It seems quite remarkable that one can construct a portfolio, with minimal volatility, diversified over many investment classes, by extracting from historical data just three parameters:

Mean, Standard Deviation, Correlation

>The Mean for every investment class and the Standard Deviation ...
Yes, of course. The Mean and Standard Deviation (which we'll call SD) are needed for every asset and the correlation between every pair of assets is needed.

>That's more than three. In fact, the number of correlations ...
That might be large. For five asset classes, that means 4x3 = 12 correlations,

>Plus five Means and five Standard Deviations!
Yes. But, of course, if the assets are completely uncorrelated ...
>Fat chance!
... then we just need the Mean and SD for each and can skip to the simpler problem, outlined in Stocks, Bonds and Volatility

>What if I've got a bunch of cash in my portfolio?
Uh ... yes. Good question. Cash has a SD of zero. In fact, the covariance with any other asset is zero. Remember the magic equations, from Part I? The first couple look like:

[1]     x₁(g₁₁-r₁)+x₂(g₂₁-r₂)+ ... +x_N(g_N1-r_N)
[2]     x₁(g₁₂-r₁)+x₂(g₂₂-r₂)+ ... +x_N(g_N2-r_N)

If the first component of your portfolio were cash ... or any asset with zero Standard Deviation ... then the annual returns g₁₁, g₁₂, etc. would be identical to r₁ (the Mean of all the returns over the past M years) so g₁₁-r₁ = 0 and g₁₂-r₁ = 0 etc.
That means zero covariance with other portfolio components. All those terms like (g₁₂-r₁)(g₄₅-r₄) would be zero.

>Can you just jump to the conclusion? Okay, it means that cash, or any risk-free asset (with zero Standard Deviation), is handled differently. On the Return-Volatility chart, it's a single point on the Return axis: (0,Rf). On the other hand, a point on the Efficient Frontier with coordinates (S,R) represents some portfolio mix with more volatile assets. If cash now becomes a component of your portfolio, the point represented by the modified portfolio is somewhere along the line (drawn in magenta) ... >Depending upon the amount of cash. Exactly. Closer to the risk-free point as the fraction of cash increases.

And the slope of that line has a name: the Sharpe Ratio. It represents the Return (relative to a risk-free return) per unit Volatility, that is:
    (MeanReturn - RiskFreeRate)/Volatility

Actually, William Sharpe (one of the Nobel laureates) called it the Return per unit of Risk ... but that's if one equates Volatility to Risk ... and we won't do that here.

>So what about points inside this frontier? I mean, to the right. I mean ... I understand. Take a gander at this picture: Point A is a point "inside" the Frontier, representing some inefficient portfolio. Now consider another point B "on" the frontier, representing an efficient portfolio. We could allocate a part of this B-portfolio to a risk-free asset, represented by point C, moving our portfolio to point D. Thus we've reduced the Standard Deviation of the original portfolio A, yet maintained the same Return as A, namely R.

>So, does anybody actually use this stuff? Does extracting some numbers from historical data provide a prescription for minimizing the future volatility of my portfolio? I mean ...
Yes, I understand. Good question. Let's see ...

We consider just three asset classes and their monthly parameters: average monthly returns: 1.185%, 1.114%, 0.789% Standard Deviations: 4.237%, 4.980%, 1.984% 3x3 covariance matrix: W shown here

>Standard Deviation ... only 4.237%? Isn't that ...?
That's a monthly SD. The annualized SD would be 4.237 SQRT(12) = 14.7% (See SQRT stuff.)
Okay, forging ahead.
If x, y and z are the fractions of our portfolio devoted to each of the three assets, we can write the pertinent equations corresponding to a prescribed average return of, say, 1.10% and generate a quadratic in x (we'll call it Q) ...

>You're kidding, right?
Here's the prescription: If we plot Q versus x, for x = 0.0, 0.1, 0.2, etc. (meaning, of course, 0%, 10%, 20% ... devoted to asset#1) there is a point beyond which a greater proportion devoted to asset#1 will give a return GREATER than the prescribed average return of 1.10%. So the curve stops there.

>What's wrong with a return greater than ...?
If you want a greater return, then prescribe a greater return. The math just does what we ask: we prescribe 1.10% - it gives 1.10% (if possible).
>And Q gives the Standard Deviation of our portfolio?
Not exactly. It gives the sum of squares of the deviations from the expected return. We'd have to divide by the number of months involved, then take the square root. However, if we just want to identify the minimum, we might as well consider Q, just as it stands.
>Aha! That's why you didn't label the Q axis!
Uh ... yes, because it's not the Standard Deviation but ...
>And you expect prople to actually do those computations ... as though they could predict the future evolution of their portfolio?
Of course, not. It's fiction ... but fun to play with.

Here's a spreadsheet. It contains the pertinent monthly data (ten year's worth: 1986 to 1996) for two dozen asset classes labelled 1 to 24 ... and you're invited to type in three of these numbers and a monthly Prescribed Average Return.

You get a chart of the magic quadratic Q (as above) and, if you click on the button, you get the Frontier ... I hope.

>You hope?
Yes ... it's a toy. Play, get a feeling for this frontier stuff ... but don't expect any great insight concerning the best mix for your portfolio. The past is ...
>... no guarantee of the future, right?
Right.
>But what if I'd like to invent my own assets, with Mean and Standard ...?
Uh ... yes, I understand. A fictitious portfolio, where you invent the parameters - including the correlations - just to answer What If questions. Then I have just the spreadsheet for you!

... still workin' on this one ... but, in the meantime, you may want to check out: another Frontier

>But what if I like big volatility? What if I like the chance of winning big? What if ... ? Yes, I understand. You'd be prepared to take on extra volatility in order to, perhaps, achieve big returns. Here's a picture of two Normal distributions of returns Both have Mean = 10% (see point A) but the red one has twice the Standard Deviation. With the blue distribution there's a 25% chance of achieving a Return greater than 20% (see point B), but with the red distribution, with the larger volatility, ... >But with the red there's a 25% chance of getting more than 30%. I'll take it!

Then forget Efficient Frontier.
>But can't I choose my volatility, then go UP to that frontier to get the maximum return?

And expect that the allocation which achieves that will be optimal, in the past and in the future?
Good luck.
That ritual is indicated in the first diagram on this page ... in dark grey