From: John Conover <john@email.johncon.com>
Subject: portfolio management
Date: Fri, 18 Sep 1998 22:20:44 -0700
Someone ask for the formalities of portfolio managment, considering
the number of stocks concurrently invested in, and the amount of time
they are held. It is really quite simple, and depends on the formula:
avg
--- + 1
rms
P = -------
2
where avg is the average of the marginal increments of an stock's
price, (also known as the fractional returns, or just returns,) rms is
the root mean square, of the marginal increments, (also known as the
standard deviation, or the square root of the variance, which is a
measure of the stock's volatility,) and P is the probability that the
stock's price will make an up movement on any given day, (also known
as the Shannon probability, or likelihood of an up movement.) You can
use a spreadsheet to calculate all three values. The prescription is,
for each and every day, subtract the stock's value on the previous
day, from the present value, and divide by the previous value. Add all
these up, and divide by the number of days to get avg. To get rms, Add
the square of all these up, divide by the number of days, and take the
square root of the result. To get P, use the formula. The gain in value
of a stock can be found by the formula:
P (1 - P)
G = (1 + rms) * (1 - rms)
There is a formula, rms = 2P - 1, which if satisfied, will result in
optimal and maximal growth. (All three of these formulas are derived
in the tsinvest(1) manual page.) If rms > 2P - 1, the growth will be
very large-for a while-and then "crash" down, (Netscape is an
example.) If rms < 2P - 1, then the growth will be stilted, (Boeing is
an example.) Both solutions will yield less long term growth than the
optimal, where rms = 2P - 1.
The only other things you need to know is that root mean square
values, (like rms,) add root mean square, ie., result = sqrt (a^2 +
b^2 + ...), and linear values, (like avg,) add linearly, ie., result =
a + b + ... The last thing is that stock prices are ergotic, which is
a technical name that means if you invest in many stocks concurrently,
it is identical to investing in one stock for many days; as long as
the number of stocks and the number of days are the same, the
probabilities will be the same.
So, if n many stocks, with identical characteristics, (ie., they all
have the same avg and rms,) are invested in at the same time:
n * avg avg
-------------- + 1 sqrt (n) --- + 1
sqrt (n) * rms rms
P(n) = ------------------ = ----------------
2 2
What this means is that investing in many stocks at the same time
increases the likelihood of your portfolio making an up movement. In
point of fact, the likelihood of a portfolio up movement is greater
than the likelihood of any of the stocks that make up the portfolio
making an up movement.
For example, a typical stock in the US equity markets has a P of 0.51,
an rms of 0.02, and an avg of 0.0004. (What this means is that, on
average, in a hundred days, the stock will make up movements for 51
days, down movements for 49, and the day-to-day fluctuations will be
less than 2% for 68.27 of those days-since one standard deviation is
68.27%. For 15.86 days, the stock will have increases that are greater
than 2%, and 15.86 days, it will have losses that are greater than
2%.) Or:
0.0004
------ + 1
0.02
P = ---------- = 0.51
2
Lets suppose we hold 10 identical stocks at the same time in the
portfolio:
0.0004
sqrt (10) * ------ + 1
0.02
P = ---------------------- = 0.5316228
2
and:
rms = 0.02 / sqrt (10) = 0.006324555
So, the portfolio chance of an up movement would have been increased
by a little over 2%, and the volatility would have been have been
decreased by about a little over a factor of 3.
Working through the gain in values for both a single stock and the
portfolio:
0.51 (1 - 0.51)
G = (1 + 0.02) * (1 - 0.02) = 1.0002
compounded daily for the stock, and:
0.5316228 (1 - 0.5316228)
G = (1 + 0.006324555) * (1 - 0.006324555)
= 1.00038
for the portfolio. Note that the growth in value of a portfolio with
ten stocks is almost double the growth in value of a portfolio with
one stock, (in point of fact, the growth in value of a portfolio with
10 stocks is larger than the sum total of the growths of the
individual stocks, and the volatility of the portfolio is about a
third of any stock in the portfolio!)
Suppose the stocks in the portfolio are held for H many days. We know
that avg adds linearly, and rms root mean square, so:
H * avg avg
-------------- + 1 sqrt (H) --- + 1
sqrt (H) * rms rms
P(H) = ------------------ = ----------------
2 2
and:
rms
So, if we were to hold a single one of our typical stocks for 10 days,
it would be the same as holding 10 stocks for one day, provided we
measured our portfolio characteristics on 10 day intervals-the rms of
the portfolio would be sqrt (H) * rms.
Combining P(n) and P(H) into a single formula:
avg
sqrt (n * H) --- + 1
rms
P(n,H) = --------------------
2
where we hold n many stocks, for H many days in the portfolio, (note
that we measure/administer/rebalance the portfolio on intervals of H
many days,) and the portfolio's rms would be sqrt (H / n) times as
large as the rms for any of the identical n many stocks in the
portfolio.
But there is a problem; the portfolio will not be volatile enough for
maximum growth. Why? Because investors in the US markets are risk
adverse, (we can measure exactly how risk adverse because when
optimal, rms = 2P - 1, which results in rms = sqrt (avg), and metrics
on the stock exchanges indicates that rms is too small, ie., investors
are reducing volatility at the expense of long term growth.) There is
a solution to this problem, and that is to reduce H to less than
unity, ie., manage the portfolio by inter-day trading, (this is what
the programmed traders do.) This is not a viable alternative for most
investors, so most investors will have to exercise the remaining
alternative, which is to minimize long term risk-and we know how to do
that-just make P unity, ie., a probability of 100% on any given day of
an up movement in the portfolio's value. And, that occurs where:
avg * sqrt (n * H) = rms
or, with our values of a typical stock:
0.0004 * sqrt (n * H) = 0.02
or n * H = 2500. Since we determined that holding 10 stocks at the
same time in our portfolio was very near optimal, H would be 250, or
about a calendar year, (there are 253 trading days in a calendar
year.)
And there you have it. For a portfolio of typical stocks in the US
equity markets, your portfolio should have a minimum of 10 stocks at
all times, (more would be better, but the advantages of more become
increasingly insignificant, and the administrative costs probably
offset any advantage,) and plan to hold each of the stocks, (unless
there is compelling reasons otherwise,) for at least a year. At the
beginning of each year, re-balance the portfolio such that there is
equal investment in each stock.
If you do so, you will end up with just under twice the portfolio
value that you think you will, every year, and on average, done
almost, but not quite, as well as the market indices.
John
BTW, 10 stocks for the long term, and re-balancing annually is an old
broker's empirical adage. It has been around Wall Street since the
turn of the century. All I did was formalize the adage. The scenario
is formidable-in the long run, it will come within a half of one
percent per year of the growth of the indices-and over the last
quarter century, would have beat every, without exception, mutual fund
in the US. The programmed traders running inter-day trading scenarios
do better. But only modestly so. (It is very easy to simulate BTW,
just get the NYSE's historical CDs, and run them through the tsinvest
program using the -d1 option on your PC. Kind of comforting when
theory, adage, and empirical results agree to many decimal places.)
--
John Conover, john@email.johncon.com, http://www.johncon.com/