Mildenhall Chapter 4 Flashcards
(14 cards)
Quantile q(p)
Formula
p = non-exceedance probability = F(x)
Continuous AND Strictly Increasing
* Quantile = q(p) = inverse F(p)
* F(q(p)) = p
What is the lowest x value that the CDF, F(X) exceeds or equal to the p-quantile
P(X < x) ≤ p ≤ P(X ≤ x) where q(p) = x
Discrete Example
* if p is in between 2 F(X) values, take the x associated with the higher one
* if p = exactly 1 F(X) value, then it can be any x between that F(X) and the next one
Value at Risk (VaR)
VaR(X) = inf {x | F(x) ≥ p } = sup {x | F(x) < p}
Lowest x value where the F(x) ≥ p where p is the percentile/p-quantile (e.g. 90th percentile)
Given that p (usually 90%, 95%, 99%, etc), what is the X loss?
You have to sort the X values in ascending order
Return Period, n
Formula
n = 1 / (1 - p) –> p = 1 - 1/n
where p is the p-quantile
e.g 75th quantile, return period = n = 4,
“1 in 4 year event”
Occurrence Probable Max Loss / Annual Aggregate VaR
Formula
VARp where we used a adjusted p = 1 + ln(p)/λ
VaRp(A) ~ VaRp, where p = 1-(1-p)/λ
Probability of at least 1 loss greater than x
1- p(0)
1 - exp(-λ * S(x))
Given X0 is thin tailed and X1 is independent thick tailed
Var_p(X)
This is an approximation
≈ E[X0] + VaR_p(X1)
For the VaRp(X1) - use aggregate VaR formula
Examples:
thin tailed - non-cat losses
thick tailed - cat losses
VaR
Subadditive
Condition and advantages
Subadditive if:
VaR(X1 + X2) ≤ VaR(X1) + VaR(X2)
Advantages:
* Simple
* Easy to explain
* Exists for all random variables
* Robust estimates
3 Ways VaR Fail to be Subadditive
- When dependence structure is of a particular, highly asymmetric form - fails because X1, X2 are dependent
- When the marginals have a very skewed distribution - fails for IID for a range of p
- When the marignals are very heavy-tailed - fails for IID for all p above a threshold
VaR Advantages/Disadvantages
Advantages
* It’s simple to explain
* Widely used by regulators, rating agencies and companies for internal risk management
* Can be estimated robustly
* Is always finite
Disadvantage
* Does not always recognize diversification (fails to be subadditive)
TVaRp(X) for Normal and Lognormal
Formula
Normal: E[X] + SD * ϕ(Zp) / (1-p)
* where Zp = norm.S.inv(p)
* where ϕ(Zp) = norm.S.dist(Zp, FALSE)
Lognormal: E[X] * norm.S.dist(σ - Zp, TRUE) / (1-p)
* where Zp = norm.S.inv(p)
FALSE = PDF | TRUE = CDF
Lognormal Mean, CV, Second Moment
- E[X] = exp(μ + SD^2/2)
- CV = sqrt(exp(SD^2)-1)
- E[X^2] = exp(2μ + 2SD^2)
Expected Policyholder Deficit
EPD, EPD Ratio, Target EPD, EPD Risk Measure
Discrete Example
EPD = E[(X- a)+] = sumproduct(probability, (X - a)+ )
Actual EPD Ratio = EPD / E[X]
Target EPD = s * E[X]
EPD Risk Measure: trial and error a new asset amount so that new EPD = Target EPD
s = expected EPD ratio | X-a can only be positive
Expected Policyholder Deficit
EPD, EPD Ratio
Continuous Example
EPD = [ 1 - F(a) ] * [ TVaRp(X) - a ]
where p = F(a)
Risk Measure
Why Regulators Prefer EPD Over VaR
- VaR ignores any losses greater than the VaR
- EPD accounts for the degree of default relative to the promised payments
