Question

Consider a position consisting of a $300,000 investment in gold and a $500,000 investment in silver. Suppose that the daily volatilities of these two assets are 1.8% and 1.2%, respectively, and that the coefficient of correlation between their returns is 0.6. What is the 10-day 97.5% VaR and ES for the portfolio? By how much does diversification reduce the VaR and ES?

Suppose in the period that you did the question above the price of 1 unit of gold and 1 unit of silver where 1250$ and 350$. The next day the prices become $1240 and $354.Calculate the 1-day 99.9% VaR and ES for the portfolio.

I already have the first half finished, I just need the second half.

Answer 1

I am answering only the second half as you have specifically asked for it.

One day return:

Gold = (1240 - 1250) / 1250 = -0.008

Silver = (354 - 350) / 350 = 0.011428571

We need to update the daily volatility of Gold & Silver. Let's update the volatility estimate using the EWMA model with λ = 0.94

(Updated daily volatility)² = λ x (old daily volatility)² + (1 - λ) x (one day return)²

(Updated daily volatility for gold)² = 0.94 x (0.018)² + (1 - 0.94) x (-0.008)² = 0.0003084

Hence, updated daily volatility for gold = 0.0003084^1/2 = 0.017561321 = 1.7561%

Similarly,

(Updated daily volatility for silver)² = 0.94 x (0.012)² + (1 - 0.94) x (0.011428571)² = 0.000143197

Hence, updated daily volatility for silver = 0.000143197^1/2 = 0.011966484 = 1.1966%

We now need to rework the portfolio's std deviation with new set of numbers.

The variance of the portfolio (in thousands of dollars) is = 300² x (1.7561%)² + 500² x (1.1966%)² + 2 x 0.6 x 300 x 1.7561% x 500 x 1.1966% = 101.3755606

Hence standard deviation = 101.3755606^1/2 = 10.06854312 (in thousands of dollars)

In order to have 99.9% var, z variable = -3.09 such that N(-3.09) = 1 - 99.9% = 0.001

Hence, The 1-day 99.9% VaR = 10.06854312 x 3.09 = 31.11179824 (in thousands of dollars) = 31,111.79824 =  31,112

and ES = 10.06854312 x e^{-3.09 x 3.09 / 2} / [sqrt(2pi) x 0.01] =  47,548

     
The standard deviation is .