Mildenhall Ch 4: Measuring Risk with Quantiles, VaR and TVaR Flashcards
Define quantile
Suppose F(x) = p then q(p) = x is known as the p-quantile.
When used as a risk measure, it is known as Value at Risk (VaR).
Identify 2 main issues when defining quantiles. Provide an example and a way to address each issue.
- The equation F(x) = p might not have a unique solution if F is not strictly increasing.
Ex: F might have a flat spot
Solution: Any of those x would be considered p-quantiles. However, it is most natural to select either first x value (lower quantile) or last x value (upper quantile) along the flat spot. - The equation F(x) = p may have no solution if F is not continuous.
Ex: F jumps from below p to above p in discrete distribution
Solution: define p-quantile such that P(X<x) =< p =< P(X =< x)
Summarize the 3 steps to compute quantiles
- Determine where a horizontal line at height p intersects graph of distribution function, including vertical segments.
- If there is a unique solution, it is the unique p-quantile.
- If there is an interval of intersection (flat spot), then:
a. Smallest value is lower p-quantile
b. Largest value is upper p-quantile
c. Any value in between is also p-quantile
Briefly describe how we can use quantile functions to simulate values of X (how is the method called)
- Randomly draw value w from uniform r.v. U
- Assume we have quantile function q- describing quantiles of X, we can simulate a value X s.t. x = q-(w)
- If we do this thousands of time, we can effectively simulate distribution of X
This simulation method is known as the inversion method.
Define Value at Risk (VaR)
The p-VaR of a loss r.v. X equals the lower p-quantile q-(p)
VaRp(X) is the smallest loss such that P(X=< VaRp(X)) >=p
State 4 advantages of VaR
- Simple to explain
- Can be estimated robustly
- Always finite
- Widely used by regulators, rating agencies & companies in their internal risk management.
State a disadvantage of VaR
It does not always recognize diversification (i.e. not always sub-additive)
Briefly explain how VaR can be expressed in terms of return period
Assuming X represents aggregate losses over 1y, a loss size VaRp(X) is expected to occur once every 1/(1-p) years.
Waiting time has geometric distribution with parameter p.
Average waiting time is 1/(1-p)
Define PML
PML estimates the largest loss that a building is likely to suffer from single fire if all protection systems function as expected
Define Maximum Foreseeable Loss (MFL)
MFL is the largest loss a building is likely to suffer from a single fire if protection system fails.
Assume CAT events follow Poisson Process, calculate the prob of 1 or more events causing a loss X or more.
1 - exp(-lambda * S(x))
Contrast n-year Occurrence PML and Aggregate PML
n-year Occurrence PML is the smallest loss X s.t. prob of one or more events causing a loss x or more in a year is at most 1/n.
Aggregate PML is the same as aggregate VaR.
Aggregate PML can be approximated using n-year occ-type PML if X is thick-tailed and n is large.
Calcuate occurrence PML
PML
= q_x * (1 - log(n/(n-1))/lambda)
= VaR_1 - log(n/(n-1))/lambda (X)
since p = 1 - 1/n
= q_x * (1 + log(p)/lambda)
= VaR_1 + log(p)/lambda (X)
1 + log(p)/lambda is known as the adjusted probability level
Calculate Agg PML
Agg Losses (A) follow Compound Poisson Process
Agg PML
= VaR(A)
can be approx as:
= VaR_1-(1-p)/lambda (X)
When X0 is thin-tailed and X1 is thick-tailed, calculate VaRp(X0+X1)
VaRp(X0+X1)
= VaRp(X)
= E(X0) + VaRp(X1)
Define thin-tailed random variable
- Log-concave distribution
OR - Bounded distribution
Define thick-tailed random variable
Subexponential distribution (aka log-convex distribution)
S(x) = k*x^alpha for some constant k
In which case is the approx for Agg PML adequate
Agg VaR approx is excellent for larger return periods
error = estimated/true - 1 decreases as n increases (thus p = 1-1/n increases as well)
True or False?
Occ PML is always larger than Agg PML.
Explain.
True since it uses larger adjusted probability.
Complete the statement:
As lambda (frequency) increases, the difference between Occ PML and Agg PML …
Explain
Increases as well because adjusted probability p increases.
Complete the statement:
As means severity increases, Occ PML… and Agg PML…
As severity increases, both Occ PML and Agg PML increase
Define a sub-additive risk measure
A risk measure is sub-additive if the risk measure of the total is less than or equal to the sum of the risk measures for the individual units
rho(X1+X2+…+Xn) =< rho(X1) + rho(X2) + … + rho(Xn)
Identify 3 common cases where VaR fails to be sub-additive.
- Dependence structure is of a particular, highly asymetric form
- The marginals have a very skewed distribution
- Marginals are very-heavy tailed
True or False?
Sub-additivity fails for all p above threshold in all 3 cases.
False,
Sub-additivity holds for large enough p under cases 1 and 2 while fails for all p above a threshold for case 3.
Complete the sentence:
VaR is sub-additive for (…) distributions
Log-concave (i.e. sufficiently thin-tailed)
Briefly explain the 1st case where VaR fails to be sub additive
Given 2 non-trivial marginal distribution, it is always possible to find dependence structure where sub-additivity fails.
Co-monotonic pairing never fails sub-additivity because it assumed 100% dependence (no diversification benefit).
Counter-monotonic pairing is the worst possible dependence structure to avoid failure of sub-additivity because it maximizes p-VaR.
Define co-monotonic pairing
Pair largest value of X1 with largest value of X2, second largest value of X1 with second largest value of X2, … 100th largest value of X1 with 100th largest value of X2.
Identify 1 advantage of co-monotonic pairing and 2 disadvantages.
Advantage:
This pairing never fails sub-additivity because it assumes 100% dependence, thus no diversification benefit. VaR(X) = VaR(X1) + VaR(X2)
Disadvantages:
1. This pairing produces the greatest variance for X = X1 + X2
2. This pairing produces the worst TVaR characteristics for total X since we are creating the highest possible sums for worst events
Define counter-monotonic pairing (crossed pairing)
Pair largest value of X1 with 100th largest value of X2, second largest value of X1 with 99th largest value of X2, … 100th largest value of X1 with largest value of X2.
Identify 2 advantages and 2 disadvantages of counter-monotonic pairing (crossed pairing).
Advantages:
1. This pairing does not have extreme right tail dependence because it does not combine worst value X1 with worst value X2. In fact, it does the exact opposite.
2. This pairing is universal & works for any non-trivial marginal distributions X1 and X2
Disadvantages:
1. This pairing produces largest VaRp(X), but not the worst way to combine 2 distributions (co-monotonic is)
2. This pairing is the worst possible dependence structure for avoiding failure of sub-additivity because it maximizes p-VaR
Briefly describe the 2nd case where VaR fails to be sub-additive
Sub-additivity can fail for iid r.v. if they are very skewed.
Independent, thin-tailed variables such as exponential distribution only fail to be sub-additive for a range of p, not all p above threshold.
Briefly describe the 3rd case where VaR fails to be sub-additive
A more serious issue is when sub-additivity fails for all p above a certain threshold which is the case for thick-tailed iid marginal distributions.
Case 3 can arise when working with cat simulation model output.
Define Tail Value at Risk (TVaR)
Conditional average of the worst (1-p) proportion of outcomes.
Calculate TVaRp(X) for a continuous distribution
TVaR
= 1/(1-p) * integral of VaRs(X) ds
between p and 1
Calculate TVaR0(X)
E(X)
Calculate TVaR1(X)
sup(X) = max(X)
State 3 TVaR characteristics
- TVaR is continuous
- It is differentiable almost everywhere (kink as jumps in F)
- It is equal to the integral of its derivative (i.e. unique)
For a normal distribution, calculate TVaR, VaR and q_x(p)
TVaR = mu + sigma * phi(Zp)/(1-p)
VaRp(X) = mu + sigma * Phi^-1(p)
q_x(p) = mu + sigma*Zp
Zp = Phi^-1(p)
For a lognormal distribution, calculate TVaR and q_x(p)
TVaR = E(X)Phi(sigma - Zp)/(1-p)
q_x(p) = exp(mu + sigmaZp)
If X is continuous and density can be factored in C(a)* x^a *g(x), calculate TVaR.
Provide distributions that satisfy this condition.
TVaR = (C(a)/C(a+1)) * (1-F(q(p);a+1)) / (1-F(q(p);a))
Gamma, Generalized Gamma, Transformed Gamma and Generalized Betta distributions are examples.
Describe the algorithm to evaluate TVaR for a discrete distribution.
- Sort outcomes into ascending order X0, X1, …, XN-1
- Find n st n smaller or equal than p*N smaller than (n+1)
- If n+1 = N, TVaR = XN-1
- If n smaller than N-1, compute
T1 = Xn+1 + … + XN-1
T2 = ((n+1) - pN)Xn
TVaR = 1/(1-p) * (T1+T2)/N
Provide a simpler way (versus algorithm) to compute TVaR
For non-fractional p-values, TVaR is simply the avg pf all of the values above p percentile.
For fractional p-values, take weighted average of Xn to XN-1 values
Describe 2 alternative risk measures to TVaR
- Conditional Tail Expectation (CTE)
E(X given X greater or equal to VaR(X)) - Worst Conditional Expectation (WCE)
max(X given A) where P(A) greater than 1-p
What is the relationship between VaR, TVaR, CTE and WCE
For continuous rv, TVaR = CTE = WCE
Not always true for discrete or mixed rv.
In general, VaR < CTE < WCE < TVaR
What is the main difference between CTE and TVaR
TVaR is a continuous function, CTE is a step function
Describe the EPD risk measure
The required amount of assets needed to obtain EPD ratio = s.
Can be found by solving for EPDs(X) in:
E((X-EPDs(X))+) = s*E(X)
Calculate EPD ratio and describe it
EPD ratio = EPD / E(X)
EPD = E((X-a)+)
EPD ratio gives the proportion of losses unpaid when X is supported by assets a.
Describe the relationship between s and EPD risk measure
Smaller s means stricter standard.
Briefly explain why regulators like EPD risk measure.
Because it accounts for degree of default relative to promised payments.
Calculate EPD from TVaR
EPD = (1-p) * (TVaR - a)
True or False?
TVaR is sub-additive.
Explain.
Always true!
TVaR of total firm is less than or equal to the sum of individual units TVaRs.
True or False?
TVaR optimally balances cost of providing capital.
Explain.
True,
It optimally balances cost of providing capital x against cost of shortfall E((X-x)+).
TVaR solves an optimization problem.
Mathematically:
TVaR = min(x + (1-p)^-1 * E((X-x)+))
Calculate A, B, TVaR and CTE
A = (1-p0)*x0
B = E((X-xo)+) = EPD with x0 assets
TVaR = (A+B)/(1-p0) = CTE
Calculate A, B, C, D, TVaR and CTE
A = (p0 - p-)Xo
B = (p+ - p0)x0
C = (1 - p+)*x0
D = E((X-x0)+)
CTE = (A+B+C+D)/(1-p-)
TVAR = (B+C+D)/(1-p0)
Thus CTE = ((p0 - p-)x0 + (1-p0)TVaR) / (1-p-)
CTE is wtd avg of x0 and TVaR
Since x0 < TVaR, CTE < TVaR
