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:
 A fraction x of our portfolio is devoted to the stock component (and y = 1  x to bonds).
 For the stock component, the Mean Annual Return and Standard Deviation are S and P.
 For the bond component, the Mean Annual Return and Standard Deviation are B and Q.
 The Pearson Correlation between the stock and bond returns is r.
 With annual rebalancing the Mean Annual Return of our portfolio is: M = x S + y B.
 The Standard Deviation SD of our portfolio is given by: SD^{2} =
x^{2}P^{2}+y^{2}Q^{2}+2 x y r P Q
Note that, as a function of x, this describes a parabola that opens UP
 We take as our Annualized Portfolio Return: R = M  (1/2)SD^{2}
See the approximation described in AM vs GM
 The Annualized Portfolio Return is then very nearly
R = x S + y B  (1/2){x^{2}P^{2}+y^{2}Q^{2}+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)P^{2} = B  (1/2)Q^{2}
>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 + Q^{2}  r P Q) / (P^{2}+Q^{2}  2 r P Q)
assuming this lies between 0 and 1

for Minimum Volatility  Percentage Stock = (Q  r*P) Q / (P^{2} + Q^{2} 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:
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
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:
SD^{2} =
x^{2}P^{2}+(1x)^{2}P^{2}+2 r x (1x) r P^{2} = 2P^{2}{ (1r)x^{2}  (1r)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 + (1x)B  (1/2)SD^{2} = S  P^{2}{ (1r)x^{2}  (1r)x + 1/2 }
and, although it gives R = S  (1/2)P^{2} 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 5050 split than 100% of either.
>zzzZZZ
Can you believe that?
>zzzZZZ
for a continuation
