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Reading 09: Common Probability Distributions (Distribution (Continuous…
Reading 09: Common Probability Distributions
Distribution
Cumulative Distribution Function (cdf)
: is the probability that a random variable is less or equal to a specific value.
\(F(x)=P(X\leq x)\)
Discrete uniform distribution
: a distribution with finite outcome number and all outcomes are equally likely
Binomial Distribution
Definition
: The probability distribution for a discrete random variable that has 2 possible outcomes
Bernoulli Random Variable
Probability of
x
success [P(sucess) = p] in
n
trial
:\(p(x)=nCx\times p^{x}\times (1-p)^{n-x} \)
Expected Value
: E(X) = np
Variance
: Var(X) = np(1-p)
Binomial Tree
2 possible outcomes each period: UP MOVE and DOWN MOVE
P(UP MOVE) + P(DOWN MOVE) = 1
Up Factor > 1 and Down Factor < 1
Beginning point (Ex: Beginning Stock Price)
Price after
n
period = \(A\times U^{n} or A\times D^{n}\)
Continuous Uniform Distribution
:
the probability of X occurring in a possible range is the length of the range relative to the total of all possible values.
if \(a\leq x_{1}\leq x_{2}\leq b\) then \(P(x_{1}\leq x\leq x_{2})=\frac{x_{2}-x_{1}}{b-a}\)
Probability distributed evenly over an interval
Expected Value = Midpoint between the largest and the smallest values (a.k.a average)
Normal Distribution
Characteristics:
The normal curve is symmetrical and bell-shaped, with a single peak at the center of the distribution.
Mean = Medium = Mode
Is completely defined by its mean and standard deviation, as skew always = 0 and kurtosis = 3
Linear combination of normally distributed random variable is also normally distributed
The tails go on forever, even if the probabilities decrease further from the mean
Multivariate & Univariate
Multivariate
Describe the probabilities for more than 1 random variable
Need means, variances, and correlation coefficients
Multivariate normal distribution for n variables
will have:
n
means
n
variances
\(\frac{n\times (n-1)}{2}\)
correlations
Univariate
: distribution that describes the probability for a single random variable.
Confident interval
Definition
: a range within which we have a given level of confidence of finding a point estimate.
For Normal distribution
68% Confidence Interval: \( \mu \pm 1.00 \) SD
90% Confidence Interval: \( \mu \pm 1.65 \) SD
95% Confidence Interval: \( \mu \pm 1.96 \) SD
99% Confidence Interval: \( \mu \pm 2.58 \) SD
Z-standardization
A normal distributed random variable X can be standardized by calculating Z-value:
Z=\( \frac{X-\mu }{\sigma }\)
Z-Table is used to find the Probability that X is \(\leq \) a given value.
\(P(X\leq x)=F(x)=F\left [ \frac{x-\mu }{\sigma } \right ]=F(Z)\) which can be found in Z-Table
mean = 0 and SD = 1
Probability distribution
Random variable
Discrete random variable
: positive probabilities associate with finite number of outcomes
Continuous random variable
: positive probability associates with a range of outcomes. Probability for each outcome = 0
Definition:
probability distribution
gives the probabilities of all possible outcomes for a random variable
Types
Discrete distribution
: Finite number of outcome
Continuous distribution
: Infinite number of outcomes
Probability function, aka p(x)
: is the probability that a discrete random variable will equal x
Tracking error
: \(R_{Portfolio}-R_{benchmark}\)
Safety-first ratio & Short-fall risk
Roy's Safety-first ratio
: \( SFRatio=\frac{E(R_{portfolio} )-R_{target}}{\sigma_{portfolio}}\)
Short-fall risk
: the Probability that a portfolio's value will fall below a specific value over a period of time.
Higher safety-first ratio means lower short-fall risk
Probability of Shortfall Risk: P(RP < RL) = F(-SFRatio)
Lognormal distribution
If x is normally distributed, then \(e^{x}\) follows a lognormal distribution
Lognormal distribution can only take positive value, thus is usually used to model asset prices.
Monte Carlo Simulation
Definition
: use randomly generated values for risk factors, based on they assumed distributions to produce a distribution of possible security values.
Limitations
: Complex + only provide answer that are no better than the assumption used
Can be used to estimate a distribution of derivatives prices or of NPVs
How to do
Specify distributions of random variables (ex: interest rate, stock prices, etc.)
Use computer random generation of variables
Price the derivatives using those values
Repeat step 2 and 3 thousands of times
Calculate mean/ variance of distribution of outcomes.
Historical Simulation
Definition
: similar to Monte Carlo, but use randomly selected past changes in risk factors to generate a distribution of possible security values.
Limitation
: Cannot consider scenarios that did not happen in the past
Advantage
: don't have to estimate distribution of risk factor
Continuous compounding
Continuously compound rate = ln(1+ Holding Period Return)