Chapter risk 6

Question 1

what sort of view is taken on risk in this chapter?

Answer 1

We want to quantify risk in monetary units so that the risk of a position can be interpreted as how much buffer capital is required to provide sufficient protection agaisnt undisirable outcomes.

Question 2

normal to see variance being used a a measure of risk. is it good?

Answer 2

No.

There are several downsides of using variance as a measure of risk.

1) variance doesnt differ between good and bad outcomes.

2) variance is not monetary value

Question 3

how do we represent a portfolio in this chapter

Answer 3

as a random variable

Question 4

sicne we measure portfolios as random variables, what do we need to do to measure the risk of the portfolio?

Answer 4

Consider the porbaiblity distribution of the portfolio

Question 5

elaborate on using portoflio random variable’s probability distribution

Answer 5

Key is that it will differ in various contexts.

An asset manager will consider the entire porbability distribution and wiegh up scenarios.

A risk controller will focus on the negative side only.

For instance, a risk controller may find some portfolio acceptable, while the asset manager does not if the returns are not there.

Question 6

what appears to be the keyword in this chapter

Answer 6

Acceptable

We are looking for acceptable portfolios

Question 7

why not compare probability distributions against each other?

Answer 7

Very difficult to do.

It is much easier to have a single number used for comparison. this is one of the motivations for using a risk measure that computes a monetary value number.

Question 8

define a risk measure

Answer 8

a function, p, so that p(X) assigns a real number to each value of X, and p(X) gives the capital requirement that we need to invest in risk free asset to consider the position as acceptable.

Question 9

when is p(X) acceptable

Answer 9

if p(X)<=0, acceptable

if p(X) > 0, not acceptable.

Question 10

elaborate on the good properties of risk measures

Answer 10

There are 6 properties.

1) Translation invariance
2) Monotonicity

3) Convexity

4) normalization

5) Positive homogeneity

6) Subadditivity

Question 11

elaborate on translation invariance

Answer 11

Refers to the fact that id we add cash to the portfolio, then the capital requirement for an accetpable portfolio should be reduced by this same amount.

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

Question 12

elaborate on monotonciity

Answer 12

if the value of a portfolio is smaller than the value of another portfolio, then this must be shown wit hthe measure.

if X1 < X2:
then
p(X1) < p(X2)

Question 13

elaborate on convexity

Answer 13

This is a risk sharing property saying that if we invest in something together, then the risk of the position when considered toegther is smaller than or equal to the risk of adding the two risks evaluated independently.

Question 14

elaborate on normalization

Answer 14

p(0) = 0

Question 15

elaborate on positive homogeneity

Answer 15

p(kX) = kp(X)
for all positive k (thus positive homogeneity)

if we double the size of the position, we double the risk, basically

Question 16

elaborate on subadditivity

Answer 16

risk sharign.

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

Question 17

how can we achieve subadditivity?

Answer 17

convexity together with positive homogeneity implies subadditivity

Question 18

what is a coherent risk measure?

Answer 18

if we have:
translation invariance,
monotonicity,
positive homogeneity,
subadditivity

then we have a coherent risk measure.

A coherent risk measure is also convex.

Question 19

what is a convex risk measure

Answer 19

must satisfy

translation invariance
monotonicity
conveixty

Question 20

what is mean-variance risk measure

Answer 20

mean variance is a risk measure that balance expected value and variance. technically it use standard deviation. it is multiplied by a number of standard deviations so one can itnerpret it as being a balance between expected return and a certain multiplicative of standard deviations, and this balance must be negative in order to produce an acceptable portoflio. Therefore, the expected value term is negative, and the variance term is positive.

Question 21

what is bad about mean variance risk measures

Answer 21

no monotoncitiy. This makes is non-convex.

Question 22

what can we say about risk measure of X when X is normally distributed?

Answer 22

any risk measure that is translation invriant and positively homogeneous must look like a mean-variance measure, for some constant c where c = R_0 p(Z).

this is becasue we can write a normally distributed variable purely by using hte mean and standard deviation of it.

So, if we know that X is normally distributed, and we assume the basic axioms, then the risk measure must satisfy the mean-variance version.

Question 23

mean variance risk measure does not have monotonicity. when is this not a big deal?

Answer 23

mistinerpreted the book, it isalways a big deal

Question 24

what is VaR

Answer 24

Value at risk.

Value at risk is the smallest amount of capital that ensures you can cover your loss X with at least alpha probability.

Question 25

give the formula for VaR

Answer 25

VaR(X) = min {m : P(mR_0 + X < 0) <= p}

Question 26

what is the common way to consider VaR

Answer 26

We borrow funds equal to the value of the portfolio at time 0, which is V_0. Then we acquire the portfolio. At time 1, we have: X = V_1 - V_0R_0 As a result, the portfolio is classified as acceptable if the difference between actual value and alternative cost value is positive.

Question 27

given an itnerpretation of VaR

Answer 27

the smallest amount of capital we need to invest in a risk free asset in order to cover our heavy losses 1-p percent of the time.

Question 27

define u-quantile

Answer 27

The u-quantile of a random variable L with distribution function F_L is defined as: F_L^{-1}(u) = min{m : F_L(m) >= u} Meaning, the u-quantile is defined as a value that is the smallest value that still makes the probability

Question 28

the formula for VaR is messy. can we make it clearer?

Answer 28

starts with: VaR(X) = min{m : P(mR_0 + X < 0) <= p} P(-X/R_0 > m) <= p 1 - P(-X/R_0 <= m) <= p P(-X/R_0 <= m) >= 1-p We can use "L" for -X/R_0 P(L <= m) >= 1-p We end up at: VaR(X) = min{m : P(L <= m) >= 1-p} So it is a basic "find the smallest value m so that the probability of L being smaller than m is larger than 1-p. L is -X/R_0, and X is the differnce between V1 and V0R0. It is common to use 0.05, 0.01 for p, so that 1-p becomes very large.

Question 29

elaborate on VaR and quantile

Answer 29

Var(X) is in the (1-p)-quantile of L, where L is the random variable.

Question 30

when is u-quantile an ordinary inverse?

Answer 30

if X is strictly increasing

Question 31

when is F_L^{-1} = m a unique value

Answer 31

if L is strictly increasing an continuous

Question 32

how is quantile function related to VaR

Answer 32

quantile function looks for this same m value. Therefore, we can use it to clear the notation. VaR(X) = F_L^{-1}(1 - p)

Question 33

what is the difference between convexity and subadditivity

Answer 33

convexity is about mixing portfolios. Subadditivity is about aggregation.

Question 34

what is the relationship between the quantile function and CDFs etc?

Answer 34

We use the uniform distribution. We consider a unifmrly distributed variable U, which is defined on (0,1). THen, if we have some distribution function F (cumulative), we have the following properties: 1) u <= F(x) if and only if F^{-1}(u) <= x 2) if F is continuous then the F(F^{-1}(u))=u 3) (Quantile transform) If U is U(0,1) distributed, then P(F^{-1}(U) <= x) = F(x) 4) Probability transform if X has CDF distr F, then F(X) is U(0,1) distributed if and only if F is continuous.

Question 35

why do we refer to mean variacne risk measures as measures and not a measure?

Answer 35

We can can multiple by adjusting the "c" constant

Question 36

define a monetary risk measure

Answer 36

A risk measure that is cash invariance AND monotoncitiy is monetary.

Question 37

what is monotoncitiy ?

Answer 37

if portfolio X always gives less than Y, then p(X) must be a larger number than p(Y).

Question 38

other word for cash invariance

Answer 38

transaltion invarianve

Question 39

important thing to remember regarding risk measures

Answer 39

thigns like variance is not monotonic. This means that risk measures that are based on variance is not necessarily monotonic. This is important to remember, because some properties are not always upheld.

Question 40

Elaborate on transaltion invariance

Answer 40

If we add cash to a position (assume in the risk free rate), this should cover part of losses. Therefore, we should have a risk measure that accounts for this. p(X + cR_0) = p(X) - c

Question 41

elaborate on solvency II

Answer 41

Solvency capital requirement is important in the insurance sector. Assuming that L_i represent the liabilities at time i, and A_i represdent the assets at time i: p(A_i - L_i) <= 0 This is the solvency requirement. We can then play around to iunderstand it better. We can use the fact that: A_1 - L_1 = difference in assets - difference in liabilities etc A_1 - L_1 = (A_1 - A_0R_0) - (L_1 - L_0R_0) + (A_0 - L_0)R_0 Then we take the risk measure of this, and use the fact that if the risk measure is monetary, then it is also cash invariant or transaltion invariant, which means that we ca npull the iniitial amount OUT like this. p(A_1-L_1) = p(SRC) - (A_0 - L_0) We know that p must be smaller than or equal to 0, so we get: p(SRC) - (A_0-L_0) <= 0 p(SRC) - A_0 + L_0 <= 0 A_0 >= p(SRC) + L_0 in other words, the assets at the beginning must be worth more than the liabilities + the risk measure of the solvency required capital. SRC is the random component. L_0 and A_0 are not random.

Question 42

what is positive homogeneity property

Answer 42

risk increase linearly with the size of the position if we double siz,e risk dobuel as well

Question 43

define a coherent riks measure

Answer 43

cash invariant, monotonicity, positive homogeneity, subadditivity

Question 44

why is subadditivity important

Answer 44

incentivize diversification it also makes financial intsitutuons not look at "splitting up" as favorable to get less capital requirements. This is crucial, because if not then the financial institutions can run on higher levels of risk.

Question 45

elaborate on liquidity risk as a property of risk measures

Answer 45

this is an issue with the positive homogeneity property, because it assumes a linear relationship between size and risk. however, in real life, when the position grow large there will be liquidity risk in addition to the sheer volume.

Question 46

what is the relationship between convex and coherent risk measures

Answer 46

all coherent measures are also convex, but not all convex measures are coherent. it is a subtle point, and it comes frm the fact that subadditivity and positive homogeneity together implies convexity property. If only subadditivie, we could separate the 2 portions, but not move the lambda factors outside. howeve,r if we have both properties, we can show the convexity property as well.

Question 47

define a convex risk measure

Answer 47

satisfying cash invariance, monotonicity, convex property. NOT positive homogeneity.

Question 48

which is stronger, convex or coherent?

Answer 48

coherent. but cohrernet doesnt include liquidity risk handling.

Question 49

why does mean-variance actually use standard deviation (sqrt var) instead of just the varinace`

Answer 49

standard deviation is subadditivity variance is not subadditivity

Question 50

what is value at risk

Answer 50

it is a quantile, statistical quantile, of the profit/losses distribution.

Question 51

what are statistical quantiles

Answer 51

quantiles are values that divide a probability distribution into equal-sized intervals. Quantiles are percentiles sort of. quantiles represent the value, the divider, but the percentile is the cumulative area.

Question 52

elaborate on the distinction between quantiles and percentiles

Answer 52

percentiles and quantiles are basically the same thing, as they refer to the same threshold. However, it seems like percentiles are used to describe a measure that relates to a percentage, meaning that its purpose is to classify regions, whereas the quantile is used to refer to the very specific value that is associated with the percentile divider. So, saying that "children above the quantile of 130 should be offered more advanced courses" and "children above the 98th percentile should be offered more advanced courses" have exactly the same meaning, it is just framed differently.

Question 53

how do we refer to soemthing being anywhere in the lower 95%, meaning that he is definitely not in the best 5%

Answer 53

The sample lies at or below the 95th percentile.

Question 54

how is the u-quantile function defined

Answer 54

as the inverse of the cumulative distribution of whatever distribution we're working with, and then with u as the argumetn. The u-quantile is associated with a value that represent a sort of percentile. F_L^{-1}(u) = min{m : P(L >= m) = u} or alternatively F_L^{-1}(u) = min{m : F_L(m) >= u}

Question 55

which of the properties holds for VaR

Answer 55

translation invariance, positive homogeneity and monotonicity

Question 56

what is the interpretation of VaR when using L as variable

Answer 56

L is for the loss. we want to keep the loss small. VaR = min{m: P(L <= m) >= 1-p}

Question 57

when using VaR, what is the 'argument'?

Answer 57

a random variable that basically track our portfolio

Question 58

give the quantile function representation of: VaR_{p}(S1-S2)

Answer 58

F_{S1-S2}^{-1}(1-p}