Stocks and Bonds       ... again

Once upon a time I wrote a tutorial on Stocks and Bonds and I went on and on and ...

>Haven't you said that before?
Yes, but I ran across a debate concerning whether annual rebalancing was to decrease volatility or increase returns.
When I looked at those earlier tutorials I saw that I had missed something that I should point out.
I also found a wee error that I corrected.

Here we want to recall the following results:

1. A fraction x of our portfolio is devoted to the stock component (and y = 1 - x to bonds).
2. For the stock component, the Mean Annual Return and Standard Deviation are S and P.
3. For the bond component, the Mean Annual Return and Standard Deviation are B and Q.
4. The Pearson Correlation between the stock and bond returns is r.
5. With annual rebalancing the Mean Annual Return of our portfolio is:       M = x S + y B.
6. The Standard Deviation SD of our portfolio is given by:       SD2 = x2P2+y2Q2+2 x y r P Q
Note that, as a function of x, this describes a parabola that opens UP
7. We take as our Annualized Portfolio Return:       R = M - (1/2)SD2
See the approximation described in AM vs GM
8. The Annualized Portfolio Return is then   very nearly   R = x S + y B - (1/2){x2P2+y2Q2+2 x y r P Q}
Note that, as a function of x, this describes a parabola that opens DOWN
Note, too, that if S = B and P = Q and r = 1 then the Annualized Portfolio Return is (very nearly) S - (1/2)P2 = B - (1/2)Q2

>And your point is?
My point is that the fractions devoted to stocks or bonds depends upon whether you want maximum annualized return or minimum volatility and the "best" stock/bond ratio will depend upon the various parameters and ...

>So what's "best"? Don't you have a magic formula?
Yes, it's here:
 for Maximum Annualized Return Percentage Stock = (S - B + Q2 - r P Q) / (P2+Q2 - 2 r P Q)   assuming this lies between 0 and 1 for Minimum Volatility Percentage Stock = (Q - r*P) Q / (P2 + Q2- 2 r P Q)   assuming this lies between 0 and 1

>Picture?
Yes. The following charts show the effect of changing the volatility of the stock component:

Here's a calculator:
 Average Stock Return S = % Average Bond Return B= % Stock Volatility P = % Bond Volatility Q = % Pearson Correlation r = % "Best" Stock Component (to maximize your Annualized return) = %   (S - B+Q2 - r P Q) / (P2+Q2 - 2 r P Q)
Note that, if S = B and P = Q, then the "best" allocation is 50% of each!

Then, of course, you may want to sleep well ... and minimize the volatility

 Average Stock Return S = % Average Bond Return B= % Stock Volatility P = % Bond Volatility Q = % Pearson Correlation r = % "Best" Stock Component (to minimize your Portfolio volatility) = %   (Q - r P) Q / (P2+Q2 - 2 r P Q)
Note that, if S = B and P = Q, then the "best" allocation is 50% of each!

>And how about minimizing volatility and maximizing return ... at the same time?
Yeah, possible ... if the Mean Returns are the same. Try that on the two calculators, above.

>Huh?
Try something like S = B = 10% and P = 30%, Q = 20%.

>And you believe all this stuff?
Of course! Don't you? I mean ... mathematics is an exact science, right? It can't lie! You can place your trust in a mathematical ...

>zzzZZZ
 Wait! Here's something interesting. Suppose we have two asset classes, like stocks and bonds, which have identical volatilities ... so P = Q. Then our Volatility (from 6, in the list above) would be: SD2 = x2P2+(1-x)2P2+2 r x (1-x) r P2 = 2P2{ (1-r)x2 - (1-r)x + 1/2 } and, although it gives SD = P when x = 0 or x = 1 (as you'd expect ... 100% asset A or 100% asset B), it has a minimum at x = 0.5 for correlations between 0% and 100%
 Suppose, too, that the Mean Returns are identical for the two asset. That is: S = B (as well as P = Q). Then we can do a similar thing with the Annualized Return approximation (from 8, above). R = xS + (1-x)B - (1/2)SD2 = S - P2{ (1-r)x2 - (1-r)x + 1/2 } and, although it gives R = S - (1/2)P2 when x = 0 or x = 1 (as you'd expect), it has a maximum at x = 0.5 for correlations between 0% and 100% ... similar to this
Can you believe that?
For assets with identical parameters you do better with a 50-50 split than 100% of either.
>zzzZZZ
Can you believe that?
>zzzZZZ