00:01Hello and welcome to Friendly Finance
00:03Anyone who reads portfolio theory is bound to come across the double summation formula for finding the standard deviation of a portfolio, and that is what we are going to be talking about in this little video
00:13The formula uses the asset weights and co-variances between asset pairs, and from time to time some of my students have asked me for a demonstration on how to open the double summation, and in this brief video, I'm going to do that for a portfolio of two assets
00:30Well, one could do it for as many assets as possible, but as will be evident in a short while, the process Goes on getting lengthier as we increase the number of assets in the portfolio, so after a while, we would start feeling the need of a computer program to do it
00:47So what I'm going to do here now is to assume that we understand all about covariances and stuff, and I'm going to go straight ahead with simplifying the double summation for a two-asset portfolio, and I'm going to leave it to you as homework to try to do it for three assets and a four-asset portfolio
01:05So now, when we think in terms of a portfolio that comprises of n assets, the first thing that we should ask ourselves is how many pairs of covariances can be
01:14Computed between the assets, let us look at a portfolio that comprises of two assets, and see how many pairs of covariances can we get
01:23We can find out the covariance between asset 1 and 1, we can do that for asset 1 and 2, we can do that for asset 2 and 1, and then for 2 and 2, so that is going to give us four pairs of covariances
01:37Uh, what we need to realize is that the covariance between 1 and 1 is simply the variance of asset one, and the covariance between 2 and 2 is simply the variance of asset two
01:48Now, in these pairs of covariances, let's call the first asset as asset I, let's give it a name, and let's call the Second asset as asset J, so that the I becomes one in this combination
02:04I is still one in this combination
02:07I becomes two in this combination, and I is still two in this combination
02:12Uh, J is one in this combination
02:14J is two in this combination
02:22J is one here, and J is two here
02:29So if we give a name as I and J to our assets, then we can write the formula for portfolio's standard deviation as this, where on the left-hand side, we have the standard deviation of the portfolio, and on the right-hand side, we have a term that has been raised to the power of half
02:37Raising it to the power of half, my friends tell us, tells us that we are taking a
02:42Square root of this term here, inside the square bracket, we have a double summation sign, about, uh, which we are going to talk shortly
02:50And then, following the double summation, we have X sub I and X sub J, which are the weights for asset I and J, respectively
02:58And then we have a sigma of I and J, which is, uh, the covariance between asset I and J
03:03So basically, what we are doing in this formula is that for all combinations of covariances between assets I and J, that is, for all pairs i, j, we are taking a sum of this term: X sub I * X sub J * covariance between I and J
03:20Now, since this sum is being taken for all I's and all J's at the same time, that is why we have Two summation signs, and that is what we call as the double summation, and to simplify a double summation is not a very tricky thing, it simply means taking the sum of a sum
03:38What we are going to do is, we are going to go to an equation editor and try to simplify this formula for a two asset portfolio
03:45So let me switch over to the equation editor here, where I have pre-written this formula for you, and we are going to now simplify it for a two asset portfolio
03:55So let us start a square bracket, inside the square bracket, let us write down the first summation as it is, so what I'm going to do is, I'm going to write I is going from 1 to
04:04Two, because it's a two asset portfolio, so n is two
04:12And inside now, what I'm going to do is I'm going to take a sum of all for all Js
04:17I'm going to take a sum of this term here, so let me write inside a small bracket, and then let me put another summation sign and say, for all Js that vary from 1 to 2, I'm going to take a sum of this term inside this small bracket
04:38So, and we should not forget to raise this thing to the power of half, because that tells us that we are taking a square root of this thing
04:44Now, let us continue further, let us start another square bracket here in this step, and let us keep this summation as it is, the I
04:54Summation, let us not, uh, disturb that for the moment
04:57Let's, let's say I is varying from 1 to 2, and inside the small bracket now, let us solve the J summation, so we are going to get rid of the summation sign here, and we are going to allow J in this term to become one first, and then we are going to allow it to become two, so let us start doing that
05:19We write an X here, and this I we keep as it is because we are not dealing with the I summation as yet, and then we write another X in the foot of which we need to write a number for J, and we are allowing J to become one, so we write a one, and then we are going to put a sigma, and in the subscript we are Going to write I, because we are not doing anything with I at all at the moment, and then for J, we are going to write A1
05:47Then we are going to put a plus sign, because we need to write this term over again and allow J to become two, so we write X, and in the foot of X, we write an I, leaving it unchanged, and then we write another X, and for J this time, we write the value two, and then we are going to put a sigma, and in the subscript, we are going to write I as it is, and for J, we are going to write a two
06:17After that, we should not forget to raise this thing to the power of half to take the square root
06:22Now let us continue simplifying, we have
06:26Only one summation left now, and that should not pose us any problem at all
06:31So inside the square bracket, let us open up the I summation
06:34First of all, for this entire thing, I is going to become one, and then for this entire thing again, I is going to become two
06:42Let us write it down
06:45So we write an X, and this time now, we are going to allow I to become one, so I becomes one
06:51We already have XJ as X1 here, so we write X, and we simply say one, and then we are going to write a summation, uh, we are going to write a sigma, and in the subscript, we are going to write one for I and one for J, and then we are going to put a plus sign, and we are going to write
07:12This item here, now X, but this time I has been allowed to become one, then another X, and this is for J, which is already two, so we write a two, and then we put a sigma, and in the foot of it, we are going to write one for I, uh, and two for J
07:37Um, it's giving me some problem here, so let me get rid of this and put a subscript here, in the foot of which I should write one for I and two for J, so I'm done with that
07:50And then I need to rewrite this term over again by allowing I to become two, so I write an X, and in the subscript this time, I say I has become two, J remains one, so I write another X1 here, then I'm going to say Sigma, and in the subscript I'm going To say two for I and one for J, then I'm going to put a plus sign, and I'm going to write this second term over again, but this time allowing I to become two
08:19So, um, I'm going to write an X, and what I need to do is to write a subscript, um, here
08:31So I go here and write a two here, and then I'm going to write another X, and in the subscript, J is two already, this one here
08:44So I'm going to go ahead and put a sigma here, and in the foot of it, I'm going to write two for I and two for J, and then I'm going to raise this entire thing to the power of half, so that we take the square root
09:02Now what I'm going to do is to write down the standard deviation
09:05Of the portfolio, one more time, let me copy it here from here and paste it here
09:12And then inside the square bracket, what I'm going to do is to simplify these terms further
09:19If you look at the first term here, this one, you will realize that X1 * X1 is going to become X1 squared, so what I'm going to write here is X, and then I'm going to write X1 squared
09:35And then this is covariance between 1 and 1, that simply means variance of asset one, so let us write here Sigma sign, and then we need to write it for the variance of one, so 1 and 2, this becomes variance of one
09:50And then let us look at the last term in this step here, X2 * X2 is going to become X squared, 2
09:57So let us do that here, X squared, 2, and then I'm going to have the covariance between two and two, which simply means the variance of asset two, and that is what I'm going to write here, variance of asset 2
10:16And then I'm going to look at my middle terms here, I'm going to put a plus sign, and I'm going to look at this term, this one here, and this one here, and if I look at them, I realize that both of them are the same, so I have the same term two times over, so I write 2 * X, and in the foot of it, I write one, and then I write another X, and in the foot of it, I write a two, and then I'm going to put a sigma, and then I'm going to write 1 and 2, and then raise
10:50This thing to the power of half, and that simplifies it for two assets
10:56Thank you very much, my friends, bye-bye.
