VAR analytical methods: Clarification

[monsieuruzairo3]

hi @David Harper CFA FRM CIPM

Following ais a question from section on Portfolio Analytical methods P2.T8.2.

2.2. A $10 million portfolio is equally invested in two currencies: $5 million in Swiss francs
(CHF) and $5 million in Japanese yen (JPY). The volatility of CHF is 10%; the volatility of the
JPY is 20%. The two currencies have a correlation of 0.30. If we assume a 95% confident
delta normal value at risk (VaR), what is the marginal value at risk (marginal VaR) of the
Swiss franc (CHF) position with respect to the two-asset portfolio that includes the CHF
position; i.e., 95% confident marginal VaR (CHF, Two-asset Portfolio)?
a) 0.106
b) 0.304
c) 0.633
d) 1.124

The answer provided:

2.2. A. 0.106
Per Jorion 7.17: Marginal VaR = Covariance / Portfolio volatility * deviate
Covariance (CHF, Portfolio) = 50% weight * 10%^2 + 50% weight * 10% * 20% * 0.30
= 0.0080;
Portfolio volatility = 12.450%; Deviate @ 95% = 1.645;
Marginal VaR = 0.0080/12.45% * 1.645 = 0.1057
Alternative, per Jorion 7.20: Marginal VaR = Portfolio VaR/W * beta (i,P)
Portfolio VaR = $1.245 million* 1.645 = $2.048 million;
Beta (CHF, Portfolio) = 0.5161
Marginal VaR = $2.048 million / $10 million * 0.5161 = 0.1057

I think the highlighted step is incorrect since weights are not squared as they should be

My approach: My Portfolio std. dev is the same as calculated above
I think used a matrix to calculate Dollar Covariance (Ra, Rp)
[Cov Ra,Rp] = [0.1^2 *5]
MVAR(A) = 1.65*.05/(.1245*10)

I saw this issue with the some of the questions from the reading too. Please suggest

KR
Uzi

 

[David Harper CFA FRM]

Hi @monsieuruzairo3

That's tempting, I love the attention to detail , but please note we want COV[CHF, Portfolio] = COV[CHF, weight(CHF)*CHF + weight(JPY)*JPY]. This is a situation where we want the covariance between a component, CHF, and the portfolio that contains the component, weight(CHF)*CHF + weight(JPY)*JPY. Such that:
COV[CHF, weight(CHF)*CHF + weight(JPY)*JPY]
= COV[CHF, weight(CHF)*CHF] + COV[CHF, weight(JPY)*JPY]
= weight(CHF)*COV[CHF,CHF] +weight(JPY)*COV[CHF,JPY]
=weight(CHF)*VAR(CHF) +weight(JPY)*COV[CHF,JPY].

Additional discussion here at the source https://forum.bionicturtle.com/threads/p2-t8-2-marginal-and-component-value-at-risk-var.4779/

Let me know if that resolves, I don't get to discuss portfolio var analytics enough ?

 

[JFZ2014]

How does COV[CHF,CHF] becomes VAR(CHF)? I am new here.

 

[David Harper CFA FRM]

@JFZ2014 I love the question, I am not new here and we too often skip this sort of thing. Variance is a special case of covariance, it just happens to be the covariance of a variable with itself. COV[CHF,CHF] = StdDev[CHF]*StdDev[CHF]*correlation[CHF,CHF] = StdDev[CHF]^2*1 = VAR[CHF], as corr(x,x) = 1.0 ... a correlation matrix is betrayed by 1.0s in the diagonal
i.e., one of our most useful formulas is cov(x,y) = sigma(x)*sigma(y)*correlation(x,y)

 

[JFZ2014]

Yes, yes, yes, thank you David. I took VAR as value at risk and forgot that it variance.