Mildenhall, AON Re

Dave Ingram, Milliman USA

Don Mango, AM Re

Risk premium for insurance product pricing l.jpg

Risk Premium for Insurance Product Pricing

Stephen MildenhallCAS/SOA ERM SymposiumWashington DC, July 2003

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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

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State of the world

Policy Payout



All of the above

Financial Consequences of policy


Risk Premium: 2000BC-today

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Standard deviation





Wang Transform

Esscher Transform


Micro-view of single risk

SD, Variance,… of what?

Which measure is appropriate?

Risk Premium

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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 Y

    • X = Y + noise

    • Every risk averter prefers Y to X (utility)

    • X has more weight in the tails

    • Var(X) > Var(Y)

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

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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

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

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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


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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

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Big Picture: moving beyond individual policy risk

States of the world relevant for one policy

Policy Payout

All states of the world

Multiple states yielding same loss L for one policy



Projection with loss of information

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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

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Three Points to Remember

  • Parameter Risk

  • Size

  • Think Big-Picture

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Pricing for Risk

David Ingram

ERM Symposium

Washington DC, July 2003

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Pricing for Risk

  • RMTF Survey of current Practices

  • Methods for Setting Risk Margins

    • Charge for Risk Capital

    • Risk Adjusted Hurdle Rates

    • Adjusted Target Calculation

    • Replication

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Methods for Setting Risk Charge

  • Judgment Methods

  • Quantitative Methods

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Judgment Methods

  • Risk Premium based on

    • Prior products

    • Market prices

    • Comfort with particular risks

    • Relative perceived risk of company products

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Quantitative Methods

  • Charge for Risk Capital

  • Risk Adjusted Hurdle Rates

  • Adjusted Target Calculation

  • Replication

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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

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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

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Risk Adjusted Hurdle Rates

  • Efficient Frontier Analysis

  • Market Analysis

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Efficient Frontier Analysis


  • Brainstorming

  • Modeling

  • Display / Identify Frontier

  • Determine Risk/Reward Trade-off Parameters

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Efficient Frontier

Efficient Frontier

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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

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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

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Market Analysis

While this is “quantitative”…

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

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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

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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

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  • 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

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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%

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Insurance Products

  • Cannot easily hedge with 100% efficiency

  • But can compare…

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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

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  • Insurance Product – VA

    • 75 Expected Return

  • Common Stock Fund A

    • 100 Expected Return

  • Common Stock Fund B

    • 110 Expected Return

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  • Work on evolving from Judgment to Quantitative

  • Quantitative methods need to be based on Pricing Risk Metric

  • Ultimately should tie to market pricing for risks

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Risk Premiums

Don Mango


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Where Are We Going?

  • Commonalities

  • Simulation Modeling

  • Explicit Valuation

  • Aggregate Risk Modeling

  • Interaction Effects

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  • Valuation of Contingent Obligations (“VALCON”)

  • Levered investment trusts

  • Strong dependencies on economic and capital market conditions

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  • 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

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  • 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.

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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

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Simulation Modeling


  • We are heading the same direction.

  • We need transparency or at least explicitness of assumptions.

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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”

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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

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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!

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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

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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 …

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Aggregate Risk Modeling


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

  • Risk Management is that Content Provider.

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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

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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

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Interaction Effects


  • 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.