Download Returns and Statistics II

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Upcoming Homework
Probability Models
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Suppose price to play = $0.85
We can draw a model of net returns:
Two-state probability model
– Two states
– Two returns
– Two probabilities
Expected Return
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The expected return to playing this game once is
1
1
E[r ]  18%  (6%)  6%
2
2
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In general, the expected return of any two-state probability model is
E[r ]  p1r1  p2 r2
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p1 and p2 are the probabilities of the two states
r1 and r2 are the returns received in the two states
Expected Return: Example
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Coin flipping game:
– Cost: $1
– If heads: $2
– If tails: $1
– Probability of heads: 0.75
What is expected return from playing?
E[r ]  0.75(100%)  .25(0%)  75%
Expected Return
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Suppose
– We don’t know the true probability model
– But we can observe past data from the game
100%
100%
0%
100%
0%
0%
100%
• Then we could estimate the expected return
• Find simple average: add-up all values and divide
by the number of values you observe
• With many observations, this would be very close to
expected return derived from the probability model
Probability Models
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Realistic probability models are very complex and
involve an infinite # of possible outcomes.
– Example: the normal distribution
To get an estimate of the expected return, it is usually
easiest to just estimate simple mean from past data if
available.
Simple probability models with only two possible
outcomes, though unrealistic, help us understand
finance theory.
Uncertainty
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Game 1:
– 10% return with 50% probability
– 20% return with 50% probability
Game 2:
– 0% return with 50% probability
– 30% return with 50% probability
Which game do you prefer?
Uncertainty
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We need a measure of uncertainty.
Both games have expected return of 15%.
How about expected deviation from mean?
Game 1 Deviations from mean:
– 10%-15%=-5% with 50% probability
– 20%-15%=5% with 50% probability
– Expected deviation from mean is zero.
Uncertainty
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Game 2 Deviations from mean:
– 0%-15%=-15% with 50% probability
– 30%-15%=15% with 50% probability
– Expected deviation from mean is zero.
The expected deviation from mean will
always be zero for any probability model.
Need a more helpful measure
Uncertainty
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How about expected squared deviation from
mean?
Game 1 squared deviations
– (-5%)2=0.0025
– (5%)2= 0.0025
– Expected squared deviation from mean is
0.0025.
Uncertainty
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Game 2 squared deviations
– (-15%)2=0.0225
– (15%)2= 0.0225
– Expected squared deviation from mean is 0.0225.
Expected squared deviations:
– Game 1: 0.0025
– Game 2: 0.0225
Uncertainty
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VARIANCE:
– Expected squared deviation from mean
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STANDARD DEVIATION:
– Square-root of the variance
Uncertainty: Example
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Coin flipping game:
– Cost: $1
– If heads: $2
– If tails: $1
– Probability of heads: 0.75
What is variance of this game?
Var[r ]  0.75(100%  75%)2  .25(0%  75%)2
 0.75(.0625)  .25(0.5625)  .1875
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What is the standard deviation?
Stdev[r ]  .1875  0.43
Uncertainty
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Suppose
– We don’t know the true probability model
– But we can observe past data from the game
The we could estimate the variance by
– Estimating expected return (simple average)
– Finding squared deviation for each outcome
– Take simple average of squared deviations
We could estimate the standard deviation as
– Square-root of estimated variance
Uncertainty
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Example: Suppose for coin flipping game we observe the following
outcomes:
– 100%, 0%, 100%, 0%
Estimated expected return: 50%
Deviations:
– 50%, -50%, 50%, -50%
Squared Deviations:
– 0.25, 0.25, 0.25, 0.25
Estimated Variance: 0.25
– Std. Deviation: .50
From True probability model:
– Expected return=75%
– Variance = 0.1875
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Std. Deviation: .4330
Variance
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We often use
s2 to represent variance
s to represent standard deviation
Later in the course we will look at how risk is
measured for portfolios that will include covariation
as well as standard deviation
What does Standard Deviation
Tell Us?
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Helps us measure likelihood of extreme outcomes.
Prob(return < 1 standard deviation from mean) = 16%
Probability of Extreme Bad
Events
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Example: Your portfolio has an expected
return of 10% with a standard deviation of
0.16 over the next year.
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What is probability that realized return is
<-22%?
Probability of Extreme Bad
Events
1. How many standard deviations is outcome from mean?
x   .22  .10
z
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 2
s
.16
-.22
.10
2 Standard Deviations (.16)
Probability of Extreme Bad
Events
2. Use excel function normsdist(z)
This function gives probability of getting z
standard deviations from mean or less.
normsdist(-2) = 0.02275 = 2.275%
Data vs. Probability Model
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Note that probability models are forward looking.
They tell us about what we should expect in the
future.
Estimates of means and variances from historical
data are backward looking. They tell us about what
happened in the past.
The hope is that the past will be indicative of the
future.
The Historical Record
Series
Lg. Stk
Sm. Stk
LT Gov
T-Bills
Inflation
Arith.
Mean%
12.49
18.29
5.53
3.85
3.15
Stan.
Dev.%
20.30
39.28
8.18
3.25
4.40
Real Rates of Return
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Suppose at the beginning of the year, the cost of a pizza is $10.00.
You have $100 in cash. You could buy 10 pizzas, but instead, you
invest the $100 in a long term gov. bond. The return on the bond is
5%. Inflation over the year is 3%.
The investment provides you a nominal income at year end of
100(1.05) = $105.
At year end, the cost of a pizza is 10.00(1.03)=$10.30.
At year end, you could buy 10.19 pizzas (105/10.3)=10.19.
Your real return is therefore only ____?%
Real Rates of Return
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C = amount of cash at beginning of period
P = price of a good at beginning of period
rn = nominal rate of return, rr = real return
i = inflation rate
The real (gross) rate of return was found above by
solving the following equation
Since
C (1  r ) / P(1  i) 1  rn
1  rr 
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C/P
1 i
Because i is small we can approximat e with rr  rn  i