Are there 2 solutions to minimum variance optimization? (calculus of random numbers)

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) + 2rcsh
d(P^2)/dr = 2
r*(h^2) + 2csh
Set d(P^2)/dr = 0 --> 2
r*(h^2) = -2csh
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)csh
P^2 = (s^2)
(r^-2) + (h^2) + 2*(r^-1)csh
d(P^2)/dr = -2
(s^2)(r^-3) - 2(r^-2)csh
Set d(P^2)/dr = 0 --> 2
(s^2)(r^-3) = - 2(r^-2)csh
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.

Solving dP/dr = 0 will yield the “optimal” r when the total value of stock is fixed.

Solving dP/dr = 0 here will yield the “optimal” r when the total value of the hedge is fixed. This is a different problem.

I think. Best of all might be to find the optimum when the total of hedge and stock combined is fixed.

I can’t see how these cases are different. If I find an optimal ratio and then discover that I have been given a different amount of stock to hedge, the same ratio should still apply.
Specifically, if I optimize in the first case, I work out that I need to hedge Stock using r * Hedge. If I subsequently discover that I have been given Stock / r instead, surely I need to use hedge (unadjusted).
But if I try to optimize for Stock/r, I get a different number for r.

The difficulty is your problem is not scale free the way you state it so your answers are not scale free and you can’t interpret them easily after scaling. If you want the answer to be scalable, you need to state the problem in a scale-free fashion. Think about the variance per dollar invested and the fraction of the money which should be allocated to the stock and the instrument.

Also stock variances are usually stated on a return basis (e.g., (20%)^2). The way you’ve set up your problem you have to be stating them on a dollar squared basis.

I’d write the problem as follows (letting s and h be the std. dev.s of returns per dollar). The fraction f of each dollar I want in the hedge instrument minimizes:

(1-f)^2 s^2 + 2f(1-f)csh + f^2h^2

and is f = (h^2 - chs)/(s^2-2chs+h^2)

Perhaps a simpler way to see that your statement of the problem is misleading you is the following. Suppose I told you c = 0 and asked what’s the best thing you could do to reduce variance when adding the hedge to the stock. You’d conclude adding none of the hedge instrument was best. If I asked you what’s the best you could do when adding stock to the hedge instrument, you’d again conclude zero. But pretty obviously 0 is not equal to 1/0.

I don’t understand why you say my problem isn’t scale free.

I have seen many texts talk about total dollar variance. I have never seen any of them refer to the units in dollars squared.

Wolfram Alpha doesn’t give a closed derivative for that, unless I typed it wrong.

I did notice that the second frame breaks down when r is 0, but I don’t think that is the same as being misleading.

The derivative is 0 = -2(1-f)s^2 + (2-4f)csh+2f h^2
= -2s^2 + 2csh +f(2s^2 - 4csh + 2h^2

The solution is (s^2 - chs)/(s^2-2chs+h^2) typo previously.
And by misleading I meant, the solution for c = 0 is obvious and your reasoning that the solutions should be the reciprocal of each other must be wrong. You’re misleading yourself to say they should give the same answer sicne they clearly don’t in one case.


Sorry been a long day for me.
This idea is interesting and I will explore it further. However, all texts on this topic I have seen (google minimum variance hedge) all give the answer in my 1st frame in the OP.

Ok, clearly something is misleading because as you say, the solutions don’t give the same answer. But I don’t get exactly where and why it is misleading. ‘Intuitively’ (to me) they should be the same. When I said ‘s’ above, I didn’t specify any actual numbers.

Lets assume that stocks are infinitely divisible.
If I have 1 stock worth 1 and I want to hedge my exposure with a hedge instrument which happens to be worth 1. I go through the steps in frame 1 and I get a hedge ratio of (lets say) 0.4
This means that if I use 40c worth of my hedge instrument, I will have minimized the variance of my portfolio.
But if I suddenly realize that I actually have $2.5 worth of Stock, surely I have no reason to think that the hedge ratio I previously calculated is invalid. 0.4 * 2.5 = 1. So to hedge my $2.5 of stock, I need $1 of the hedging instrument.
What I don’t get is if I optimized it in terms of frame 2, I would get a different hedge ratio. What is frame 2 actually optimizing for, if not the minimum variance of the final portfolio?

If you scale up the stock position, the answer does scale in proportion.

Let’s try it this way. When you add a short position in the instrument to the stock you are doing two things: removing some risk of the stock and adding some uncorrelated risk inherent in the instrument. The former depends on the covariance the latter depends on only the instrument’s variance. If you simply wanted to remove the common risk but did not care about the uncorrelated risk of the instrument, you would hold even more (a bigger short position) of the instrument. But you do care about this uncorrelated risk as it adds to variance so you push that answer towards zero.

When you do the second problem thinking about a fixed position in the instrument, you again push the amount of stock towards zero due to its uncorrelated risk. But pushing the stock position towards zero is the same relatively speaking as pushing teh instrument position away from zero which is exactly what you’re seeing.

Could I not come at it from the other side (having very low uncorrelated risk and not removing enough common risk) and say that I am pushing it away from zero? After all, isn’t that what optimization is: finding the sweet spot? I realize I may have completely missed your point. If so, what does ‘that’ refer to in the last sentence in the quote, if not the ratio?

It’s getting late where I am and I need to sleep so I will only see your response in the morning, but please complete the following sentence: “The procedure in the first frame of the OP finds the combination of Stock and Hedge that has the minimum possible variance. The procedure in the second frame of the OP finds the combination of Stock and Hedge that …”

Thanks in advance.

OldGuy provides a theoretical explanation of the point I tried to make. The key point made by OldGuy, I think, is that, in this case the expression to be minimized must be a pure number; i.e. you must explicitly or implicitly divide the variance by square-dollars. For example, in P^2 = (s^2) + …, the right-side is a pure number when the quantity of stock is fixed.

Thus I already answered your latest question.

As I stated earlier, the first procedure finds the Hedge that minimizes variance when Stock is fixed. The second procedure finds the Stock that minimizes variance when Hedge is fixed. It may seem intuitive that these are the same problem, but they aren’t. You can see the same effect in simpler problems.

I can’t quite fill in those blanks because that’s not exactly what either maximization does. State the problem as follows. Holding n shares and m hedging instruments what is the variance?

var = n[sup]2[/sup]s[sup]2[/sup] + 2nmcsh + m[sup]2[/sup]h[sup]2[/sup]

What minimizes variance? Well obviously n = m = 0. Not interesting.

Your first problem: Among all portfolios with n = 1 what m* = m*(1) minimizes variance?

Your second problem (though not directly) Among all portfolios with m = 1 what n* = n*(1) minimizes variance?

Generalized first problem: For any fixed n what m*(n) minimizes variance? Ans: m*(n) = nm*(1). It scales just as you want.

Now suppose I fix n at the optimal answer to question two that is n*(1). Shouldn’t the answer to choose m to minimize variance when n = n*(1) give an answer of m = 1? Answer no.

Why not? Because the sets you’re searching for the minimum in are different in the two problems. I’ll try to draw you a picture, but I don’t promise. I think I’ll have to send it to you privately. I don’t think I can do it here.


Saffer, I sent you an email with an explanation via images. I hope it helps.

I know this is a bit late but finally the penny has dropped for me on this.
I was browsing through some of my previous threads and found this and started to think about it again.
My confusion results (as was pointed out by OldGuy) from the fact that I was stating the problem in dollars rather than returns. And just because the minimum variance of a fixed dollar amount of my stock and a variable dollar amount of hedge leads to r dollars of hedge, that does not mean that if I start with r dollars of hedge and find the minimum variance for a variable amount of dollars of stock, I will get to the same value.
I fell a bit silly for not getting it first but I suspect that if the exact sentence above was used, I would have got it immediately.
So really it’s your guy’s fault for not explaining properly. :stuck_out_tongue:

My phrasing was not the exact one you wanted, but I think it was close.