Question

14.1 Consider a position consisting of a $100,000 investment in asset A and a $100,000 investment in asset B. Assume that the daily volatilities of both assets are 1%  and that the coefficient of correlation between their returns is 0.3. What are the five-day 97% VaR and ES for the portfolio?

14.23 The calculations in Section 14.3 assume that the investments in the DJIA, FTSE 100, CAC 40, and Nikkei 225 are $4 million, $3 million, $1 million, and$2million,respectively.How do the VaR and ES  change if the investments are $3 million, $3 million, $1 million, and $3 million, respectively? Carry out calculations when (a) volatilities and correlations are estimated using the equally weighted model and (b) when they are estimated using the EWMA model. What is the effect of changing λ from 0.94 to 0.90 in the EWMA calculations? Use the attached spreadsheets data for your calculation.

Answer 1

14.1). Daily volatility for both assets = volatility*investment = 1%*100,000 = 1,000

Portfolio variance = [(daily volatility for asset A)^2 + (daily volatility for asset B)^2 + (2*daily volatility for asset A*daily volatility for asset B*correlation)

= (1,000)^2 + (1,000)^2 + (2*1,000*1,000*0.3) = 2,600,000

Standard deviation (SD) = variance^0.5 = 2,600,000^0.5 = 1,612.45

Since 97% confidence is required, N-1(0.03) = 1.881

So, 1-day VaR = SD*1.881 = 1,612.45*1.881 = 3,032.69

5-day VaR = (5^0.5)*1-day VaR = 5^0.5*3,032.69 = 6,781.30

Expected shortfall (ES) = where

mean = 0 and SD = 5-day SD = 5^0.5*1,612.45 = 3,605.55; Y = N-1(0.03) = 1.881; X = 0.97

ES = 3,605.55*exp(-1.881^2/2)/((2*3.14)^0.5*(1-0.97)) = 8,174.45