Minimum variance portfolio...

[Tater10]

I can't figure out the minimum variance of situations like this:

Security A Standard Deviation 14%/year.
Security B Standard Deviation 8%/year.
Security C Standard Deviation 10%/year.

Correlation coeff between A&B = .4
Correlation coeff between A&C = .25
Correlation coeff between B&C = .2

What % of each security do you own to have the lowest standard deviation portfolio? Also, what if I could only be long securities (have weights greater than 1)?

I can figure out the total variance of the portfolio given % of each security. I'm hoping there is a general way to do this, as I've never been a calculus freak.

Thanks in advance...

[mosdef]

Hey dude,

Here's a short answer - I'm in a hurry.

Let x be the % of Sec A, y the % of Sec B.

Write Var(x,y)=the variance of the portfolio in terms of x and y. You should get a quadratic in x and y.

Let Var_x(y) be the variance with x fixed (so it's just a function of y) and Var_y(x) be similarly defined.

The minimum variance with be where Var_x(y) and Var_y(x) both have derivative 0. So solve Var_x'(y)=0 and Var_y'(x)=0. This will give you 2 equations and 2 unknowns: solve for x and y.

If you have limitations (no negative holdings for example) then you would need to revise the above if the minimum falls outside the "boundaries". This is done by testing along the boundaries.

This is a pretty incomplete answer. Someone else may provide more help. If not let me know how it goes.

Mosdef

[edtost]

solve it as a linear program.

unfortunately, i'm really bad at that, and am far away from my books on the subject...hopefully someone else will chime in. i think you're looking for something like the "simplex method".

[alThor]

[ QUOTE ]

I can figure out the total variance of the portfolio given % of each security. I'm hoping there is a general way to do this, as I've never been a calculus freak.


[/ QUOTE ]

Do what mosdef said. Write variance as a function of the shares (you only need two variables, since the third is 1-x-y).
Take the derivative of VAR with respect to x and set equal to zero.
Take the derivative of VAR with respect to y and set equal to zero.
Solve those two equations you get.

alThor

PS Ed, unfortunately variance is not linear.