AIM 3: Evaluate when VAR or CFAR is the more appropriate measure of risk.

1、Cash flow at risk (CAR) is most relevant to a nonfinancial firm that:

A) holds substantial liquid financial assets.
 
B) has unused borrowing capacity.
 
C) depends solely on its cash flow to fund new projects.
 
D) has ready access to capital markets.

The correct answer is C

CAR is most relevant to a nonfinancial firm that depends solely on its cash flow to fund new projects.

AIM 4: Compute the VAR and CFAR impact as well as the expected net gain on projects that have various scopes relative to the size of the firm and its other projects.

1、A firm has a portfolio of traded assets worth $200 million with a VAR of $20 million. The standard deviation of the return on the portfolio is 0.50. The firm is considering the sale of a position worth $1 million in an asset that has an expected return of 6 percent and a covariance of return with the portfolio of 0.20. The position that would be added has an expected return of 10 percent and a covariance of return with the portfolio of 0.40. The VAR is based on a 95 percent confidence level.
The impact of the trade on the volatility of the portfolio is an increase of:

A) 0.002. 
 
B) 0.004.
 
C) 0.006. 
 
D) 0.008.

The correct answer is A

The beta, relative to the overall portfolio, of the proposed position is βip = 0.40/0.502 = 1.60, and the beta of the replaced position is βjp= 0.20/0.502 = 0.80. The volatility impact of the trade is equal to ( βip – βjp) × Δw × Vol(Rp)= (1.60 – 0.80) × (1/200) × 0.50 = 0.002.

The correct answer is A

The impact on the expected return is [E(Ri) – E(Rj)] × Δw = (0.10 – 0.06) × (1/200) = 0.0002.

The correct answer is B

Impact on VAR = –[E(Ri) – E(Rj)] × Δw × W + ( βip – βjp) × 1.65 × Vol(Rp) × Δw × W = –(0.10 – 0.06) × (1/200) × 200 + (1.60 – 0.80) × 1.65 × 0.50 × (1/200) × 200 = 0.62.

2、A firm has a portfolio of traded assets worth $100 million with a VAR of $15 million. The standard deviation of the return on the portfolio is 0.50. The firm is considering the sale of a position worth $1 million in an asset that has an expected return of 6 percent and a covariance of return with the portfolio of 0.20. The position that would be added has an expected return of 10 percent and a covariance of return with the portfolio of 0.50. The VAR is based on a 95 percent confidence level.

The impact of the trade on the volatility of the portfolio is an increase of:

A) 0.004.
 
B) 0.002. 
 
C) 0.008.
 
D) 0.006. 

The correct answer is D

The beta, relative to the overall portfolio, of the proposed position is βip = 0.50/0.502 = 2.00, and the beta of the replaced position is βjp=0.20/0.502=0.80. The volatility impact of the trade is equal to (βip - βjp) x Δw x Vol(Rp) = (2.00 – 0.80) x (1/100) x 0.50 = 0.006.

The correct answer is B

The impact on the expected return is (E(Ri)-E(Rj)) x Δw = (0.10 - 0.06) x (1/100)=0.0004 = 0.04%

The correct answer is C

Impact on VAR = -(E(Ri)-E(Rj)) x Δw x W + (βip - βjp) x 1.65 x Vol(Rp) x Δw x W = - (0.10 -0.06) x (1/100) x 100 + (2.00 - 0.80) x 1.65 x 0.50 x (1/100) x 100 = 0.95.

3、Suppose that a trader has a portfolio of $5 million that is a portion of his firm’s total portfolio of $200 million. The beta of the trader’s return with the return of the firm is 1.20. The contribution of the trader to the firm value at risk (VAR) of $100 million is:

A) $5.0 million.
 
B) $3.0 million. 
 
C) $2.5 million.
 
D) $20.0 million.

The correct answer is B

Contribution of trader to VAR = wi x β x VAR(portfolio) = (5/200)(1.2)100 =3.

4、For a trade small relative to portfolio size, which of the following tends to increase the VAR of the portfolio? Purchasing:

A) an asset that has a higher expected return relative to the portfolio than the asset sold.
 
B) an asset that has a lower beta relative to the portfolio than the asset sold.
 
C) an asset that has a higher beta relative to the portfolio than the asset sold. 
 
D) a low volatility asset.

The correct answer is C

A trade increases VAR if the asset bought has a higher beta coefficient with respect to the portfolio than the asset sold, or if it has a lower expected return.

5、Suppose a trader has a portfolio of $20 million of his firm’s total portfolio of $500 million. The beta of the trader’s return with the return of the firm is 0.50. The contribution of the trader to the firm VAR of $100 million is:

A) $4 million.
 
B) $8 million.
 
C) $2 million.
 
D) $50 million.

The correct answer is C

Contribution of trader to VAR = wi × β × VAR(portfolio) = ($20 / $500)(0.5)$100 = $2 million.

6、 firm has a portfolio of traded assets worth $400 million with a VAR of $30 million. The standard deviation of the return on the portfolio is 0.60. The firm is considering the sale of a position worth $2 million in an asset that has an expected return of 6 percent and a covariance of return with the portfolio of 0.40. The position that would be added has an expected return of 10 percent and a covariance of return with the portfolio of 0.80. The VAR is based on a 95 percent confidence level.

The impact of the trade on the volatility of the portfolio is a(n):

A) increase of 0.0022.
 
B) increase of 0.0044.
 
C) decrease of 0.0080.
 
D) increase of 0.0033.
 

The correct answer is D

The beta, relative to the overall portfolio, of the proposed position is βip = 0.80/0.602 = 2.22, and the beta of the replaced position is βjp= 0.40/0.602=1.11. The volatility impact of the trade is equal to (βip - βjp) × Δw × Vol(Rp) = (2.22 – 1.11) × (2/400) × 0.60 = 0.0033.

The correct answer is A

The impact on the expected return is (E(Ri)-E(Rj)) × Δw = (0.10 - 0.06) × (2/400) ×$400,000,000 = $80,000.

 The correct answer is B

Impact on VAR = -{(E(Ri)-E(Rj)) × Δw × W} + [(βip - βjp) × 1.65 × Vol(Rp) × Δw × W] = - {(0.10 -0.06) × (2/400) × 400} + [(2.22 - 1.11) × 1.65 × 0.60 × (2/400) × 400] = 2.12.

A) I and II.
 
B) I, II, and IV.
 
C) I, II, and III.
 
D) II, III, and IV.
 

The correct answer is B

The VAR impact of a trade in a diversified portfolio can be calculated based on the asset betas, the size of the trade, and the expected returns of each assets. Correlation between the assets does not enter into the calculation. Note that if the portfolio were not well-diversified, standard deviation would be the relevant measure of risk rather than beta, and correlation would explicitly enter the calculation.

8、For a trade that is small relative to portfolio size, which of the following tends to decrease the VAR of the portfolio?

Purchasing an asset that has a higher beta relative to the portfolio than the asset sold.
Purchasing an asset that has a higher expected return relative to the portfolio than the asset sold.
Purchasing an asset that has a lower beta relative to the portfolio than the asset sold.
Purchasing a low volatility asset.
A) I and II only.
 
B) II and III only.
 
C) III only.
 
D) II, III, and IV only.

The correct answer is B

A trade decreases VAR if the asset bought has a lower beta coefficient with respect to the portfolio than the asset sold, or if it has a higher expected return. The impact of volatility alone cannot be determined without knowledge of the correlation with other portfolio assets.

AIM 5: Explain how additional CFAR costs impact the capital budgeting decision.

1、Raymor Corporation has an expected cash flow of $150 million with volatility of $50 million. It is considering a project that costs $50 million and has a volatility of $20 million. The cash flow of the new project has a correlation of 0.70 with the cash flow of existing projects. The impact of the new project on cash flow at risk (CAR), calculated at the 95 percent confidence level, is closest to:

A) $30.52 million.
 
B) $36.28 million.
 
C) $98.37 million.
 
D) $25.69 million. 

The correct answer is D

The new cash flow volatility is [Var(CE)+Var(CN)+2Cov(CE,CN)]1/2 = [502+202+2(.7)(50)(20)]1/2 =65.57. ΔCAR = ΔVolatility x 1.65 = 1.65 x (65.57 –50) =25.69.

AIM 6: Describe the process of CFAR and VAR allocations to the existing activities of the firm.

1、If the standard deviation of the market’s returns is 5.8%, the standard deviation of a stock’s returns is 8.2%, and the covariance of the market’s returns with the stock’s returns is 0.003, what is the beta of the stock?

A) 1.07.
 
B) 0.89.
 
C) 0.05.
 
D) 1.12.

 The correct answer is B

The formula for beta is: (Covstock,market)/(Varmarket), or (0.003)/(0.058)2 = 0.89.

AIM 9: Explain the limitations on project selection and the use of derivative instruments as ways to decrease VAR/CFAR.

A) commissions on derivative trades.
 
B) costs of customization.
 
C) credit risk premiums in derivative prices. 
 
D) problems of moral hazard.

The correct answer is A

Commissions on listed derivatives are quite small compared to the other costs listed.