Estimating Portfolio VaR with Student t Returns
Summary
The document considers a three-day, 99% value-at-risk estimate for a portfolio split equally between two stocks, given an annualized covariance matrix and joint Student t returns with five degrees of freedom. It asks whether portfolio returns can be combined as they would be under a normal model.
The answer calculates portfolio volatility from the asset variances, covariance, and portfolio weights, scales it to three trading days using a 250-day year, and multiplies by the stated Student t critical value. It notes that weights should reflect each position’s share of total portfolio value, which requires stock prices as well as share counts. The explanation is a brief recipe rather than a fully worked VaR calculation; it does not supply prices or a final numerical estimate. It also leaves assumptions about the Student t scale and the treatment of tail probability implicit, so the stated critical value should be checked against the intended VaR convention.
Key ideas
- Portfolio variance depends on both asset variances and their covariance, weighted by portfolio value shares.
- Annualized volatility can be scaled to a three-day horizon using the square root of the horizon fraction under standard time-scaling assumptions.
- The response uses a Student t critical value with five degrees of freedom to represent heavier tails than a normal model.
- Share counts alone do not determine portfolio weights; the stock prices and total portfolio value are also needed.
Tags
Full text
# The VaR of a portfolio with Student t returns
# 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?
## Answer by AlRacoon (score 2)
https://quant.stackexchange.com/a/44290
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.