Chapter 22: Value at Risk

Question 1

What is VaR?

Answer 1

VaR is a fairly simple risk measure. It is concerned with the left tail of the probability distinction (i.e. losses). You could say that VaR asks the question: “how bad can things get?”. However, VaR does not tell you how much you can expect to lose if things do go bad.

Question 2

What is the aim of the VaR measure?

Answer 2

When using the VaR measure one is interested in making a statement of the following form:

• I am X percent certain there will not be a loss of more than V dollars in the next N days.

The variable V is the VaR of the portfolio. It is a function of two parameters: the time horizon (N days) and the confidence level (X%). It is the loss level over N days that has a probability of only (100 - X)% of being exceeded.

When N days is the time horizon and X% is the confidence level, VaR is the loss corresponding to the (100 - X)th percentile of the distribution of the gain in the value of the portfolio over the next N days.

Question 3

How is a N-day VaR measure calculated?

Answer 3

See below formula

Question 4

How is VaR denoted mathematically?

Answer 4

1. Choose a time interval from t to T, e.g. 10 days.
   2: Let R be the return of the portfolio with value process V over the interval from t to T, that is: R = V(T) / V(t) - 1.
   3: Define the loss of the portfolio over the interval from t to T, that is: L = -V(t) * R.
2. Choose a confidence level (alpha), for example alpha = 95%.

The [t, T]-period VaR at a alpha% confidence interval, which is denoted by VaR_alpha(L), is defined as below:

Remember that x is the portfolio loss - “we are 95% sure that the portfolio loss over the next 10 days will not exceed the number x”

Question 5

VaR can be estimated historically, but what is the problem with doing so?

Answer 5

1: We can never be certain the history will repeat itself.
2: How large should the sample be for us to say something useful about the VaR?
3. Has the portfolio changed over time?

Question 6

How do you calculate the VaR of a portfolio?

Answer 6

1: The first step is to calculate the variance of the portfolio using the below equation. In this equation alpha_i and alpha_j is the amount invested in asset i and asset j, and sigma_i and sigma_j are the standard deviations of the returns of asset i and asset j.
2: Calculate the standard deviation from the variance.
3: Calculate the 1-day VaR of the portfolio. This is done by multiplying the standard deviation of the portfolio with the correct percentile - if it’s the 97,5th percentile it’s 1,96 and if it’s the 99th percentile it’s 2,326.
4: After this you can calculate the 10-day VaR of the portfolio by saying 1-day VaR * the square root of 10.
5: Now you can calculate the 10-day VaR for asset i and asset j. This is done by firstly calculating the change in the value of the investment in asset i and asset j respectively: std. dev. asset i * square root 10 * 1,96 and then the same for asset j.
6: Finally you can calculate the diversification benefit. This is done by adding the 10-day VaR for asset i and asset j and then subtracting the 10-day VaR of the portfolio that was calculated in step 4.