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Fin 500Q –  Quantitative Risk Management

Homework #4 Solutions

1. An investor has a $1000 portfolio of stocks, each of which has a normal distribution of returns. We are told the annual variance on the return of his portfolio is 0.3. The mean return of each of the assets is 10%. The betas of the three assets with respect to his portfolio are 1 .5, −0.75, and 0.25, respectively.

(a) Find the annual VaR0 .05 . You can use the fact that z0 .05 = − 1.65.

Answer: His annual VaR is: − 1000 · (0.10 − 1.65 · ^0.3) = 803.74.

(b) Find the contribution of the 3rd asset to the overall VaR. You may use the facts that the portfolio

weights sum to 1 and that 对i(3)=1 βi wi  = 1.

Answer: Solving the equations:

1.5 · 0.5 − 0.75 · w2 + 0.25 · w3      =   1 0.5 + w2 + w3      =   1,

we obtain w2  = −0.125 and w3  = 0.625. The marginal VaR per dollar invested in asset 3 is ∆VaR3  = −(0.10 − 1.65 · 0.25 · ^0.3) = 0.1259.

The contribution of the third asset to the portfolio VaR is ∆VaR3  · w3  = 0.0787.  For the full $1000 position, the contribution is 78.71.

(c) Using the marginal VaR, find the approximate increase in the VaR if the investor increases his position in the 2nd asset by 20 dollars.

Answer: The marginal VaR per dollar invested in asset 2 is

∆VaR2  = −(0.10 − 1.65 · (−0.75) · ^0.3) = −0.7778.

By increasing investment in asset 2 by $20, the VaR will approximately decline by 15.56.  Note that by increasing his position in asset 2, the investor lowers his overall VaR. This is because asset 2 is a negative beta asset, that is, its return has a negative covariance with the overall portfolio of the investor.

2. Consider the Barings case discussed in class.   Suppose the volatilities of the JGB and the Nikkei are 0.3% and 1.2% per month, respectively, and the correlation between JGB and the Nikkei is 0.4. Suppose Leeson’s portfolio was long $30b in JGB and long $10b in the Nikkei. Please calculate the following quantities:

Note:  please  see  the  attached spreadsheet for  all  computations,   “barings .hw5.xls” .   You  may  want  to verify the formulas in the Excel file .

(a) The VaR0 .05  for each of the assets.

Answer: VaR of JGB = 148.50. VaR of Nikkei = 198.00.

(b) The VaR0 .05  of the portfolio.

Answer: Variance of the position is 31140. The VaR is 291.17.

(c) The β of each asset with Leeson’s portfolio.

Answer:  βJ  = 0.5318 and βN  = 2.4046.  The portfolio weights are wJ  = 0.75 and wN  = 0.25. The weights satisfy the necessary condition: wN  · βN  + wJ  · βJ  = 1.

(d) The marginal VaR, also called ∆VaR, for each asset. Answer: ∆VaRJ  = 0.0039 and ∆VaRN  = 0.0175.

(e) The component VaRs and the contribution of each position to the VaR of the portfolio. Answer:  Component VaR of JGB is 116.13, with a contribution of 39.9 percent.  Component VaR of Nikkei is 175.04, with a contribution of 60.1 percent.

(f) Using the marginal VaR, find the VaR of the portfolio if Leeson was to increase the amount invested in the Nikkei by $400 million.

Answer: The change in the VaR is ∆VaRN  · 400 = 7.001.

3. Data for the S&P 500 and two stocks, IBM and General Electric (GE), from 1980 to 2020 are collected in the spreadsheet“sp ibm ge data.xls”. There are 10340 prices for each. An investor wants to evaluate

the VaR of an investment of $1000 in an equally weighted portfolio of the three assets ( in each).     Note:  please  see  the  attached spreadsheet for all  computations,   “sp ibm ge daily.xls” .   You  may want to verify the formulas in the Excel file .

(a) Suppose we assume that all means, variances and covariances are constant over time.  Find the VaR5%  of the portfolio. Conduct the unconditional, independence, and joint tests for the VaR at 95% confidence level.

Answer: We compute VaR5%  = 23.10. We get the following LR statistics:

LRU  = 55.87,    LRI  = 56.47,    LRTOT  = 112.34.

Hence, we reject unconditional coverage, independence of violations, and the joint restrictions for the constant volatility model.

(b) Now assume that variances and covariances are time varying, and the RiskMetrics method is valid. Use the recursive formula to compute the portfolio variance, and find the VaR5%  for each day in the sample. Conduct the unconditional, independence, and joint tests for the VaR at 95% confidence level.

Answer: We get the following LR statistics:

LRU  = 3.35,    LRI  = 0.54,    LRTOT  = 3.89.

Hence, we do not reject unconditional coverage, independence of violations, and the joint re- strictions for the RiskMetrics model at 95% confidence.