Download Get Rich Slowly, Almost Surely Modeling stock prices

Get Rich Slowly, Almost Surely
DAVID BENKO
University of South Alabama
Department of Mathematics and Statistics, ILB 325
Mobile, AL 36688
[email protected]
AMS (MSC2010) Classification Number: 91G99
You see a Martian walking on the street. What do you do?
i) seek shelter
ii) take some pictures
iii) introduce yourself and offer him/her/it a ride
iv) call your broker to buy lots of stocks.
In this article, we will discuss the risk associated with stock investments. The above unusual question
will be answered, too.
The recent volatility of the stock market has left most investors stunned. In 1952 only 4% of US
population owned stocks but today about half of the American households own stocks. Despite this,
most investors have no education in either economics or the stock market. They instead rely on the
recommendations of their financial advisors. The adviser typically shows the client a chart of the stock
market’s history, commenting that short-term volatility is normal, but stocks have been going up in the
long run. The chart is very convincing and it is easy to accept the “buy-and-hold” argument. The
problem when looking at that chart is twofold:
a) Stock prices behave very much like a random walk, and therefore future performance is independent
of the past. As Warren Buffett once said: “If past history was all there was to the game, the richest
people would be librarians.”
b) Over the past 20 years, the stock price of Microsoft has increased more than any other. Hence,
showing the chart of Microsoft as a typical example would be misleading. By the same logic, looking only
at the chart of the US stock market is meaningless. In the past century the United States was one of the
fastest growing economies in the world. Other countries were not so lucky, as their economies grew more
slowly.
Modeling stock prices
The simplest way to think about a stock price is to assume that it represents the value of the company
per share in 30 years. That is, F uture V alue = N · P , where N is the number of shares outstanding and
P is the current share price. We assume that in 30 years the company will go bankrupt, its assets will
be sold and stock holders will get the proceedings. (This is a realistic assumption, since most companies
do go bankrupt after being in business for a few decades.) Because 30 years is a long time, even a small
change in the annual growth expectation of the company will have a huge impact on its future value and,
therefore, a huge impact on the current share price. For example, if the growth expectation is changing
from 3% to 3.5%, then the share price will change from P to (1.035/1.03)30 P , which is a jump of about
16%.
This explains why sometimes stocks can move up and down so rapidly. For example the stock price of
a large steel manufacturing company was around $200 in June, 2008, and four month later it had fallen
to $30. (A more sophisticated method is the discounted cash flow model [1, p. 15] where future dividends
or cash flows are used instead of the future value of the company.)
In order to gain a better understanding of the dynamics of the stock market, we will need to consider a
model that incorporates randomness. In this model w(t) will denote a Brownian motion. (See Figure 1).
The reader may think of it as a random continuous function with the property that for any fixed√t ≥ 0,
w(t) is a random variable having normal distribution with mean zero and standard deviation t. In
particular w(0) = 0. (To get familiar with the Brownian motion, one may think of the discrete Brownian
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motion: each year we flip a coin and if it is head we win $1 otherwise we lose $1. If
√ w(n) denotes the
total gain during the first n years, then w(n) has mean zero and standard deviation n.)
Figure 1: A possible realization of w(t), generated by a computer
Let us consider a stock which is worth $1 at time t = 0. A simple but frequently used model for the
stock price is the geometric Brownian motion:
P (t) = eµt+σw(t) ,
(see [7, p. 57]) where t is the time in years, µ is a constant, called drift factor and σ is the volatility of
the stock price. That is, σ is the standard deviation of the change in log P (t) during one year. Although
this model is simple, it is very natural and it is also used in the celebrated Black-Scholes option pricing
model.
√
2
Since for fixed t the random variable w(t) has density function e−z /(2t) / 2πt, z ∈ R, we can calculate
the expected value of P (t) as follows:
E(P (t)) = E(eµt+σw(t) ) =
µt
=e e
(tσ)2
2t
Z
R
(z−tσ)2
2t
e−
√
2πt
z2
e− 2t
eµt+σz √
dz
2πt
R
Z
dz = e(µ+
σ2
2
)t
= eRt ,
where we introduced
σ2
.
2
This means that at time t our average gain is eRt − 1, since our original investment was 1 dollar.
A bank offering interest rate r compounded continuously would give us ert − 1 gain which is (almost)
risk free. The treasury bill with 3 month maturity is considered to be a risk free bond. We will use the
letter r for the interest rate of the treasury bill. Investors wish to be compensated for taking risk when
buying stocks, so R > r must hold - at least theoretically. (Since we do not know the exact value of R,
it may happen that R ≤ r, in this case the stock is overvalued.)
The difference R − r(> 0) is called the equity risk premium.
R := µ +
Everything is relative!
If our investment was $1 in a stock and the stock price drops, say, to $0.80, it is very common that the
broker calls it a “paper loss” and he claims that it is not a real loss unless we sell the stock. Nothing is
further from the truth. Just think of a casino: if you lost some chips it is a real loss, even though you
did not cash out!
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True, if you don’t sell and wait t years, the stock may come back to $1 and you may feel satisfied
that you waited. But keep in mind that if you took your losses and put the 80 cents in treasuries, after
t years you would have 0.80ert dollars, which can be more than $1 depending on how large t is.
It is clear now that it is not enough to calculate our gain on an investment. The real question is if
our investment is growing better or worse than the risk free treasury.
Definition The value of the investment of $1 at time t relative to the risk free treasury bill is
Rel(t) =
P (t)
= e(µ−r)t+σw(t) .
ert
We can think of Rel(t) as our “normalized money” (see Figure 2). At time t if Rel(t) < 1 we will
simply say that we lost money, and if Rel(t) > 1 we will simply say that we gained money. Similarly, if,
say, Rel(t) = 0.5 we will simply say that we lost half of our money. We will omit to say “relative to the risk
free treasury bill” since ert is the money we deserve risk free, so that must be the base of comparison - and
not the money lying under the pillow earning no interest. The expected value is E(Rel(t)) = e(R−r)t > 1,
because we assumed that R > r.
Using R = µ + σ 2 /2, we have µ − r = R − r − σ 2 /2, and so
Rel(t) = e(R−r−σ
2
/2)t+σw(t)
.
2
Observe that Rel(t) has drift R − r − σ /2. Its importance will be discussed later on.
Figure 2: A possible outcome during 20 years
More time, less risk?
Let 0 < t be a fixed time, our investment horizon. (E.g. t = 20 years.) Let us find the probability that
Rel(t) is in the interval (a, b), where 0 ≤ a ≤ b ≤ +∞.
P (a < Rel(t) < b) = P (a < e(R−r−σ
2
/2)t+σw(t)
< b)
ln a − (R − r − σ 2 /2)t
w(t)
ln b − (R − r − σ 2 /2)t √
√
=P
< √ <
.
σ t
t
σ t
√
√
2
Here w(t)/ t is a random variable with standard normal distribution. Its density is e−x /2 / 2π, so the
probability is
Z
−x2
1
P (a < Rel(t) < b) = √
e 2 dx,
2π I
3
2
/2)t ln b−(R−r−σ 2 /2)t
√
√
where the integration takes place on the interval I = ln a−(R−r−σ
.
,
σ t
σ t
Further, it follows that
Z
ln a − (R − r − σ 2 /2)t
−x2
1
√
P (a < Rel(t)) = √
,∞ ,
e 2 dx, where I =
σ t
2π I
and
Z
−x2
1
ln b − (R − r − σ 2 /2)t √
P (Rel(t) < b) = √
.
(1)
e 2 dx, where I = − ∞,
σ t
2π I
In order to apply these formulas we need to find a proper value for the risk premium R − r and
the volatility σ. For the Standard and Poors 500 index (which contains 500 large companies) historical
data is available for both the risk premium and volatility. In addition, diversification is important when
investing. So instead of a single stock we will assume that we invest in the Standard and Poors 500 index.
In real life R, σ and even r are not constant values. Recently r has been extremely low: r = 0.001
(i.e., 0.1 percent), but we cannot assume that this will be typical in the future. In the United States
during 1926−2005 the stock index of large companies grew with an annualized rate of R = 0.104 (10.4%!)
on average, and on average r = 0.037 was the treasury rate [2, p. 35]. This would give R − r = 0.067.
However during 1802-1925 the risk premium was only 2.6%, [6, p. 18].
In the past century historical market returns in the world averaged 3.8% plus the rate of inflation
[2, p. 343 and 348]. If we assume that for the world r was about the same as the inflation, we get an
estimate for the historical risk premium of the world:
R − r = 0.038.
We will use this number in our calculations, since even between economists there is a vigorous debate
about the value of the equity risk premium.
It is also reasonable to set
σ = 0.20,
since this was the (annualized) standard deviation of the Standard and Poors 500 stock index in the
past 10 years. (As a curiosity we mention that during October-November, 2008, the volatility was a
remarkable 0.76!)
Financial advisors often claim that we should ignore the volatility of stocks because stocks beat bonds
on the long run; at least historically they did during almost any 20-year period, [6, p. 28]. Let us examine
if this claim is supported by our model.
Let t = 20 years. The expected value E(Rel(20)) = e0.038·20 = 2.14 suggests that stocks are great
investments since we will have on average more than twice as much money as we would with risk-free
Treasury bills. But this is only an average. By (1),
Z v
−x2
1
P (Rel(20) < 1) = √
e 2 dx,
2π −∞
ln 1 − (0.03 − 0.202 /2)20
√
= −0.4025,
0.20 20
so the probability of losing money is P (Rel(20) < 1) = 0.34 = 34%. (We used the NORMSDIST(-0.4025)
function in Excel to evaluate the above integral.)
We also calculated the expected value E(Rel(t)) and the probabilities P (Rel(t) ≤ 0.5), P (Rel(t) < 1)
and P (Rel(t) ≥ 2) for the time periods t = 5, 20, 40 and 100 years, see below.
with v =
5 years
20 years
40 years
100 years
E(Rel(t))
1.21
2.14
4.57
44.70
P (Rel(t) ≤ 0.5)
0.04
0.12
0.13
0.11
P (Rel(t) < 1)
0.42
0.34
0.28
0.18
P (Rel(t) ≥ 2)
0.09
0.35
0.51
0.71
Observe that investing for a longer time does not really save us from the risk of losing money (even
though our expected value is increasing with time). In fact, it is interesting that P (Rel(t) ≤ 0.5) is
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increasing with t for a while - just the opposite one may suspect. Actually, it is increasing until t = 38.5
years and then it is decreasingly approaching to zero.
There is also some good news: the probability of losing money is decreasing with time, because
R v(t) −x2
R − r − σ 2 /2 = 0.018 is positive. Indeed, we have seen that P (Rel(t) < b) = √12π −∞ e 2 dx, where
v(t) =
R − r − σ 2 /2 √
ln b
√ −
t.
σ
σ t
(2)
For b = 1, v(t) is strictly decreasing and hence P (Rel(t) ≤ 1) is strictly decreasing, too.
Let now 0 < b < 1. One can also analyse P (Rel(t) < b) using v 0 (t). It turns out that P (Rel(t) < b)
is strictly increasing on (0, − ln b/(R − r − σ 2 /2)] and decreasing on [− ln b/(R − r − σ 2 /2), ∞). That is
how we got the 38.5 years mentioned above.
Can we swim against the drift?
With our choices R − r = 0.038 and σ = 0.20 we have R − r − σ 2 /2 = 0.018 > 0. But can the drift
R − r − σ 2 /2 be negative? Obviously if the risk premium R − r is less than 0.202 /2 = 0.02, then the drift
is negative. But even with R − r = 0.038 we can have a negative drift, if we invest in something which
have a high volatility.
From (2) we conclude: Let R − r − σ 2 /2 be negative and let 0 < b ≤ 1. Then P (Rel(t) ≤ b) is a
strictly increasing function. In other words, the more we wait the more likely it is to lose a certain portion
of our money. Moreover, this probability is approaching to 1 (!), since (2) implies that:
Observation. For any 0 < b,

if R − r − σ 2 /2 > 0
 0,
0.5,
if R − r − σ 2 /2 = 0
lim P (Rel(t) ≤ b) =
t→∞

1,
if R − r − σ 2 /2 < 0.
One can go even futher and show that if R − r − σ 2 /2 < 0, and c is small (namely, if 0 < c <
|R − r − σ 2 /2|), then P (limt→∞ [Rel(t)ect ] ≤ 1) = 1.
That is, Rel(t) tends to zero exponentially fast almost surely. (Here, almost surely means with
probability 1.) This implies that for large t with very high probability, Rel(t) will be almost zero.
However, with a small probability, Rel(t) will be huge, since our expected value E(Rel(t)) = e(R−r)t is
large. This very much resembles a lottery, where we have a tiny chance of winning a huge amount of
money, but we lose the price of the ticket almost certainly. (But for the lottery, the expected gain is
negative.)
Now let us consider a hypothetical stock market where market participants are investing their money
in a lump sum only once, and their time horizon is infinity. Even if the drift R − r − σ 2 /2 is slightly
negative, people will be bankrupt in the distant future (maybe in a million years) almost surely. But if
the drift is positive, almost surely they will be billionaires (or googolplexionaires) because almost surely
Rel(t) tends to infinity exponentially fast. In this hypothetical world the drift should be zero, making
the risk premium to be exactly σ 2 /2 = 2%!
Here is a discrete example to demonstrate another way that large volatility can cause lottery-like
behavior (if the risk premium is insufficient). Suppose we invest $100 for three years, and each year
we can either double our money or lose 90 percent of it, with the same probability. After one year our
balance will be either $10 or $200. After two years our balance will be $1 or $20 (if our previous balance
was $10), or $20 or $400 (if our previous balance was $200). Finally, after three years, our balance will
be one of the following, each with the same 1/8 probability: $0.1, $2, $2, $2, $40, $40, $40, or $800. In
one case, we have a huge $700 gain, and in the rest we have big losses. Although our expected gain is
$15.76, the final outcome is lottery-like.
The sure way to win: swim dynamically!
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So far we just considered the buy-and-hold strtegy: we invest $1 today and see what happens in the
future. Now let us allow dynamic strategies, so we are allowed to put money in stocks and also take it
out, any time as we wish. Our first impression is that it does not matter: if the drift R − r − σ 2 /2 is
negative, we are still losing money on the long run with high probability.
Suprisingly, it is not the case! Notice that for the buy-and-hold strategy,
Rel(t) = e(R−r−σ
2
/2)t+σw(t)
→ 0,
if and only if (R − r − σ 2 /2)t + σw(t) → 0. This gives an idea to play a strategy which will maximize
not E(Rel(T )) but the expected growth rate E(ln Rel(T )), where T is fixed (say, T = 20 years).
Also, if our total wealth is $1,000 and we win $200 on the stock market, we will be extreamly happy.
But if we have 1 million dollars, we shoul win $200,000 to have the same happiness. It follows that
happiness can be measured better by the logarithm of our wealth, and not our total wealth. In economics
they use the term “logarithmic utility function” for this.
Suppose we have wealth H and we invest P dollars in stocks and H − P dollars in bonds for one day.
We use the binomial model: if the market goes up (with 0.5 probability), we will have αP + h(H − P )
wealth, where α > 1, and if the market goes down, we will have βP + h(H − P ) wealth, where 0 < β < 1.
Here, h is a constant slightly bigger than 1, it reflects the interest rate for one day. The expected value
of the logarithm of our wealth is:
1
[ln(αP + h(H − P )) + ln(βP + h(H − P ))].
2
(3)
Now write P = ρH. If we plug this into (3), we see that we need to maximize the product
H 2 αρ + h(1 − ρ) βρ + h(1 − ρ)
for ρ ∈ [0, 1]. The answer we get for ρ is clearly independent of H.
Thus we have to play a constant-mix strategy: any time we should risk a certain portion of our money
in stocks, and the rest we keep in bonds. Say, 60% in stocks, 40% in bonds. Of course, we have to do
occasional rebalancing to keep the stock portion of our portfolio close to the optimum. But what is the
optimal portion ρ?
Dynamic/continuous rebalancing is when we rebalance every second or even more often. It is impossible to perfectly implement it in practice but stochastic calculus can be used to find our wealth at time
t:
2
H(t) = e((1−ρ)r+ρR−(ρσ) /2)t+ρσw(t) .
(4)
Notice that H(t) is a geometric Brownian motion. As before, our relative money is defined as: Rel(t) =
H(t)/ert . Finally, from (4),
1
E(ln Rel(t)) = ρ(R − r) − (ρσ)2 t,
(5)
2
since E(w(t)) = 0 for any fixed t. (5) is a quadratic equation in ρ and it is maximal if
ρ=
R−r
.
σ2
(6)
Of course, if this number is bigger than 1, then we invest 100% in stocks, so in fact, ρ = 1 in that case.
If we plug (6) back to (4), we get
1
Rel(t) = e 2 (
R−r 2
R−r
σ ) t+ σ w(t)
Rel(t) = e(R−r−σ
2
,
/2)t+σw(t)
,
if
R−r
≤ 1,
σ2
if
R−r
> 1,
σ2
But this is a big suprise because the drift is now either the positive number (1/2)((R − r)/σ)2 or
the positive number R − r − σ 2 /2 (> σ 2 /2 > 0). So even if the drift R − r − σ 2 /2 is negative for the
buy-and-hold strategy, with the constant-mix strategy given at (6) we can achieve a positive drift! Our
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expected return will be less than with the buy-and-hold strategy but our wealth is not going to converge
to 0 with probabiltity 1. Just the opposite: it will converge to infinity with probability 1!
Formula (6) is called the Kelly criterion because J. L. Kelly, Jr discovered it while studying transmission rates of noisy communication channels, [4]. E.O. Thorp successfully used it for finding the optimal
bets in blackjack, and also in investing. The Kelly criterion is missing from most financial books but we
believe it deserves more attention.
People often say: no risk no gain. Let us accept it as a fact, that the biggest gain can be achieved
only by taking the biggest risk. More precisely: the bigger the risk is, the bigger the expected happiness
should be. So a person with a 100% stock portfolio should have greater expected happiness (i.e., greater
expected logarithmic wealth), than a person who has a 90% stock, 10% bond portfolio. But this implies
that the optimal stock ratio given at (6) should be equal to 1. Thus, by this logic, the risk premium
should be exactly
R − r = σ 2 = 0.22 = 0.04,
a very interesting consequence of the more risk more expected happiness assumption.
The historical risk premium for the world was about 0.038 - which is suprisingly close to 0.04. If we
use R−r = 0.038 and σ = 0.2, the optimal Kelly ratio is ρ = 0.038/0.22 = 0.95, so 95% of our investments
should be in stocks. Although this is indeed the optimal strategy maximizing expected happiness, it is
probably too risky strategy for most of us. So maybe our utility function measuring happiness is not ln x
but something else?
Let t = 40 years and let us invest $1. We follow the 50% in stocks and 50% in bonds strategy,
constantly rebalanced (instead of the 95% in stocks strategy).
Now let us try to compare the risk of losing $0.25 with the corresponding risk associated with the
buy-and-hold strategy. We have to be careful because we are comparing apples with oranges - these two
investing strategies have different expected returns: for the constant-mix strategy E(Rel(t)) = eρ(R−r)t =
2.14, wheras for the buy-and hold strategy E(Rel(t)) = e(R−r)t = 4.57. So instead of $1 we invest only
$0.47 in stocks with the buy-and-hold strategy, and keep the rest in bonds. Then the expected relative
value of this portfolio will be $2.14, too.
The relative value of the bonds gives no gain or loss, so in order to lose $0.25 the stocks must go down
from $0.47 to $0.22. That is, the stocks goes down to 46.5% of their original value. The probability of
this can be calculated the same way as the other probabilities calculated in this paper. The answer is
0.12.
On the other hand, since (4) is a geometric Brownian motion, it is easy to find the probability of a
loss of $0.25 for the constant-mix strategy. It is 0.09. So the constant-mix strategy may reduce the risk.
Back to the Martian
Let us now return to the puzzle about the Martian at the beginning of the paper. If we see a Martian
it may be a good idea to buy lots of stocks because of the following reason. Stocks offer better expected
returns than bonds (if they are priced correctly). But people do not keep all their savings in stocks
because of the associated risk. Consequently, if the risk was significantly reduced, many investors would
flock to stocks to get the better returns. This would increase the price of the stocks until the risk would
be high again. If we see a Martian, we can assume that there is a working economy on the planet it
is coming from. They may have some kind of a stock market as well. So eventually we will be able to
buy extraterrestrial stocks and the aliens will be able to buy terrestrial stocks. Since their economy is
completely independent of ours, we can hugely decrease the risk of our portfolio if we keep half of our
money in terrestrial stocks and half in extraterrestrial stocks. Because of the decreased risk, the price of
stocks of both planets will shoot up until the risk will be high again.
This logic seems pointless as we may never see a Martian. Until we recognize that one of the reasons
stocks went up so much in the past is that people were able to diversify away some of their risks by buying
international stocks. But the economies of the world got more and more interconnected by globalization.
So much so that buying international stocks would not decrease our risk very much. We need another
planet...
Final remarks. The model we used is imperfect because it assumes that R, r and σ are constants.
Even small changes in these parameters can result large changes in the probabilities we calculated. Also,
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stock returns have “flat tails” suggesting they are not lognormally distributed.
This author has no idea if stocks will be rising or falling in the coming decades. He keeps 50 percent
of his retirement account in stocks and 50 percent in bonds.
Interested readers can read about a similar topic, the St. Petersburg paradox on Wikipedia. Enjoyable
readings for beginning investors are A Random Walk Down Wall Street and The Random Walk Guide
To Investing, both by B. G. Malkiel.
REFERENCES
1. A. A. Adams, P. M. Booth, D. C. Bowie, and D. S. Freeth, Investment Mathematics, Wiley, 2002,
ISBN 0-471-99882-6
2. W. N. Goetzmann and R. G. Ibbotson, The Equity Risk Premium: Essays and Explorations, Oxford
University Press, USA, 2006, ISBN: 0195148142
3. I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, 2nd ed., Springer, 2008,
ISBN: 0387976558
4. J.L. Kelly, Jr., A new interpretation of information rate., Bell. System Tech. J. 35 (1956), 917926.
5. B. G. Malkiel, A Random Walk Down Wall Street, W. W. Norton & Company, Inc., revised and
updated edition, 2007, ISBN 0393330338
6. J. J. Siegel, Stocks for the Long Run, 3rd ed., McGraw-Hill, 2002, ISBN 0-07-137048-X
7. R. J. Williams, Introduction to the Mathematics of Finance, AMS, 2006, ISBN 0-8218-3903-9
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