So I was reading about minimum variance optimization and I noticed the problem could be framed in 2 ways. I was expecting the 2 ways to lead to the same answer but they don’t.

For this discussion, the following notation is used:

s^2: The variance of the stock we want to hedge.

h^2: The variance of the instrument we want to use as a hedge.

P^2: The variance of the portfolio resulting from the combination of the stock and hedging instrument.

c: The correlation between the stock and hedging instrument.

r: The value we are solving for, indicating the ratio between the stock and hedging instrument.

1st, the usual way this problem is framed, starting with the formula for the sum of variances.

P^2 = (s^2) + (r^2)*(h^2) + 2*r*c*s*h
d(P^2)/dr = 2*r*(h^2) + 2

*c*s

*h*

Set d(P^2)/dr = 0 --> 2r*(h^2) = -2

Set d(P^2)/dr = 0 --> 2

*c*s

*h*

r = -cs/h

r = -c

So this tells us that if we make a portfolio consisting of some value of the stock and r times that value of the hedge instrument then we will have minimized the variance of the portfolio.

(Value of Stock) = r * (Value of Hedge)

or equivalently

(Value of Stock) / r = (Value of Hedge)

This is the same ratio, no? What happens if I frame the problem in this way?

P^2 = (s^2)*(1/r^2) + (h^2) + 2*(1/r)*c*s*h
P^2 = (s^2)*(r^-2) + (h^2) + 2*(r^-1)

*c*s

*h*

d(P^2)/dr = -2(s^2)

d(P^2)/dr = -2

*(r^-3) - 2*(r^-2)

*c*s

*h*

Set d(P^2)/dr = 0 --> 2(s^2)

Set d(P^2)/dr = 0 --> 2

*(r^-3) = - 2*(r^-2)

*c*s

*h*

r = -s/(ch)

r = -s/(c

The 2 values of r are different.

Incidentally, I have set up a spreadsheet with 10000 random iterations of a stock and hedge across various hedge ratios. When I set the portfolio to be (Stock) + r * (Hedge) then the minimum variance is at the first ratio from above. When I set the portfolio to be (Stock) / r + (Hedge) then the minimum variance is at the second ratio from above. For each case, there is only 1 minimum.

When you plug in the optimum r from the **first frame** into the portfolio variance equation from the **first frame**, the result is: (s^2)*(1-c^2)
When you plug in the optimum r from the second frame into the portfolio variance equation from the first frame, the result is (s^2)*(1-c^2)/(c^2). Since c is always less than 1, this result is always larger than the first result.

So I can follow the math all the way through, but I don’t intuitively get it. If you find an optimum ratio between these then that should be the optimum ratio regardless of the scale. I’m sure there is something really stupid I am missing here.