Portfolio Theory
Dr. Andrea Rigamonti
andrea.rigamonti@econ.muni.cz
Lecture 3
Content:
• Statistics notions
Statistics notions
A random variable (r.v.) is one whose value depends on the
outcome of a random experiment.
• A discrete r.v. can only take certain specific values.
• A continuous r.v. can take any value (inside a certain interval)
A probability distribution is a function that assigns a certain
probability that each possible outcome will occur.
• For a discrete r.v. a probability mass function (or probability
distribution function) is the function that links each outcome
with the corresponding probability.
• For a continuous r.v. we have a probability density function
(pdf), and the probability of the events is given by the area
under a function (i.e. by its integral).
Statistics notions
The most popular continuous distribution is the
Gaussian (or normal) distribution. Characteristics:
• Symmetric
• Unimodal
• Completely described by its mean µ and variance σ2
• Any linear transformation of a normally distributed r.v.,
or any linear combination of independent normally
distributed r.v., is itself normally distributed.
The pdf of a normally distributed r.v. is given by:
𝑓 𝑦 =
1
2𝜋𝜎
𝑒−(𝑦−𝜇)2/2𝜎2
Statistics notions
The pdf of a normal r.v. has a bell shape:
• The area under the pdf measures the probability
• Probability is measured in the range [0,1], so the entire area
under the curve (the sum of probability of all events) is 1
• The probability of an exact value is zero (a line has area 0)
• We can compute the probability of an interval
Statistics notions
A standard normal distribution is a normal distribution with
mean µ = 0 and standard deviation 𝜎 = 1.
Given a normally distributed r.v. 𝑌, a standard normally
distributed r.v. is given by:
𝑍 =
𝑦 − 𝜇
𝜎
~𝑁(0,1)
The cumulative distribution function (or cumulative density
function), cdf, of a continuous r.v. measures the probability
that the r.v. is less (or equal to) a certain value.
The cdf 𝐹(𝑦) and is given by the integral of the pdf:
𝐹 𝑦 = 𝑃 𝑌 ≤ 𝑦 = න
−∞
𝑦
𝑓(𝑡) ⅆ𝑡
Statistics notions
The cdf of a normally distributed r.v. has a sigmoid shape:
• It is given by:
F 𝑦 = 𝑃 𝑌 ≤ 𝑎 = ‫׬‬−∞
𝑎 1
2𝜋𝜎
𝑒−(𝑦−𝜇)2/2𝜎2
ⅆ𝑦
• There are no exact analytical solutions for this integral
• The values of the cdf is therefore computed numerically for
the standard normal and provided into tables
Statistics notions
The average value of n financial returns 𝑅𝑖 is known as
measure of location or measure of central tendency.
• Mode: the most frequently occurring value
• Median: the middle value when the elements are arranged
in ascending order
• Arithmetic mean:
𝑅 𝐴 =
1
𝑛
෍
𝑖=1
𝑛
𝑅𝑖
• Geometric mean:
𝑅 𝐺 =
𝑛
ෑ
𝑖=1
𝑛
(1 + 𝑅𝑖) − 1
Statistics notions
With log returns 𝑅 𝐴 and 𝑅 𝐺 are the same
With simple returns 𝑅 𝐺 < 𝑅 𝐴
The geometric mean accounts for compounding and is less
affected by outliers BUT it cannot be used as estimate for
future returns
The most important measures of spread are:
• The range: max – min
• The interquartile range: 𝑄3 − 𝑄1
where 𝑄3 is the
3
4
(𝑛 + 1) 𝑡ℎ value and 𝑄1 is the
1
4
(𝑛 + 1) 𝑡ℎ
Statistics notions
• The variance:
𝜎2 =
1
𝑛 − 1
෍
𝑖=1
𝑛
(𝑅𝑖 − 𝑅 𝐴)2
• The standard deviation σ (square root of the variance)
• The coefficient of variation: 𝐶𝑉 =
𝜎
𝜇 𝐴
The bar in 𝜎2 indicates a (sample) estimate. We divide by
𝑛 − 1 to correct for the loss of a degree of freedom. The
formula for the population variance is:
𝜎2
=
1
𝑛
෍
𝑖=1
𝑛
(𝑅𝑖 − μ 𝐴)2
where μ 𝐴 is the “true” mean.
Statistics notions
In practice, true parameter values are generally not available
and we use estimates. To keep the notation light, the bar is
usually not used when there is no risk of confusion.
Higher moments are important when data are not gaussian
• The skewness measures the asymmetry around the mean:
𝑠 =
1
𝜎3
1
𝑛 − 1
෍
𝑖=1
𝑛
𝑅𝑖 − 𝑅 𝐴
3
• The kurtosis measures the fatness of the tails and how
peaked at the mean the distribution is:
𝑘 =
1
𝜎4
1
𝑛 − 1
෍
𝑖=1
𝑛
𝑅𝑖 − ത𝑅 4
Statistics notions
Normal vs positive (or right) skewed distribution
Normal (k=3) vs leptokurtic (k>3) distribution
Statistics notions
Measures of association evaluate links between variables:
• The covariance between two variables 𝑋 and 𝑌 is:
σ 𝑋,𝑌 = 𝐶𝑜𝑣(𝑋, 𝑌) =
1
𝑛 − 1
෍
𝑖=1
𝑛
(𝑋𝑖 − 𝑋)(𝑌𝑖 − 𝑌)
• The (Pearson) correlation between 𝑋 and 𝑌 is:
ρ 𝑋,𝑌 = 𝐶𝑜𝑟𝑟 𝑋, 𝑌 =
𝐶𝑜𝑣 𝑋, 𝑌
𝑆𝐷 𝑋 𝑆𝐷 𝑌
=
σ 𝑋,𝑌
𝜎 𝑋 𝜎 𝑌
A negative (positive) covariance means that the variables
move, on average, in the opposite (same) direction.
Statistics notions
The correlation takes value between -1 (perfect negative
correlation) and 1 (perfect positive correlation).
σ 𝑋,𝑌 or ρ 𝑋,𝑌 equal to zero indicate no linear correlation,
but not necessarily independence.
Only if 𝑋 and 𝑌 are joint normally distributed, zero
covariance implies that they are independent.
Statistics notions
The variance-covariance matrix, or simply covariance matrix,
is a symmetric matrix that contains all the variances (on the
diagonal) and covariances (off the diagonal) of the data on
which it is estimated:
∑=
𝜎1
2 𝜎1,2 … 𝜎1,𝑛
𝜎2,1
⋮
𝜎2
2
⋮
…
⋱
𝜎2,𝑛
⋮
𝜎 𝑛,1
𝜎 𝑛,2 … 𝜎 𝑛
2
=
𝜎1
2 𝜎1,2 … 𝜎1𝑛
𝜎1,2
⋮
𝜎2
2
⋮
…
⋱
𝜎2,𝑛
⋮
𝜎1,𝑛
𝜎2,𝑛 … 𝜎 𝑛
2
(Σ is the standard notation to indicate the covariance matrix;
do not confuse it with the sum symbol!)
Statistics notions
The relationship between a dependent variable and an
explanatory variable can be described with the linear
regression model with a single regressor:
𝑦𝑖 = 𝛼 + 𝛽𝑥𝑖 + ε𝑖
This equation describes a line that fits the data, plus a
disturbance (or error) term ε𝑖.
To fit the data we choose the parameters 𝛼 and 𝛽 using
the Ordinary Least Squares (OLS): take each vertical
distance from the point to the line, square it and then
minimize the total sum of the areas of squares.
Statistics notions
• For each data point 𝑖 we denote the fitted value obtained from
the regression line as ො𝑦𝑖.
• ොε𝑖 denotes the residual: ොε𝑖 = 𝑦𝑖 − ො𝑦𝑖
• The OLS method minimizes the residual sum of squares (RSS):
෍
𝑖=1
𝑛
ොε𝑖
2
= ෍
𝑖=1
𝑛
(𝑦𝑖 − ො𝑦𝑖)2
Statistics notions
The values of 𝛼 and 𝛽 selected by minimizing the RSS are ො𝛼 and መ𝛽.
The equation of the fitted line is therefore:
ො𝑦𝑖 = ො𝛼 + መ𝛽𝑥𝑖
The OLS estimators for the single regressor case are:
መ𝛽 =
σ𝑖=1
𝑛
(𝑥𝑖 − ҧ𝑥)(𝑦𝑖 − ത𝑦)
σ𝑖=1
𝑛
(𝑥𝑖 − ҧ𝑥)2
ො𝛼 = ത𝑦 − መ𝛽 ҧ𝑥
If 𝐸 ε𝑖 = 0, 𝑣𝑎𝑟 ε𝑖 = 𝜎2 < ∞, 𝑐𝑜𝑣 ε𝑖, ε𝑗 = 0, 𝑐𝑜𝑣(ε𝑖, 𝑥𝑖) = 0,
then estimators ො𝛼 and መ𝛽 determined by OLS are BLUE: Best Linear
Unbiased Estimator.
Statistics notions
The multiple linear regression model with 𝑘 regressors (and
𝑘 − 1 explanatory variables) has the form
𝑦𝑡 = 𝛽1 + 𝛽2 𝑥2𝑖 + 𝛽3 𝑥3𝑖 + ⋯ + 𝛽 𝑘 𝑥 𝑘𝑖 + ε𝑖
• Each 𝛽1, 𝛽2, … , 𝛽 𝑘 is known as partial regression coefficient,
and represents the partial effect of the given explanatory
variable on the explained variable, after holding constant (or
eliminating the effect of) all other explanatory variables.
• The regressor 𝑥1𝑖 is not written explicitly, as it is always
equal to 1.
• The intercept 𝛽1 𝑥1𝑖 = 𝛽1is not an explanatory variable,
although for notational convenience 𝑘 might be referred to
as the “number of explanatory variables”.
Statistics notions
The model can be written in matrix form as
𝒚 = 𝑿𝜷 + 𝜺
where 𝒚 and 𝜺 are a 𝑛 x 1 vectors, 𝜷 is a 𝑘 x 1 vector, and 𝑿 is a
𝑛 x 𝑘 matrix.
• Written in extended form, the equation describing the model is
𝑦1
𝑦2
⋮
𝑦𝑛
=
1
1
𝑥12 𝑥13
𝑥22 𝑥23
⋯ 𝑥1𝑘
⋯ 𝑥2𝑘
⋮
1
⋮ ⋮
𝑥 𝑛2 𝑥 𝑛3
⋮ ⋮
⋯ 𝑥 𝑛𝑘
𝛽1
𝛽2
⋮
𝛽 𝑘
+
ε1
ε2
⋮
ε 𝑛
• The RSS has to be minimized with respect to all the elements of 𝜷
𝑅𝑆𝑆 = ො𝜺′ො𝜺 = ොε1 ොε2 ⋯ ොε 𝑛
ොε1
ොε2
⋮
ොε 𝑛
= ොε1
2
+ ොε2
2
+ ⋯ + ොε 𝑛
2
= ෍
𝑖=1
𝑛
ොε𝑖
2
Statistics notions
• The coefficient estimates are given by
෡𝜷 =
መ𝛽1
መ𝛽2
⋮
መ𝛽 𝑘
= 𝑿′ 𝑿 −1 𝒚
• The variance of the error terms is given by 𝑠2 =
ො𝜺′ො𝜺
𝑛−𝑘
. We have
𝑛 − 𝑘 degrees of freedom because 𝑘 parameters were estimated.
• The covariance matrix of ෡𝜷 is given by
𝑣𝑎𝑟 ෡𝜷 = 𝑠2 𝑿′ 𝑿 −1
• The standard errors of the coefficients in ෡𝜷 are thus given by the
square roots of the terms on the leading diagonal of 𝑣𝑎𝑟 ෡𝜷 .