CFA Level 1 (2009) - 1

Study Session 2 Cross-Reference to CFA Institute Assigned Reading #7 - Statistical Concepts and Market Returns

Figure 4: Frequency Polygon of Intelco Stock Return Data

7

5

3

Interval Midpoints

LOS 7.d: Define. calculate, and interpret measures of cemral tendency, including the population mean, sample mean, arithmetic mean, weighted average or mean (including a portfolio return viewed as a weighted mean), geometric mean, harmonic mean, median, and mode.

Measures of central tendency identify the center, or average, of a dara set. This central point can then be used to represent the typical, or expected, value in the data set.

To compute the population mean, all the observed values in the population are summed (2::X) and divided by the number of observations in the population, N. Note that the population mean is unique in that a given population only has one mean. The population mean is expressed as:

The sample mean is the sum of all the values in a sample of a population, 'EX, divided by the number of observations in the sample, n. It is used to make inferences about the population mean. The sa.mple mean is expressed as:

n

L:Xi

x: = i=l n

Note the use of 11, the sample size, versus N, the population size.

Study Session 2

Cross-Reference to CFA Institute Assigned Reading #7 - Statistical Concepts and Market Returns

Example: Population mean and sample mean

Assume you and your research assistant are evaluating the stock ofAXZ Corporation. You have calculated the stock returns for AXZ over the last 12 years to develop the data set shown below. Your research assistant has decided to conduct his analysis using only the returns for the five most recent years, which are displayed as the bold numbers in the data set. Given this information, calculate the population mean and the sample mean.

Data set: 12%,25%, 34%, 15%, 19%, 44%, 54%, 33%, 22%, 28%, 17%, 24%

Answer:

j..I. = population mean = 12+25+ 34+15+19 +44 + 54+33 +22 + 28+17 +24

The population mean and sample mean are both examples of arithmetic means. The arithmetic mean is the sum of the observation values divided by the number of

observations. It is the most widely used measure of central tendency and has the following properties:

•All interval and ratio data sets have an arithmetic mean.

•All data values are considered and included in the arithmetic mean computation.

•A data set has only one arithmetic mean (i.e., the arithmetic mean is unique).

•The sum of the deviations of each observation in the data set from the mean is always zero.

The arithmetic mean is the only measure of central tendency for which the sum of the deviations from the mean is zero. Mathematically, this property can be expressed as follows:

11

sum of mean deviations = 2)Xi - X) = 0 i=l

Example: Arithmetic mean and deviations from the mean

Compute the arithmetic mean for a data set described as:

Data set: [5,9,4, 10]

Study Session 2 Cross-Reference to CFA Institute Assigned Reading #7 - StatisticaJ Concepts and Market Returns

Answer:

The arithmetic mean of these numbers is:

x= 5+9+4+10 = 7 4

The sum of the deviations from the mean (of 7) is:

n

E(Xj - X) = (5 -7) +(9-7) + (4 -7) + (10 -7) = ~2+2"-3-1-3=0

~ .

Unusually large or small values can have a disproportionate effect on the computed value for the arithmetic mean. The mean of 1, 2, 3, and 50 is 14 and is not a good indication of what the individual data values really are. On the positive side. the arithmetic mean L1ses all the information available about the observations. The arithmetic mean of a sample from a population is the best estimate of both the true mean of the sample and the value of the next observation.

The computation of a weighted mean recognil,es that different observations may have a disproportionate influence on the mean. The weighted mean of a set of numbers is computed with the following equation:

n

Xw = LWiXj = (wIX] + w2X2 + ... + wnX)

i=!

Example: Weighted mean as a portfolio return

A portfolio consists of 50% common stocks, 40% bonds, and 10% cash. If the return on common stocks is 12%, the return on bonds is 7%, and the return on cash is 3%, what is the portfolio return?

Answer:

xw = w stock Rsrock + wbonds Rbonds +w cash Rcash

Xw = (0.50 x 0.12) + (0.40 x 0.07) + (0.10 x 0.03) = 0.091, or 9.1%

The example illustrates an extremely important investments concept: the return for a portfolio is the weighted average ofthe returns ofthe individual assets in the portfolio. Asset weights are market weights, the market value of each asset relative to the market value of the entire portfolio.

The median is the midpoint of a data set when the data is arranged in ascending or descending order. Half the observations lie above the median and half are below. To

Study Session 2

Cross-Reference-to CFA Institute Assigned Reading #7 - Statistical Concepts and Market Returns

determine the median, arrange the data from the highest to the lowest value, or lowest to highest value, and find the middle observation.

The median is 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.

,' . ;, J •• ',- ' - "

>i'hitisthe medianreturnTorfive:portfo!io'managers with 1O-year annualiiedto~:.

30%,25%,23%,21%,15%

Then, seleer the observation that has an equal number of observations above and below it-the one in the middle. For the given data set, the third observation, 23%, is the median value. .;

Consider that while we calculated the mean of 1,2,3, and 50 as 14, the median is 2.5. If the data were 1, 2, 3, and 4 instead, the arithmetic mean and median would both be 2.5.

The mode is the value that occurs most frequently in a data set. A data set may have more than one mode or even no mode. When a distribution has one value that appears most frequently, it is said to be unimodal. When a set of data has two or three values that occur most frequently, it is said to be bimodal or trimodal, respectively.

Study Session 2

Cross-Reference to CFA Institute Assigned Reading #7. - Statistical Concepts and Market Returns

Example: The mode

What is the mode of the following. data set?

Data set: [30%, 28%, 25%, 23%,28%, 15%, 5%]

Answer:

The mode is 28% because it is the value appearing most frequently.

The geometric mean is often used when calculating investment returns over multiple periods or when measuring compound growth rates. The general formula for the geometric mean, G, is as follows:

]\;O(C rh;1[ thi~ ("(lllJtion has a solurion only if rhe producr lInder rhl: radical sign is non-

11 Cf:,;l[ I",,'.

\Vhen cllcularing the geomerric mean for a returns dara ser, it is necessary to add 1 to each value under the radical and then subtract] from the result. The geometric mean return

CJ'1be computed using the following equation:

l-r-Rc; =~(l+Rl)X(l+R2)X... x(I+Rn)

where:

R( = the rerum for period t

Example: Geometric mean return

For the last three years, the returns for Acme Corporation common stock have been

-9.34%, 23.45%, and 8.92%. Compute the compound annual rate of returu9y.erthe 3-year period.

Answer:

1+R G = </(-0.0934 +1) X (0.2345 +1) x (0.0892 +1)

1+ RG = </0.9066 X 1.2345 X 1.0892 = ~1.21903 = (1.21903/,3 = 1.06825

Rc = 1.06825 - 1 = 6.825%

Solve this type of problem with your calculator as follows:

• On the TI, enter 1.21903 [yX) 0.33333 [=), or 1.21903 [y") 3 [l/x) [=)

•On the HP, enter 1.21903 [ENTER) 0.33333 [y"], or 1.21903 [ENTER] 3 [l/x) [yX)

Note that the 0.33333 represents the one-third power.

Study Session 2

Cross-Reference to CFA Institute Assigned Reading #7 - Statistical Concepts and Market Returns

An investor purchases $1,000 of srock each month, and over the last three months the prices paid per share were $8, $9, and $10. What is the average cost per share for the shares acquired?

Answer:

The previous example illustrates the interpretation of the harmonic mean in its most common application. Note that the average price paid per share ($8.93) is less than the

For values that are not all equal: harmonic mean < geometric mean < arithmetic mean. This mathematical fact is the basis for the claimed benefit of purchasing the same dollar amount of mutual fund shares each month or each week. Some refer ro this practice as "dollar cost averaging."

LOS 7 .e: Describe_J calculate and interpret quartiles, quintiles, deciles, and percentiles.

Quantile is the general term for a value at or below which a stated proportion of the data in a distribution lies. Examples of quantiles include:

•Quartiles-the distribution is divided into quarters.

•Quintile-the distribution is divided into fifths.

•Decile-the distribution is divided into tenths.

•Percentile-the distribution is divided into hundredths (percents).

Note that any quantile may be expressed as a percentile. For example, the third quartile partitions the distribution at a value such that three-fourths, or 75%, of the observations fall below that value. Thus, the third quartile is the 75th percentile.

Study Session 2 Cross-Reference to CFA Institute Assigned Reading #7 - Statistical Concepts and Market Returns

The formula for the position of the observation at a given percentile, y, with n data points sorted in ascending order is:

L = (n+1)L

y 100

Quantiles and measures of central tendency are known collectively as measures of location.

Example: Quartiles

What is the third quartile for the following distribution of returns?

8%,10%,12%,13%,15%,17%,17%,180/0,19%,23%,24%

Answer:

The third quartile is the point below which 75% of the observations lie. Recognizing that there are 11 observations in the data set, the third quartile can be identified as:

75

Ly = (l1+I)x- = 9 100

When the data is arranged in ascending order, the third quartile is the ninth data point from the left, or 19%. This means that 75% of all observations lie below 19%.

As you will see in the next example, if L is not a whole number, linear interpolation must be used to find the quantile.

Example: Quartiles

What is the third quartile for the following distribution of returns?

8%,10%, 12%, 13%, 15%, 17%, 17%, 18%, 19%,23%,24%,26%

Answer:

With 12 observations in this data set, the third quartile can be identified as:

75

Ly =(12+1)X-=9.75 100

This means that when the data is arranged in ascending order, the third quartile (75th percentile) is the ninth data point from the left, plus 0.75 x (distance between the 9th and 10th data values). Specifically, the third quartile is [19 + 0.75 x (23 - 19)] = 22%, indicating that 75% of all observations lie below 22%.

Study Session 2

Cross-Reference to CFA Institute Assigned Reading #7 - Statistical Concepts and Market Returns

LOS 7.f: Define, calculate, and interpret 1) a range and a mean absolute deviation, and 2) the variance and standard deviation of a population and of a sample.

Dispersion is defined as the variability around the central tendency. The common theme in finance and investments is the tradeoff between reward and variability, where the central tendency is the measure of the reward and dispersion is a measure of risk.

The range is a relatively simple measure of variability, but when used with other measures it provides extremely useful information. The range is the distance between the largest and the smallest value in the data set, or:

range = maximum value - minimum value

Example: The range

What is the range for the 5-year annualized total returns for five investment managers if the managers' individual returns were 30%, 12%,25%,20%, and 23%?

Answer:

range = 30 - 12 = 18%

The mean absolute deviation (MAD) is the average of the absolurc values of the deviations of individual observations from the arithmetic mean.

n

I:!Xj-X!

MAD = -,-i~_-.:....I _ n

The 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.

Example: MAD

Whatis the MAD of the investment returns for the five managers discussed in the preceding example? How is it interpreted?

Answer:

annualized returns: [30%, 12%, 25%, 20%, 23%]

x= [30+12+25+20+23] =22%

5

MAD =[8 + 10 + 3+2 + 1] = 4;8%

5

This result can be interpreted to mean that, on average, an individual return will deviate ±4.B% from the rneanreturnof22% ..

Study Session 2

Cross-Reference to CFA Institute Assigned Reading #7 - Statisrical Concepts and Marker Returns

The population variance is defined as the average of the squared deviations from the mean. The population variance (a 2) uses the values for all members of a population and is calculated using the following formula:

N

l:(X j -IJi

N

Example: Population variance, a 2

Assume the 5-year annu:tlized tot:tl returns for the five investment managers used in the earlier example represent all of the managers at a small investment firm. What is the population variance. of returns?

Interpreting this result, we can say that the average variation from the mean rerum is 35.60% squared. Had we done the calculation using decimals instead of whole percents, the variance would be 0.00356. What is a percent squared? Yes, this is nonsense, but let's see what we can do so that it makes more sense.

As you have just seen, a major problem with using the variance is the difficulty of interpreting it. The computed variance, unlike the mean, is in terms of squared units of measurement. How does one interpret squared percents, squared dollars, or squared yen? This problem is mitigated through the use of the standard deviation. The population standard deviation, a, is the square root of the population variance and is calculated as follows:

0=

N

Example: Population standard deviation, a

Using the data from the preceding examples, compute the population standard deviation.

Answer:

(30 - 22)2 + (12 - 22)2 + (25 - 22)2 + (20 - 22)2 + (23 - 22)2

a=

5

= .J35.60 = 5.97%

5wdy Session 2

Cross-Reference to CFA Institute Assigned Reading #7 - Statistical Concepts and Market Returns

Calculated with decimals instead of whole percents, we would get:

cr2 = 0.00356 and cr = ";0.00356 = 0.05966 = 5.97%

Since the population standard deviation and population mean are both expressed in the same units (percent), these values are easy to relate. The outcome of this example indicates that the mean return is 22% and the standard deviation about the mean is 5.97%. Note that this is greater than the MAD of 4.8%, a result (cr > MAD) that holds in general.

The sample variance,?, is the measure of dispersion that applies when we are evaluating a sample of n observations from a population. The sample variance is calculated using the following formula:

The most noteworthy difference from the formula for population variance is that the denominator for s~ is n - 1, one less than the sample size n, where G~ uses the emire population size N. Another difference is the use of the sample mean, X, instead of the population mean, 1.1. Based on the mathematical theory behind statistical procedures, the use of the emire number of sample observations, n, instead of n - 1 as the divisor in the computation of 52, will systematically underestimate the population parameter, a 2 , particularly for small sample sizes. This systematic undtrestimation causes the sample variance to be what is referred to as a biased estimator of the population variance. Using n - 1 instead of n in the denominator, however, improves the statistical properties of 51 as an estimator of cr2. Thus,?, as expressed in the equation above, is considered to be an unbiased estimaror of cr2.

Example: Sample variance

Assume that the 5-year annualized total returns for the five investment managers used in the preceding examples represent only a sample of the managers at a large investment firm. What is the sample variance of these returns?

Answer:

-[30+12+25+20+23]

Thus, the sample variance of 44.5(%2) can be interpreted ro be an unbiased estimator of the population variance. Note that 4-4.5 "percent squared" is 0.00445 and you will get this value if you put the percem returns in decimal form [e.g., (0.30 - 0.22)2, etc.].