SASF 2003 CFA® REVIEW
Class Notes
Level I
Section(s) A & B
Date of Class:
Jan. 7 (9), 2003
Topic:
Instructor:
Phone:
Email:
Economics
John Veitch, Ph.D, CFA
(415) 422-6271
veitchj@usfca.edu
AIMR Study Session(s): 2 Investment Tools –
Quantitative Methods
The Security Analysts of San Francisco
The Association for Investment Management and Research (AIMRsm) does
not endorse, promote, review, or warrant the accuracy of the products or
services offered by organizations sponsoring or providing CFA® exam preparation materials or
programs, nor does AIMR verify pass rates or exam results claimed by such organizations.
“CFA,” “Chartered Financial Analysts,” and “AIMR” are marks owned by AIMR.”
Slides used in class available on my website - www.usfca.edu/economics/veitch/
Study Session 2
Investment Tools
Quantitative Methods
Reading Assignments
1. Quantitative Methods for Investment Analysis, Richard A. DeFusco, Dennis W.
McLeavey, Jerald E. Pinto, and David E. Runkle (AIMR, 2001)
A. “The Time Value of Money,” Ch. 1
B. “Statistical Concepts and Market Returns,” Ch. 3
C. “Probability Concepts,” Ch. 4
D. “Common Probability Distributions,” Ch. 5
Note: Candidates are responsible for the practice problems at the end of the chapters.
Answers to chapter problems are found at the end of each chapter.
Learning Outcomes - (*) means new LOS for 2003
1. A. “The Time Value of Money”
The candidate should be able to
a) calculate the future value (FV) and present value (PV) of a single sum of money;
Calculate the Future Value, FVN, of a Current Value, PV, which earns an interest rate of r per period to
be received N periods in the future - FVN  PV (1  r ) N
Calculate the Present Value, PV, of a Future Value, FVN, to be received N periods in the future if
investments earn an interest rate of r per period - PV 
1
FVN
(1  r ) N
PLEASE NOTE: You are responsible for mastering the use of your financial calculator to solve
these problems. The SASF CFA Review course does not provide training in either of the two
approved calculators.
b) calculate an unknown variable, given the other relevant variables, in single-sum
problems;
1. Calculate Future Value, FVN
FVN  PV (1  r )
N
Given - Current Value = PV
Periods in the future = N
Interest Rate per period = r
2. Calculate Present Value, PV
PV 
1
FVN
(1  r ) N
Given – Future Value = FVN
Periods in the future = N
Interest Rate per period = r
2
b) calculate an unknown variable, given the other relevant variables, in single-sum
problems;
3. Calculate interest rate, r
(1  r ) N 
FVN
Given - Current Value = PV
Future Value = FVN
becomes
PV
1
FV
 N 1
r N
PV 

4. Calculate number of periods, N
ln( 1  r ) N  ln 

N  ln 

FVN
FVN
Periods in the future = N
Given - Current Value = PV

PV 
Future Value = FVN
  ln( 1  r )
PV 
Interest Rate per period = r
c) calculate the FV and PV of a regular annuity and an annuity due;
Regular or ordinary annuity is a finite set of sequential cash flows, all with the same value A, which
has a first cash flow that occurs one period from now.
FV
Re g
N
PV Re g
 1  r N  1
 A1  r   A1  r     A1  r   A1  r   A

r


1 

1

1
1
1
1
1  r N 


A
A



A

A

A
1  r  1  r 2
r


1  r N 1
1  r N




N 1
N 2
1
0
An annuity due is a finite set of sequential cash flows, all with the same value, which has a first
cash flow that is paid immediately.
FV
Due
N
PV Due
 1  r N  1
Re g
 A1  r   A1  r     A1  r   A1  r 
  1  r FVN
r


1 

1
N

1
1

1  r  

 A
A 
A  A1  r 
 1  r PV Re g
N 1
1  r 
r


1  r 




N
N 1
1
d) calculate an unknown variable, given the other relevant variables, in annuity
problems;
There are three variables in any annuity problem: the interest rate per period rp, the number of periods
in the annuity N, and the present value PV, or future value of the annuity FVN.
3
e) (*) show the equivalence between present value and discounted future value;
This is immediate from an understanding of the LOS in Part a).
f) calculate the PV of a perpetuity;
Perpetuity is a perpetual annuity or a set of never-ending set of sequential cash flows, all with the
same value A, with the first cash flow occurring one period from now.
PV Perpetuity 
 
1
1
1  A
A
A



A



2
t 
1  r  1  r 
r
t 1  1  r  
g) calculate an unknown variable, given the other relevant variables, in perpetuity
problems;
Given any two of the variables in the formula, it is possible to calculate the value of the third variable - PV Perpetuity 
A
r
h) calculate the FV and PV of a series of uneven cash flows;
It is possible to directly calculate the FV or PV of an uneven stream of cash flows, Fi where i = 1, 2,
… n are the cash flows in each period.
FV  F1 1  r   F2 1  r     Fn1 1  r   Fn 1  r 
1
1
1
1
PV 
F1 
F2   
Fn 1 
F
2
n 1
1  r 
1  r 
1  r 
1  r n n
n 1
n 2
1
0
It may also be possible to break out uneven cash flows into one or more annuities plus a few
standalone cash flows. See the problems at the end of this chapter in the assigned text for lots of
examples with answers to these types of problems.
i) calculate time value of money problems when compounding periods are other than
annual;
Future Value = PVN
 r 
PV  FVN 1  S 
 m
 m N
Periods per year = m
Years in the future = N
Quoted Annual Interest Rate = rS
j) distinguish between the stated annual interest rate and the effective annual rate;
Stated Annual interest rate or quoted interest rate = m x rp where rp is the periodic interest rate times
the number of periods in a year, m.
It does not account for the effects of compounding within the year.
Effective Annual interest rate – is the amount to which a unit of currency will grow to in a year with
interest on interest (compounding) taken into account = (1+rp)m - 1
4
k) calculate the effective annual rate, given the stated annual interest rate and the
frequency of compounding;
Given a stated periodic interest rate, rP, equal to the stated annual rate, rS, divided by the number of
periods m in a year, the effective annual interest rate is -
Effective annual rate  1  rp   1
m
Assuming continuous compounding at the stated interest rate, rS, the effective annual rate is -
Effective annual rate  e rS  1
l) draw a time line, specify a time index, and solve problems involving the time value of
money as applied to mortgages, credit card loans, and saving for college tuition or
retirement.
See the problems at the end of this chapter in the assigned text for lots of examples with answers to
these types of problems.
B. “Statistical Concepts and Market Returns”
The candidate should be able to
a) differentiate between a population and a sample;
A Population is defined as all members of a specified group.
A Sample is a subset of a defined population.
b) explain the concept of a parameter;
A Parameter is a descriptive measure or characteristic of a defined population. It is thus the actual true
measure of the population’s characteristic(s).
c) explain the differences among the types of measurement scales;
Nominal Scale: categorizes each member of the population or sample using an integer for each
category. It is the weakest level of measurement with no implied ranking or intensity.
Ordinal Scale: each member of the population or sample is placed into a category and these categories
are ordered with respect to some characteristic. It is a stronger level of measurement because it allows
ordering across members. Think Letter Grades.
Interval Scale: each member is assigned a number from a scale. This scale provides a ranking across
members and assurance that differences between scale values are equal. Scale values can thus be
added or subtracted in a meaningful way. Think temperature either Celsius or Fahrenheit.
Ratio Scale: All the characteristics of interval scales plus a true zero point. Allows computation of
meaningful ratios, as well as addition and subtraction. Think rates of asset returns or height.
d) define and interpret a frequency distribution;
Frequency Distribution: is a tabular display of data summarized into a relatively small number of
intervals. The frequency distribution is the list of intervals together with the corresponding measures
of frequency for the variable of interest.
5
e) define, calculate, and interpret a holding period return;
Holding Period Return: is expressed in percent terms, i.e. independent of currency units, and is
calculated over a period of time.
Holding Period Return = Rt
Rt 
Pt  Pt 1  Dt
Pt 1
Share Price end of time t = Pt
Share Price end of time t-1 = Pt-1
Cash Distributions during period t = Dt
Holding Period Return, Rt, consists of capital gains over the period plus distributions during the period
divided by the beginning price (distribution yield).
f) define and explain the use of intervals to summarize data;
An interval is a set of values in which observations on a random variable’s outcomes may fall. The
set of intervals must be mutually exclusive and exhaustive, that is each observation must fall into
one and only one interval.
The frequency with which observations fall into each interval is used to construct the frequency
distribution for a random variable’s outcomes.
g) calculate relative frequencies, given a frequency distribution;
A frequency distribution shows the absolute number of observations in each interval. A
relative frequency divides each corresponding absolute frequency by the total number of
observations. Thus a relative frequency distribution shows the percentage of total
observations in each interval.
h) describe the properties of data presented as a histogram or a frequency polygon;
A histogram is the graphical equivalent of a frequency distribution; it is a bar chart where
continuous data on a random variable’s observations have been grouped into a frequency
distribution.
A frequency polygon is the line graph equivalent of a frequency distribution; it is a line graph
that joins the frequency for each interval, plotted at the midpoint of that interval.
6
i) define, calculate, and interpret measures of central tendency, including the
population mean, sample mean, arithmetic mean, geometric mean, weighted mean,
median, and mode;
Measures of central tendency summarize the location on which the data are centered.
Population Mean: calculated as

1
N
N
X
i 1
i
where there are N members in the population and each observation is Xi i =1, 2, …N.
Sample Mean: calculated as
X 
1 n
 Xi
n i 1
where there are n observations in the sample and each observation is Xi i =1, 2, …n. It is also
the arithmetic mean of the sample observations.
n
Weighted Mean: calculated as
X w   wi X i
i 1
where there are n observations, each observation is Xi, and the weight associated with each
observation is wi i =1, 2, …n. If wi = 1/n, then this is the sample mean. If wi is the probability
of Xi occurring then this is the expected value.
1
G  n X 1  X 2    X n or ln G  ln  X 1  X 2    X n 
Geometric Mean: calculated as
n
where there are n observations and each observation is Xi.
i) define, calculate, and interpret measures of central tendency, including the
population mean, sample mean, arithmetic mean, geometric mean, weighted mean,
median, and mode;
Median: calculated as the middle observation in a group that has been ordered in either
ascending or descending order. In an odd-numbered group this is the (n+1)/2 position. In an
even numbered group it is the average of the values in the n/2 and (n+1)/2 positions.
Mode: is the most frequently occurring value in the distribution. A distribution may have one,
more than one, or no mode.
j) distinguish between arithmetic and geometric means;
Geometric mean is always less than or equal to the arithmetic mean because of Jensen’s
inequality. In general the difference between the two measures increases with the variability
in the period-by-period observations. Geometric mean is particularly useful in calculating
returns over multiple periods when interest rate compounding is a factor.
k) (*) describe and interpret quartiles, quintiles, deciles, and percentiles;
Percentiles: First rank the observations, from lowest to highest. Then divide the ranked
distribution into equal size sections with each section equal to the desired percentage of the
total observations. i.e.
Quartiles: divide the data into four sections with each section containing 25% of the data.
Quintiles: divide the data into five sections with each section containing 20% of the data.
7
Deciles: divide the data into 10 sections with each section containing 10% of the data.
Percentiles are useful as a way to rank a large number of observations into a smaller, more
tractable, set of ranked groups based on the distribution of the observations in the sample.
l) define, calculate, and interpret (1) a portfolio return as a weighted mean, (2) a
weighted average or mean, (3) a range and mean absolute deviation, and (4) a
sample and a population variance and standard deviation;
Range: is the difference between the maximum and minimum values in a dataset.
Mean Absolute Deviation: MAD 
1 n
 X i  X is the average of the data’s absolute
n i 1
deviations from the mean.
1 N
1 N
2


 X i 2   2 is the average of the
X





i
N i 1
N i 1
population’s squared deviations from the mean. The population standard deviation is simply
the square root of the population variance.
Population Variance:  2 
n
1
X i  X 2 is the average of the sample data’s squared

n  1 i 1
deviations from the sample mean. The sample standard deviation is simply the square root of
the sample variance.
Sample Variance: s 2 
m) calculate the proportion of items falling within a specified number of standard
deviations of the mean, using Chebyshev’s inequality;
Chebyshev’s Inequality: Let k be any positive constant greater than 1. The proportion of observations
within k standard deviations of the mean is at least [1 – 1/k2].
For example, if k=2 then the proportion of observations within 2 standard deviations of the mean is at
least 1 – 1/22 = .75
n) define, calculate, and interpret the coefficient of variation;
Coefficient of variation, CV  s
X
. CV shows relative dispersion. If X is returns on an asset then CV
shows the amount of risk (measured by sample standard deviation s) for every % of mean return on the
asset. The lower an asset’s CV, the more attractive it is in risk per unit of return.
o) define, calculate, and interpret the Sharpe measure of risk-adjusted performance;
Sharpe measure, SM 
r
p
 rf

 p . SM is a more precise return-risk measure as it takes into
account an investor can earn the risk-free rate, rp, without bearing any risk. Hence a portfolio’s risk
(measured by its standard deviation p) must be compared to its return in excess of the risk-free rate
rp  r f . The higher is SM, the better the return-risk tradeoff on the portfolio for an investor.


8
p) describe the relative locations of the mean, median, and mode for a nonsymmetrical
distribution;
See next LOS for definitions of each type of skewness below.
i.
Symmetrical distribution: Mean = Median = Mode
ii.
Positively-skewed distribution: Mean > Median > Mode
iii.
Negatively-skewed distribution: Mean < Median < Mode
q) define and interpret skewness and explain why a distribution might be positively or
negatively skewed;
A frequency distribution that is not symmetric is called skewed.
A positively-skewed distribution is characterized by many small losses but a few extremely large
gains, and so it has a long tail on the right side of the distribution.
A negatively-skewed distribution is characterized by many small gains but a few extremely large
losses, and so it has a long tail on the left-hand side of the distribution.
Skewness arises as a result of the properties of asset prices and returns. For example a share price can
never be negative – there is a lower limit on the asset’s returns (-100%) but no theoretical limit on its
upper limit – therefore an asset’s return may be positively-skewed.
r) define and interpret kurtosis and explain why a distribution might have positive
excess kurtosis;
A frequency distribution that is more or less peaked than a Normal distribution is said to exhibit
kurtosis. If the distribution is more peaked than a Normal (i.e. exhibits “fat tails”) it is leptokurtic. If it
is less peaked than a Normal it is called platykurtic.
Positive excess kurtosis, i.e. a leptokurtic distribution, means that large positive and negative
deviations from the mean have higher probabilities for occurring than they would under a Normal
distribution. If an portfolio’s returns are leptokurtic then its true risk is higher than the risk suggested
by an analysis that assumes returns are Normally distributed. This is important for Value at Risk
(VAR) calculations that must assume distributions for asset returns in a portfolio.
s) (*) describe and interpret measures of skewness and kurtosis;
N
Excess Kurtosis measure is approximately equal to 
X
1
i
X
i
X
4
3
n
s4
Measure is free of units but is always positive regardless of sign of the deviation of the
observation from the mean. A distribution with “fat tails”, higher numbers of large deviations,
is leptokurtic and will have an excess kurtosis measure that is positive. The Normal
distribution has excess kurtosis equal to zero.
i 1
n
Relative Skewness measure is equal to 
n
 n  1 n  2 
 X
i 1
3
s3
Measure is free of units but preserves sign of the deviation of the observation from the mean.
A distribution that has many small positive deviations from the mean but a few large negative
9
deviations from the mean is a negatively-skewed distribution has a relative skewness measure
that is negative.
t) explain why a semi-logarithmic scale is often used for return performance graphs.
A semi-log scale graphs the log of the asset’s price on the vertical axis and time (arithmetic scale) on
the horizontal axis. Equal vertical movements thus reflect equal percent changes in the asset’s price,
i.e. its return. Thus it is easy to compare returns across a variety of periods with a semi-log graph,
while it is difficult to do so if arithmetic scales were used on both axes.
C. “Probability Concepts”
The candidate should be able to
a) (*) define a random variable, an outcome, an event, mutually exclusive events, and
exhaustive events;
A random variable is a quantity whose outcome is uncertain. Mutually exclusive events mean one and
only one event can occur at any time. Exhaustive events one of the events must occur, i.e. that the
listed events cover all possible outcomes
b) explain the two defining properties of probability;
Two defining properties of Probability.
i.
Probability of any event E is a number between 0 and 1, 0  PE   1.
ii.
Sum of the probabilities of any list of mutually exclusive and exhaustive events equals 1.
c) distinguish among empirical, a priori, and subjective probabilities;
Empirical probability is when the probability of an event occurring is estimated from data, usually in
the form of a relative frequency.
A priori probability is when probability of an event is deduced by reasoning about the structure of the
problem itself.
These first two approaches to probability are sometimes referred to as objective probabilities because
they should not vary from person to person.
Subjective probability is when the probability of an event is based on a personal assessment without
reference to any particular data.
d) describe the investment consequences of probabilities that are inconsistent;
Inconsistent probabilities create profit opportunities because investors can buy and sell assets at the
resulting inconsistent prices in ways that allow them to achieve profits on average. These buying and
selling decisions should eliminate inconsistent prices, and probabilities, in the market.
e) distinguish between unconditional and conditional probabilities;
Unconditional or marginal probability, P(A), is the probability of event A occurring without
reference to any other event.
Conditional probability, P(A|B), is the probability of event A occurring given that event B is
known to already have occurred. P(A|B) = P(AB)/P(B) if P(B) ≠ 0. Conditional probabilities
10
are important in tests of market efficiency, where event B is some piece of public or private
information that becomes available to the market at some point of time.
f) define a joint probability;
Joint probability, P(AB), is the probability of both event A and event B occurring together.
g) calculate, using the multiplication rule, the joint probability of two events;
Multiplication Rule for probabilities - Joint probability, P(AB), is
P(AB) = P(A|B) P(B) = P(B|A) P(A)
h) calculate, using the addition rule, the probability that at least one of two events will
occur;
Addition Rule for probabilities – Given events A and B, the probability that A or B occurs is
equal to:
P(A or B) = P(AU B) = P(A) + P(B) - P(AB)
If you don’t see this result then construct a Venn diagram of Events A and B that share some
overlap. The sum P(A) + P(B) counts P(AB) twice, so it must be subtracted.
i) distinguish between dependent and independent events;
Definition of Independent Events – Two events A and B are independent if and only if:
P(A|B) = P(A) or equivalently P(B|A) = P(B)
If two events are dependent, then the occurrence of one of the events is related to the
probability of the occurrence of the other event.
j) calculate a joint probability of any number of independent events;
Multiplication Rule for Independent Events - Joint probability of independent events A1, A2,
… Am is:
P(A1A2…Am) = P(A1)P(A2)… P(Am-1)P(Am)
Think about calculating the probability of getting 10 heads on ten coin flips.
k) calculate, using the total probability rule, an unconditional probability;
Total Probability Rule - Probability of event A is:
i.
P(A) = P(A|S) P(S) + P(A|SC) P(SC)
ii.
P(A) = P(A|S1) P(S1) + P(A|S2) P(S2)… + P(A|Sm) P(Sm) where S1, S2, … , Sm
are mutually exclusive and exhaustive events.
l) define, calculate and explain expected value, variance and standard deviation;
Expected Value of a random variable is the probability-weighted average of the possible
outcomes of the random variable. Expected Value of random variable X is calculated as:
m
EX    Pxi xi
i 1
11
Variance of a random variable is the expected value of squared deviations from the random
variable’s expected value.
Standard deviation is the square root of the variance. Used as a measure of risk shows
dispersion of possible outcomes around expected level of outcomes.
 2   2  X   E  X  EX 2


m) explain the use of conditional expectation in investment applications;
It is important to use all relevant information in making forecast of investment returns and
risk. We refine our expectations as new information becomes available, leading to conditional
expectations, and the expected value of a random variable given an event or scenario S is.
E X S   Px1 S x1  Px 2 S x 2    Px n S x n   Pxi S xi
n
i 1
n) calculate an expected value using the total probability rule;
Expected value using the Total Probability Rule - Probability of event A is:
i.
E(X) = E(X|S) P(S) + E(X|SC) P(SC)
ii.
E(A) = E(X|S1) P(S1) + E(X|S2) P(S2)… + E(X|Sm) P(Sm) where S1, S2, … , Sm
are mutually exclusive and exhaustive events.
Allows us to state unconditional expected value in terms of conditional expected values for a
set of mutually exclusive and exhaustive events.
o) define, calculate, and interpret covariance;
Covariance between two random variables X and Y is defined as
cov X , Y     X , Y    XY  E X  EX Y  EY 
A negative covariance between X and Y means that when X is above its mean its is likely that
Y is below its mean value. If the covariance of the two random variables is zero then on
average the values of the two variables are unrelated. A positive covariance between X and Y
means that when X is above its mean its is likely that Y is above its mean value.
The covariance of a random variable with itself, its own covariance, is equal to its variance.
cov X , X     X , X    XX  E X  EX  X  EX 
p) explain the relationship among covariance, standard deviation, and correlation;
Correlation between two random variables X and Y measured as:
  X , Y   corr ( X , Y )  cov X , Y  
X Y
Correlation takes on values between –1 and +1. Correlation is a standardized measure of how
two random variables move together, i.e. correlation has no units associated with it.
A correlation of 0 means there is no straight-line (linear) relationship between the two
variables. Increasingly positive (negative) correlations indicate an increasingly strong positive
(negative) linear relationship between the variables. When the correlation equals 1 (-1) there
is a perfect positive (negative) linear relationship between the two variables.
12
q) calculate the expected return and the variance for return on a portfolio;
Portfolio consisting of two assets A and B, wA invested in A. Asset A has expected return rA and
variance  A2 . Asset B has expected return rB and variance  B2 . The correlation between the two
returns is  AB .
Portfolio Expected Return: (There is an error in the book!!!!)
ErP   wA rA  1  wA rB
Portfolio Variance:
 2 rP   w A2  A2  1  w A 2  B2  2w A wB CovrA , rB  or
 2 rP   w A2  A2  1  w A 2  B2  2w A wB  AB A B
r) calculate covariance given a joint probability function;
Covariance between two random variables RA and RB using the joint probability function for RA and
RB is: (There is an error in the book!!!)
covRiA , R Bj    PRiA , R Bj RiA  ER A R Bj  ER B 
i
j
s) calculate an updated probability, using Bayes’ formula;
Bayes’ formula uses the occurrence of an event to infer the probability of the scenario generating it.
Essentially the individual is updating her or his beliefs concerning the causes that may have produced
the new observation.
One begins with a set of prior probabilities for an event of interest and upon receiving the new
information, the rule for updating your probability of the event is:
Updated probability of the event given the new information =
Probability of the New Information given Event x Prior Probability of Event
Unconditional Probability of the New Information
t) calculate the number of ways a specified number of tasks can be performed using
the multiplication rule of counting;
Multiplication Rule of Counting: If one thing can be done n1 ways, and a second task can be done,
given the first, n2 ways and a third task can be done, given the first two things, n3 ways, and so on for k
things total, then the total number of ways the k things can be done is n1 x n2 x n3 x …x nk
u) solve counting problems using the factorial, combination, and permutation notations;
Factorial Notation: n-factorial = n! = n x (n-1) x (n-2) x (n-3) x.. x 2 x 1.
Multinomial Formula: (used for labeling problems in assigning k different labels to n members, with
n1 labels of the first type, n2 labels of the second type, etc with n = n1 + n2 +…+ nk)
n!
n1!n2 !   nk !
13
u) solve counting problems using the factorial, combination, and permutation notations;
Combination Formula: (used for problems in choosing r objects from n total objects, where the order
of the r objects listed does not matter)
n
 n
n!
C r    
 r  n  r !  r!
Permutation Formula: (used for problems in choosing r objects from n total objects, where the order of
the r objects listed does matter)
n
Pr 
n!
n  r !
v) distinguish between problems for which different counting methods are appropriate;
Use the following set of questions to determine which approach to take.
i.
Does the problem have a finite number of outcomes? If yes, then one of the approaches
here may work. If no, then number of outcomes is infinite and these counting tools do not apply.
ii.
Are we assigning n members to n slots? If yes then use n factorial, n! If no then continue.
iii.
Do we need the number of ways to assign 3 or more labels to each member of a group? If
yes then use the multinomial formula. If no, then continue.
iv.
Do we need to count the number of ways to choose r objects from a total of n objects
when the order of the r objects does not matter? If yes, then use the combination formula. If no,
then continue.
v.
Do we need to count the number of ways to choose r objects from a total of n objects
when the order of the r objects does matter? If yes, then use the permutation formula. If no then
continue.
vi.
Can the multiplication rule be used? If not then count the possibilities one by one.
w) calculate the number of ways to choose r objects from a total of n objects, when the
order in which the r objects is listed does or does not matter.
Combination Formula: (used for problems in choosing r objects from n total objects, where the order
of the r objects listed does not matter)
n
 n
n!
C r    
 r  n  r !  r!
D. “Common Probability Distributions”
The candidate should be able to
a) explain a probability distribution;
Probability distribution specifies the probabilities of the possible outcomes of a random variable.
b) distinguish between and give examples of discrete and continuous random
variables;
14
Discrete random variable can take on at most a countable number of possible values, such as coin flip
or rolling dice.
Continuous random variable can take on an uncountable (infinite) number of possible values, such as
asset returns or temperatures.
c) describe the range of possible outcomes of a specified random variable;
Clearly this depends on the random variable that is specified. For example
i.
Coin toss for example takes on {Head, Tail}.
ii.
Rolling a die takes on (1,2,3,4,5,6}.
iii.
Share returns (in percent) lie in the interval [-100, +∞ ).
d) define a probability function and state whether a given function satisfies the
conditions for a probability function;
Probability function specifies the probability that the random variable takes on a specific value: P(X =
x). To determine if a given function is a probability function it must fulfill the two key properties in
the next LOS.
e) state the two key properties of a probability function;
Two Key Properties of a Probability Function.
i.
0 ≤ p(x) ≤ 1 because a probability lies between 0 and 1.
ii.
The sum of probabilities p(x) over all values of X equals 1.
f) define a cumulative distribution function and calculate probabilities for a random
variable, given a cumulative distribution function;
Cumulative Distribution function specifies the probability that the random variable X is less than or
equal to a particular value x, P(X ≤ x). For a discrete random variable this is the sum of the
probabilities for all values less than or equal to x.
g) (*) define a probability density function;
Probability Density function (pdf) specifies the probability that a continuous random variable takes on
a specific value.
i.
Pdf of a number is a function 0  p  x   1 .
ii.
Integral over the range of the r.v. equals 1,
x
 p  x  dx  1 .
x
h) define a discrete uniform random variable and calculate probabilities, given a
discrete uniform probability distribution;
Discrete Uniform Random Variable: The uniform random variable X takes on a finite number of
values, k, and each value has the same probability of occurring, i.e. P(xi) = 1/k for i = 1,2,…,k.
Examples of simple uniform random variables:
i.
Coin flip:
Prob(head) = ½
Prob(tail) = ½
ii.
Die: Prob(Die shows 1) = 1/6, Prob(Die shows 2) = 1/6, …Prob(Die shows 6)=1/6
15
i) define a binomial random variable and calculate probabilities, given a binomial
probability distribution;
Bernoulli random variable is a binary variable that takes on one of two values, usually 1 for success or
0 for failure. Think of a single coin flip as an example of a Bernoulli r.v.
Binomial random variable: X ~ B(n, p) is defined is the number of successes in n Bernoulli random
trials where p is the probability of success on any one Bernoulli trial. The probability distribution for a
Binomial random variable is given by:
 n
n!
n x
n x
px n, p     p x 1  p  
p x 1  p 
x!n  x !
 x
The distribution is symmetric when p = .5, but otherwise it is skewed.
j) calculate the expected value and variance of a binomial random variable;
Binomial X ~ B(n,p) Expected Value:
E X   np
Binomial X ~ B(n,p) Variance:
 2  X   np1  p 
16
k) construct a binomial tree to describe stock price movement and calculate the
expected terminal stock price;
Evolution of Share Price over the Week
Present
Tuesday Close
Wednesday Close
Prob. of up = p = 0.6
Stock Price = S = $100
Amt of Move = $2
Thursday Close
p = 0.216
S = $106
p = 0.36
S = $104
Possible Movements of Stock Price
Over the Week
p = 0.144
S = $102
p=
S=
0.6
$102
p = 0.144
S = $102
p = 0.24
S = $100
p = 0.096
S = $98
S=
$100
p = 0.144
S = $102
p = 0.24
S = $100
p = 0.096
S = $98
p=
S=
0.4
$98
p = 0.096
S = $98
p = 0.16
S = $96
p = 0.064
S = $94
Expected value
Variance
Standard deviation
$100.40
$3.84
$1.96
$100.80
$7.68
$2.77
$101.20
$11.52
$3.39
17
l) describe the continuous uniform distribution and calculate probabilities, given a
binomial probability distribution;
Continuous Uniform Random Variable: X is a random variable that has equal probabilities for taking
on values in the interval [a , b].
Continuous Uniform Probability Density:
 1

f x    b  a
 0
for a  x  b
otherwise
Continuous Uniform Cumulative Distribution:
 0
x  a
F x   
b  a
 1
Continuous Uniform r.v. Expected Value:
xa
for a  x  b
xb
E  X   a  b
2
Continuous Uniform r.v. Variance:
 2  X   b  a  12
2
Second part of the LOS is badly written!!! I believe what they mean here is that if an event occurs
when the value of the continuous random variable falls below a critical level, say Xmin, then you can
use the Cumulative distribution function for the uniform r.v. X to calculate the probability the event
will occur for use in a binomial formula as p, the probability of success. This is accomplished as:
pEvent Occurs   F  X min  
X min  a
ba
See example 5-7 on page 242 of the text for an example of this calculation.
m) explain the key properties of the normal distribution;
Normal distribution is a continuous, symmetric probability distribution that is completely described by
two parameters: its mean, μ, and its variance, σ2. Written as N(μ, σ2).
i.
ii.
The normal distribution is said to be bell-shaped with the mean showing its central
location and the variance showing its “spread”.
A linear combination of two or more Normal random variables is also normally
distributed.
n) construct confidence intervals for a normally distributed random variable;
Confidence Intervals for a Normally distributed random variable X ~ N(μ, σ2)
50% Confidence Interval
P(X within the range μ ± (2/3) σ) = .50 or 50%
68% Confidence Interval
P(X within the range μ ± σ) = .68 or 68%
90% Confidence Interval
18
P(X within the range μ ± 1.645 σ) = .90 or 90%
or
P(X within the range X  1.645 ) = .90 or 90% if using sample measures
95% Confidence Interval
P(X within the range μ ± 1.96 σ) = .95 or 95%
or
P(X within the range X  1.96 ) = .95 or 95% if using sample measures
99% Confidence Interval
P(X within the range μ ± 2.58 σ) = .99 or 99%
or
P(X within the range X  2.58 ) = .99 or 99% if using sample measures
o) define the standard normal distribution and explain how to standardize a random
variable;
Standard Normal distribution is a Normal distribution with mean μ=0, and variance σ2=1. A Standard
Normal random variable is usually written as Z ~ N(0, 1).
General Normal random variable X ~ N(μ, σ2) can be standardized to a Standard Normal random
variable Z as Z 
X 

. The resulting variable has mean zero and variance equal to 1.
p) calculate probabilities using the standard normal probability distribution;
You can calculate the probabilities of a normal random variable X ~ N(μ, σ2) taking on a range of
specified values, say a < X < b, directly as the area under the normal curve using the cumulative
Normal distribution function as N(a < X < b| μ, σ2) = N(X < b| μ, σ2) - N( X < a| μ, σ2) .
You should be able to show what this looks like using a diagram of the Normal distribution.
q) distinguish between a univariate and a multivariate distribution;
Univariate distribution describes the probability behavior of a single random variable.
Multivariate distribution describes the probability behavior for a group of related random variables.
r) explain the role of correlation in the multivariate normal distribution;
Multivariate Normal distribution for n related random variables is completely defined by the means of
each normal variable, the variance of each normal variable, and the n(n-1)/2 distinct correlations
between the random variables. These correlations describe how the probability behaviors of the
random variables are related to one another, i.e. how their deviations from their respective means are
related across the variables on average.
s) define shortfall risk;
Shortfall risk is the risk that a portfolio’s value or return will fall below some specified minimum
acceptable level over some specified period. Take Rmin as the minimum acceptable level for portfolio
returns. Shortfall risk is P(Rp < Rmin), where Rp is the portfolio’s random return.
t) calculate the safety-first ratio and select an optimal portfolio using Roy’s safetyfirst
criterion;
Roy’s Safety-First Criterion is measured as SFRatio = [E(Rp) – Rmin]/2
19
Three steps to choosing an optimal portfolio using the safety-first criterion:
i.
Calculate each portfolio’s SFRatio.
ii.
Evaluate the probability from the standard normal distribution at this level, i.e. the
probability the portfolio will deliver less than Rmin that is calculated from the Standard Normal
Cumulative distribution as N(-SFRatio).
iii.
Choose the portfolio with the lowest probability, i.e. the highest SFRatio value.
u) explain the relationship between the lognormal and normal distributions;
A random variable is lognormally distributed if the natural log of the random variable follows a
Normal distribution.
v) distinguish between discretely and continuously compounded rates of return;
Continuously compounded rate of return, given the holding period yield, is the natural log of 1 + the
holding period return or equivalently the natural log of the ending price divided by the beginning
price.
w) calculate a continuously compounded return, given a specific holding period return;
Given specific holding period return, Rt,t+1, associated continuously compounded return, rt,t+1, is:
P
rt ,t 1  ln 1  Rt ,t 1   ln  t 1 
Pt 

x) explain Monte Carlo simulation and historical simulation and describe their major
applications;
Monte Carlo simulation involves the use of a computer and models of the world to find approximate
solutions to complex problems.
In finance Monte Carlo simulation generally involves identifying risk factors associated with the
problem and specifying probability distributions for them. Repeated random sampling from these
distributions is then used to simulate the risk factors. It can be used to evaluate how sensitive a model
is to changes in the specification of its parameters. It can also be used to experiment with changes in
policies or conditions to investigate the impacts on important risk factors.
Historical simulation involves the use of a repeated sampling from historical data series to establish
the behavior of important risk factors in generating returns.
Main limitation of historical simulation is that it can only reflect risks that exist in the historical series
being used. It is therefore difficult to use it for “what-if” experiments or to investigate the impact of
“rare” events.
20
J. Veitch: Level I Study Session #2 Questions
BEST STUDY TIP: You can only learn this stuff by doing it. There are lots of
examples in the text, DO THEM!!!!
1.
Which measurement scale allows values to be added or subtracted in a meaningful way?
A.
B.
C.
D.
2.
Interval scale.
Nominal scale.
Ordinal scale.
None of the above.
An analyst developed the following probability distribution of the rate of return for a
common stock:
Scenario
Recession
Normal
Boom
Probability
0.20
0.60
0.20
Rate of return
-0.05
0.10
0.25
The standard deviation of the rate of return is closest to:
A.
B.
C.
D.
3.
Which of the following orderings is correct for a distribution that is negatively skewed?
A.
B.
C.
D.
4.
0.0090.
0.1003.
0.0949.
0.0010.
Mean = Median = Mode
Mean < Median < Mode
Mean < Mode < Median
Mean > Median > Mode
Company A’s returns exhibit a variance of 0.0225 and a mean return of 0.12. Assume the
risk free rate is 0.07, what is the Sharpe ratio and Coefficient of Variation for Company A?
A.
B.
C.
D.
Sharpe ratio
0.333
2.222
3.00
0.333
Coef. of Variation
0.80
0.1875
1.25
1.25
21
5.
An individual deposits $100,000 at the beginning of each of the next 10 years, starting
next year, into an account paying 8 percent interest compounded annually. The present
value of the amount of money in the account at the end of 15 years will be closest to:
A.
B.
C.
D.
6.
An individual deposits $100,000 today into an account paying 8 percent interest
compounded annually. They wish to know the number of years it will take for the account
balance to reach $250,000. The number of years required is closest to:
A.
B.
C.
D.
7.
$736,000.
$368,004.
$501,654.
$454,383.
The correlation coefficient between two random variables is best described as:
A.
B.
C.
D.
9.
10.
12.
22.
8.
An individual sells their consulting company for a series of payments: $100,000 at the end
of the next three years, followed by $50,000 per year for the next seven years. If the
interest rate is 6%, the present value of the deal is closest to:
A.
B.
C.
D.
8.
$981,815.
$671,008.
$1,250,000.
$855,948.
A measure of how the two random variables move together.
The variation of the deviations from their respective means.
A measure of the non-linear association between two variables.
A standardized measure of how the two random variables move together.
The unconditional probability that a company will have negative earnings is 30%. An
analyst calculates the probability of a company having negative earnings in a recession is
30%. The probability of a recession is 60% and the probability of a boom is 40%.
Calculate the probability that the company will have negative earnings in a boom:
A.
B.
C.
D.
20%.
24%.
30%.
70%.
22
10.
An investment strategy has an expected return of 10 percent and a standard deviation of 12
percent. If investment returns are normally distributed, the probability of earning a return
more than 30 percent is closest to:
A.
B.
C.
D.
11.
A portfolio manager has a list of 12 recommended stocks to hold in her portfolio. She is
constrained to hold exactly 6 stocks in her portfolio but can choose among any on the
recommended list. How many possible portfolios are possible given these constraints?
A.
B.
C.
D.
12.
12!.
6!.
12!/6!.
12!/(6! 6!)
A company’s earnings each quarter are distributed as a continuous uniform random
variable on the interval (-50, 100). Assuming that earnings each quarter are independent of
the previous quarter, what is the probability that the company reports three consecutive
quarters of negative earnings?
A.
B.
C.
D.
13.
5%.
16%.
10%.
2.5%.
12.5%.
1.5625%.
6.75%.
3.70%.
You receive the following information regarding two portfolios and your customer’s
requirements:
Expected Return
Std. Deviation of
Return
Portfolio A
10%
Portfolio B
20%
10%
20%
In addition you are told the risk-free return is 5% and your customer’s minimum required
return is 6%. On the basis of Roy’s Safety First measure (SFR) which of the following
statements is correct?
A. Your customer is better off with Portfolio A because its SFR of 0.40 is better than the
SFR of 0.70 for Portfolio B.
B. Your customer is better off with Portfolio B because its Roy’s Safety First Measure of
0.70 is better than the SFR of 0.40 for Portfolio A.
C. Your customer is better off with Portfolio A because its Roy’s Safety First Measure of
0.50 is better than the SFR of 0.75 for Portfolio B.
D. Your customer is better off with Portfolio B because its Roy’s Safety First Measure of
0.75 is better than the SFR of 0.50 for Portfolio A.
23
14.
An asset has an expected return equal to 8% and a standard deviation 12%. What is the
standardized value of an actual return of 14% on the asset?
A.
B.
C.
D.
15.
6%.
1.5
0.50
0.5%.
You are looking at an asset with a mean return of 15% and a standard deviation of 5%.
Using Chebyshev’s inequality as an approximation, what is proportion of the asset’s
returns should fall in the interval (0%, 30%)?
A.
B.
C.
D.
75%.
33%.
89%.
11%.
J. Veitch: Level I Study Session #2 Answers
1.
A.
LOS Level I Study Session 2-1.B.c
2.
C.
LOS Level I Study Session 2-1.C.l & .m
Calculate expected return as: .2(-.05) + .6(.10) + .2(.25) = 0.10
Calculate variance as: 2 = .2(-.05 - .10)2 + .6(.10 - .10)2 + .2(.25 - .10)2 = 0.009
Calculate standard deviation as:  =sqrt(0.009) = 0.0949



3.
B.
LOS Level I Study Session 2-1.B.o
4.
D.

LOS Level I Study Session 2-1.B.m & .n
Calculate CV  s  0.0225 0.12  0.15 0.12  1.25
X

Calculate SR 

LOS Level I Study Session 2-1.A.c
A
1  100, 000 
1 
Simple1-year annuity PV  1 

1 
  $671, 008
N
r  1  r  
.08  1.0810 




5.
6.
B.
B.
r
p
 rf

0.12  0.07
 0.05
 0.333
p 
0.15
0.0225
LOS Level I Study Session 2-1.A.b

Use formula for PV of single payment and solve for N. PV 
1
1  r N
FV
24
7.
1

  N ln 1.08 or N = 11.9

$100,000 

LOS Level I Study Session 2-1.A.g
Can find PV of this cash stream as PV of 10-year annuity of $50,000 per year plus
PV of annuity of $50,000 for 3 years.
C.
1.08
N
10 year
i. PVannuity

3 year
ii. PVannuity

$250,000 or ln 250
100
50,000 
1 
1 
  $368,004
.06  1.0610 
50,000 
1 
1 
  $133,650
.06  1.063 
iii. PVCashFlows = $501,654
8.
D.
9.
C.
LOS Level I Study Session 2-1.C.p & .q





10.
A.
LOS Level I Study Session 2-1.D.m


11.
D.

12.
LOS Level I Study Session 2-1.C.k
This is a total probability question where you must solve for one of the
conditional probabilities in the calculation.
Event A = Negative Earnings, Event S = recession, Event SC = boom
P(A) = .30, P(S) = .60, P(SC) = .40, P(A|S) = .30, P(A|SC) = ?
P(A) = P(A|S)P(S) + P(A|SC)P(SC) or .30 = (.30)(.60) + P(A|SC)(.40)
P(A|SC) = [.30 - .18]/.4 = .30 or 30%
D.

X 
30  10
 1.667 .

12
P(Z > 1.65) = 5%. Probability of being more than 1.65 st. dev. from its mean is
5% for a Normal distribution.
Calculate the standardized normal r.v. -- Z 

LOS Level I Study Session 2-1.D.v
Note that in constructing the portfolio the order of the companies chosen is not
important, therefore the correct formula is the combination formula.
LOS Level I Study Session 2-1.D.k
This is the continuous uniform r.v. to binomial r.v. example I mentioned in class.
i. Earnings are continuous uniform r.v. on (-50, 100).
ii. Probability of negative earnings is given by cumulative distribution -P(X<0) = [0 – (-50)]/[100 – (-50)] =1/3
iii. New binomial r.v. is sign of earnings. P(earnings negative) = 1/3
P(earnings positive) = 2/3
iv. Prob. of three consecutive quarters negative earnings = {1/3)(1/3)(1/3) =
.0370 or 3.70%
25
13.
B.




14.
C.


15.
C.


LOS Level I Study Session 2-1.D.s
Roy’s Safety First Ratio for a portfolio is SFRp = [E(Rp) – Rmin]/p. Note that
larger values are better here because they indicate that the minimum return is a
larger number of standard deviations away from the expected return, hence
returns below the required minimum are less likely to occur.
Portfolio A SFR = [10 – 6]/10 = 0.40
Portfolio B SFR = [20 – 6]/20 = 0.70
Hence Portfolio B is preferred over Portfolio A on the basis of the SFR criterion.
LOS Level I Study Session 2-1.D.n
Standardizing a random variable involves calculating Z = (X – )/, where X is
the actual value of the variable.
In this example Z = (14 – 8)/12 = 6/12 = 0.50.
LOS Level I Study Session 2-1.B.l
Chebyshev’s Inequality states that approximately (1 – 1/k2)% of observations fall
within k standard deviations of the mean, as long as k > 1.
Here k = 3, so (1-1/9) = 81% of observations should lie in the interval.
26
Permission to print questions & answers from past AIMR Study Guides has been granted as indicated by the following statements.
Reprinted with permission from the 1994 Level I CFA® Study Guide. Copyright (1994), Association for Investment
Management and Research, Charlottesville, VA. All rights reserved.
Reprinted with permission from the 1995 Level I CFA® Study Guide. Copyright (1995), Association for Investment
Management and Research, Charlottesville, VA. All rights reserved.
Reprinted with permission from the 1996 Level I CFA® Study Guide. Copyright (1996), Association for Investment
Management and Research, Charlottesville, VA. All rights reserved.
Reprinted with permission from the 1997 Level I CFA® Study Guide. Copyright (1997), Association for Investment
Management and Research, Charlottesville, VA. All rights reserved.
Reprinted with permission from the 1998 Level I CFA® Study Guide. Copyright (1998), Association for Investment
Management and Research, Charlottesville, VA. All rights reserved.
Reprinted with permission from the 1999 Level I CFA® Study Guide. Copyright (1999), Association for Investment
Management and Research, Charlottesville, VA. All rights reserved.
Reprinted with permission from the 2000 Level I CFA® Study Guide. Copyright (1999), Association for Investment
Management and Research, Charlottesville, VA. All rights reserved.
Reprinted with permission from the 2001 Level I CFA® Study Guide. Copyright (2000), Association for Investment
Management and Research, Charlottesville, VA. All rights reserved.
Reprinted with permission from the 2002 Level I CFA® Study Guide. Copyright (2001), Association for Investment
Management and Research, Charlottesville, VA. All rights reserved.
Reprinted with permission from the 2003 Level I CFA® Study Guide. Copyright (2002), Association for Investment
Management and Research, Charlottesville, VA. All rights reserved.
27