Question 1

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.

Accepted Answer

VaR is a dollar amount of loss.While (a)is not a complete definition of VaR,it is the only answer that is not outright incorrect,as are cases (b),(c),and (d).VaR is calculated using both the first and second (and more)moments of return,not just one of them.It is not expressed as a percentage.

Question 2

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

Accepted Answer

To calculate the 99% Value at Risk (VaR) for a two-asset portfolio, we first need to determine the portfolio's expected return and standard deviation. The expected return of the portfolio ($E(R_p)$) can be calculated as a weighted average of the individual assets' returns, given equal investment, this simplifies to the average of the two returns:$$E(R_p) = \frac{10\% + 15\%}{2} = 12.5\%$$The portfolio's standard deviation ($\sigma_p$) can be calculated using the formula for two assets:$$\sigma_p = \sqrt{w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\sigma_1\sigma_2\rho}$$Given $w_1 = w_2 = 0.5$, $\sigma_1 = 20\%$, $\sigma_2 = 30\%$, and $\rho = 50\%$, we have:$$\sigma_p = \sqrt{0.5^2(0.2)^2 + 0.5^2(0.3)^2 + 2(0.5)(0.5)(0.2)(0.3)(0.5)}$$$$= \sqrt{0.01 + 0.0225 + 0.015} = \sqrt{0.0475} = 0.218$$The portfolio's standard deviation is 21.8%.To find the 99% VaR, we use the formula:$$VaR = -[E(R_p) - Z_{\alpha}\sigma_p]$$For a 99% confidence level, $Z_{\alpha}$ is approximately 2.33 (from standard normal distribution tables). The portfolio's value is $20 ($10 in each asset), so we calculate the VaR in terms of portfolio value:$$VaR = -[(12.5\% - 2.33 \times 21.8\%) \times 20]$$$$= -[0.125 - 0.50794] \times 20$$$$= -[-0.38294 \times 20]$$$$= 7.6588$$Rounding to two decimal places, the 99%-VaR of the portfolio is approximately $7.66.

Question 3

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

Accepted Answer

C

Question 4

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

Accepted Answer

There is a 1% probability of losing your entire investment and a 5% probability of losing $30 or worse.The 98%-VaR is therefore $30,that is,there is a 98% probability that you will lose at most $30 of your investment.

Question 5

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.

Accepted Answer

By the linear homogeneity property,the VaR will exactly double.

Question 6

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.

Accepted Answer

Value-at-Risk (VaR) is defined as the dollar amount of loss that will not be exceeded with a certain confidence level over a specific period. It is a measure of the risk of loss on a portfolio. Choice B accurately describes VaR as the dollar loss corresponding to a particular probability in the left tail of the profit-and-loss distribution, which is where losses are found.

Question 7

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.

Accepted Answer

Simulation is slow relative to the analytic delta-normal method.Assuming that the past gives a good picture of the future is not always a safe assumption.There is no parameterization of the simulation as the method simply draws samples from past history.The big benefit is that it does not require the restrictive assumption that returns are normal,which is known to be empirically invalid.

Question 8

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.

Accepted Answer

The delta-normal method is parametric.

Question 9

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

Accepted Answer

There is a 1% probability of losing your entire investment and a 5% probability of losing $30 or worse.The 95%-VaR is therefore $30,that is,with probability 0.95,your losses will be at most $30.

Question 10

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) $133$

Accepted Answer

The answer of A portfolio has a current value of...

Question 11

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

Accepted Answer

There is a 2% probability of losing your entire investment and a 6% probability of losing $30 or worse.The 5%-VaR is therefore the entire investment.

Question 12

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

Accepted Answer

There is a 2% probability of losing your entire investment and a 6% probability of losing $30 or less.The 90%-VaR is therefore $30,that is,with probability 0.90,your losses on the portfolio will be less than $30.

Question 13

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

Accepted Answer

The 95%-VaR (Value at Risk) represents the maximum loss with a 95% confidence level. Given the 0.02 probability of getting nothing from each bond, the chance of both bonds failing is 0.02 * 0.02 = 0.0004 (0.04%). Since the correlation between the bonds is zero, their outcomes are independent. The 95% confidence level means we are looking for scenarios that cover up to 95% of outcomes, leaving 5% in the tail. The scenario where both bonds fail is well outside the 95% confidence interval (it's in the worst 0.04% of outcomes), so the 95%-VaR is when one bond fails, which happens with a probability greater than 5% but less than 100%. Therefore, the maximum loss at the 95% confidence level is the loss of one bond, which is $100 (since you get nothing from one bond but still get the $110 from the other, resulting in a net loss of $100 from the initial $200 investment).

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 99%-VaR in this scenario is
A)$30
B)$40
C)$70
D)$100

Accepted Answer

There is a 1% probability of losing your entire investment and a 5% probability of losing $30 or worse.The 99%-VaR is therefore $100.

Question 15

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

Accepted Answer

The probability that the portfolio will be worth less than 800 after one year is the area under the normal distribution curve to the left of 800. The z-score for 800 is (800 - 100) / 100 = -2. The area under the normal distribution curve to the left of -2 is 0.0228.

Question 16

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) $133$

Accepted Answer

The 99% Value at Risk (VaR) is calculated as the mean plus the Z-score corresponding to the 99% confidence level times the standard deviation. For a 99% confidence level, the Z-score is approximately 2.33. Thus, VaR = 100 + (2.33 * 100) = 100 + 233 = 333. However, since VaR is typically expressed as a loss from the current value, we subtract this from the current portfolio value of $1000, resulting in a VaR of 1000 - 333 = $667. Therefore, the correct answer reflects the loss, which is -$333, but since this specific value isn't an option and considering the context of the provided choices, the closest correct interpretation of the methodology would lead to the selection of D) 133, acknowledging a misunderstanding in the calculation explanation. The correct calculation for the loss from the current value should actually focus on the negative impact, thus directly applying the Z-score methodology to the profit/loss distribution, not the final portfolio value.

Question 17

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.

Accepted Answer

The remaining choices are not always valid.

Question 18

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.

Accepted Answer

VaR does not consider the shape of losses in the left tail of the P&L distribution as well as the shape of losses outside the left tail of the P&L distribution, making it deficient as a risk measure. Therefore, both choices A and B are correct.

Question 19

The value-at-risk of a portfolio is
A)Always positive.
B)Always negative.
C)May be positive or negative.
D)Always non-negative.

Accepted Answer

The value-at-risk of a portfolio may be positive or negative. A positive value-at-risk indicates that the portfolio is expected to gain value, while a negative value-at-risk indicates that the portfolio is expected to lose value.

Question 20

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.

Accepted Answer

VaR is not a return measure,only a risk measure.

Quiz 20: Value at Risk

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

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?

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

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

Value-at-Risk (VaR) is most closely defined as

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

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

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?

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 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 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?

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

VaR as a risk measure has the following deficiency:

The value-at-risk of a portfolio is

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