In a recent tutorial we were talking about some kind of "best" allocation between stocks and bonds, given the Mean return and Volatility of each component. >Stocks and bonds or large cap and small or domestic and foreign or ... Yes, yes. Any two assets. But the interesting thing was, if the Mean Return (say S) and Volatility (say P) were the same (for the two components) then we could get an annualized return greater than either and, at the same time, a volatility less than either. >Huh? Have you forgotten already? We assume: Each component has a Mean Return = S and a Standard Deviation = P. The Pearson Correlation between the two assets is r. We allocate a fraction x to the first asset and y = 1-x to the second. We call the Annualized Return and Standard Deviation of our portfolio R and SD. Depending upon the allocation x and correlation r, we get something like the situation illustrated in Figure 1 where R can be maximized at the same allocation where the Volatility SD of our portfolio is minimized, namely x = 0.5 (or 50% in each asset). >And for any correlation r between 0% and 100%? Well, I did the charts in Fig. 1 with positive correlation (implying that the two assets tend to go up or down together). However, it's even better if the correlation is negative.

Figure 1

For example, check out Figure 2 where we consider a correlation between -50% and +50% (meaning that r goes from -0.50 to +0.50). The best cases are when there is a negative correlation (so when one asset goes up or down the other tends to go down or up). Remember that, in the earlier tutorial we used an approximation for the Annualized Return, namely:       Annualized Return = Mean Return - (1/2) (Standard Deviation)2 so, for us, that means       R = S - (1/2) SD2 where S is the Mean Return for each asset (assumed to be the same) and if we use the expression for SD which we got in that earlier tutorial, we can write:       R = S - 2P2{ (1-r) x (x-1) + 1/2 } Notice the parabolic nature of this curve (as a function of x).

Figure 2

>I asume that you also reduce the volatility.
Indubitably.
>So you're looking for two stocks with the same return, the same volatility and a negative correlation.
Exactly.
>Good luck!
Aah, but I have one. It's General Electric.
>Yeah ... GE and what?
And GE.
>Huh?

Okay, here's what we do: We look at the weekly closing price for GE stock over the past ten years - about 500 prices. We shift this collection of 500 prices by 1 week. We calculate the weekly returns for each of these two sets. We calculate the correlation between these two sets of returns: this week's return vs next week's return ... over ten years We repeat this for a 2-week shift, then a 3-week shift etc. We get Figure 3 You'll have noticed that the Mean and Volatility of the two sets of returns are the same, eh? >Of course! We're talking about the same stock! Exactly!

Figure 3

>And you've got a negative correlation.
Indubitably.
>And that's for GE stock.
And other stuff:

>And what good is it?
I have no idea ... yet.
>But you have to buy the first asset at this week's close and the same asset at next week's close. Right?
Or at last week's close ... yes. So let's consider the following:

• We consider a certain basket of stocks tracked by both an Exchange Traded Fund (ETF) and a Mutual Fund.
• We can buy and sell the ETF at any time (like a stock) ... so we only trade at the opening price.
• The Mutual Fund, on the other hand, tracks the closing price.
• We pray that the Mean and Volatility of daily returns for the opening and closing prices are the same ... or at least close!
• We also pray that the correlation isn't 100%

In this case we have a Mean daily return for the ETF of -0.0143% and for the Mutual Fund -0.0045%
... and Volatilities (respectively) of 1.33% and 1.34% so that ...
>So the annualized return for the ETF is worser, right?
Yes, since we're using that approximation: Mean - (1/2)Volatility² and the ETF has the more negative Mean and the larger Volatility.
(Note that the right-most dot in last two charts corresponds to 100% ETF).

>And the correlation is negative? Well ... not exactly, but it's pretty small as shown in Figure 4 where we've shown the opening and closing prices for a few successive market days so we could see that it's often the case that when one is up (or down) the other is down (or up) ... and that accounts for the small correlation of 2.91%. >Yeah, so does this work for other stocks? Well, we'd really need a Mutual Fund that follows a single stock ... and that ain't easy to come by. But a basket of stocks, like an index, that's possible, eh?

Figure 4

For example here's the S&P 500:

and the TSE

>And the distribution of returns - from open to open and ...? And close to close? Like so (for the TSE daily returns over the past five years ... some 1250 returns) which shows the number of returns in each small interval from -5.0% to 4.5% and we note that the distribution is pretty similar. >As you'd expect, eh? Yes.

Here's some more, with different time intervals:

>And are you trading every day?
Uh ... well, we're rebalancing every day. That'd cost a bundle, eh?

>Yeah. And will it work in the future as it has in the past?
Uh ... I'd check but ... uh, it's broke.

Motivated by a suggestion of Rob G, here's a neat scenario to illustrate this remarkable strategy

• We earlier got this result (assuming assets A and B with Mean Returns S and B and Standard Deviations P and Q:

  for Maximum Annualized Return	Percentage Asset A = (S - B + Q2 - r P Q) / (P2+Q2 - 2 r P Q)   assuming this lies between 0 and 1
  for Minimum Volatility	Percentage Asset A = (Q - r*P) Q / (P2 + Q2- 2 r P Q)   assuming this lies between 0 and 1

• To simultaneously minimize Volatility and maximize Annualized Return, we need S = B. (That is, the same Mean Return.)
• So any two portfolios (or assets) with the same Mean Return (call it M) can be mixed to provide a portfolio with minimum Volatility and maximum annualized Return.
• We adopt the (approximate) formula: R = M - (1/2)SD² that relates the Annualized Return to the Mean and Standard Deviation.
• On a R vs SD plane we plot this relation:
  
• Our two portfolios (or assets) are represented by two points on this graph (since they're assumed to have the same Mean return M).
  Of course, we now need to determine some "optimal" mix of assets A and B !!
  But, although SD may change, any mix will have the same Mean Return M so will lie on this curve !!
• However, there's also this thing called the Efficient Frontier which plots Mean Return vs Standard Deviation.
  
  "Risk" is often taken to be the Standard Deviation (for some unknown reason) so we can call this the Risk-return plane.
• Suppose the Efficient Frontier has the equation M = f(SD).
  Then (to make the vertical axis the Annualized Return R) we plot, instead, f(SD) - (1/2)SD² vs SD. That gives:
  
• Now we superimpose the first and last of these charts:
  

>And you really believe this stuff?
Uh ... not really, but others might. Rob & I are gonna make a fortune selling a CD with this strategy.
As Rob says:

"For $9.99 our experts will send you a free video showing how to find the
infinite number of previously unknown portfolios residing on the locus of wealth"

>Gimme a break ...