Here's the problem.
We want to split our investments between two asset classes, A and B, say Large Cap and Small Cap stocks ...

>Or Growth and Value, or Equity and Income, or ...
Pay attention. Assume the Average Annual Returns for each asset are a and b where a 12.3% return means a = 0.123 and so on. If a fraction x of our portfolio is devoted to asset A, and the balance, 1-x, is devoted to asset B, then G, the Mean Return for our portfolio will be

(1)       G = a x + b (1-x)

>Assuming that you rebalance your portfolio every year, right?
Yes, in order to maintain the x and (1-x) fractions.
If the Standard Deviations for each asset are P and Q, respectively, then SD, the Standard Deviation for our portfolio is given by:

(2)       SD² = P² x² + Q² (1-x)² + 2 x (1-x) R

>R? What's R?
That's the Covariance between the two assets, A and B.

>You're kidding. Am I supposed to know what ...?
Okay, pay attention. If the annual returns of asset A, over N years, are a₁, a₂, a₃ ... a_N then, for this asset:
Mean Return = a = (1/N)Σ a_n
(Standard Deviation)² = P² = (1/N)Σ (a_n - a)²

>And Σ means add them all up, right?
Right. Now, we're ready for the Covariance.

Covariance = R = (1/N)Σ (a_n - a)(b_n - b)

>I assume that b_n stands for the annual returns of asset B, right?
Right. Now, notice something interesting. Those things like (a_n - a) and (b_n - b) are the deviations from the average returns. If the two assets were correlated, then when the first return is larger (or smaller) than its average it's likely that the second return is larger (or smaller) than its average and the product (a_n - a)(b_n - b) is more likely to be positive so adding all these terms is likely give a positive sum and ...

>I think you're trying to tell me that R is likely to be positive when ...
Yes, when the assets are correlated. However, when they are NOT correlated, it's likely to be ZERO, the positive deviations cancelling with the negative.

>A picture is worth a thousand ... Okay, here's a picture: It assumes that asset A has a Mean Return a = 8% and a Standard Deviation P = 20% and asset B has Mean b = 15% and Standard Deviation Q = 35% and we assume no correlation between asset classes and we devote a certain percentage of our portfolio to asset A and look at the Mean Return and Standard Deviation of our portfolio using the magic formulas (1) and (2) above, with R = 0 and we notice that ... >A mix of each is better than either, right? If your objective is to minimize the Volatility (or Standard Deviation) then, yes, a combination of two uncorrelated assets is better than 100% of either.

>But I'm more interested in the Annualized Return, not the Standard Deviation.
Okay, let's use the following approximation for AR the Annualized Portfolio Return (see Average vs Annualized), namely
AR = G - SD²/2 in terms of G, the Mean Portfolio Return, and SD, the Portfolio Standard Deviation:

(3)       AR = a x + b (1-x) - (1/2){ P² x² + Q² (1-x)² + 2 x (1-x) R }

>A picture is worth a thousand ... Okay, here's another picture: Here it's again assumed that asset A has a Mean Return a = 8% and a Standard Deviation P = 20% and asset B has Mean b = 15% and Standard Deviation Q = 35% and we again assume no correlation between asset classes and ... >So a combination is again better than either, eh? For this example of no correlation, but notice that we'd want more of asset A if we want a smaller Volatility, since asset A has the smaller Volatility, but ...

>More of asset B is we want a bigger gain ...
Because asset B has the larger Mean Return.

>But a mix is still better than either.
Remember we're still talking no correlation.

>So start talking about correlation!
Okay.

Here's a set of Portfolio stuff where we assume a Correlation Coefficient from r = -1.0 (meaning when one return goes up, the other goes down, in perfect synchronism) to r = +1.0 (meaning when one return goes up, the other goes up as well).

>I thought we were talking about Covariance and now you talk about ...
The Pearson Correlation Coefficient, r, is related to the Covariance like so:

Covariance = r P Q

(See Pearson Correlation Coefficient.)

Here's some interesting pictures with correlation, where (as before),
asset A has a Mean Return a = 8% and a Standard Deviation P = 20% and
asset B has Mean b = 15% and Standard Deviation Q = 35%:

>Mamma mia! The Minimum Volatility goes up and Maximum Annualized Return goes down!
You said it.

>I assume you have a magic formula for getting the Maximum Annualized Return.
Yes, it's like so:

Maximum Annualized Return (approximately), obtained for : x = (1/2) + (1/2) { Q2 - P2 + 2(a - b) } / { P2 + Q2 - 2r P Q } where a and P are the Mean Annual Return and Standard Deviation for asset A b and Q are the Mean Annual Return and Standard Deviation for asset B r = Covariance/(PQ) is the Pearson Correlation Coefficient

>Approximately? Why approximately?
We're using that approximate expression for Annualized Return, formula (3), remember? Besides, parameters like Mean Return and Standard Deviation are based upon historical data and we can't ...

>Yeah, yeah, I know. The past is no guarantee of the future.
You're getting smarter.

>Well ... that's a terrible formula.
Okay, use this handy dandy calculator to get the "Optimal" parameters (yielding the largest Annualized Return).
The Annualized Returns displayed below actually use a somewhat different approximation (noted here), namely:
(1+Annualized_Return)² = (1+Mean_Return)² - Standard_Deviation²:

Mean Annual Return of asset A = %       Standard Deviation of asset A = % Mean Annual Return of asset B = %       Standard Deviation of asset B = % Correlation Coefficient (a number between -1 and =1) = 'Optimal' Percentage of asset A = %     approximately

Note: Beware of putting a = b and P = Q and r = 1 ... 'cause there's a ZERO denominator and y'all get NaN% which says: "That means both assets are the same, eh?"

>Yeah, but I haven't the faintest idea of what's a reasonable value for that Pearson Correlation Coefficient.
Look! Here's a bunch of values for the total U.S. stock market and other asset classes:

And then there's this:

and this correlation between some sectors (using 1950-2000 data):

>You're haven't talked about bonds or U.S. vs Canada or ....

See also Correlations and Pseudo-periodicity