PART I. VAR and ES

1. If the daily, 95% confidence level value at risk (VaR) of a portfolio is

correctly estimated to be $10,000, one would expect that:

a. In 1 out of 20 days, the portfolio value will decline by $10,000 or less.

b. In 1 out of 95 days, the portfolio value will decline by $10,000 or less.

c. In 1 out of 95 days, the portfolio value will decline by $10,000 or

more.

d. In 1 out of 20 days, the portfolio value will decline by $10,000 or

more.

2. How many exceptions in backtesting a 90% VaR would be expected over

a 250-day trading year?

a. 10

b. 15

c. 25

d. 50

3. Suppose that you have two investments, each of which has a 0.9%

chance of a loss of $10 million and a 99.1% chance of a loss of $1 million.

These two investments are independent of each other.

(1) What is the VaR for one of the investments at the 99% confidence

level?

(2) What is the expected shortfall for one of the investments at the

99% confidence level?

(3) What is the 99% VaR for a portfolio consisting of the two

investments?

(4) What is the 99% expected shortfall for a portfolio consisting of the

two investments?

(5) Verify that VaR does not satisfy the subadditivity condition in this

example whereas expected shortfall does.

4. Suppose that the daily change in the value of a portfolio is normal with a

mean of zero and a standard deviation of $2 million. Calculate the

following VaRs: (1) one-day 97.5% VaR, (2) five-day 97.5% VaR, and (3)

five-day 99% VaR. (Hint: The left-tailed z-value is 1.96 for the 97.5%

confidence level.)

FINA 4210 2

PART II. Historical Simulation

(Please turn in your Excel file. You can submit Parts I and II

separately or combine them into one excel file.)

Find the attached file named “Hist_Simulation_HW2.” You will use this file to

calculate VaR under the historical simulation approach. The first worksheet

“data” shows historical data for 501 trading days for three indices, S&P 500

index, DJIA index, and BofA Merrill Lynch Total Bond Return Index. The value

of the investment in each index on May 20, 2020 is as follows:

Index Portfolio Value (‘000s)

S&P 4,000

DJIA 3,000

BofA Merrill Lynch 3,000

Total 10,000

1. Go to the second worksheet “500 Scenarios” and calculate the values

of the three indices on 05/21/2020 for 500 scenarios. For example,

Scenario 1 in the first row shows the values of the indices on

05/21/2020, assuming that their percentage changes between

05/20/2020 and 05/21/2020 are the same as they were between

05/23/2018 and 05/24/2018. You will repeat this for Days 1 through

500.

2. Once you calculate the values of the indices for 500 scenarios, you can

calculate the value of your portfolio. For example, the value of the

portfolio under Scenario 1 is

4,000 ×

2,964.61

2,971.61 + 3,000 ×

24,517.79

24,575.9

+ 3,000 ×

1,299.99

1,300.520 = 9,982.262

Therefore, a loss from the portfolio under Scenario 1 is (10,000

– 9.982.262) × 1,000 = $17,738. Please calculate the values of

the indices/portfolio and losses for 500 scenarios.

3. Copy the portfolio losses to the third worksheet “Ranked

Losses_VaR and ES.” Since the losses in the second worksheet

include equations, you should copy the values only (copy the

values of losses, go to the third worksheet, right-click on a cell,

and click “Paste Special” and “Values”). Rank the losses and find

the fifth worst loss (1% of the 500 losses). What is the one-day

99% VaR of your portfolio?

4. What is the one-day 99% expected shortfall based on the loss

distribution in Question 3?