Hi,
Just wondering on Reading 54, the Value Added = risk*IR - lambda*risk^2
How do they go from that equation to VA(optimal)=IR^2/(4*lambda)?
(I’m getting a negative value as optimal VA using differentiation)
Any help is appreciated thanks.
You should first calculate W* which is the optimal residual risk and then using that calculate the optimal VA
Optimal W is IR/(2*lambda)
VA = IR * W* - lambda * W* ^2
inputing that into VA = IR * IR/(2*lambda) - lambda * (IR^2/(4*lambda^2))
VA = IR^2 /(2*lambda) - IR^2/4*lambda = IR^2/4*lambda
You should first calculate W* which is the optimal residual risk and then using that calculate the optimal VA
Optimal W is IR/(2*lambda)
VA = IR * W* - lambda * W* ^2
inputing that into VA = IR * IR/(2*lambda) - lambda * (IR^2/(4*lambda^2))
VA = IR^2 /(2*lambda) - IR^2/4*lambda = IR^2/4*lambda
The optimal W is basically the first differential (which gives the maxima) of the function. So if you difference w.r.t ‘w’ you get the max VA
I’ve gotten to the second last step but I can’t get to the final step. This is as far as I can get:
IR^2/(2*lambda) - IR^2/(4*lambda) = [IR^2/(2*lambda)][1-(1/(2*lambda))]
IR^2/(2*lambda) - IR^2/(4*lambda) = 2 * IR^2/(4*lambda) - 1 * IR^2/(4*lambda) = ( 2 - 1 ) * IR^2/(4*lambda) = IR^2/(4*lambda)
Thanks MFEMSF, got it now!
You’re welcome!