COMM 472: Quantitative
Analysis of Financial Decisions
Part 1:
Fundamental Properties of Returns
Equity Returns
• Throughout this section, we will make
extensive use of the dynamic properties of
equity returns:
– Returns are dynamic in the following sense:
what happens to returns today may, and in fact
does, effect how returns behave tomorrow.
Preliminary Concepts and
Notation
• We’ll focus our attention on equity returns.
– Random returns are the result of one or both of
the following effects:
• Future dividends (usually random),
• Future prices (usually random).
R t 1
D t 1  Pt 1

Pt
Preliminary Concepts and Notation
• Note that returns are the sum of two components:
– The Dividend Yield:
Dt 1
DYt 
Pt
– The Capital Gain:
cg
t 1
R
Pt 1

Pt
Preliminary Concepts and
Notation
• Note:
– This definition is of a gross return (i.e. net
return = gross return - 1).
– The return relates to a price change over some
specific period of time. That time period can be
an instant, a day, a month, a year, etc.
depending on the application. (We’ll spend a lot
of time in this course talking about the effect of
time horizon on returns.)
Preliminary Concepts and
Notation
• Returns can also be stated in their logarithmic
form:
rt 1

ln(R t 1 )

ln( D t 1  Pt 1 )  ln( Pt )
• So that:
Rt 1  e
rt 1
Multiperiod Returns
• Returns over long horizons show the effect
of compounding:
– For simple returns (no dividends):
R t,t  T
Pt  T Pt 1 Pt  2 Pt  3
Pt  T



Pt
Pt Pt 1 Pt  2
Pt  T 1
– Long-horizon returns are the product of shorthorizon returns.
Multiperiod Returns
• For log returns (no dividends), long-horizon
returns are the sum of short-horizon
returns, since:
Rt ,t T
 Rt 1Rt  2 Rt 3  Rt T
rt 1 rt 2 rt 3
rt T
 e e e e
rt 1 rt  2  rt 3 rt T

e
Multiperiod Returns
• When we choose whether to work with
simple or log returns we make a tradeoff:
– Simple returns are more convenient when aggregating
returns of stocks in portfolios. (The weighted average
of simple returns is the simple return on the portfolio.
This is not true for log returns.)
– Log returns are more convenient when aggregating
returns of a stock or index across time (For statistics, it
is more convenient to work with sums than products.
For example, the sum of normally distributed log
returns is normal whereas the product is not.)
The Distribution of Returns
• Returns are usually random. We can add
some structure to the return process if we
specify a distribution for the returns.
• One common specification is that single
period log returns are normally distributed.
• Is this a reasonable assumption?
Estimating Moments of Returns
• Random variables can be characterized by
their “moments”.
• Typically, we work with the first two
moments – the mean and the variance.
ˆ
ˆ
2

1
rt

T
1
2
ˆ

(rt   )

T
Other Moments
• Higher moments are sometimes of interest:
Sˆ
Kˆ
1
3

(r  ˆ )
3  t
Tˆ
1
4

(r  ˆ )
4  t
Tˆ
– These moments are called the skewness and kurtosis.
For a standard normal distribution (mean 0, variance 1)
the skewness is 0 and the kurtosis is 3.
Dynamic Portfolio Strategies
Part 2:
Dynamic Properties of Returns
Statistical Properties of Long
Horizon Returns
• The properties of long-horizon returns can
be derived from their short-run properties.
• We make use of the following properties of
expectations and variances:
E(rt 1  rt  2 )
var(r t 1  rt  2 )


E(rt 1 )  E(rt  2 )
var(r t 1 )  var(r t  2 )
 2 cov( rt 1 , rt  2 )
Case 1: IID Normal Returns
• If “1-period” log returns are independent
and identically normally distributed then
the mean and variance of the returns
distribution grows proportionally with
the horizon.
– Remember that:
rt , t  T  rt 1  rt  2  rt 3    rt  T
Long-Run Mean Returns
• So:
E(rt , t  T )


• Notice that:
E (rt 1 )  E (rt  2 )    E (rt T )
TE (rt 1 )
– No one return provides information about any
other (returns are independent).
– Each return has the same mean (identically
distributed).
Long-Run Variance
var(r t , t  T )

var( rt 1 )  var( rt  2 )    var( rt  T )

T var (rt 1 )
• Notice that:
– Returns are uncorrelated so that no covariance
terms show up (independent).
– All variances are the same (identically
distributed).
IID Return Dynamics
• This model of return dynamics, although
very simple, has been the mainstay of
“Dynamic Asset Pricing” for many years.
– Black-Scholes pricing is based on this model of
returns.
• Question: Does the data support this
hypothesis?
Empirical Properties of LongHorizon Returns
• Are returns independent?
– Independence has broader implications, but we
will focus on whether or not returns are
uncorrelated.
• Two methods of testing:
– Directly measure “autocorrelations” of returns.
– Examine “variance ratios”.
Empirical Properties of Long
Horizon Returns
• Why do we worry about independence of
returns?
– Dependencies, as we will see, have dramatic
effects on long-run return properties.
– One example: “time diversification”
• If returns from one time period are negatively
correlated with those from another, then long
horizon returns will be less risky than would be
predicted if they were assumed independent.
Autocorrelation
• Definition:
– The autocorrelation coefficient measures the
correlation between two random variables from
a time series (eg. two returns on an index).
– The autocorrelation must be specified with
respect to some “lag length” (the time between
measurements of the random returns):
 k  
cov( rt , rt  k )
var( rt ) var( rt  k )
Autocorrelation
• We’ll assume throughout that return series
are “covariance-stationary”:
– One-period returns at all dates have the same
variance.
– The covariance between returns at different
dates depends only on the lag (k):
cov( rt , rt k ) cov( rt -k , rt )
 k  

var( rt )
var( rt )
Estimating Autocorrelations
• Correlation coefficients can be calculated
by using sample averages:
ˆ (k )


ˆ (k ) 
cov(rt , rt  k )
T -k
1
(rt  ˆ )( rt  k  ˆ )

T t 1
ˆ (k )
2
ˆ
Testing Significance
• In order to assess the statistical significance
of these autocorrelations we need to know
the sampling distributions of the statistics.
– If we assume returns are IID:
~
T ˆ (k )
N (0,1)
~ (k )

T ~
 (k ) ~
T-k

T-k
2
ˆ
ˆ (k ) 
1


(k )
2
(T - 1)
N (0,1)

Testing for IID Returns with
Autocorrelations
• One problem with testing for IID returns
using autocorrelations is that it is not clear
what lags to use to test for zero
autocorrelation.
– If returns are IID all autocorrelations should be
zero.
• One solution is to use a statistic that
summarizes many autocorrelations.
Portmanteau Statistic
• The Q-statistic simply sums the squares of
many autocorrelation statistics:
m
Qm  T  ˆ (k )
2
k 1
• This statistic tests for zero autocorrelation at
all of m lags, giving power to test against a
broad variety of alternative hypotheses for
return dynamics.
Testing for IID Returns with the
Portmanteau Statistic
• The Q-statistic has a “chi-squared”
distribution with m degrees of freedom.
• This distribution can be used to determine,
statistically, whether or not the statistic is
significantly different from zero.
Variance Ratio Statistics
• We saw earlier that if returns are IID, the variance
of long-horizon returns is proportional to the
horizon. This result serves as the basis for using
“Variance Ratios” to test whether or not returns
are IID.
• Definition:
– The q-horizon variance ratio statistic is the ratio of the
variance of the q-period return to the variance of the 1period return, divided by q.
Variance Ratio Statistics
VR( q) 
var(r t , t  q )
q var( rt )
• Note that for IID returns this statistic should
be identically 1 for all horizons.
Variance Ratio Statistics
• eg. With two-period IID returns:
VR( 2) 



var(r t , t  2 )
2 var( rt )
var( rt 1 )  var( rt  2 )  2 cov( rt 1 , rt  2 )
2 var( rt )
2 var( rt )  2 cov( rt 1 , rt  2 )
2 var( rt )
1
Variance Ratio Statistics
• As an alternative, suppose returns are
autocorrelated:
VR( 2) 



var(r t , t  2 )
2 var( rt )
var( rt 1 )  var( rt  2 )  2 cov( rt 1 , rt  2 )
2 var( rt )
2 var( rt )  2 cov( rt 1 , rt  2 )
2 var( rt )
1   (1)
Variance Ratio Statistics
• This is an important result that has
implications for portfolio choice:
– The investing horizon can have dramatic
effects on the risk-return relationship.
• In particular, if returns are negatively
autocorrelated, long horizon investors will
face less variable equity returns than will
short horizon investors.
Testing for IID Returns with
Variance Ratio Statistics
• The null hypothesis is that returns are IID normal:
rt     t with  t IID N (0,1)
• We’ll work with 2n+1 log prices to determine the VR(2)
statistic:
1 2n
ˆ 
rt  k

2n k 1
2n
1
2
ˆ
ˆ a2 
(r


)

t k
2n k 1
2n
1
2
ˆ
ˆ b2 
(p

p

2

)

t 2k
t  2 k -2
2n k 1
Testing for IID Returns with
Variance Ratio Statistics
•
Note that:
1. With log returns, the mean return is simply
the geometric average return (the last log
price minus the first log price divided by 2n).
2. The mean of the two period return is twice the
one period return.
3. Only n two-period returns are used to
estimate the two-period return variance.
Testing for IID Returns with
Variance Ratio Statistics
• The variance ratio statistic is normally
distributed:
ˆ
VR(2) 
;
ˆ
2
b
2
a
~
2n VR(2)  1 N 0,2
Testing for IID Returns with
Variance Ratio Statistics
• The general case:
ˆ
ˆ a2

1 nq
rt  k

nq k 1

1 nq
2
ˆ
(r


)

tk
nq k 1
ˆ b2 (q) 
1 n
2
ˆ
(p

p

q

)

t  qk
t  qk -q
nq k 1
Testing for IID Returns with
Variance Ratio Statistics
• An improvement (correct bias and overlap):
 a2 (q) 
1 nq
2
ˆ
(r


)

t k
nq  1 k 1
 (q) 
1 nq
2
ˆ
(p t  k  p t  k -q  q )

m k q

q
q(nq  q  1) 1  
 nq 
2
c
m

Variance Ratio Statistics:
Distribution of Test Statistics
• The variance ratio statistics are normally
distributed:
2
ˆ

b (q)
ˆ
VR (q ) 
;
2
ˆ a
 c2 (q)
VR (q ) 
;
2
a


nq VRˆ (q )  1 ~ N 0,2(q  1) 
 2(2q  1)( q  1) 
~

nq VR (q )  1 N  0,
3q

