00:01
So here we are given some information about two securities, k and l.
00:04
For k, we know that the mean of k is 13%, and the standard deviation of k is equal to 19%.
00:13
For l, we know that the mean of l is equal to 10%, and the standard deviation of l is equal to 16%.
00:22
So now what i want to do is form a portfolio.
00:26
And i'm going to call my portfolio p, and i'm going to call my portfolio p.
00:30
My portfolio as alpha k plus one minus alpha l.
00:36
And this alpha here is the sheer of my portfolio invested in k.
00:42
Right.
00:43
So for example, if alpha is equal to 0 .4, 40 % of my portfolio will be in k and 60 % of my portfolio will be in l.
00:52
We want to think about the risk -free rate, right? so what i want to do is think about the variance of this p.
01:00
I want to think about the variance of this p, and i want to set this equal to zero, because that would be risk -free, right? if i can get the variance to nothing or equivalently the standard deviation to nothing, then i can, right, can have eliminated the variance completely.
01:20
So i can say the variance of alpha -k plus one minus alpha -l, the sum following the standard rules of variance, is equal to, right, the variance of alpha k plus the variance, the variance of alpha -l, plus twice the co -variance of alpha -k.
01:54
Oh, sorry, it's not alpha -l, it's one minus alpha -l.
01:59
I should be careful.
02:01
One minus alpha -l, plus twice the co -variance.
02:05
Of alpha k 1 minus alpha l.
02:10
Now that's a little bit squeezed there, sorry.
02:13
But this is just a standard rule of variance, right? rule of variance.
02:21
Variance.
02:23
And so now we can simplify this a little bit more.
02:27
Variance is a second moment thing.
02:29
So this always comes out to the alpha squared times the variance of k, plus 1 minus 0 .5.
02:38
Alpha squared outside the variance of l plus alpha 1 minus alpha twice the covariance of k and l.
02:48
Now we have to use one more thing here, right? and the thing that we have to use is the perfectly negatively correlated.
02:57
So we know the correlation coefficient here all the way back up here is minus one.
03:03
But what i don't have is an expression for the correlation coefficient.
03:07
So what i need to do is divide through here, right? and remember, we're setting this equal to zero to get the correlation coefficients.
03:15
So what i can do here is zero is equal to alpha squared...