2008 investment (round 1) question 36/41

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HI David,

A $20 million portfolio consists of only two equally-weighted and uncorrelated positions in Assets A & B. Asset A ($10 million) has a volatility of 10% and Asset B (also $10 million) has a volatility of 20%. At 99% confidence, what is an approximation of the incremental VaR given an additional investment of $1 million in Asset B?​

I'm not sure how you calculuated the marginal var of Asset B. The question says that the positions are uncorrelated (so rho =0? right).


1st approach: ($0.4)/($2.24)*2.326 = 0.416.
2nd approach: ($5.20 = portfolio VaR)/($20 portfolio)*(1.6 = beta) = 0.4161.

Q1: Can you explain how you got the 0.4?
Q2: How did you calculate 5.2 for the port var and beta 1.6?

Thanks in advance,

John

 

[David Harper CFA FRM]

Hi John

The question is more difficult than an actual exam question but I am glad you highlighted what is the hard part of it. Specifically, I think the hard part is computing the beta of a position relative to the portfolio. Per Friday's briefcast, beta is covariance (position, portfolio)/variance (portfolio) but the numerator here is non-trivial to figure.

As the question is based on Jorion 7.2 (Portfolio). I just added a second page to the EditGrid/XLS on the member page; the second page/tab contains the illustration using the numbers from this question, if you find a direct examination helpful.


The first approach (row 25) uses the matrix notion (p 171, Jorion Chapter 7) where marginal VaR = Dollar covariance / Portfolio Volatility * 2.33. Dollar covariance (this is my term, I am not sure Jorion gives it a name) = (Sigma)*(x) = Covariance matrix * Position vector. In this case, for asset B, sigma*x = 0*$10+.04*10 = 0.4.

The second approach (row 26), as you note, uses marginal VaR = porfolio VaR/W * beta. IMO, each approach has a hard part as you either need to the dollar covariance (1st) or the beta (2nd). The $5.2 portfolio VaR = 2.33 (99% confidence) * portfolio volatility ($2.2). The portfolio volatility is SQRT[w'*Sigma*w]. But as this is two assets, you can get directly with SQRT[$10^2*10%^2+$10^2*20%^2+ (2)($10)($20)(0)

The beta (see p 171) = (W) * (sigma)(x)/[x'*sigma*x]. However, to me it is more intuitive to see this as Cov(position B, portfolio)/Variance(portolio). We just need the covariance (position B, portfolio). (Which, btw, is the same hard step in Stulz cash flow VaR for small project)

covariance (b returns, P returns) = COV(B, 50%*A+50%*B) = (50%)COV(B,A) + (50%)COV(B,B) = 50%*[COV(A,B)+VAR(B)] = 50%*[0 + 0.04] = 0.02. Now, beta = cov (b, p)/var(p) = 0.02/[(2.24/$20)^2] = 1.6. Re this denominator, note I converted from dollar to returns volatility as this entire covariance expression is in returns (%) terms not dollar terms.

Note here one potential confusion relates to dollars versus returns. And specifically, 0.4 "dollar covariance" = 0.02 returns covariance * $20 portfolio!

Hope that helps!...David