1.5 Common Probability Distribution
1.5 Common Probability Distribution
1.5 Common Probability Distribution
PROBABILITY DISTRIBUTION
The set of probabilities for the possible outcomes of a random variable is
called a probability distribution.
The underlying foundation of most inferential statistical analysis is the concept
of a probability distribution.
The focus in the investments arena is on four probability distributions.
1. Uniform
2. Binomial
3. Normal
4. Lognormal
0.25
0.2
0.2
0.15
0.15
Probability(X)
0.1
0.05
Probability (X)
0.1
0.05
0
1
0
0
PROBABILITY FUNCTIONS
The possible outcomes of a random variable and their associated
probabilities are collectively known as a probability function.
By convention, discrete random variable probability functions are denoted as
p(x) and known as probability mass functions (pmf).
Continuous random variable probability functions are denoted as f(x) and are
known as probability density functions (pdf).
Probability functions have two very important properties:
1. Any and all individual probabilities described by the probability function take
on a value between 0 and 1 (including 0 and 1).
2. The sum of all probabilities described by the probability function is equal to
1.
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12
13
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Variance
Bernoulli, B(1,p)
p(1 p)
Binomial, B(n,p)
np
np(1 p)
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Stock price
moves up
Stock price
equals uS
Stock price
moves down
Stock price
equals dS
Stock price today is denoted S. If u(d) is 1 plus the rate of return when the stock
price moves up (down), then uS (dS) is the end of period price for the stock and
the diagram depicts a single Bernoulli trial.
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18
uudS
udS
dS
uddS
ddS
dddS
Stock price today is denoted S. If u(d) is 1 plus the rate of return when the stock
price moves up (down), then uS (dS) is the end of period price for the stock,
and the diagram above depicts a series of Bernoulli trials that represents the
movement in stock price as a binomial random variable.
The probability of an up move is known as the up transition probability, and
that of a down move is known as the down transition probability.
Note that this tree recombines. In other words, udS = duS, etc.
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0.1
0.05
0
0
21
22
23
25
26
27
28
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STANDARD NORMAL
A normal distribution with a mean of 0 and standard deviation of 1 is called
standard normal.
The prevalence of the normal distribution has led to a process whereby
probability tables that have been calculated for a standard normal distribution
can be used to make probability statements for any normally distributed
variable.
This process is known as standardizing and is accomplished by:
1. Taking the observation(s) of interest and subtracting the mean of that
observations observed distribution;
2. Dividing the result by the observed distributions standard deviation.
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SHORTFALL RISK
The risk that portfolio value will fall below a minimum acceptable level
across a specified time horizon.
We can define a threshold below an acceptable minimum threshold level, RL.
Roys safety-first criterion states that the optimal portfolio is one that minimizes
the probability of a return below RL.
If returns are normally distributed, then we can calculate Roys safety-first ratio
as
Portfolio
RL= 0.0375
The optimal portfolio will be the one with the
highest SFRatio because that portfolio
E(R)
0.25
0.11
.014
0.27
0.08
.020
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You are researching asset allocations for a client with an $800,000 portfolio.
Although her investment objective is long-term growth, at the end of a year
she may want to liquidate $30,000 of the portfolio to fund educational
expenses. If that need arises, she would like to be able to take out the
$30,000 without invading the initial capital of $800,000.
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2.
3.
Portfolio
RL = 0.0375
E(R)
0.25
0.11
.014
0.27
0.08
.020
SFRatio
0.787 0.906
0.513
0.216 0.182
0.880
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where and are the parameters for the normal distribution underlying it.
Key result: If a stocks continuously compounded return is approximately
normally distributed, then its future stock prices are lognormally
distributed.
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CONTINUOUS COMPOUNDING
If the frequency with which we calculate interest is infinitely large or the
time interval infinitely small, we are using continuous time compounding.
A continuously compounded return is one in which time is viewed as
continuous; with discrete compounding, time advances in finite intervals.
A continuously compounded return is calculated as
or
or
If the holding period return from time = 0 to time = T is 0.056, then the
continuously compounded return for time (0,T) can be determined.
- If the holding period return is 0.056, then a stock that begins with a price of
$1 ends the period with a price of $1.056.
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SUMMARY
Four probability distributions are commonly used in investments applications:
the uniform, the binomial, the normal, and the lognormal distributions.
The uniform distribution is generally used in simulation applications, including
Monte Carlo analysis.
The binomial forms the basis for many option pricing applications.
The normal distribution, with its attractive statistical properties, is prevalent
throughout investments and is used as a central distribution for modeling
returns.
The lognormal distribution is generally used to model asset prices for which the
normal, with its theoretical infinitely negative lower bound, is unsuited.
- If returns are normally distributed and continuously compounded, then asset
prices can be shown to be lognormally distributed.
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BERNOULLI EXAMPLE
Evaluating Block Brokers
- View each trade as a Bernoulli trial.
- What are probabilities of a successful trade with each broker?
- Binomial assumes
- Probability is constant for all trials.
- Trials are independent.
- Probabilities are based on the counting from the prior chapter.
Nov 2003
Profitable Losing
BB001
BB002
BERNOULLI EXAMPLE
1. If you are paying a fair price on average in your trades
with a broker, what should be the probability of a
profitable trade?
2. Did each broker meet or miss that probability
expectation?
3. Under the assumption that the prices of trades were fair:
a. Calculate the probability of three or fewer profitable
trades with Broker BB001.
b. Calculate the probability of five or more profitable
trades with Broker BB002.
BERNOULLI EXAMPLE
1. Probability is 50/50 if the trades are fair.
2. BB001 misses at 0.25; BB002 exceeds expectations at
0.625.
3. You need the cdf for three or fewer for BB001, and five
or greater for BB002.
a. For BB001, this probability is 0.0730.
b. For BB002, this probability is 0.3633.
Note: The magnitude of trade losses/gains may also be
important, and that is masked by the classification as
profitable or not.