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Jorion, Incremental VaR - Best hedge (equation 7.24-26)

emilioalzamora1

Well-Known Member
Hi @David Harper CFA FRM,

I wanted to start a new topic which has not yet been discussed in great detail.

Jorion mentions on page 170 that a particular new trade involves a position in one risk factor (asset). The portfolio value changes from the old value of W to a new value of W(p+a) = W + a, where a is the amount invested in asset(i).

The variance of the dollar returns on the new portfolio is then written as:

σ^2(p+a)+ W^2(p+a) = σ^2*(p)W^2 + a^2*σ^2(i) + 2aWcov(i,p)

The notation is a bit clumsy at first glance, particularly the left-hand side, σ^2(p+a)+ W^2(p+a), and this is why I would like to disentangle it a bit:

Basically the left-hand side of the equation is simply the portfolio variance incl. the new asset (i), the notation in Jorion for this is quite bad! We can then rewrite the equation in our familiar form of the two-asset portfolio variance having:

σ^2(newp) = W^2*σ^2(p) + a^2*σ(i) + 2aWcov(i,p)

where σ^2(i) = variance of the new asset (i)
where σ^2(p) = variance of the 'old' existing asset (p)

This is simply equation 7.24 in Jorion who goes on differentiating this with respect to a (in 7.25) leading to:

σ^2(newp) = 2aσ^2(i) + 2Wcov(i,p)

Setting this to zero and solving for a should yield:

0 = 2aσ^2(i) + 2Wcov(i,p)
- 2Wcov(i,p)= 2aσ^2(i)
- 2Wcov(i,p) / 2σ^2(i) = a

Simplyfying:

- Wcov(i,p) / σ^2(i) = a

It therefore can be concluded that the 'new' minimized portfolio variance (after having addded the new position), the weight (a), Jorion rather defines it in $-terms, can be achieved following the below condition:

- Wcov(i,p) / σ^2(i) = a

@David Harper CFA FRM, can you please confirm whether this derivation for 'a' is in line with yours?
 
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