Download Session #1 Statistical Concepts and Market Returns

Session #1
Statistical Concepts and Market
Returns
Dr. Himanshu Joshi
Summarizing Data using Frequency
Distribution
• A Frequency distribution is a tabular display of
data summarized into a relatively small
number of intervals.
• Frequency distribution help in the analysis of
large amount of statistical data, and they work
with all types of measurement scales.
Frequency Distribution
• Rates of returns are the fundamental units
that analyst and portfolio managers use for
making investment decisions, and we can use
frequency distribution to summarize rate of
return.
Holding Period Return
Rt = (Pt – Pt-1 + Dt)/Pt-1
Where:
Pt = price per share at the end of period t.
Pt-1 = price per share at the end of time t-1, the time immediately preceding time period t
Dt = cash distribution received during time period t.
Holding Period Return
Equation for holding period return can be
used to define the holding period return on
any asset for a day, week, month, or year
simply by changing the interpretation of the
time interval between successive values of
the time index, t.
Common Stock: the distribution is a dividend.
Bonds: the distribution is a coupon payment.
Holding Period Return: Two Important
Characteristics
Element of Time: Holding period return
has an element of time attached to it. For
example: if a monthly time interval is used
between successive observations for price,
then the rate of return is monthly figure.
No Currency Unit Attached: It has no
currency unit attached to it. The return will
hold regardless of the currency in which
prices are denominated.
Continuously Compounded Returns
• Rt = 100% x ln (Pt/Pt-1)
Construction of Frequency Distribution
Sort the Data in ascending Order.
Calculate the range of the data, defined as Range = Maximum Value – Minimum Value.
Decide on the number of intervals in the frequency distribution, k.
Determine interval width as Range/k
Determine the interval by successively adding the interval width to the minimum value, to determine
the ending point of intervals, stopping after reaching an interval that include maximum value.
Count the number of observations falling in each interval.
Construct a table of intervals listed from smallest to largest that show the number of observations
falling in each interval.
Construction of a Frequency
Distribution
• The actual number of observations in a given
interval is called the absolute frequency, or
simply frequency. The frequency distribution
is the list of intervals together with the
corresponding measure of frequency.
Construction of a Frequency
Distribution: An Example
• Suppose we have 12 observations sorted in
ascending order: -4.57, -4.04,-1.64, 0.28, 1.34,
2.35, 2.38, 4.28, 4.42, 4.468, 7.16, and 11.43.
• Range = Maximum – Minimum = 11.43 – (4.57) = 16
• If we decide number of intervals to be 4.
• Then interval width = range/k = 16/4 =4
Construction of a Frequency
Distribution
• Table 1. Endpoints of Intervals
-4.57+4 =
-0.57
-0.57+4 =
3.43
3.43 + 4=
7.43
7.43 +4=
11.43
• Table 2. Frequency Distribution
Interval
Absolute
Frequency
A
-4.57<=Observation < -0.57
3
B
-0.57<=Observation<3.43
4
C
3.43<=Observation<7.43
4
D
7.43<=Observation<11.43
1
S&P 500, 1926-2002, Total Annual
Return
Frequncy
8
7
6
5
4
3
2
1
0
Frequncy
(-30%) to (-28%)
(-28%) to (-26%)
(-26%) to (-24%)
(-24%) to (-22%)
(-22%) to (-20%)
(-20%) to (-18%)
(-18%) to (-16%)
(-16%) to (-14%)
(-14%) to (-12%)
(-12%) to (-10%)
(-10%) to (-8%)
(-8%) to (-6%)
(-6%) to (-4%)
(-4%) to (-2%)
(-2%) to (0%)
0% to 2%
2% t0 4%
4% to 6%
6% to 8%
8% to 10%
10% to 12%
12% to 14%
14% to 16%
16% to 18%
18% to 20%
20% to 22%
22% to 24%
24% to 26%
26% to 28%
28% to 30%
30% to 32%
32% to 34%
34% to 36%
36% to 38%
38 % to 40%
40% to 42%
42% to 44%
S&P 500, 1926-2002, Total Monthly
Returns
Absolute Frequency
200
180
160
140
120
100
80
60
Absolute Frequency
40
20
0
Interpretation of Frequency
Distribution
Exercise on Construction of Frequency
Distribution
Relative Frequency
• The relative frequency is the absolute
frequency of each interval divided by the total
number of observations.
Using Pivot Table/Grouping to Create
Frequency Distribution
Step#1. Select/activate any cell in the data base, and
insert a pivot table.
Step#2. Drag and Drop Return field in Row Labels and
Value Boxes.
Step#3. Select any return in the row area of the pivot
table and open the Grouping Dialog box by right clicking
on the cell.
Step#4. Excel has already entered the lowest and highest
values in the Return field in the starting and ending at
boxes. It has also suggested a step increment in the box
next to By. Select the Interval Gap and Click OK.
Frequency Distribution Using Excel
Pivot Tables
Count of Return 2007 (%)
-8.33--6.33
1
-2.33--0.33
4
-0.33-1.67
7
1.67-3.67
11
3.67-5.67
6
5.67-7.67
7
7.67-9.67
4
9.67-11.67
6
11.67-13.67
7
13.67-15.67
3
15.67-17.67
8
17.67-19.67
2
19.67-21.67
8
21.67-23.67
5
25.67-27.67
1
27.67-29.67
1
31.67-33.67
2
37.67-39.67
1
39.67-41.67
1
43.67-45.67
1
Grand Total
86
Total
12
10
8
6
Total
4
2
0
-8.33--6.33
-2.33--0.33
-0.33-1.67
1.67-3.67
3.67-5.67
5.67-7.67
7.67-9.67
9.67-11.67
11.67-13.67
13.67-15.67
15.67-17.67
17.67-19.67
19.67-21.67
21.67-23.67
25.67-27.67
27.67-29.67
31.67-33.67
37.67-39.67
39.67-41.67
43.67-45.67
Row Labels
Measures of Central Tendency
1. Arithmetic Mean: Analysts and portfolio
managers often want one number that
describes a representative possible outcome
of an investment decision. The arithmetic
mean is by far the most frequently used
measure of the middle or centre of data.
Definition: Sum of Observations/Number of
Observations.
Arithmetic Mean
• Population Mean
μ = ∑i=1N Xi / N
N= Number of observations in population.
• Sample Mean
X- = ∑i=1n Xi /n
n= Number of observations in sample.
Cross Sectional vs. Time Series Mean
• Cross Sectional Mean: Sample might be the
2013 return on equity (ROE) for 30 companies
in the BSE Sensex. In this case we calculate
mean ROE in 2013 as an average across 30
largest market cap firms in BSE.
• When we examine the characteristics of some
units at a specific point in time, we are
examining Cross-Sectional-Data.
Cross Sectional vs. Time Series Mean
• If our sample consists of historical monthly
returns on BSE Sensex for the past five years,
then we have a time series data.
• Panel Data: Mix of Cross-Sectional Data and
Time Series Data.
Properties of Arithmetic Mean
• Arithmetic mean can be linked to the centre of
gravity of an object.
• Deviations from the arithmetic mean forms the
foundation for more complex concepts of risks
like: variance, skewness, kurtosis, and
heteroscedasticity.
• Advantage of Arithmetic Mean: it uses all
information about size and magnitude of the
observation.
• Limitation: Sensitivity to extreme values, e.g.,
Returns = 2%, 3%, 6%, and -80%.
The Geometric Mean (G.M)..
• The geometric mean is most frequently used
to average rates of change over time to
compute the growth rate of a variable.
• In Investment and valuation, we frequently
use the G.M to average a time series of rates
of return on an asset or a portfolio or to
compute growth rate of financial variables
such as earning or sales.
The Geometric Mean..
• G = n√X1. X2. X3……..Xn
with Xi >= 0 for i= 1,2,3,….n
The above equation has a solution, and geometric
mean exists, only if the product under the radical
sign is non-negative.
Q. Is that O.K for Stock returns?
Q. What can be most negative value of Stock
Return?
Geometric Mean Return Formula
• RG = [∏t=1T (1+Rt)1/T] – 1
Example.. GM and AM..
Q. As a mutual fund analyst, you are examining,
as of early 2016, the most recent five years to
total returns for two U.S large-cap value
equity mutual funds:
Year
SLAX
PRFDX
2002
16.2%
9.2%
2013
20.3%
3.8%
2014
9.3%
13.1%
2015
-11.1%
1.6%
2016
-17.0%
-13.0%
GM and AM
Fund
Arithmetic
Mean
Geometric Mean
Difference (BPS)
SLASX
3.54%
2.43%
111
PRFDX
2.94%
2.53%
41
•Geometric Mean Return is less than the Arithmetic Mean Return. In fact GM is
always less than the AM.
•The only time AM and GM are equal when there is no variability in the
observations- that is when all the observations in the series are same.
•In general, the difference between AM and GM increases with the variability in
the period by period observations.
•Returns of SLASX are more variable than the PRFDX, and consequently the
spread between AM and GM is larger for SLASX.
•Although SLASX has higher AM return, PRFDX has the higher GM return.
How the Analyst should interpret this
result?
•
AM vs. GM Example..
• A hypothetical investment in a single stock
costs $100. one year later, the stock is trading
at $200. at the end of the second year, the
stock price falls back to the original purchase
price of $100. No dividends are paid during
the period of past two years.
• The arithmetic mean return reflects the
average of the one-year returns, while
geometric return is compound rate of return.
Solve for AM and GM
Quartile (1/4), Quintile (1/5), Deciles
(1/10) and Percentiles (1/100)
• Let y be the value at or below which y percent
of the distribution lies, or the yth percentile.
• For example, P18 is the point below which 18
percent of the observations lie; 100-18 =82%
are greater than P18.)
• The formula for the position of a percentile in
an array with n entries sorted in ascending
order is: Ly = (n+1) * (y/100)
Quartile and Percentile
• Calculate the 10th and 90th percentiles.
• Calculate 1st, 2nd, and 3rd quartiles.
• What is the median?
Quintiles in Investment Practice
• Investment analysts use quintiles every day to rank
performance-performance of portfolios and stocks.
• We can illustrate the use of quintiles, in particular
quartiles, in investment research using the example of
Bauman, Conover, and Miller (1998).
• That study compared the performance of international
growth stocks to value stocks.
• Typically growth stocks are defined as those for which
the market price (MPS) is relatively low in relation to
EPS, BPS, and DPS.
• Growth stocks on the other hand, have comparatively
high prices in relation to those same measures.
Quintiles in Investment Practice
• The Bauman et al. classification criteria were
following valuation measures: price to earnings,
price to cash flow, price of book value, dividend
yield (D/P).
• They assigned one fourth of the of the total
sample with the lowest P/E to quartile 1, and the
1/4th with the highest P/E to Quartile to Q4. The
stocks with the second highest P/E formed Q3
and stocks with second lowest P/E formed Q2.
Interpretation..
• Small company portfolio had a median market cap of $46.6
million AND the large company portfolio had a median
value of $2472.3 million.
• Large companies were 50 times larger than small
companies, yet their mean returns were less than the half
those of small companies.
• Overall, Bauman et al. found two effects. First, international
value stocks outperformed international growth stocks.
Second, international small stocks outperformed
international large stocks.
• Next step for analysis: how value and growth stocks
performed while controlling for size?
• This involved constructing 16 different value/growth and
size portfolios (4x4), they found that value stocks
outperformed growth stocks except when market
capitalization was very small.
Measures of Risk
1. Mean Absolute Deviation (MAD) =
∑i=1n |Xi – X-|/n
2. Population Variance and Standard Deviation
3. Sample Variance and Standard Deviation.
Note: Standard Deviation is dispersion from the
arithmetic mean, and not the geometric
mean.
Semi-variance & Semi-deviation
• Variance or standard deviation of returns take account of
returns above and below mean, but investors are
concerned only with the downside risk, for example,
returns below the mean.
• Semi-variance Computation:
1. Calculate sample mean
2. Identify observations that are smaller than mean
(discarding observations equal to or greater than mean);
suppose n* are smaller than the mean.
3. Compute the sum of squared negative deviations from
mean (using n* observations that are smaller than mean)
4. Divide the sum of squared negative deviations by n* -1.
∑for all Xi<X- (Xi – X-)2/(n2 – 1)
Coefficient of Variation
• We may sometimes find it difficult to interpret
what Std. Deviation means in terms of the
relative degree of variability of different set of
data, however, either because the data sets
have markedly different means or because the
datasets have different units of measurement.
• The coefficient of Variation is useful in such
situations:
• CV = s/X- or CV = SD/Mean
CV
The Sharpe Ratio
• Sh= (RP- - RF)/Sp
• Where RP = Return on Portfolio, RF = Risk Free
Return, Sp = S.D of the portfolio
• Although CV was designed as a measure of
relative dispersion, its inverse reveals
something about return per unit of risk
because the standard deviation of returns is
commonly used as measure of investment
risk.
Sharpe Ratio
• Finance theory tells us that in the long run,
investors should be compensated with
additional mean return above risk free rate for
bearing additional risk, at least if the risky
portfolio is well diversified.
Sharpe Ratio
Moments and Central Moments
• The shape of a probability distribution can be
described by the “moments” of the
distribution.
• Raw moments are measured relative to an
expected value raised to the appropriate
power.
• The first moment is the mean of the
distribution, which is expected value of
returns: E(Rk) = ∑ni=1 pi x Rik
Moments and Central Moments
• Central moments are measured relative to the
mean (i.e., central around the mean). The kth
central moment is defined as:
• E(R- μ)k = 𝒏𝒊=𝟏 𝑷𝒊 𝑹𝒊 − 𝝁 𝒌
• Since central moments are measured relative
to the mean, the first central moment equals
zero and is, therefore, not typically used.
The Second Central Moment
(Variance)
• The second central moment is the variance of
the distribution, which measures the
dispersion of data.
• Variance = σ2 = E[(R – μ)2]
• Since the moments higher than the second
central moment can be difficult to interpret,
they are typically standardized by dividing the
central moment by σk.
The Third Central Moment (Skew)
• The third moment measures the departure from
symmetry in the distribution. This moment will
be equal to zero for a symmetric distribution
(such as normal distribution).
• Third central moment = E [ (R – μ)3]
• The skewness statistics is the standardized third
central moment. Skewness refers to the extent to
which the distribution of data is not symmetric
around its mean.
• Skewness = E [(R – μ)3]/σ3
The Fourth Central Moment
• The fourth central moment measures the
degree of clustering in the distribution.
• Fourth central moment = E[(R - μ)4]
• The kurtosis statistics is the standardized
fourth central moment of the distribution.
Kurtosis refers to the degree of peakedness or
clustering in the data distribution and is
calculated as:
• Kurtosis = E[(R - μ)4]/σ4
The Fourth Central Moment (Kurtosis)
• Kurtosis for the normal distribution equals
roughly 3. therefore, the excess kurtosis for
any distribution equals:
• Excess Kurtosis = Kurtosis - 3
Symmetry and Skewness in Returns
Distribution
• Mean and Variance may not adequately describe
an investment’s distribution of returns.
• In calculation of variance, for example, the
deviations around the mean are squared, so do
not know whether large deviations are likely to
be positive and negative.
• One important characteristic of interest to analyst
is the degree of symmetry in return distribution.
Normal Distribution
• Its mean and median are equal.
• It is completely described by two parametersits mean and variance.
• Roughly 68.3% of its observations lie between
+/- 1 SD from mean, 95.5% of its observations
lie between +/-2 SD from mean and 99.7% of
observations lie between +/-3 SD from mean.
• A distribution not normal is called Skewed.
Normal Distribution
Skewness
• Nonsymmetrical distributions may be either
positively or negatively skewed and result
from occurrence of outliers in the data set.
• Outliers are observations with extraordinary
large values, either positive or negative.
•
A Positive Skewness
• A Positively skewed distribution is characterized by
many outliers in the upper region, or right tail.
• The mean is affected by the outliers, is a positively
skewed distribution, there are large positive
outliers which will “pull” the mean upward, or
more positive.
Exp. 100 products, 99 sold for Rs.1,00,000, and 1
sold for Rs. 10,00,000
• Mode = Rs. 1,00,000, Median = Rs.1,00,000.
• Mean
=
99,00,000+10,00,000/100=
1,09,00,000/100 = Rs. 1,09,000
Positively Skewed Distribution
A Negatively Skewed Distribution
• A negatively skewed distribution has a
disproportionately large number of outliers on
the lower side of the tail (left tail).
• For a negatively skewed, unimodal distribution,
the mean is less than the median, which is less
than the mode. In this case large, negative outlier
tend to “Pull” the mean downwards (to the left).
• Exp. 100 products, 99 sold for 1,00,000 and 1 sold
for 10,000. Mean = {99,00,000 + 10,000}/ 100 =
99,100, while mode and median = 1,00,000.
Negatively Skewed Distribution
Skewness
• Positive Skewness: A Return distribution with
positive skew has frequent small losses and a
few extreme gains. It has a long tail on its right
side. Mode< Median<Mean
• Negative Skewness: A Return distribution with
negative skew has frequent small gains and
few extreme losses. Mean<Median<Mode
Skewness of a Sample
• SK = [n/(n-1)(n-2)] x ∑i=1n (Xi – X-)3
S3
Q. Calculating Skewness for a Mutual Fund.
Skewness..
Skewness
Interpretation of Skewness..
• Based on this small sample of mutual fund,
the distribution of annual returns for the fund
appears to be approximately symmetric (very
slightly negatively skewed).
• The negative and position deviations are
equally frequent and large positive deviations
approximately offset negative deviations.
Kurtosis
• Another way in which a return distribution might differ
from a normal distribution is by having more returns
clustered closely around the mean (being more Peaked)
and more returns with large deviations from the mean
(having fatter Tails).
• Relative to normal distribution, such a distribution has a
greater percentage of small deviations from mean return
(more small surprises) and a greater percentage of
extremely large deviations from the mean return (more big
surprises).
• Most investors would perceive a greater chance of
extremely large deviations from the mean as increasing
risk.
Kurtosis
• Kurtosis is the measure that tells us when a
distribution is more or less peaked than a
normal distribution.
• A distribution that is more peaked than
normal is called leptokurtic. (more frequent
extremely large surprises)
• A distribution that is less peaked than the
normal is called platykurtic.
• Identical to normal distribution – mesokurtic.
Kurtosis
Kurtosis
Excess Kurtosis
Interpretation of Kurtosis
• The Distribution seems to be mesokurtic,
based on sample excess kurtosis close to zero.
With skewness and excess kurtosis both close
to zero, annual returns seems to have been
normally distributed.
Kurtosis- Interpretation
• Most research about the distribution of the
securities returns has shown that returns are not
normally distributed.
• Actual securities returns tend to exhibit both
skewness and kurtosis.
• Skewness and kurtosis are critical concept of risk
management because when securities returns
are modeled using an assumed normal
distribution, the predictions from the models will
not take into account the potential for extremely
large, negative outcomes.
Skewness-Kurtosis- Interpretation
• In fact, most risk managers put very little
emphasis on the mean and standard deviation
of a distribution and focus more on the
distribution of the returns in the tails of the
distribution – that is where the risk is.
• In general, greater positive kurtosis and more
negative skew in returns distributions
indicates increased risk.
Covariance and Correlation
• Covariance (Ri, Rj) = E{[Ri – E(Ri)][Rj – E(Rj)]}
• Correlation (Ri, Rj) = Covar (Ri, Rj) /σ(Ri) σ(Rj)
Coskewness and Cokurtosis
• Previously, we identified moments and central
moments for mean and variance. In a similar
fashion, we can identify cross central
movements for the concept of covariance.
• The third central movement is called
Coskewness and fourth one is called
Cokurtosis.
Coskewness and Cokurtosis
Stock Returns
Stocks
Time
1
2
3
1
0.00%
-2.40%
-12.60%
-12.60%
-1.20%
-12.60%
2
-2.40%
-12.60%
-5.30%
-5.30%
-7.50%
-5.30%
3
-12.60%
2.40%
0.00%
-2.40%
-5.10%
-1.20%
4
-5.30%
-5.30%
-2.40%
12.60%
-5.30%
5.10%
5
2.40%
0.00%
2.40%
0.00%
1.20%
1.20%
6
5.30%
5.30%
5.30%
5.30%
5.30%
5.30%
7
12.60%
12.60%
12.60%
2.40%
12.60%
7.50%
Mean
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
Median
0.00%
0.00%
0.00%
0.00%
-1.20%
1.20%
Mode
#N/A
#N/A
#N/A
4 Portfolio A
#N/A
#N/A
Portfolio B
#N/A
Variance
0.0064
0.0064
0.0064
0.0064
0.0050
0.0050
Skewness
0.0000
0.0000
0.0000
0.0000
0.9538
-0.9538
Kurtosis
0.4931
0.4931
0.4931
0.4931
0.3141
0.3141
Covriance 1,2
0.0031
Covariance 3,4
0.0031
Coskewness and Cokurtosis
• Although returns vary over time, the mean,
variance, skewness and kurtosis of all stocks
returns are same under this scenario.
• In addition, the covariance between returns of
stock 1,2 and stock 3,4 are same.
• By combining stock 1,2 in portfolio A, and stock
3,4 in portfolio B, we find that return of portfolio
A and B have the same mean and variance.
• However, these combined returns do not have
the same skewness (i.e., the Coskewness
between stocks in the portfolio is different).
Coskewness and Cokurtosis
• The reason for this difference is that the ranking
of returns over time (e.g., from best to worst) is
different for each stock, and when combined in a
portfolio, these differences skew the portfolio
returns distributions.
• For example, the worst return for Stock 1
occurred during time period 3, but in portfolio A,
the worst return occurred during time period 2.
• Similarly, the best return for stock 4 occurred
during the time period 4, but in portfolio B, the
best return occurred during the time period 7.
Coskewness and Cokurtosis
• From a risk management point of view, it is helpful to know
that the worst outcome in Portfolio B is 1.7 times greater
than the worst outcome in Portfolio A.
• So, although the mean and variance of these portfolios are
equal, shortfall risk expectations can differ depending upon
time period.
• This is important information to know, however most risk
models choose to ignore the effects of Coskewness and
Cokurtosis. The reason being that as the number of
variables increase, the number of Coskewness and
Cokurtosis terms will increase rapidly, making data more
difficult to analyze.
• Practitioners instead opt to use more tractable risk models,
such as GARCH, which captures the essence of Coskewness
and Cokurtosis by incorporating time-varying volatility
and/or time varying correlation.
Using GM and AM
Why GM is appropriate for making investment
statements about past performance?
Why arithmetic mean is appropriate for making
investment statements in a forward looking
context?