Asset Concentration
... a continuation of Part II 
We stare at various asset classes, like so
and decide (for example) to concentrate our portfolio on Small Cap Value (SV) stocks
in the hope that we can capture the returns associated with this class.
Figure 1 shows that, for the period since 1928 where
the Mean Return was 18.8% and the SD was 32.0% and the Annualized return was 14.2%,
we note that, very nearly (see
Annualized vs Mean):
(1+Annualized Return)^{2} =
(1+Mean Return)^{2}  SD^{2}
>Exactly?
Well, this Magic Formula is an approximation, but let's
put Mean Return = 0.188 (corresponding to 18.8%) and SD = 0.320 (corresponding
to 32.0%) and we'd get:
(1+Mean Return)^{2}  SD^{2}
= 1.188^{2}  0.320^{2} = 1.309
 Figure 1

so we'll look at SQRT(1.309) = 1.144 and compare to the exact 1+Annualized Return
which is 1.142 so the approximation would give 14.4% ...
>Instead of the exact Annualized Return of 14.2%, right?
Right. The approximation is pretty good, eh? We're using:
Annualized Return = SQRT[(1+Mean Return)^{2}  SD^{2}]  1
... approximately
>What's this got to do with ...?
Asset Concentration? Good question.
What an investor wants is a good Annualized Return. Forget Standard Deviation or Mean Return.
She wants Annualized Return.
>But they're related, right? You just said so.
Yes, so consider this:
Don't avoid investments just because they have large Volatility
(Volatility = SD^{2}) because they could be associated with large Annualized Returns.
For example, if we're happy with an 8% Annualized return, then we could invest in a stock
(or a basket of stocks) which has a SD of 10% or 20% or 30% or 40% ... so long as the
Mean Return increases appropriately, as shown in Figure 2.
>Like 8.5% or 9.8% or 12.1 % or ...
Yes.
>What's this got to do with Asset Concentration?
 Figure 2

Figure 3

Well, let's compare investments that have the same Annualized Return (according to the
Magic Formula), but some have
(High Mean Return & High Volatility) and others have (Low Mean Return & Low Volatility).
If we assume (for convenience) that the returns are Normally distributed, then we'd expect
10% of returns to exceed Mean + 1.28*SD and we'd expect
25% of returns to exceed Mean + 0.67 SD and ...
>And 40% to exceed Mean + 0.25 SD ... as per blue dots in Figure 3?
Yes. So now, suppose we're willing to accept the higher volatility for the
40% chance to
get those extra returns, above Mean + 0.25 SD.

We have, for a given, prescribed Annualized Return, say A, that the Mean
(M) and Standard Deviation (S) satisfy (approximately, according to
our Magic Formula):
(1+M)^{2} = C + S^{2}
where C = (1+A)^{2}
For various Cvalues this describes a collection of hyperbolas ... as shown in Figure 2.
This extra Annualized Return depends upon our choice of Mean and Standard Deviation for our
investments, but it we consider just those assets which give, say, an 8% Annualized
Return then the pertinent curve is shown in red, in Figure 2.
At S = 20%, then M = 9.8% and this combination would give an
Annualized Return of 8%.
>Approximately!
Yes, approximately, according to our Magic Formula.
 Figure 4

However, if we increase the Mean and Standard Deviation of our investments  maintaining the
expected Annualized Return of 8% (as per the red curve in
Figure 2)  then M + 0.25 S increases so the return we might expect to get,
40% of the time, increases dramatically ... as in Figure 4.
>And our chance of losing bigtime is greater too, eh?
Sure, but ...
>And I ask again: What's this got to do with Asset Concentration?
Well, we need to pick some percentage, like 40%, and
decide what Asset Class is our best choice ... in order to achieve these extra returns,
40% of the time. Once we identify the Asset Class we put a
BIG fraction of our money into that Class. That's ...
>Asset Concentration?
Why ... uh, yes.
>And does that mean lots of Small Cap Value?
We'll see. In the meantime, stare raptly at these pictures
(where SV means Small Cap Value and
LG means Large Cap Growth and TSM = Total Stock Market and ...
>Yeah, I get it, and I suppose you're talking about "passive" investing ...
Yes, where our portfolio mimics some "Index" which follows some particular asset class and
we expect to get the returns associated with that restricted, concentrated Index and ...
>So, where are the pictures?
and, more recent stuff:
>So which do you like best? SV or SG or ...?
Well, let's look at which asset class gives the largest value of
Mean + 0.25 Standard Deviation = M+S/4.
We're not frightened by large volatility but would like to be reassured, when
the volatility IS large, that there's a good chance to make extra gains so we look at
Mean + 0.25 Standard Deviation to see how large that is and
for the last fifty years, we get figure 5:
>What's the light blue line mean?
We plot a bunch of lines, like:
M+S/4 = 10%
M+S/4 = 15%
M+S/4 = 20% etc..
These lines move UP as the percentage increases: 10%, 15%, 20%, etc.
The last line (which doesn't lie above the points) is
M+S/4 = 25.9%
... and it passes through SV as shown in Figures 5 & 6.
>So that means SV is best?
Well, yes, for the past fifty years, but, for the past ten years or so ...
it's been Large cap Growth!
 Figure 5
Figure 6

>So what's your conclusion?
Uh ... gimme some time to cogitate. In the meantime, see if you can follow the gyrations
of Mean & Standard Deviation (over the previous 10 years) for Small Cap Value:
