# The VaR of a portfolio with Student t returns

A portfolio consists of 300 stocks,150 of A and 150 of B, their annualized covariance matrix is as following: $$\begin{pmatrix} 0.09 & 0.018\\ 0.018 & 0.04 \end{pmatrix}$$

Thoese two stocks are jointly distributed as Student t with 5 degrees of freedom. What is the 3-day 99% VaR for this portfolio?

If thoese two stocks are distributed normally, I can directly compute the distribution of the return of the portfolio, but when stocks are distributed as student t, can I still do it the same way?

## 1 Answer

The std. deviation of your 2 asset portfolio can be determined by applying the formula below:

$$\sigma_p= (Variance_aW^2_a+Variance_bW^2_b+2W_aW_b(cov_{ab}))^.5$$

where W are the weights, sigma is the standard deviation, and cov is the covariance between the asset returns. (From your covariance matrix, the variance of asset a is 0.09; Variance of asset b is 0.04 and Covariance of a,b is 0.018).

Determine the weights of each asset by multiplying the position size (150) by the corresponding stock price and divide it by the the total portfolio value.

Since your covariance matrix is annualized, turn this into a 3 day volatility by multiplying the volatility by $$(3/250)^.5$$ Note: This assumes 250 trading days in a year

Multiply this 3-Day volatility by the 99% critical value of a t-distribution with 5 degrees of freedom, which is 3.365

As you mentioned, the distribution is a t-dist which is similar to a normal distribution but more peaked. As such, the distribution can be described by the volatility and the degrees of freedom.

• I did not check the calculation but did you consider in your answer that the variance of a standard student-t is not 1 but $\frac{n}{n-2}$ where $n$ is the degrees of freedom? for small $n$ this matters.
– Ric
Feb 25 '19 at 20:59
• @ Richard I just assumed the variance-covariance matrix accounted for the degrees of freedom. I guess one could adjust these tor account for the degrees of freedom via your formula. Feb 25 '19 at 21:46
• Thank you so much! I have a further question: If I were going to find the VaR using Monte Carlo simulation, say 10000 samples, I come up with two available approaches: The first one is to generation 10000 pairs of correlated normal ramdom numbers using cholesky factorization, then I can get the return of each paired sample (10000 samples in total) and calculate quantile. The second is to find the volatility of the portfolio first (as what you did above), then generate 10000 single sample and calcualte quantile. Will these two approaches get to the same result? Feb 25 '19 at 22:56