# Value at Risk

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1. Consider a firm with a trading book valued at $100 million. The return of these assets is distributed normally with a yearly standard deviation of 25 percent. The firm can liquidate all of the assets immediately. How much capital should the firm have so that 99 days out of 100, the firm's return on assets is high enough that, after liquidating its portfolio, it would have capital left?

2. Consider the firm in problem 1. Now the firm is in a situation where it cannot liquidate its portfolio for five days. How much capital does it need so that 95 days out of 100, it ends the period with positive capital if it has to liquidate its portfolio?

3. A firm has a trading book composed of two assets with normally distributed returns. The first asset has an annual expected return of 10 percent and an annual volatility of 25 percent. The firm has a position of $100 million in that asset. The second asset has an annual expected return of 20 percent and an annual volatility of 20 percent, as well. The firm has position of $50 million in that asset. The correlation coefficient between the returns of these two assets is 0.2. Compute the 5 percent annual VaR for that firm's trading book.

4. Growth Inc. has a yearly cash flow at risk of $200 million. With an increase in equity, Growth Inc. has to serve as a cushion against losses and has a net cost for the firm of 12 percent per year. Growth Inc. can expand the scale of its activities by 10 percent. The firm wants to increase its equity capital so that it can absorb a cash flow shortfall equal to its CaR, after expanding its activities so that its probability of default after experiencing such a shortfall would be the same as if it had not expanded its activities. How much equity capital does it have to raise? How much must the project earn before taking into account the capital required to protect it against losses in order to be profitable?

5. Consider the choice of two mutually exclusive projects by Innovate Ltd. The first project is a scale-expanding project. By investing $100 million, Innovate Ltd. expects to earn $20 million a year net of funding costs. The project is infinitely lived so there is no depreciation. This project also increases cash flow at risk by $50 million. The second project requires no initial investment and is expected to earn $25 million. This project increases the cash flow at risk by $200 million. Under which conditions will the first project be more advantageous than the second project, assuming that neither project has systematic risk?

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1. Consider a firm with a trading book valued at $100 million. The return of these assets is distributed normally with a yearly standard deviation of 25 percent. The firm can liquidate all of the assets immediately. How much capital should the firm have so that 99 days out of 100, the firm's return on assets is high enough that, after liquidating its portfolio, it would have capital left?

We have to calculate VaR at (1-99/100) = 1%

Z value corresponding to this probability = 2.33

yearly volatility = 25%

Number of trading days in a year = 252

therefore daily volatility= 1.57% =25.% / √252

Value of the asset= $100 million

Therefore VaR= $3.66 million =2.33 x 1.57% x $100.

Answer: Capital= $3.66 million

2. Consider the firm in problem 1. Now the firm is in a situation where it cannot liquidate its portfolio for five days. How much capital does it need so that 95 days out of 100, it ends the period with positive capital if it has to liquidate its portfolio?

We have to calculate VaR at (1-95/100) = 5%

Z value corresponding to this probability = 1.65

yearly volatility = 25%

Number of trading days in a year = 252

Number of days= 5

therefore 5 day volatility= 3.52% =25.% / √(252 / 5)

Value of the asset= $100 million

Therefore VaR= $5.81 million =1.65 x 3.52% x $100.

Answer: Capital= $5.81 million

3. A firm has a trading book composed of two assets with normally distributed returns. The first asset has an annual expected return of 10 percent and an annual volatility of 25 percent. The firm has a position of $100 million in that asset. The second asset has an annual expected return of 20 percent and an annual volatility of 20 percent, as well. The firm has position of $50 million in that asset. The correlation coefficient between the returns of these two assets is 0.2. Compute the 5 percent annual VaR for that firm's trading ...

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