CFA 2018 smart summary, study session 03, reading 10 1
Bạn đang xem bản rút gọn của tài liệu. Xem và tải ngay bản đầy đủ của tài liệu tại đây (80.58 KB, 3 trang )
2017 Study Session # 3, Reading # 10
“COMMON PROBABILITY DISTRIBUTIONS”
Probability Distribution
Describes the probabilities of all possible
outcomes for a random variable.
Sum of probabilities of all possible
outcomes is 1.
Random Variable
Distribution
Discrete
Finite (measurable) # of possible
outcomes.
P(x) cannot be 0 if ‘x’ can occur.
We can find the probability of a
specific point in time.
Probability Density Function (PDF)
It is used for continuous distribution.
Denoted by f(x).
Discrete uniform random variable
Probability Function
Probability of a random variable being equal
to a specific value.
Properties:
0 ≤ p(x) ≤ 1
Σ p(x) = 1
Continuous
Infinite (immeasurable) # of possible
outcomes.
P(x) can be zero even if ‘x’ can occur.
We cannot find the probability of a
specific point in time.
Cumulative Distribution Function (CDF)
Calculates the probability of a random
variable ‘x’ taking on the value less than or
equal to a specific value of ‘x’.
F(x) = P (X ≤ x)
All outcomes have the same probability.
Uniform Probability Distribution
Discrete
Has a finite number of specified outcomes.
P(x)×k. K is the probability for ‘k’ number of
possible outcomes in a range.
cdf: F(xn) = n.p(x).
Continuous
Defined over a range with parameters ‘b’
(upper limit) & ‘a’ (lower limit).
cdf: It is linear over the variable’s range.
Properties:
P ( x ≤ a) = 0 & P (x ≥ b) = 1
ି
P( a < x < b) =
௫మ ି௫భ
Binomial Distribution
Properties:
Two outcomes (success & failure).
‘n’ number of independent trials.
Probability of success remains constant.
p(x) =
݊!
௫ (1 − )ି௫
ሺ݊ − ݔሻ!. !ݔ
Binomial Tree
Shows all possible combinations of up & down
moves over a number of successive periods.
Node: Each of the possible values along the
tree.
U is up-move factor.
D is down-move factor (1/U).
p is probability of up move.
(1-p) is probability of down move.
Copyright © FinQuiz.com. All rights reserved.
2017 Study Session # 3, Reading # 10
Normal Distribution
Properties of Normal Distribution:
Symmetric distribution
Mean = Median = Mode
Skewness = 0
Kurtosis = 3 & Excess Kurtosis = 0
Range of possible outcomes lie between -∞ to + ∞
Asymptotic to the horizontal axis
Described by two parameters i.e. Mean and Variance or (standard deviation)
When S.D ↑ (↓), the curve flattens (steepens)
Smaller the S.D, more the observations are centered around mean.
Not appropriate to use for options.
Not appropriate to use to model asset prices.
Central Limit Theorem ⇒
Sum and mean of large no. of independent
variables in approximately normally distributed.
Linear combination of two or more normal random variables is also normally
distributed.
Confidence Interval
Range of values around the expected value
within which actual outcome is expected to be
some specified percentage of time.
Confidence
Interval
x ± 1s
x ± 1.65s
x ± 1.96s
x ± 2s
x ± 2.58s
x ± 3s
%
68.%
90.%
95%
95.45%
99%
99.73%
Applications of Normal Distribution
Shortfall Risk
Risk that portfolio value will fall below
some minimum level at a future date.
Safety First Rule focuses on Shortfall
Risk.
Roy’s Safety First Criterion
Optimal portfolio minimizes the
probability that the return of the
portfolio falls below some
minimum acceptable level.
Minimize P(RP < RL).
SFRatio =
[ܧሺܴ ) − ܴ ሿ
σ
Choose the portfolio with greatest
SFRatio.
Sharpe Ratio
= [E (Rp) – Rf] / σp
Portfolio with the highest Sharpe ratio
minimizes the probability that its
return will be less than the Rf
(assuming returns are normally
distributed).
Managing Financial Risk
value ofvariable
losses (in money terms) expected over
Lognormal Value at risk (VAR) ⇒minimum
A random
of probability.
Distribution a specified time period at a specified
whoselevel
natural
log has normal distribution
⇒use of
of techniques to estimate losses in
Properties Stress testing/scenario analysiscannot
beset
negative
extremely worst combinations of
or scenarios.
is events
completely
described by mean and variance
Log Normal distribution
is more appropriate to use to model asset prices
is used in Black Scholes Merton Model
Compounds Rate
of Return
Discrete:
Daily, annually, weekly, monthly compounding
Continuous
ln(S1/S0) = ln(1+HPR)
These are additive for multiple periods.
Effective annual rate based on continuous
compounding is given as:
EAR = e Rcc-1
Copyright © FinQuiz.com. All rights reserved.
2017 Study Session # 3, Reading # 10
4. Monte Carlo Simulation
Use of a computer to generate a large number of random samples from a probability distribution
Uses
It is used to:
Plan and manage financial risk.
Value complex securities
Estimate VAR
Examine model's sensitivity to changes in
the assumptions.
Simulation Procedure for Stock Option Valuation
Step 1: Specify underlying variable
Step 2: Specify beginning value of underlying variable
Limitations
Complex procedure.
Highly dependent on assumed distributions.
Based on a statistical rather than an
analytical method.
Random Number Generator
An algorithm that generates uniformly
distributed random numbers between 0 and 1.
Step 3: Specify a time period
Step 4: Specify regression model for changes in stock price
Step 5: K random variables are drawn for each risk factor using
computer program/ spreadsheet
Step 6: Estimate underlying variables by substituting values of random
observations in the model specified in Step 4.
Step 7: Calculate value of call option at maturity and then discount back that
value at time period 0
Step 8: This process is repeated until a specified number of trials ‘I’ is completed.
Step 9: Finally, mean value and S.D. for the simulation are calculated
Historical Simulation or Back Simulation
Based on actual values & actual
distribution of the factors i.e., based on
historical data.
Drawbacks
Cannot be used to perform “what if’
analysis.
History does not repeat itself.
Copyright © FinQuiz.com. All rights reserved.
