Statistical estimation of expectation and SD of returns:
Past returns data
Good estimate if
1.
States of the market and associated returns are constant over time
2.
Large enough sample
Challenges: time varying market states, choice of sample period, etc.
23

Estimate of Expected Returns
Arithmetic Average of rate of return:
It is an unbiased estimate of expected returns.
Geometric (time-weighted) average return:
TV
= Terminal Value of the Investment
g
= geometric average rate of return
24
r
ˆ
E
(
r
)
1
n
r
(
t
)
t
1
n

Estimate of Variance and SD
Estimate of variance:
Estimate of standard deviation:
They are unbiased estimates of variance and standard deviation.
25
ˆ
2
1
n
1
[
r
(
t
)
t
1
n
r
]
2
ˆ
1
n
1
[
r
(
t
)
t
1
n
r
]
2

Probability Distribution of States
26
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Probability distribution (Density plot)
Probabilities (bar
heights) sum up
to 1
Possible HPR
realizations

Probability Distribution of States
27
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
0
0.05
0.1
0.15
0.2
0.25
A finer distribution
Recession
Recession
Mild recession
Normal
Boom

The Normal Distribution
28
Example: Normal Distribution with mean return 10% and
standard deviation 20%

Why the Normal Distribution?
A good approximation:
The mean of a large number of independent movements is well
approximated by a Normal Distribution
Central Limit
Theorem
Convenience:
Normal distribution is symmetric
A Normal Distribution is completely characterized by its mean and
standard deviation
The weighted average of two normally distributed variables is still
normally distributed
29

Deviations from Normal
Distribution
What if excess returns are not normally distributed?
Standard deviation is no longer a complete measure of risk
Sharpe ratio is not a complete measure of portfolio performance
Need to consider skewness
and kurtosis
Skewness: a measure of asymmetry. Normal has zero skewness.
Kurtosis: a measure of tails (fat or not). Normal has zero kurtosis.
30
Skew
average
(
R
R
_
)
3
ˆ
3
Kurtosis
average
(
R
R
_
)
4
ˆ
4
3

Normal and Skewed Distribution
31
Risk averse investors care more about extreme negative values than
extreme positive values
When skewness > 0, SD overestimates risk

Normal and Fat-Tailed
Distribution
32
SD underestimates the likelihood of extreme values

Value at Risk (VaR)
Value at risk (VaR): the quantile of a distribution below which lies
q % of the possible values of that distribution, usually 5%
Interpretations:
5% VaR means 95% of returns will exceed this VaR, only 5% will worse
than VaR
Measure of downside risk, worst loss that will be suffered with given
probability
VaR of Normal Distribution:
VaR (0.05, normal) = Mean + (-1.65) SD
since for standard normal distribution,
P
(
X
<-1.65) = 0.05.

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- Fall '09
- Xia
- Normal Distribution, Standard Deviation, Interest, Interest Rate