Value at Risk from Delta of a single asset portfolio

I am trying to figure out the following, for me unfamiliar type of question:

Given is a single asset portfolio: the Delta of the portfolio is 15, the value of the asset is 10 and the daily volatility is 2.2%. From this, I have to calculate the one-day 98% VaR of the portfolio.

I have not encountered a situation where the Delta is directly related to the VaR so I am not sure how I should approach this problem. Help is very much appreciated.

• What is the delta of a portfolio? For an option, it is the derivative of the option price w.r.t. the underlying, but in the context of a portfolio this doesn't make much sense. – pbr142 Apr 1 '14 at 11:13

Given that by delta means that if the price goes up by 0.01% i.e. one basis point, you gain 15 and vice versa if the price goes down by one basis point. You know that the daily standard deviation is 2.2%, than again you know that $220*15 = 3300$ is the standard deviation of your portfolio. So, since we are using a normal distribution you can look at a table which describes the standard deviate for $\alpha=100\%-98\%=2\%$. If I remember correctly the standard deviate for $\alpha=2\%$ is $2.05$ and that gives you the loss measure called $VAR=-3300*2.05=-6765$
• Thank you for your help. The standard deviate for $\alpha = 2%$ is in fact $2.05$. Also, to compute the $VaR$, shouldn't I add the mean $\mu$ to the equation like: $VaR = 10 + 3300*2.05$? – Radda Apr 2 '14 at 9:57
• Thank you for correcting the standard deviate Radda. You say in the question that the value of the asset is 10 (to me that means that the underlying market price is 10). You said nothing of a mean in your question. The mean is not the same as the asset value. If you had a mean daily return for the underlying (average daily return of one unit of the asset) you would have to calculate the number of units. From our definition of the delta, the delta for 1 unit of the underlying is 0.001 so the question is now how many units is your portfolio i.e. $0.001*x=15$ gives $x=15000$ units – steinbitur Apr 2 '14 at 10:40
• If you had given a daily expected return $\mu$ then the VAR loss would be given as $VAR=15000*10*\mu-3300*2.05$ – steinbitur Apr 2 '14 at 10:42
• and let's say that the daily expected return is 5 basis point's then the VAR would be $VAR=150000*0.0005-3300*2.05=-6690$ – steinbitur Apr 2 '14 at 10:47