Deck 20: Value at Risk

Question 1

The value-at-risk of a portfolio is

A) Always positive.
B) Always negative.
C) May be positive or negative.
D) Always non-negative.

Answer 1

May be positive or negative.

Question 2

You invest $100 each in two bonds. Each bond will pay you $110 at the end of the year with probability 0.98 and nothing with probability 0.02. The correlation between the bonds is zero. In this scenario, the 98%-VaR of your portfolio is

A) $- 20$ .
B) Zero.
C) $90
D) $200

Answer 2

$90

Question 3

Which of the following best characterizes the mathematical properties of the risk measure VaR?

A) It is a dollar quantity.
B) It is a measure of first moment of returns.
C) It is a measure of the second moment of returns.
D) It is a percentage loss rate.

Answer 3

It is a dollar quantity.

Question 4

You invest $100 each in two bonds. Each bond will pay you $110 at the end of the year with probability 0.98 and nothing with probability 0.02. The correlation between the bonds is zero. In this scenario, the 95%-VaR of your portfolio is

A) $- 20$ .
B) Zero.
C) $90
D) $200

Question 5

Value-at-Risk (VaR) is most closely defined as
(a) The probability of a specified level of negative return of a portfolio in the left tail of its profit-and-loss distribution.
(b) The dollar loss at which a pre-specified cumulative probability occurs in the left tail of the profit-and-loss distribution.
(c) The probability of a negative return below a specified threshold in the left tail of the distribution.
(d) The negative return rate for which a pre-specified probability lies in the left tail of the distribution.

Question 6

Consider a two-asset portfolio invested with $10 in each asset. The mean returns of the two assets are $10 \%$ and $15 \%$ . The correlation of returns is 50%. The standard deviation of returns is 20% and 30%, respectively. What is the 99%-VaR of this portfolio?

A) 6.22
B) 7.66
C) 8.40
D) 10.58

Question 7

A portfolio has a current value of $1000. The annual profit $X$ is distributed normally with mean 100 and standard deviation 100. How much capital is adequate for the portfolio at a 95%-VaR?

A) $- 133$
B) $- 65$
C) $65$
D) $133$

Question 8

Monte Carlo is widely-used approach for computing VaR. Relative to other methods which of the following is a benefit of using this approach?

A) It is fully flexible in parameterizing the forward-looking distribution.
B) It always requires very few parameters.
C) It always uses normality.
D) It is the fastest approach to computing VaR.

Question 9

Which of the following is not a valid statement about VaR?

A) It is not a general measure of risk but a measure of extreme risk only.
B) It is an ingredient in computing capital requirements.
C) It is a single metric comparing return performance across portfolios.
D) It is a one-sided measure of risk.

Question 10

You invest $100 in a corporate bond. You estimate that with probability 0.94, the corporation will pay back the promised amount of $110 at the end of one year; with probability 0.04, the corporation will default and the recovered amount will be $70; and with probability 0.02, the corporation will default and you will recover nothing. The 95%-VaR in this scenario is

A) $30
B) $40
C) $70
D) $100

Question 11

The delta-normal method for computing VaR has many advantages. Which of the following is not a characteristic of the delta-normal approach?

A) VaR can be calculated analytically.
B) It is simple to apply because it uses the normal probability distribution.
C) It is a non-parametric method.
D) It can be used to compute risk-decompositions.

Question 12

If a portfolio is doubled in size, keeping its portfolio structure (holdings proportions) the same as before, the VaR will

A) Remain the same.
B) Double.
C) Increase but be less than double its previous value.
D) More than double.

Question 13

A portfolio has a current value of $1000. The annual profit $X$ is distributed normally with mean 100 and standard deviation 100. What is the probability that the portfolio will be worth less than 800 after one year?

A) 0.0014
B) 0.0062
C) 0.0228
D) 0.1587

Question 14

You invest $100 in a corporate bond. You estimate that with probability 0.95, the corporation will pay back the promised amount of $110 at the end of one year; with probability 0.04, the corporation will default and the recovered amount will be $70; and with probability 0.01, the corporation will default and you will recover nothing. The 98%-VaR in this scenario is

A) $30
B) $40
C) $70
D) $100

Question 15

VaR as a risk measure has the following deficiency:

A) It does not consider the shape of losses in the left tail of the P&L distribution.
B) It does not consider the shape of losses outside the left tail of the P&L distribution.
C) Neither of the above.
D) Both of the above.

Question 16

Historical simulation as a method of computing VaR has the following major benefit in comparison to the delta-normal method:

A) It is a faster approach.
B) It uses past returns to forecast future returns.
C) It requires the same number of parameters as the delta-normal method.
D) It does not assume normality of the P&L return distribution.

Question 17

You invest $100 in a corporate bond. You estimate that with probability 0.94, the corporation will pay back the promised amount of $110 at the end of one year; with probability 0.04, the corporation will default and the recovered amount will be $70; and with probability 0.02, the corporation will default and you will recover nothing. The 90%-VaR in this scenario is

A) $30
B) $40
C) $70
D) $100

Question 18

A portfolio has a current value of $1000. The annual profit $X$ is distributed normally with mean 100 and standard deviation 100. What is the 99%-VaR?

A) $- 133$
B) $- 65$
C) $65$
D) $133$

Question 19

You invest $100 in a corporate bond. You estimate that with probability 0.95, the corporation will pay back the promised amount of $110 at the end of one year; with probability 0.04, the corporation will default and the recovered amount will be $70; and with probability 0.01, the corporation will default and you will recover nothing. The 99%-VaR in this scenario is

A) $30
B) $40
C) $70
D) $100

Question 20

You invest $100 in a corporate bond. You estimate that with probability 0.95, the corporation will pay back the promised amount of $110 at the end of one year; with probability 0.04, the corporation will default and the recovered amount will be $70; and with probability 0.01, the corporation will default and you will recover nothing. The 95%-VaR in this scenario is

A) $30
B) $40
C) $70
D) $100

Question 21

You invest $100 each in two bonds. Each bond will pay you $110 at the end of the year with probability 0.98 and nothing with probability 0.02. The correlation between the bonds is zero. In this scenario, the 99%-VaR of your portfolio is

A) $- 20$ .
B) Zero.
C) $90
D) $200

Question 22

Which of the following measures of risk does not have the linear homogeneity property?

A) VaR.
B) Expected shortfall.
C) Variance.
D) Standard deviation.

Question 23

Given two portfolios $P _ { 1 }$ and $P _ { 2 }$ , which risk measure $R ( )$ does not always satisfy the "sub-addivity" property (i.e., that $R \left( P _ { 1 } + P _ { 2 } \right) \leq R \left( P _ { 1 } \right) + R \left( P _ { 2 } \right)$ where $R ( )$ is the measure of portfolio risk)?

A) VaR.
B) Expected shortfall.
C) Variance.
D) Standard deviation.

Question 24

Identifying the risk contribution of an asset to a portfolio is more difficult than identifying its return contribution because

A) Risk does not always increase when you add assets to a portfolio.
B) Risk, unlike return, is not additive.
C) Risk is a property of portfolios, not one of individual assets.
D) Some assets have no risk.

Question 25

The expected shortfall (ES) measure does not satisfy the following coherence property of risk measures:

A) Linear homogeneity.
B) Sub-additivity.
C) Translation invariance.
D) None of the above.

Question 26

"Subadditivity" is the requirement of a coherent risk measure that

A) The risk of a subset of portfolio holdings be strictly less than the risk of the entire portfolio.
B) The risk of the combination of two portfolios be at most the sum of the risks of the two portfolios.
C) The risk of the combination of two portfolios be at least the sum of the risks of the two portfolios
D) The sum of the risks of each constituent of the portfolio equals the total risk of the portfolio.

Question 27

Worst-case scenario analysis develops a measure that computes, say, for one year's returns
(a) The worst possible outcome for a portfolio in repeated trials of generating one-year portfolio outcomes.
(b) The mean, over repeated trials of generating one-year portfolio returns, of the worst possible outcome in each trial.
(c) The worst possible mean outcome for a portfolio in repeated trials of generating one-year portfolio outcomes.
(d) The meanest possible outcome for a portfolio in repeated trials of generating one-year portfolio outcomes.

Question 28

Consider a two-asset portfolio invested with $10 in each asset. The mean returns of the two assets are $10 \%$ and $15 \%$ . The correlation of returns is 50%. The standard deviation of returns is 20% and 30%, respectively. What are the risk-contribution proportions of each asset to the 99%-VaR of this portfolio?

A) 33:67
B) 36:64
C) 48:52
D) 50:50

Question 29

VaR fails the following requirement of a "coherent" risk measure:

A) Linear homogeneity.
B) Monotonicity.
C) Subadditivity.
D) Translation invariance.

Question 30

"Monotonicity" is the requirement of a risk-measure that if Portfolio A dominates Portfolio B (in the sense of always doing at least as well as B in every state of the world and strictly better in some states), then the risk of Portfolio A should be less than the risk of Portfolio B. Which of the following statements is correct?

A) Standard deviation (SD) fails to satisfy monotonicity.
B) Value-at-Risk (VaR) fails to satisfy monotonicity.
C) Expected shortfall (ES) fails to satisfy monotonicity.
D) All three of these portfolio risk-measures (SD, VaR, and ES) fail monotonicity.

Question 31

VaR-bases risk decomposition is the calculation that allocates the total VaR of a portfolio to each of its assets or subportfolios. Which of the following statements is most valid?

A) Risk decomposition is based on the fact that VaR is linearly homogeneous in its constituents.
B) Risk decomposition assumes that the assets in a portfolio are distributed multivariate normal.
C) Risk decomposition depends on the sequence in which assets are added to an existing portfolio.
D) Risk decomposition is not possible if Monte Carlo simulation is used to compute VaR.

Question 32

Consider a $900 portfolio with three assets, each held in equal value. The VaR of the portfolio is such that an increase in $1 of any of the asset holdings results in a $0.05 increase in VaR. The VaR of this portfolio is approximately equal to:

A) $5
B) $15
C) $45
D) $135

Question 33

Which of the following risk measures is not translation invariant (i.e., does not satisfy the property that if we add a risk-free asset to a portfolio with a return of $r$ , the risk of the portfolio should come down by the extent of this addition)?

A) VaR.
B) Expected shortfall.
C) Kurtosis.
D) Standard deviation.

Question 34

If every position in a portfolio is doubled in size, the risk contribution of the original portion of the portfolio, as measured by VaR, will

A) Remain the same as before.
B) Double.
C) Halve.
D) More than double.