2. What can we say about returns?
Cannot be perfectly predicted — are random.
Ancient Greeks:
Would have thought of returns as determined by
Gods or Fates (three Goddesses of destiny)
Did not realize random phenomena exhibit
regularities(Law of large numbers, central limit th.)
Did not have probability theory despite their
impressive math
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3. Randomness and Probability
Probability arouse of gambling during the
Renaissance.
University of Chicago economist Frank Knight
(1916) distinguished between
Measurable uncertainty (i.e.games of
chance):probabilities known
Unmeasurable uncertainty (i.e.finance):
probabilities unknown
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4. Uncertainty in returns
At time t, Pt+1 and Rt+1 are not only unknown,
but we do not know their probability
distributions.
Can estimate these distributions: with an
assumption
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5. Leap of Faith
Future returns similar to past returns
So distribution of Pt+1 can estimated from
past data
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6. Asset pricing models (e.g. CAPM) use the
joint distribution of cross-section {R1t, R2t,…
RNt} of returns on N assets at a single time t.
Rit is the returns on the ith asset at time t.
Other models use the time series {Rt, Rt-1,…
R1} of returns on a single asset at a
sequence of times 1,2,…t.
We will start with a single asset.
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7. Common Model:IID Normal Returns
R1,R2,...= returns from single asset.
1. mutually independent
2. identically distributed
3. normally distributed
IID = independent and identically distributed
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8. Two problems
The model implies the possibility of
1.
unlimited losses, but liability is usually
limited Rt ≥-1 since you can lose no more
than your investment
1 + Rt(k) = (1 + Rt)(1 + Rt-1) ... (1 + Rt-k+1) is
2.
not normal
Sums of normals are normal but not
products
But it would be nice to have normality, so
math is simple
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9. The Lognormal Model
Assumes: rt = log(1 + Rt)* are IID and normal
Thus,we assume that
rt =log(1 + Rt) ~ N(µ,σ2)
So that 1 + Rt = exp(normal r.v.) ≥ 0
So that Rt ≥ -1. y 10
9
This solves the first problem y=e^{x} 8
7
6
5
4
3
(*): log(x) is the natural logarithm of x. 2
1
-3 -2 -1 0 1 2 3
x
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10. Solution to Second Problem
For second problem:
1 + Rt(k) = (1 + Rt)(1 + Rt-1) ... (1 + Rt-k+1)
log{1 + Rt(k)} = log{(1 + Rt) ... (1 + Rt-k+1)}
=rt + ... + rt-k+1
Sums of normals are normal (See Lecture Notes 2)
⇒ the second problem is solved
Normality of single period returns implies
normality of multiple period returns.
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11. Louis Jean-Baptiste Alphonse
Bachelier
The lognormal distribution goes back to Louis
Bachelier (1900).
dissertation at Sorbonne called The Theory of
Speculation
Bachelier was awarded “mention honorable”
Bachelier never found a decent academic
job.
Bachelier anticipated Einstein’s (1905) theory
of Brownian motion.
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12. In 1827, Brown, a Scottish botanist, observed the
erratic, unpredictable motion of pollen grains under
a microscope.
Einstein (1905) — movement due to bombardment
by water molecules — Einstein developed a
mathemetical theory giving precise quantitative
predictions.
Later, Norbert Wiener, an MIT mathematician,
developed a more precise mathematical model of
Brownian motion. This model is now called the
Wiener process.
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13. Bachelier stated that
“The math. expectation of the speculator is
zero” (this is essentially true of short-term
speculation but not of long term investing)
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14. Example 1
A simple gross return (1 + R) is lognormal~ (0,0.12)
– which means that log(1 + R) is N(0,0.12)
What is P(1 + R < 0.9)?
Solution:
P(1 + R < 0.9) = P{log(1 + R) < log(0.9)}
P{log(1 + R) < -0.105} (log(0.9)= -0.105)
P{ [log(1 + R)-0]/0.1 < [-0.105-0]/0.1}
P{Z<-1.05}=0.1469
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15. Matlab and Excel
In MATLAB, cdfn(-1.05) = 0.1469
In Excel:NORMDIST(-1.05,0,1,TRUE)=0.1469
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16. Example 2
Assume again that 1 + R is lognormal~
(0,0.12) and i.i.d. Find the probability that a
simple gross two-period return is less than
0.9?
Solution:log{1 + Rt(2)} = rt + rt-1
[ Rmbr Lec-2: if Z=aX+bY µZ=a µX +b µY σZ2=a2 σX2 +b2 σY2+2abσXY]
2-period grossreturn is lognormal ~ (0,2(0.1)2)
So this probability is
P(1 + R(2) < 0.9)=P(log[1 + R(2)]<log0.9)=
P(Z<-0.745)=0.2281
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17. Let’s find a general formula for the kth period
returns. Assume that
1 + Rt(k) = (1 + Rt)(1 + Rt-1) ... (1 + Rt-k+1)
log {1 + Ri} ~ N(µ,σ2) for all i.
The {Ri} are mutually independent.
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23. Geometric Random Walk
Therefore if the log returns are assumed to be i.i.d
normals, then the process {Pt:t=1,2,...} is the
exponential of a random walk.We call it a geometric
random walk or an exponential random walk.
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25. If r1,r2...are i.i.d N(µ,σ2) then the process is
called a lognormal geometric random walk
with parameters (µ,σ2).
As the time between steps becomes shorter
and the step sizes shrink in the appropriate
way, a random walk converges to Brownian
motion and a geometric random walk
converges to geometric Brownian motion;
(see Stochastic Processes Lectures.)
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