Stulz - Var Impact of a small project

[sathyat]

Hi David,
I am going through the excel sheet for Var impact of a small project. In one of the steps you calculate the covariance between the asset and the portfolio.

I don't understand how you came up with this formula for calculating covariance between an asset and the portfolio. Can you please explain it a bit further ?

Regards,
Sathya

 

[sathyat]

Another question regarding the same excel. Can you please explain how you came up with the formula for 'Marginal Cost of VAR Per dollar of VAR' ?

Regards,
Sathya

 

[David Harper CFA FRM]

Hi Sathya,

On the first (covariance btwn asset and portfolio), it is a step that Stulz relegates to a footnote: kudos for your detailed examination. The step (property of covariance) is not covered to my knowledge in Gujarati or Quant; i.e., i don't see how the exam could query through the whole sequence here. Nevertheless the key property is:

COV(X1 + X2 + X3, Z) = COV(X1,Z) + COV(X2,Z) + COV(X3,Z)
see http://mathworld.wolfram.com/Covariance.html
(but again, this is not an FRM exam concept, just FYI)

So, in the example the three-asset portfolio is the SUM OF the three assets. And we have
COV(portfolio, asset1) = COV(asset1 + asset2 + asset3, asset1)
= COV(asset1, asset1) + COV(asset2, asset1) + COV(asset3, asset1)
= VAR(asset1) + COV(asset2, asset1) + COV(asset3, asset1)

But actually the portfolio return = (1/3)(asset1+asset2+asset3). So substitue in (1/3) and the (1/3) just "falls out front" and we get:
= (1/3)[VAR(asset1) + COV(asset2, asset1) + COV(asset3, asset1)]

On the second, I find this easier to just see this as an input variable; i.e., what is the marginal dollar impact of VaR. The formula is based on Stulz incredibly convoluted p96 - 97 (I mean, it's really impossible to follow) which reduces to:

An input assumption that a $1 decrease in VaR allows firm to reduce equity by $1.11
An input assumption that firm value change = 10% of equity change; i.e., reduce equity by $1 implies value increase of +10%
Ergo, firm value change = positive (1.11)*(10%) of VaR decrease, or about $0.11 per VaR descrease

I would very loosely paraphrase as, "a dollar decrease in VaR only adds $0.11 due to the incremental benefit of more debt/less equity."

David