Risk measures

Question 1

elaborate on types of risk

Answer 1

we have differnet types of risk:

market risk: movements in financial market

credit risk: issuer risk refers to the risk that the custiemr will default

liquidity risk

operational risk

The idea is that “risk” is not well defined. What we mean by risk depends on the context.

Question 2

elaborate on what we should do we risk

Answer 2

We need to define a risk measure, which is a way of quantifying the risk.

Typically, we quantify risk in monetary units. The value would represent a capital buffer that we need.

Question 3

What are solvency 2, basel 3 etc

Answer 3

These are regulations placed on financial institutions that they must follow to ensure a certain way to deal with losses etc.

Solvency 2 is the insurance sector.

Basel 3 is for banks

Question 4

what VaR is used by Solvency 2?

Answer 4

99.5% over one year

Question 5

what is the potnetial pitfall with these regulations?

Answer 5

They say that banks etc must use VaR, but not “how” they should compute it. Makes inconsistent results

Question 6

elaborate on risk measures

Answer 6

A risk measure is a function that transforms the random variable X into a single number. X is a random variable representing our portfolio’s value.

The point of this is that this “value” that is spit out is a quantification of the risk.

It is “the minimum capital required that needs to be added to the initial posiiton to make the final position acceptabel”.

“Acceptable” refers to referrring to a probability of portfolio being non-profitable.

If p(X)<= 0, our portfolio is acceptable.
if p(X) > 0, our portfolio is not acceptable, it requires more capital.

Essentially, we need to add cash to shift the entire distribution.

Question 7

simplest example of a risk measure

Answer 7

an absolute lower bound

Question 8

elaborate on the absolute lower bound as a risk measure

Answer 8

“the portfolio is acceptable if it is surely larger than some given value”.

p(X) = min (a >= 0 : X + aR_0 >= C) = ( c - x) / R_0

if C=0, we get the case where p(X) = - x/R_0, which means that we need to invest “x” in risk free asset in order to guarantee no losses at time 1.

Question 9

elaborate on the mean-variance risk measures

Answer 9

p(X) = - E[X / R_0] + c sqrt(Var[X / R_0])

Question 10

what is “standard deviation premium principle”?

Answer 10

Same as mean-variance

Question 11

what is the risk measure we typically want?

Answer 11

coherent

Question 12

what is a monetary risk measure?

Answer 12

Needs to satisfy the properties of “monotonicity” and “cash invariance”

Question 13

define a monotonic risk measure

Answer 13

if X2 <= X1, then p(X1) >= p(X2)

Question 14

define cash invariance

Answer 14

p(X + cR_0) = p(X) - c

basically, adding cash to the position should reduce the “minimum capital required to make our portfolio acceptable”.

Question 15

according to solvency 2, what is needed to make portfolio acceptablr?

Answer 15

p(A1 - L1) <= 0

A1 is the assets
L1 is the liabilities

It is possible to make some maths here and get:

p(A1 - L1) = p((A1 - A0R0) - (L1 - L0 R0)) - (A0 - L0)

So basically, since p(A1 - L1) <= 0, we need:

A0 >= L0 + SCR, where SCR is the Solvency Capital Requirement

Question 16

define a coherent riks measure

Answer 16

Requires cash invariance, monotonicity, positive homogeneity, and subadditivitry

Question 17

Define positive homogenetity

Answer 17

p(kX) = kp(X)

This basically just means that risk increase linearly with the size of the position.

If we double the position size, we double the risk as well.

Note that in some cases, this is wrong. For instance in extremely large positions.

Question 18

Define subadditivity

Answer 18

p(X1 + X2) <= p(X1) + p(X2)

Meaning, the risk of two assets together should never be larger than the risk of holding both of them individually. Basically, diverisification should rewarded.

Question 19

Define a convex measure

Answer 19

A convex measure satisfy:
- cash invariance
- monotonicity
- convexity properties

All coherent are convex. But not all convex are coherent

Question 20

define convex proeprty

Answer 20

Question 21

what happens if we have a convex measure that satisfies positive homogeneity?

Answer 21

We get a coherent measure

Question 22

What proeprties can create convexity?

Answer 22

subadditivity + positive homogeneity

Question 23

does coonvexity imply positive homogeneity and subadditivity?

Answer 23

no, not alone

Question 24

What properties does the mean-variance risk measure have?

Answer 24

it satisfies:
- cash invariance
- convexity
- positive homogeneity
- subadditivity

Because it is lacking monotonicity, we do not have that it is either a convex risk measure or a coherent risk measure.

Question 25

what happens if we take the mean-variance measure and instead of using the standard deviation, we use the variance?

Answer 25

Standard deviation is subadditivite, variance is not

Question 26

Introduce VaR

Answer 26

Value at Risk is a statistical technique that allows us to estimate hte potential losses in a certain time . Definition: VaR at some level 'p' is a statistical quantile of the profit/losses of a distribution. If X is the value of our position, then: VaR_p (X) = min {m : P(mR_0 + X < 0) <= p} The smallest value 'm' such that the when compounded with the ris kfree rate and added to the portfolio value, the probability that this sum is less than 0 is smaller than or equal to the level p. In other words, we are adjusting the value m that we need ot invest in risk free returns so that we make the probability of default be lower than or equal to that specific probability p.

Question 27

if X is positive regardless, what is VaR?

Answer 27

Negative, indicating that we do not need ot invest anything to mke the position acceptabel

Question 28

what is the relationship between CDF and VaR

Answer 28

The inverse of the CDF gives the quantile, which we know is the VaR

Question 29

elaborate on the "better way" or "alternative way" of looking at VaR

Answer 29

consider the loss distribution isntead of teh entire payoff distirbuiton. We start with the regular VaR: VaR = min {m : P(mR0 + X < 0) <= p} We look at the prob: P(mR0 + X < 0) <= p We also have that "L = -X/R0" P(mR0 < -X) <= p P(m < -X/R0) <= p m is the value from the p-quantile P(m < L) <= p We want it the other way around since L is the random variable. We use that P(m < L) = 1 - P(m >=L) We get: 1 - P(L <= m) = p P(L <= m) = 1 - p So, now we have the expression that "the probability that our loss is less than or equal to 'm' is for instance 95%". 'm' is still the value at risk, but now we write it as a function of the loss function ratehr.

Question 30

elaborate on the properties of VaR

Answer 30

cash invariant monotonicity positive homogeneity IT DOES NOT HAVE: - subadditivity - convexity As a result, VaR does not reward diversification

Question 31

is VaR always not subadditive?

Answer 31

In general, we dont have subadditivity. however, in certain cases, we do have subadditivty.

Question 32

name example of case where VaR is subadditive

Answer 32

When VaR is applied to normally distributed returns, we have subadditivity.

Question 33

What is VaR when we consider it on normally distributed returns?

Answer 33

VaR is still the negative of the inverse cumulative distribution. Since we are considering a normal distribution, we have the inverse as well. The formula is provided in the sheet Note that this is about value at risk for the returns over the given time period.

Question 34

when the returns are normally distributed, what happens to the prices?

Answer 34

They are log normally distributed

Question 35

elaborate on downsides of VaR

Answer 35

one need to consider a time hgorizon one need to consider the value for p often, the lack of subadditivity can cause a lack of incentive to diversify Also, VaR does not give an answer to the size of the losses.

Question 36

perhaps a better risk measure than VaR

Answer 36

Expected shortfall

Question 37

elaborate on expected shortfall

Answer 37

The biggest (one of the biggest) downfalls to value at risk is that it doesnt account for losses larger than the VaR. To correct this, one may compute the average value at risk for all p smalelr than the reference value.

Question 38

give the alternative way of defining expected shortfall

Answer 38

Question 39

nice property of ES

Answer 39

it is coherent

Question 40

how to find empirical CDF

Answer 40

we want to, for each x_k, count the instances in our sample where "X<=x_k", and divide on "n" where 'n' is the number of sample points. When we do this for each x_k, we approximate the empirical CDF.

Question 41

what is to formula to find which value to use in a sample when we want a specific quantile, like 0.05-quantile, from some sample?

Answer 41

compute the subscript and we're good

Question 42

if we use L = -X/R_0 isntead of the regular expression, what do we get in regards to qunatile order?

Answer 42

1-p. If we use p=0.05, we need to use 0.95 now

Question 43

how do compute empirical quantile when using L=-X/R_0 from a sample

Answer 43

Question 44

what is the purpose of using confidence intervals in this context?

Answer 44

Error estimates for our VaR and ES estimates

Question 45

elaborate on using confidence intervals in the empiricial context

Answer 45

We want the following;: P(A <= quantile(p) <= B) = q This entails: P(quantile <= A) = (1-q)/2 P(quantile >= B) = (1-q)/2 IF we do not know the distribution of the desntityy we use to compute the quantile, then it is very difficult to find an exact expression for the confidence interval. However, we can do it empirically.

Question 46

elaborate on double sided confidence intervasl

Answer 46

Sometimes, we want to find an interval where we choose A and B so that each side of hte parameter we are estimating has the same probability: P(A <= theta <= B) = q would then be the same as: P(A >= theta) = P(B <= theta) = (1-q)/2