18.2 Risk Measures

3 major class of risk measures

  1. Moment-based (e.g. mean, variance)

  2. Tail-based (e.g. VaR, TVaR)

  3. Probability Transformation (e.g. WTVaR, Mean of Transformed distn)

Definition 18.1 Loss here is not just loss from claims, but “negative profit” after considering income (e.g. premium), expenses etc

  • \(Y =\) Losses

  • \(\rho(Y) =\) Risk Measure

18.2.1 Moment-Based Measures

\(\mathrm{E} \left[ Y^k \right]\) is the kth moment of \(Y\) which represent losses

  • 1st moment = Mean

  • 2nd central moment = Variance = \(\mathrm{E}\left[ (Y-\mathrm{E}[Y])^2 \right]\)

    • S.d. is preferred as it has the same units as the loss

    • Semi-s.d. only captures unfavorable deviations

    • Quadratic risk measures are not enough to capture market attitudes to risk (they do not capture skewness)

  • 3rd moment = Skewness

Exponential Moments

\[\mathrm{E}\left[Y e^{cY/\mathrm{E}[Y]}\right]\]

  • Captures all the moment using taylor expansion

  • Scaled to $ but captures the effect of large losses on the risk exponentially

  • Does not exist unless there is a maximum possible loss

18.2.2 Tail-Based Measures at \(\alpha\) Percentile

Table 18.1: Tail-Based Measures
Risk Measure Name Description
\(VaR\) Value at Risk \(\alpha\)th Percentile
\(TVaR\) Tail Value at Risk Avg loss above \(\alpha\)th percentile
\(XTVaR\)1 XS Tail Value at Risk \(TVaR - Mean\)
\(EPD\)2 Expected Policyholder Deficit \((TVaR - VaR) \cdot (1 - \alpha)\)
Value of Default Put Option3 Market cost of insuring for losses over \(VaR_{\alpha}\)

  1. \(XTVaR\) focuses on funding losses above the mean

  2. \(EPD\) = unconditional expected value of defaulted losses if capital = \(VaR_{\alpha}\)

    • If there was no risk premium required, this is the cost of insuring for losses over \(VaR_{\alpha}\)

    • First part of the equation is the expected loss given default and the second part is to get to the unconditional value

    • Expected loss of insuring the company for losses excess of their capital amount (here it’s \(VaR_{\alpha}\))

  3. Value of default put is great than EPD, it includes an additional risk premium (market cost of insuring for losses over \(VaR_{\alpha}\))

    • Value of insuring against default

    • Estimate using option pricing

18.2.3 Probability Transforms

To recognize that large losses are worse for the firm in more than a linear way (e.g. losses twice as big is > twice as worst)

Change the loss distribution function by putting more weight (add probability) to the worst losses

  • e.g. Esscher transform
\[\begin{equation} f^*(y) = k \cdot e ^{y/c} \: f(y) \tag{18.1} \end{equation}\]

  • Lower \(c\) \(\Rightarrow\) Higher losses from the transformed distn

    • Can get non sensical results for \(c\) that’s too low

  • \(EPD\) on a transformed distn can give us the value of the default put

    • The largest losses will have additional weight (to account for the risk premium we discussed above)

Remark.

  • Most asset pricing formulas like CAPM and Black-Scholes can be expressed as transformed means
    \(\therefore\) transformed means are a promising risk measure for determining the market value of risk

  • Complete market is where any risk can be sold, but we work in incomplete markets

  • Theory of pricing in incomplete markets favors:

    • Minimum Martingale Transform (MMT)

    • Minimum Entropy Martingale Transform (MET)

    • These give reasonable approximations of reinsurance prices

  • Mean under then Wang transform closely approximates market prices for bonds and CAT bonds

  • We can calculate other risk measures (e.g. VaR) on the transformed distribution

    • \(WTVaR\) (Weighted TVaR) is \(TVaR\) on a transformed distribution

    • EPD on transformed distribution can use to estimate value of default put

18.2.4 Generalized Moments

Includes more than just \(\mathrm{E}[Y^k]\), can include all of the above risk measures (Not just expectations of powers of the variable)

TVaR expressed in generalized moments

\[TVaR_{\alpha} = \mathrm{E} \left[ Y \cdot \left( Y > F^{ -1}_Y(\alpha) \right) \big| Y > F^{ -1}_Y(\alpha) \right]\]

  • \((Y > b) = \begin{cases} 1 & : Y > b\\ 0 & : \text{otherwise} \end{cases}\)

  • Average of \(Y\) when \(Y\) > than the \(\alpha^{th}\) percentile: \(F^{ -1}(\alpha)\)

Spectral Measures

  • This includes many tail measures that can be written in the following form

\[\rho(Y) = \mathrm{E} \left[ Y \cdot \eta \left( F(Y) \right) \right]\]

  • \(\eta \geq 0\)

  • Multiply the loss by a non-negative scalar \(\eta(Y)\) when taking the expectations

  • TVaR is a spectral measures

Blurred VaR

  • Takes the distance from the target percentile as the weighting function

\[\eta(p) = \mathrm{exp} \left \{ - \theta (p-\alpha)^2 \right \}\]

  • Give more weight to losses near \(\alpha^{th}\) percentile while still using the whole distribution

  • \(\theta\) controls how quickly the weight drops of as we get further from \(\alpha\)

  • Takes several results and averages them together

Which risk measures to use

  • \(TVaR\) at a low percentile is preferred as it captures all risks above this level

    Set at level where we begin to loss capital

    • High percentile is not recommended as it ignores all risk below that percentile

  • Try to get close to market value of risk

    • \(WTVaR\) with the MET is promising, or

    • Exponential moment \(\mathrm{E}[Y e^{cY/\mathrm{E}[Y]}]\)