Stocks and Bonds       ... what Ratio ?

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

>Don't you always go on and ...
Pay attention. The earlier tutorial is here, but, although I talked about "what mix" to minimize standard deviation (=Volatility) or volatility per unit gain and there were pictures and ...

I never discussed the effect on annualized gain if you added bonds to your portfolio.

>You must have, 'cause I remember your going on and on ...
Okay, here's what we'll do ...

>Wait! Before you do this math mumbo jumbo, has a bond component done any good, in the past?
 Okay, as a measure of the efficacy of including some bonds, let's start with some portfolio invested in the S&P 500 and Government Long Bonds and we withdraw 5% each year, this amount increasing with inflation ... and we use the actual annual returns for the two components selected at random from the annual returns for the years 1928 to 2000 and, for inflation, we use a 40-year sequence of inflation rates (the 40-year sequence chosen from the same time period: 1928 - 2000) and we do this a thousand times, a la Monte Carlo, and we see how often our portfolio would have survived for 40 years and ... >A picture is worth a thousand ... Okay. Check it out Note, too, that if we used the actual returns from 1950 to 2000, in the order in which they actually occurred (as well as inflation rates), and we tracked the worst rate of withdrawal (greater than which our portfolio would NOT have lasted 30 years), we'd get the second chart which shows that greater Bond components gave higher rates of withdrawal.

>So 10% bonds is good, eh?
Can I continue?
>Be my guest.
Okay, here's what we'll do ...

• The Mean (annual) Stock Return is S and the Mean Bond Return is B.
• The Standard Deviation (of annual returns) for the Stock is P and for bonds it's Q.
• A fraction x of our portfolio is devoted to Stocks, the balance y = 1 - x is for Bonds.
• The Mean Return of our portfolio is then
[1]     R = x S + y B (assuming annual rebalancing).
• The Standard Deviation of our portfolio is then given by
[2]     V2 = x2P2+y2Q2+2 x y r P Q
where r is the (Pearson) correlation between our Stocks and our Bonds.
• We can estimate our annualized return using the Magic Formula:
[3]     Annualized Return = Mean Return - Volatility2/2
• We stick in the Mean Return, R, from [1] and the Volatility, V, from [2] and we get:
[4]     Annualized Return = x S + y B - {x2P2+y2Q2+2 x y r P Q}/2
• We can now put y = 1 - x and recognize [4] as the equation of a parabola which has a maximum ... somewhere:
[5]     Annualized Return = a + bx - c x2   where
a = B-Q2, b = S-B+Q2-rPQ and c = (1/2)(P2+Q2 - 2rPQ)
• The maximum of a + bx - c x2 occurs at x = b/2c which, for us means:
[6]     "Best" Stock Fraction = {S-B+Q2- rPQ} / {P2+Q2 - 2rPQ}
• The optimal bond component is then calculated from y = 1 - x:
[7]     "Best" Bond Fraction = 1 - {S-B+Q2- rPQ} / {P2+Q2 - 2rPQ}

>zzzZZZ
Okay, here are some pictures, where the maximum annualized gain (in the range of 0% to 100% bonds) is the big dot:

>zzzZZZ
Here! Play with this ... but remember that our parabola could have a maximum at a negative x-value and that just means that, in the range "0" to "1" (or 0% to 100%) we'd choose 0% and also the volatility will be out in left field if the maximum is negative and the answer(s) are rounded to the nearest integer and we're using that estimate [3] for the annualized return and ... well, play:

 Average Stock Return = % Average Bond Return = % Stock Volatility = % Bond Volatility = % Pearson Correlation = % "Best" Bond Component = %

Or, if you'd prefer an online Excel spreadsheet ... click!

In place of [3], above, we could use what I think to be a better approximation and get a somewhat different result, like so:

•     [3a]     (1+Annualized Return)2 = (1+Mean Return)2 - Volatility2
• We stick in the Mean Return, R, from [1] and the Volatility, V, from [2] and we get:
[4a]     (1+Annualized Return)2 = (1+x S + y B)2 - {x2P2+y2Q2+2 x y r P Q}2
• We put y = 1 - x and recognize the right-side of [4a] as a 4th degree polynomial %@\$\$%&#
• That explains why we used the simpler estimate [3] ... as math-types are wont to do
However, there's a faint gray line in the chart (when you do the online spreadsheet thing).
That's the 4th degree polynomial ...

>Huh? I can't see it.
Stick in a 50% stock volatility, then you'll see it.

>Okay, but that equation [3] is pretty weird, eh?
Well, consider this:

(1+x)(1-x) = 1-x2 is
[3] Annualized Return = Mean Return - (1/2)Volatility2
in disguise! For two returns x and -x we have
Mean Return = x + (-x) = 0 and
Volatility2 = x2
Now we need to know that SQRT(1+z) = 1 + (1/2) z (approximately, for small z)
Then
Annualized Return = SQRT[ (1+x)(1-x) ] - 1 = SQRT[ 1 - x2 ] - 1
= [ 1 - (1/2) x2 ] - 1 (approximately) = 0 - (1/2)x2
which is equation [3], eh?

>zzzZZZ

Or, you can play with this online Excel spreadsheet ... click!
to see the difference between the approximations [3] and [3A], in calculating the Annualized Return.