Hi

If you can find this derivation helpful:

Marginal Var=d(Varp)/d(wi*Vp) is the as per definition where Varp is the Var of the portfolio and the wi*Vp is the $ invested in the ith position.

=>Marginal Var=d(z*sigma(p)*Vp)/d(wi*Vp)=(z*Vp/Vp)*d(sigma(p))/dwi (as z and V are constants per se so take them out)

**=>Marginal Var=(z)*d(sigma(p))/dwi ....E1)**

Also we know that, standard deviation of portfolio^2

=sigma(p)^2=w1^2*sigma(1)^2+....+wi^2*sigma(i)^2+....+wn^2*sigma(n)^2 +2*wi*(w1*Cov(i,1)+w2*Cov(i,2)+....+wn*Cov(i,n))+Other Covariance terms (considering that portfolio has n positions)

differentiate w.r.t wi both sides of above equation to get:

d(sigma(p)^2)/d(wi)=2 wi*sigma(i)^2+2*(w1*Cov(i,1)+w2*Cov(i,2)+....+wn*Cov(i,n))

2*sigma(p)* d(sigma(p))/d(wi)=2 (w1*Cov(i,1)+w2*Cov(i,2)+....+wi*sigma(i)^2+....+wn*Cov(i,n))

**sigma(p)* d(sigma(p))/d(wi)=(w1*Cov(i,1)+w2*Cov(i,2)+....+wi*Cov(i,i)+....+wn*Cov(i,n))... E2)** (as sigma(i)^2=Cov(i,i))

Also we know that portfoio return=Rp=w1*R1+w2*R2+....+wi*Ri+...+wn*Rn

=>Cov(Ri,Rp)=w1*Cov(R1,Ri)+w2*Cov(R2,Ri)+....+wi*Cov(Ri,Ri)+...+wn*Cov(Rn,Ri) (the (Ri,Rp) is same as(i,p) notation used above)

=>Cov(i,p)=w1*Cov(1,i)+w2*Cov(2,i)+....+wi*Cov(i,i)+...+wn*Cov(n,i)

Since from equation 2) w1*Cov(1,i)+w2*Cov(2,i)+....+wi*Cov(i,i)+...+wn*Cov(n,i)=sigma(p)* d(sigma(p))/d(wi)

**=>** sigma(p)* d(sigma(p))/d(wi)=Cov(i,p)

=>**d(sigma(p))/d(wi)=Cov(i,p)/sigma(p) ...3)**

Put value of **d(sigma(p))/d(wi) in Equation 1) to get**

=> **Marginal Var=z*****Cov(i,p)/sigma(p)**

Thus **Marginal Var=z*****Cov(i,p)/sigma(p)**

thanks

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