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Figure 5.9
Probability of investment outcomes after 25 years with
a lognormal distribution (approximated from a binomial tree)
0
.005
.01
.015
.02
.025
.03
.035
.04
.045
.05
0
20
40
60
80
100
120
140
160
180
200
Probability of Outcome
Investment Outcome (truncated at $200)
Tail
= $0.00002
Tail
= $10,595,634
to (1 1 r
1
)(1 1 r
2
) 2 1, which is not
normally distributed. Perhaps the
normal distribution does not qualify
as the simplifying distribution we
purported it to be. But the lognor-
mal distribution does! What is this
distribution?
Technically, a random variable X
is lognormal if its logarithm, ln( X ),
is normally distributed. It turns
out that if stock prices are “instan-
taneously” normal (i.e., returns
over the shortest time intervals are
normally distributed) then their
longer-term compounded returns and
the future stock price will be log-
normal.
20
Conversely, if stock prices
are distributed lognormally, then
the continuously compounded rate
of return will be normally distrib-
uted. Thus, if we work with continuously compounded (CC) returns rather than effec-
tive per period rates of return, we can preserve the simplification provided by the
normal distribution, since those CC returns will be normal regardless of the invest-
ment horizon.
Recall that the continuously compounded rate is r
CC
5 ln(1 1 r ), so if we observe
effective rates of return, we can use this formula to compute the CC rate. With r
CC
nor-
mally distributed, we can do all our analyses and calculations using the normally distrib-
uted CC rates. If needed, we can always recover the effective rate, r , from the CC rate
from: r 5 e
r
CC
2 1.
Let’s see what the rules are when a stock price is lognormally distributed. Suppose the
log of the stock price is normally distributed with an expected annual growth rate of g
and a SD of s . When a normal rate compounds by random shocks from instant to instant,
the fluctuations do not produce symmetric effects on price. A positive uptick raises the
base, so the next tick is expected to be larger than the previous one. The reverse is true
after a downtick; the base is smaller and the next tick is expected to be smaller. As a
result, a sequence of positive shocks will have a larger upward effect than the downward
effect of a sequence of negative shocks. Thus, an upward drift is created just by volatil-
ity, even if g is zero. How big is this extra drift? It depends on the amplitude of the ticks;
in fact, it amounts to half their variance. Therefore m, the expected continuously com-
pounded expected rate of return, is larger than g. The rule for the expected CC annual
rate becomes,
E(r
CC
) 5 m 5 g 1
½
s
2
(5.21)
20
We see a similar phenomenon in the binomial tree example depicted in Figure 5.9 . Even with many bad
returns, stock prices cannot become negative, so the distribution is bounded at zero. But many good returns can
increase stock prices without limit, so the compound return after many periods has a long right tail, but a left tail
bounded by a worst-case cumulative return of 2 100%. This gives rise to the asymmetric skewed shape that is
characteristic of the log-normal distribution.
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C H A P T E R
5
Risk, Return, and the Historical Record
155
With a normally distributed CC rate, we expect that some initial wealth of $ W
0
will com-
pound over one year to W
0
e
g1
½
s
2
5 We
m
, and hence the expected effective rate of return is
E(r) 5 e
g1
½
s
2
2 1 5 e
m
2 1
(5.22)
If an annual CC rate applies to an investment over any period T, either longer or shorter
than one year, the investment will grow by the proportion r (T ) 5 e
r
CC
T
2 1. The expected
cumulative return, r
CC
T , is proportional to T , that is, E(r
CC
T ) 5 mT 5 gT 1 ½
s
2
T and
expected final wealth is
E(W
T
) 5 W
0
e
mT
5 W
0
e
(g1
½
s
2
)T
(5.23)
The variance of the cumulative return is also proportional to the time horizon:
Var( r
CC
T ) 5 T Var( r
CC
),
21
but standard deviation rises only in proportion to the square root
of time: s( r
CC
T ) 5
"TVar(r
CC
) 5 s
" T.
This appears
to offer a mitigation of investment risk in the long run: Because
the expected return increases with horizon at a faster rate than the standard deviation, the
expected return of a long-term risky investment becomes ever larger relative to its standard
deviation. Perhaps shortfall risk declines as investment horizon increases. We look at this
possibility in Example 5.11.
21
The variance of the effective annual rate when returns are lognormally distributed is: Var( r ) 5 e
2 m
(
e
s
2
2 1).
22
In some versions of Excel, the function is NORM.S.DIST(z, TRUE).
An expected effective monthly rate of return of 1% is equivalent to a CC rate of
ln(1.01) 5 0.00995 (0.995% per month). The risk-free rate is assumed to be 0.5% per
month, equivalent to a CC rate of ln(1.005) 5 0.4988%. The effective SD of 4.54%
implies (see footnote 21 ) a monthly SD of the CC rate of 4.4928%. Hence the monthly
CC risk premium is 0.995 2 0.4988 5 0.4963%, with a SD of 4.4928%, and a Sharpe
ratio of .4963/4.4928 5 0.11. In other words, returns would have to be .11 standard
deviations below the mean before the stock portfolio underperformed T-bills. Using the
normal distribution, we see that the probability of a rate of return shortfall relative to
the risk-free rate is 45.6%. (You can confirm this by entering 2 .11 in Excel’s NORMSDIST
function.
22
) This is the probability of investor “regret,” that after the fact, the investor
would have been better off in T-bills than investing in the stock portfolio.
For a 300-month horizon, however, the expected value of the cumulative excess
return is .4963% 3 300 5 148.9% and the standard deviation is 4.4928
"300 5 77.82,
implying a whopping Sharpe ratio of 1.91. Enter 2 1.91 in Excel’s NORMSDIST function,
and you will see that the probability of shortfall over a 300-month horizon is only .029.
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