Modern Portfolio Theory (MPT) was introduced by Harry Markowitz with a paper Portfolio Selection (1952 Journal of Finance). Thirtyeight years later he shared a Nobel Prize with Merton Miller and William Sharpe for what has become a theory for portfolio selection. The Efficient Frontier evolved from this analysis.
Here's the problem.
We want to split our portfolio between three types of investment, say Large Cap and Small Cap and Foreign or maybe ... >Or maybe Growth and Value and Income, or maybe ...
We suppose that the returns for each are r_{1}, r_{2} and r_{3}
and the Standard Deviations are S_{1}, S_{2} and S_{3}. (1) x + y + z = 1 because they'd add to our entire portfolio, like 30% devoted to the first and 50% to the second and 20% for the third and 0.3+0.5+0.2=1
Suppose we want a return of R from our portfolio. Then, because of the fractions x, y and z devoted to components with returns r_{1}, r_{2} and r_{3} we assume the return of our portfolio would be given by: (2) x r_{1} + y r_{2} + z r_{3} = R like 30%, 50% and 20% means 0.3(r_{1})+ 0.5(r_{2})+ 0.2(r_{3}) = R, our portfolio return
>We assume? We want to select the fractions x, y and z so as to satisfy these equations, yet minimize the Volatility (or Standard Deviation) of our portfolio. For each specified value of R we find the minimum Volatility, and plot R versus this Minimum Volatility. >Suppose I choose R = 100% Anyway, we'd get something like this (where, in this example, we show the appropriate percentages of each portfolio component):
This curve is called the Efficient Frontier. Well, actually, it's just the upper part that's the efficient part ... where the return is maximized. In fact, you'll also get this curve if you pick a Volatility (instead of a return R) and ask for the maximum return with this prescribed Volatility. What the chart says, is:
If our portfolio components were uncorrelated, the problem of finding the least volatile mix is simpler. (See Stocks & Bonds.) However, we'll now assume that there is some correlation between components (a la Modern Portfolio Theory).
>Why don't you just say "years" and I'll understand that it could be "months" or "minutes" or ... Suppose the annual returns of component "1", over M years are g_{11}, g_{12}, ... g_{1M} and the returns of component "2" are g_{21}, g_{22}, ... g_{2M} etc. etc., then (remember how to calculate Standard Deviation, calculating the average of the squares of deviations from the Mean?) these N Standard Deviations (for the N portfolio components) are given by: S_{1}^{2} = (1/M){ (g_{11}  r_{1})^{2}+ (g_{12}  r_{1})^{2}+ ... + (g_{1M}  r_{1})^{2} } for component "1"Okay. If, for the first of the past M years, the N portfolio components had returns g_{11}, g_{21}, ... g_{N1} then our portfolio had a return of x_{1}g_{11}+x_{2}g_{21}+ ... +x_{N}g_{N1} so the deviation from R, our Mean Return over the past M years, is, for this first year: x_{1}g_{11}+x_{2}g_{21}+ ... +x_{N}g_{N1}  R = x_{1}g_{11}+x_{2}g_{21}+ ... +x_{N}g_{N1}  { x_{1}r_{1}+x_{2}r_{2}+ ... +x_{N}r_{N}} which can be written:
and, for the second year we'd get the deviation: ... etc. etc. ...
and, for year M we'd get the deviation: Then SD^{2} for our portfolio is the sum of the squares of all the above, divided by M ... namely the Average square of these M annual deviations. It can be expressed as the product or three matrices, namely X^{T}WX where X is the Nx1 matrix with components x_{1}, x_{2} etc. and X^{T} is its transpose (an 1xN matrix) and W is an NxN covariance matrix with elements containing the various product terms arising from the squaring and that'd be ... >zzzZZZ Anyway, to make a long story short, we assume we KNOW all about the historical info on the components  meaning we know all these crosscorrelations (hence the matrix W, as in X^{T}WX)  and we want to select the fractions x_{1}, x_{2}, ... x_{N} so that:
