Mildenhall Ch 4: Measuring Risk with Quantiles, VaR and TVaR

Question 1

Define quantile

Answer 1

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).

Question 2

Identify 2 main issues when defining quantiles. Provide an example and a way to address each issue.

Answer 2

1. 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.
2. 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)

Question 3

Summarize the 3 steps to compute quantiles

Answer 3

1. Determine where a horizontal line at height p intersects graph of distribution function, including vertical segments.
2. If there is a unique solution, it is the unique p-quantile.
3. 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

Question 4

Briefly describe how we can use quantile functions to simulate values of X (how is the method called)

Answer 4

1. Randomly draw value w from uniform r.v. U
2. Assume we have quantile function q- describing quantiles of X, we can simulate a value X s.t. x = q-(w)
3. If we do this thousands of time, we can effectively simulate distribution of X

This simulation method is known as the inversion method.

Question 5

Define Value at Risk (VaR)

Answer 5

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

Question 6

State 4 advantages of VaR

Answer 6

1. Simple to explain
2. Can be estimated robustly
3. Always finite
4. Widely used by regulators, rating agencies & companies in their internal risk management.

Question 7

State a disadvantage of VaR

Answer 7

It does not always recognize diversification (i.e. not always sub-additive)

Question 8

Briefly explain how VaR can be expressed in terms of return period

Answer 8

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)

Question 9

Define PML

Answer 9

PML estimates the largest loss that a building is likely to suffer from single fire if all protection systems function as expected

Question 10

Define Maximum Foreseeable Loss (MFL)

Answer 10

MFL is the largest loss a building is likely to suffer from a single fire if protection system fails.

Question 11

Assume CAT events follow Poisson Process, calculate the prob of 1 or more events causing a loss X or more.

Answer 11

1 - exp(-lambda * S(x))

Question 12

Contrast n-year Occurrence PML and Aggregate PML

Answer 12

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.

Question 13

Calcuate occurrence PML

Answer 13

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

Question 14

Calculate Agg PML

Answer 14

Agg Losses (A) follow Compound Poisson Process

Agg PML
= VaR(A)
can be approx as:
= VaR_1-(1-p)/lambda (X)

Question 15

When X0 is thin-tailed and X1 is thick-tailed, calculate VaRp(X0+X1)

Answer 15

VaRp(X0+X1)
= VaRp(X)
= E(X0) + VaRp(X1)

Question 16

Define thin-tailed random variable

Answer 16

1. Log-concave distribution
   OR
2. Bounded distribution

Question 17

Define thick-tailed random variable

Answer 17

Subexponential distribution (aka log-convex distribution)

S(x) = k*x^alpha for some constant k

Question 18

In which case is the approx for Agg PML adequate

Answer 18

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)

Question 19

True or False?
Occ PML is always larger than Agg PML.
Explain.

Answer 19

True since it uses larger adjusted probability.

Question 20

Complete the statement:
As lambda (frequency) increases, the difference between Occ PML and Agg PML …
Explain

Answer 20

Increases as well because adjusted probability p increases.

Question 21

Complete the statement:
As means severity increases, Occ PML… and Agg PML…

Answer 21

As severity increases, both Occ PML and Agg PML increase

Question 22

Define a sub-additive risk measure

Answer 22

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)

Question 23

Identify 3 common cases where VaR fails to be sub-additive.

Answer 23

1. Dependence structure is of a particular, highly asymetric form
2. The marginals have a very skewed distribution
3. Marginals are very-heavy tailed

Question 24

True or False?
Sub-additivity fails for all p above threshold in all 3 cases.

Answer 24

False,
Sub-additivity holds for large enough p under cases 1 and 2 while fails for all p above a threshold for case 3.

Question 25

Complete the sentence: VaR is sub-additive for (...) distributions

Answer 25

Log-concave (i.e. sufficiently thin-tailed)

Question 26

Briefly explain the 1st case where VaR fails to be sub additive

Answer 26

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.

Question 27

Define co-monotonic pairing

Answer 27

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.

Question 28

Identify 1 advantage of co-monotonic pairing and 2 disadvantages.

Answer 28

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

Question 29

Define counter-monotonic pairing (crossed pairing)

Answer 29

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.

Question 30

Identify 2 advantages and 2 disadvantages of counter-monotonic pairing (crossed pairing).

Answer 30

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

Question 31

Briefly describe the 2nd case where VaR fails to be sub-additive

Answer 31

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.

Question 32

Briefly describe the 3rd case where VaR fails to be sub-additive

Answer 32

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.

Question 33

Define Tail Value at Risk (TVaR)

Answer 33

Conditional average of the worst (1-p) proportion of outcomes.

Question 34

Calculate TVaRp(X) for a continuous distribution

Answer 34

TVaR = 1/(1-p) * integral of VaRs(X) ds between p and 1

Question 35

Calculate TVaR0(X)

Answer 35

E(X)

Question 36

Calculate TVaR1(X)

Answer 36

sup(X) = max(X)

Question 37

State 3 TVaR characteristics

Answer 37

1. TVaR is continuous 2. It is differentiable almost everywhere (kink as jumps in F) 3. It is equal to the integral of its derivative (i.e. unique)

Question 38

For a normal distribution, calculate TVaR, VaR and q_x(p)

Answer 38

TVaR = mu + sigma * phi(Zp)/(1-p) VaRp(X) = mu + sigma * Phi^-1(p) q_x(p) = mu + sigma*Zp Zp = Phi^-1(p)

Question 39

For a lognormal distribution, calculate TVaR and q_x(p)

Answer 39

TVaR = E(X)*Phi(sigma - Zp)/(1-p) q_x(p) = exp(mu + sigma*Zp)

Question 40

If X is continuous and density can be factored in C(a)* x^a *g(x), calculate TVaR. Provide distributions that satisfy this condition.

Answer 40

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.

Question 41

Describe the algorithm to evaluate TVaR for a discrete distribution.

Answer 41

1. Sort outcomes into ascending order X0, X1, ..., XN-1 2. Find n st n smaller or equal than p*N smaller than (n+1) 3. If n+1 = N, TVaR = XN-1 4. If n smaller than N-1, compute T1 = Xn+1 + ... + XN-1 T2 = ((n+1) - p*N)*Xn TVaR = 1/(1-p) * (T1+T2)/N

Question 42

Provide a simpler way (versus algorithm) to compute TVaR

Answer 42

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

Question 43

Describe 2 alternative risk measures to TVaR

Answer 43

1. Conditional Tail Expectation (CTE) E(X given X greater or equal to VaR(X)) 2. Worst Conditional Expectation (WCE) max(X given A) where P(A) greater than 1-p

Question 44

What is the relationship between VaR, TVaR, CTE and WCE

Answer 44

For continuous rv, TVaR = CTE = WCE Not always true for discrete or mixed rv. In general, VaR < CTE < WCE < TVaR

Question 45

What is the main difference between CTE and TVaR

Answer 45

TVaR is a continuous function, CTE is a step function

Question 46

Describe the EPD risk measure

Answer 46

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)

Question 47

Calculate EPD ratio and describe it

Answer 47

EPD ratio = EPD / E(X) EPD = E((X-a)+) EPD ratio gives the proportion of losses unpaid when X is supported by assets a.

Question 48

Describe the relationship between s and EPD risk measure

Answer 48

Smaller s means stricter standard.

Question 49

Briefly explain why regulators like EPD risk measure.

Answer 49

Because it accounts for degree of default relative to promised payments.

Question 50

Calculate EPD from TVaR

Answer 50

EPD = (1-p) * (TVaR - a)

Question 51

True or False? TVaR is sub-additive. Explain.

Answer 51

Always true! TVaR of total firm is less than or equal to the sum of individual units TVaRs.

Question 52

True or False? TVaR optimally balances cost of providing capital. Explain.

Answer 52

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)+))

Question 53

Calculate A, B, TVaR and CTE

Answer 53

A = (1-p0)*x0 B = E((X-xo)+) = EPD with x0 assets TVaR = (A+B)/(1-p0) = CTE

Question 54

Calculate A, B, C, D, TVaR and CTE

Answer 54

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