Question 1

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)?

Accepted Answer

VaR (Value at Risk) does not always satisfy the sub-additivity property, meaning that the risk of a combined portfolio can be greater than the sum of the risks of the individual portfolios. This is because VaR does not fully account for the distribution of losses beyond a certain threshold.

Question 2

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)?

Accepted Answer

Kurtosis is not translation invariant because it measures the "tailedness" of the distribution of returns, which is not affected by adding a constant (risk-free asset return) to all returns. Translation invariance applies to measures that assess risk based on deviations from the mean or expected value, which is not what kurtosis does.

Question 3

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

Accepted Answer

Variance does not have the linear homogeneity property, which means it does not scale linearly with the size of the portfolio.

Question 4

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

Accepted Answer

VaR fails the subadditivity requirement of a "coherent" risk measure. This means that the VaR of a portfolio may not be equal to the sum of the VaRs of its individual positions. Therefore, VaR is not considered a "coherent" risk measure.

Question 5

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

Accepted Answer

By the linear homogeneity property,the VaR will exactly double.The capital required will also double.And the risk contribution of the original component of the portfolio will remain the same.

Question 6

Worst-case scenario analysis develops a measure that computes,say,for one year's returns

Accepted Answer

Worst-case scenario analysis involves calculating the average of the worst outcomes across multiple trials, which aligns with option B.

Question 7

"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?

Accepted Answer

Standard deviation fails to satisfy monotonicity because it measures the spread of returns around the mean, not the actual risk of loss, and can assign higher risk to a portfolio with better outcomes.

Question 8

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?

Accepted Answer

The risk-contribution proportions are determined by the marginal VaR of each asset, which depends on the standard deviation and correlation. Asset 2, with a higher standard deviation and mean return, contributes more to the portfolio's risk.

Question 9

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

Accepted Answer

The answer of You invest $100 each in two bonds.Each...

Question 10

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:

Accepted Answer

Since each asset is held in equal value in a $900 portfolio, each asset represents $300 of the portfolio. An increase of $1 in any of the asset holdings increases the VaR by $0.05. Therefore, for each asset, the increase in VaR proportional to its value in the portfolio is $300 * $0.05 = $15. Since there are three assets, the total increase in VaR for the portfolio is $15 * 3 = $45.

Question 11

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

Accepted Answer

Return contribution of an asset can easily be calculated by multiplying the asset's weight in the portfolio by its expected return. However, risk contribution of an asset is not solely based on its weight in the portfolio. The correlation between the asset and other assets in the portfolio also affects its risk contribution. The risk of a portfolio is not a simple sum of risks of individual assets, making it challenging to identify an asset's risk contribution. Hence, option B is the correct answer.

Question 12

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?

Accepted Answer

See Chapter 20 for a proof that risk decomposition is derived using the linear homogeneity property.It does not require normality.Risk decomposition in fact shows that the sequence in which assets are added to a portfolio is irrelevant,and the same decomposition is arrived at no matter the sequence.Risk decomposition requires computing first derivatives of total VaR with respect to each asset dollar weight in the portfolio.These first derivatives may also be computed in a simulation setting.

Question 13

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

Accepted Answer

ES is a coherent risk-measure.

Question 14

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

Accepted Answer

Subadditivity requires that the risk of a combined portfolio is less than or equal to the sum of the risks of the individual portfolios. In other words, the risk of the combination should not be greater than the sum of the risks of the individual portfolios. This is why choice B is the correct answer.

Quiz 20: Value at Risk

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) ?

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) ?

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

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

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

Worst-case scenario analysis develops a measure that computes,say,for one year's returns

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

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:

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

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?

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

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

Quiz 20: Value at Risk

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 rrr ,the risk of the portfolio should come down by the extent of this addition) ?

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

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

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

Worst-case scenario analysis develops a measure that computes,say,for one year's returns

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

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:

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

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?

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

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

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) ?