session 2 - reading 7 statistical concept and market return

measurement scales -

NOIR nominal scale - observations are classified with no particular order. example is assining the number 1 to a muncipal bond fund 2 to corporate bond fund.. ordinal scale - every observation is assigned to one of several categories. then categories are ordered with respect to a specified characterstic. interval scale - provides relative ranking, like ordinal scales plus the assurance that difference between scale values are equal. weakness of this measure is that a measurement of zero does not necessarily indicate the total absence of what we are measuring. this means that interval scale based ratios are meaningless. 30 degree is not three times as hot ratio scale- represent the most refined level of measurement. provides ranking and equal differences between scale values, and they also have a true zero point as the origin. example is money

sample variance formula s^2

(X sub i - X bar)^2 / n-1

normal distribution has excess kurtosis equal to

0

arithmetic mean is the sum of observation values divided by he number of observations. it is the most widely used measure of central tendency and has the following propertie

1. all interval ratio data sets have an arithmetic mean 2. all data values are considered and included in the arithmetic mean computation 3. a data set has only one arithmetic mean (the arithmetic mean is unique) 4. the sum of the deviations of each observation in the data set from the mean is always zero

steps to construct a frequency distribution

1. define the interval (must be mutually exclusive) 2. tally the observations 3. count the observations

skewness affects the location of the mean, median and mode of a distribution

1. for a symmetrical distribution, the mean, median, and mode are equal 2. for a positively skewed, unimodal distribution, the mode is less than the median, which is less than the mean. the mean is affected by outliers; in a positively skewed distribution, there are large, positive outliers which will tend to "pull' the mean upward or more positive. an example is 100 homes, 99 sell for 100,000 and one for 1000000. the median adn the mode will be 100.000 but th emean will be 109000. hence the mean is pulled upward. 3. for a negatively skewed unimodal distribution, the mean is less than the median, which is less than the mode. in this case, there are large, negative outliers that tend to pull the mean downward (to the left)

two limiations of the sharpe ratio

1. if two portfolios have negative sharpe ratios, it is not necessarily true that the higher sharpe ratio implies superior risk adjusted performance. increasing risk moves a negative sharpe ratio closer to zero (ie higher). 2. sharpe ratio is useful when standard deviation is an appropriate measure of risk however, investment strategies with option characteristics have asymmetric return distribuions, reflecting a large probability of small gains coupled with a small probability of large losses. in such cases, standard deviation may underestimate risk and produce sharpe ratios that are too high.

difference between the population variance and the sample variance

1. s^2 is n-1 where o^2 uses the entire population size N 2. use of the sample mean, X bar, instead of the population mean, u. if you use n instead of n - 1 as teh divisor in the computation of s^2, will systematically underestimate the population parameter, o^2, particularly for small sample sizes. this systematic underestimation causes the sample variance to be what is referred to as a biased estimator of the population variance. using n-1 instead of n is the denominator, however, improves the statistical properties of s^2 as an estimator of o^2. this s^2 as expressed in the equation above, is considered to be an unbiased estimator of o^2.

suppose you are comparing the annual returns distribution for a real estate portfolio with a mean of XX% and an annual returns distribution for a real estate portfolio with a mean of XX%. a direct comparison between the dispersion of the two distributions is not meaningful because of the relatively large difference in their means. to make a meaningful comparison,, a relative measure of dispersion must be used. relative dispersion is the amount of variability in a distribution realtive to a reference point or benchmark. relative dispersion is commonly measured with the coefficient of variation (CV) ...

CV = s sub x / X bar = standard deviation of x / average value of x this formula enables us to make a direct comparison of dispersion across different sets of data. in an investment setting, the CV is used to measure the risk (variability) per unit of expected return (mean). CV is variation per unit of return..

Measures of dispersion

Indicates the riskiness of an investment Range Mean absolute deviation Variance

Measures of central tendency

Indication of an investments expected return identify the center, or average of a data set. Arithmetic mean Geometric mean Weighted mean, median or mode

what is absolute frequecy

actual number of observations that fall within a given interval

mean absolute deviation (MAD)

average of the absolute values of the deviations of individual observations from the arithmetic mean. MAD = ( x sub i - X bar ) / n computation of the MAD uses the absolute values of each deviation from the mean because the sum of the actual deviations from the arithmetic mean is zero. o = sigma o > MAD holds in general

positively skewness

characterized by many outliers in the upper regio, or right tail. a postively skewed distribution is said to be skewed right because of its relatively long upper (right) tail.

what is population variance? what is the formula

defined as the average of the squared deviations from the mean. formula is o^2 = (x sub i - u ) ^ 2 / N

population

defined as the set of all possible members of a stated group. example: stocks on NYSE

leptokurtic

describes a distribution that is more peaked than a normal distribution

statistical methods falls into one of two categories

descriptive statistics or inferrential statistics

negatively skewness

distribution has a diproportionately large amount of outliers that fall within its lower (left) tail. a negatively skewed distribution is said to be skewed left because of its long lower tail.

sample skeweness

equal to the sum of the cubed deviations from the mean divided by the cubed standard deviations and by the number of observations. (1/n) (summation of X sub i - X bar ) ^3 / s^3 denomiator is always positive. numerator can be positive or negative, depending on whether observations above the mean or observations below the mean tend to be further form the mean or average. when a distribution is right skewed, sample skewness is positive because the deviations above the mean are larger on average. a left skewed distribution has a negative sample skewness.

to interpret kurtosis, - it is measured realtive to the kurtosis of a normal distribution, which is 3. positive values of excess kurtosis indicate a distribution tha is leptokurtic (more peaked, fat tails), whereas negative values indicate a platykurtic distribution (less peaked, thin tails). excess kurtosis values than exceed 1.0 in absolute value are considered large.

excess kurtosis = sample kurtosis - 3

variance is difficult to interpret because it's squared. this problem is mitigated from the use of the standard deviation. the population standard deviation o is the square root of the population variance

formula is on page 134 o = (X- u ) ^2 / N

quantile

general term for a value at or below which a stated proportion of the data in a distribution lies. quartiles quintile decile percentile

geometric vs arithmetic

geometric mean of past annual returns is the appropriate measure of past performance. it gives us the average annual compound return. to estimate multi year returns, the geometric mean is appropriate measure. arithmetic mean is statistically the best estimator of the next year's return given

histogram

graphical presentation of the absolute frequency distribution. allows us to quickly see where most of the observations are concentrated.

leptokurtic distribution has excess kurtosis equal to

greater than 0

for values that are not all equal, write order of the harmonic, geometric and arithmetic means

harmonic mean < geometric mean < arithmetic mean

mesokurtic

has the same kurtosis as a normal distribution

what does it mean to have a distribution with excess kurtosis?

if it has eithr mmore or less kurtosis than the normal distribution. the computed kurtosis for all normal distributions is three. sometimes reported as -3.

measure of location formula

l sub y = (n+1) (y/100)

platykurtic distribution has excess kurtosis equal to

less than 0

kurtosis

measure of the degree to which a distribution is more or less "peaked' than a normal distribution

parameter

measure used to describe a characteristic of a population. many exist but investment analysis utilizes just a few - mean return and standard deviation of return

sharpe measure

measures excess return per unit of risk. r - rf / o r = portfolio return rf= risk free return o = standard deviation of portfolio returns

median and why is it important?

midpoint of a data set when the data is arranged in ascending or descending order. important because the arithmetic mean can be affected by extremely large or small values (outliers) when this occurs, the median is a better measure of central tendency than the mean because it is not affected by extreme values that may actually be the result of errors in the data.

frequency polygon

midpoint of each interval is plotted on the horizontal axis, and the absolute frequency for that interval is plotted on the vertical axis

mode unimodal bimodal trimodal

occurs most frequently in a data set. unimodal is one value bi - 2 values tri - 3 values

write formula for population mean "u'

page 126

write formula for sample mean "X bar'

page 126

what are infrential statistics?

pertains to procedures used to make forecasts, estimates or judgments about a large set of data on the basis of the statistical characterstics of a smaller set (a sample)

what is the excess return part of the sharpe ratio?

r sub p - rf excess return on portfolio p measures extra reward that investors receive for exposing themselves to risk. portfolios with large positive sharpe ratios are preferred to portfolios with smaller ratios because it is assumed that rational investors prefer return and dislike risk.

range formula

range = maximum value - minimum value simple measure of variability

platykurtic

refers to a distribution that is less peaked, or flatter than a normal distribution

skewness

refers to the extent to which a distribution is not symmetrical

how is relative frequency calculated?

relative frequency is calculated by dividing the absolute frequency of each return interval by the total number of observations.

chebyshev's inequality

states that for any set of observations, whether sample or population data and regardless of the shape of the distribution, the percentage of the observations that lie within k standard deviations of the mean is at least 1 - 1/k^2 for all k >1. * importance of chebyshev's inequality is that it applies to any distribution. if we know the underlying distribution is normal, we can even be more precise about the percentage of observations that will fall within 2 and 3 standard deviatons of the mean.

sample

subset of the population of interest

cumulative absolute frequency and cumulative relative frequency

summing the absolute or relative frequencies starting at the lowest interval and progressing through the highest. look on page 124

frequency distribution

tabular presentation of statistical data that aids the analysis of large data sets. summarize statistical data by assigning it to specified groups or intervals

statistics

the word is used to refer to data (eg the average return on XYZ stock was 8% over the last ten years)

weighted mean

this concept recognizes that different observations may have a disproportionate influence on the mean.

harmonic mean

used for certain computations, such as the average cost of shares purchased over time. recite formula - page 131 = N / summation of 1/X sub i

sample statistic

used to measure a characteristic of a sample

what are descriptive statistics?

used to summarize the important characteristics of large data sets.

when is the geometric mean used?

when calculating investment returns over multiple periods or when measuring compound growth rates