Stocks vs Bonds and Volatility :
Part I

I keep hearing (reading?) that one should diversify, holding both stocks
and bonds (or, maybe a Balanced Mutual Fund) in a portfolio. The idea is to reduce
the risk. (Of course, one is never certain what is meant by "risk":
see Risk stuff.)
In order to avoid a prolonged discussion of "What is risk?", we'll just consider
reducing the volatility ... for now.
Anyway, I first looked at two mutual funds: a Canadian Equity Fund and (managed by the same
company) a Bond Fund. Assuming that we are interested in avoiding too much volatility  as measured
by the Standard Deviation (SD)  we avoid equating volatility to risk (!) 
we might vary the percentage of Stocks and Bonds. The result (for
these particular mutual funds) is like so:
Note that as the percentage of stock fund increases, the gain increases
... and so does the volatility. One is tempted to conclude that, if we want to sleep soundly,
we should hold some Bonds in our portfolio.
If we ask:
"Pick a number from 1 to 10 which measures your tolerance to volatility, with 10 being
the most tolerant"
 Fig. 1

then we could probably pick a percentage (from 0 to 100) and suggest that that percentage
should be in stocks.
>How about a percentage = "100  (your age)", in stocks?
You've been reading too many comic books. Anyway ...
Fig. 2 
>Why mutual funds? Why not consider stocks & bonds? That's what you said above, when ...
Okay, let's consider the S&P500 index and ... let's see, an Index fund which follows the
Lehman Bros. Aggregate Bond Index (U.S. government and corporate bonds).
Again, as we increase
the percentage of stock, the gain and the volatility (or Standard Deviation) increases ...
sort of.
>Sort of?
Well, as you can see from Fig. 2, the volatility goes down at first, as you increase the stock
component, then it goes up.

>What's the graph of Gain vs Standard Deviation look like? When one
goes up so does the other. That means ...
Okay, for both the Equity + Bond Funds and S&P500 + Bond Index Fund, the graphs look like this:
Fig. 3
>Look at that! The wiggle! First down then up and, with enough stock, the graphs look like straight lines! Doesn't that surprise you?
Hmmm ... it surely does. A wiggle  and a linear relationship? Seems unlikely, but maybe that'll give
me something to work on ... mathematically speaking ...
>Please don't.
Here goes:
We assume that a stock share and a bond fund unit each cost $1.00 (to start) and we invest $1.00 in a portfolio
that consists of a fraction x of stock and the balance, 1x, in bonds. Our portfolio
is then worth x + (1x) = 1 dollar. Note that x and (1x)
also represent the original number of units (in our portfolio) of stocks and bonds, respectively.
Let's call the sequence of monthly stock and bond prices s_{0}, b_{0}
and s_{1}, b_{1} etc.
Our portfolio begins with stock worth s_{0} per share and
bonds worth b_{0} per unit, and, a month later, the stocks and bonds
are worth s_{1} and b_{1} etc. etc. ... and they're worth s_{n} and b_{n}
after n months when our portfolio of is worth:
P_{n} = xP_{n} + (1x)P_{n}
where xP_{n} and (1x)P_{n} are the stock and bond components.

At a price of s_{n} per share, xP_{n} dollars must correspond to
xP_{n}/s_{n} shares of stock
 At a price of
b_{n} dollars per unit, (1x)P_{n} dollars must correspond to
(1x)P_{n}/b_{n} units of bond fund.
We conclude that, when we rebalance
our portfolio each month (in order to maintain the x to (1x) stock to bond
ratio) we must have, after n months:
(1)  u = xP_{n}/s_{n} shares of stock 
(2)  v = (1x)P_{n}/b_{n} units of bond fund

(3)  P_{n} = us_{n} + vb_{n} (the value of our portfolio)

If, over the next month, the stock and bond prices change by dollar amounts
Δs_{n} and Δb_{n}
then our portfolio changes by a dollar amount
ΔP_{n}
= u Δs_{n} +
v Δb_{n}
so the fractional change (or monthly Gain) in our portfolio is:
ΔP_{n}/P_{n}
= u Δs_{n}/P_{n} +
v Δb_{n}/P_{n}
>I take it we can this multiply by 100 to get the percentage Gain.
You got it.
Now we substitute for u and v from Equations (1) and (2) above, and get:
ΔP_{n}/P_{n}
= x Δs_{n}/s_{n} +
(1x) Δb_{n}/b_{n}
For convenience, we'll call the monthly stock and bond gains G_{n} and g_{n}
so
(4)  G_{n} = Δs_{n}/s_{n} (the monthly stock Gain)

(5)  g_{n} = Δb_{n}/b_{n} (the monthly bond Gain)

(6)  ΔP_{n}/P_{n}
= xG_{n} +
(1x)g_{n} (the monthly portfolio Gain)

The Mean (or Average) of our monthly Gains, over n months, is then:
A_{n} = xA(G_{n}) + (1x)A(g_{n})
where A_{n} is our Average monthly portfolio Gain and A(G_{n}) and A(g_{n}) are the Means of the
monthly stock and bond Gains.
The Standard Deviation, for our Gains (over n months, namely SD_{n}) is then obtained from:
SD^{2}_{n} 
=(1/n)Σ{xG_{n}
+ (1x)g_{n}  A_{n}}^{2} 

=(1/n)Σ{x[G_{n}A(G_{n})]
+ (1x)[g_{n}  A(g_{n}]}^{2} 
where we've substituted A_{n} = xA(G_{n}) + (1x)A(g_{n}).
Now we do the squaring and get:
(7) ...... 
SD^{2}_{n} 
=x^{2}
{(1/n)Σ[G_{n}A(G_{n})]^{2}
}
+ (1x)^{2}
{(1/n)Σ[g_{n}A(g_{n})]^{2}
}
+ 2x(1x)
{(1/n)Σ[G_{n}A(G_{n})]
[g_{n}  A(g_{n})]}

The guys in {red} and {blue}
are just the squares of the Standard Deviations of the stock and bond portions of our
portfolio (over n months) ... and the guy in {green} is just
... uh ...
>Garbage?
Yeah, garbage ... sort of. (For more highly correlated portfolio components, see
Efficient Frontier.)
If we stare at the product [G_{n}A(G_{n})][g_{n}A(g_{n})], inside the
{green} summation, it's the deviation (from their Means) of
the stock and bond Gains. If they're not correlated
>(fat chance!) then we can ignore this last
summation since the positive and negative terms should cancel ... over the long run.
Anyway, we'll do that ... for the time being.
>Are Bonds and Stocks unrelated? If stock prices go up, don't bond
prices ...?
Here's a chart of the monthly returns: BondIndex vs the S&P500
You'll notice that they're lightly correlated ...
>Lightly correlated? Is that a technical term?


... and the blue line is the "best fit" regression line and its slope, namely 0.0254,
is called the beta^{*} and measures the relation between this BondIndex and the S&P500 and
we can see that it's small. In fact, it suggests that bonds may be expected to go down when stocks
go up ... on average ... over the long term ... sometimes, but not always ...
>Can't you say anything with any certainty?
Yes. If we were to plot S&P500 vs S&P500 the regression
line would have a slope of 1.000, meaning S&P500 was in perfect synch with the S&P500.
*Normally, one plots the excess
returns of two investments (here: BondIndex & SP500) over some riskfree return.
In fact, it's that negative correlation, between Bonds and Stocks, which is responsible for
the "wiggle" we saw earlier, in Fig.3. Adding some stocks to an allbond portfolio decreases
the volatility. That's because, when the bond gain goes up (or down) the stock component tends
to go down (or up). It keeps our portfolio gains in check. The portfolio gains don't vary as
widely as a portfolio with just bonds. The volatility is lessened.
>I assume you can prove that  mathematically.
Sure! In Equation (7) above we just find the derivative with respect to x at
x = 0 to show that it starts off negative if stock and bond
prices are related by
G_{n} = α + βg_{n}
and
A(G_{n}) = α + βA(g_{n})
with β (that's beta) stuck in as a negative number
so that ...
>Can we move on?
Okay.
Let's plot SD versus x, from the last of the equations above (namely Eq. 7) 
where we've replaced the summations by convenient symbols.
We get the magic formula:
SD = {
A x^{2} +
B (1x)^{2} +
Cx(1x)
}^{1/2}
If we pick the actual numbers for the S&P500 and Bond Index Fund (namely their Standard
Deviations and their dependence upon x = the percentage of stock), we get the green curve:
Recall from an earlier chart (Fig. 2), that the SDs are 1.9% (actually, 1.88%, for 0% stock)
and 3.8% (actually, 3.79%, for 100% stock).
If we now use these SDvalues for A and
B and guess at a wee value of
C which gives a reasonable fit (namely 0.015%),
then use our magic formula for SD, we get the red curve.
Our purpose here is to illustrate that the value of C
is rather small (compared to A and B).

Fig. 4

One interesting thing to note:
If we put C = 0, implying a complete lack of correlation
between stocks and bonds, then the Standard Deviation of the stock/bond combination is
SD = SQRT{
A x^{2} +
B (1x)^{2} } which is
smaller than either
SQRT{A}
or SQRT{B} so ...
>I assume you can prove that.
Yes. Do you want to see the proof?
>Definitely not!
Okay. Since P = SQRT{A}
and
Q = SQRT{B}
are the Standard Deviations of stocks and bonds, respectively, we note that a combination of
stocks and bonds has a smaller volatility (or Standard Deviation) than either.
>That's assuming that they are NOT correlated, right?
Yes. It's true if their returns are completely independent or, at least, if their returns have very small correlation.
>Which happens ... how often?
Rarely.
Fig. 5 
Okay, we'll just ignore the C "garbage" term and ...
>What!
We're assuming complete lack of correlation between stocks and bonds, we're working up a
mathematical model, an approximation, an estimate of the real world, a ...
>Okay ...
Okay, pay attention.
The above equation (with C = 0), can be rewritten like so:
SD^{2} =
A x^{2} +
B (1x)^{2}
and that's the equation of an hyperbola.

An example hyperbola is shown above,
where we've changed percentages into just plain fractions. Notice that, for larger stock
fractions, the curve approaches a straight line ... an asymptote.
>And if you didn't toss out that C term? What then?
It'd still be an hyperbola. But stare at Fig. 5a where I've stuck in the wee value
(namely 0.015%) for C. I've coloured the corresponding curve
red:
SD = {
A x^{2} +
B (1x)^{2} +
Cx(1x)
}^{1/2}
and I've superimposed upon it a
blue curve  the one which has
C = 0, namely:
SD = {
A x^{2} +
B (1x)^{2}
}^{1/2}
See? Can't hardly tell the difference. The pair just gives a purple curve (because
red and blue give
purple, eh?).
So, ignoring C (which is related to the correlation) is a reasonable thing to do.
 Fig. 5a

>That's neat, but what're those numbers 64 and 25 in the equation,
on the chart, in Fig. 5? Are they ...?
I said that was an example (with numbers that exaggerate the hyperbolic nature)
... though you'd get that curve with Stock and Bond SDs of 8 and 5 percent.
Remember, those are the squares of the SDs for stocks and bonds, respectively,
and the curve starts at SD = 5 and ends at SD=8 and 5^{2} = 25 and 8^{2} = ...
>Yeah, yeah, I get it. But what about SD versus Gain,
instead of SD vs Stock Fraction?
Before we get to that, do you notice anything familiar about the graph in Fig. 5a?
>No.
It says that, starting with a 100% bond portfolio, the volatility goes down as you start
adding some stock.
>Do you believe that?
I think it would depend upon the particular stock, the particular bond fund,
the state of the economy, interest rates, time of year, the weather in Bermuda, the price of a cup
of coffee, the ...
>May I interrupt, please? What if we rebalance just once a year instead of once a ...?
Fig. 6

If you wish to maintain a 40% stock portfolio, for example, and are determined to sell/buy
stock and buy/sell bonds, every year (instead of monthly), so the stock percentage remains 40%, you'd find that
the stock percentage varies between rebalancing  as you'd expect.
Fig. 6 shows this, where the stock percentage is bang on once a year because of the
rebalancing (see the green dots on the "40% stocks" graph), but wanders off, in between.
>Especially in '92 when stocks took off. The stock fraction really got cut!
Actually, I assume we'd rebalance at the beginning of each year and 1991 was
great for stocks and the rebalancing on Jan, 1992, influenced all of 1992.

>So ... does all this math stuff tell us what percentage we should
have ... in stocks and bonds?
And what about talking about Gains?
And all this is for a fixed number of months, ten year's worth.
What if you change ...?
Good questions.
Let's change the number of months.
The earlier graph (the blue one,
in Fig. 3) was over 120 months.
(It's shown here again, in blue). That's ten years.
Anyway, Fig. 7 shows that the relationship remains pretty well linear, above 40% stock.
 Fig. 7

Let's continue ...
>You ignored that C number. What if you leave it in. What if ...?
You mean SD = {
x^{2}A +
(1x)^{2}B +
Cx(1x)
}^{1/2} ?
Okay, we'll write:
SD^{2} =
x^{2}P^{2} +
(1x)^{2}Q^{2} +
2r x(1x)PQ
where P and Q are the Standard Deviations of the two assets and we've let
C = 2 r PQ. (As it turns out, r always lies between 1 and +1.)
Then SD^{2} goes from Q^{2} at x = 0 to P^{2} at x = 1.
Further, the SD^{2} curve is a parabola and its slope at x = 0 is 2Q^{2} + 2 r PQ = 2Q(rP  Q ).
Further, its slope at x = 1 is 2P^{2}  2 r PQ = 2P(P  r Q).
Suppose that P is larger than Q (as in Figure 8).
Note that, if r is zero ... or very small ... then the curve is decreasing at x = 0 but increasing at x = 1 (as in Figure 8).
However, if r is sufficiently large (greater than Q/P), then the slope
is positive throughout the range of x (namely from x = 0 to x = 1 ) ... and we don't get a minimum (as we did in Figure 4).
>Then r should be small ... but who IS that guy?
Who is r? He's the Pearson Correlation.
>Aah, now I see why you're talking about uncorrelated stocks and bonds ... so you can get a minimum, eh?
You got it
 Fig. 8

for Part II.
