CFA 7: Statistical Concepts and Market Returns

An analyst gathered the following information about the return distributions for two portfolios during the same time period: Skewness Kurtosis Portfolio A −1.3 2.2 Portfolio B 0.5 3.5 The analyst stated that the distribution for Portfolio A is more peaked than a normal distribution and that the distribution for Portfolio B has a long tail on the left side of the distribution. Which of the following is most true? The statement is not correct in reference to either portfolio. The statement is correct in reference to Portfolio A, but the statement is not correct in reference to Portfolio B. The statement is not correct in reference to Portfolio A, but the statement is correct in reference to Portfolio B.

A is correct. The analyst's statement is not correct in reference to either portfolio. Portfolio A has a kurtosis of less than 3 meaning that it is less peaked than a normal distribution (platykurtic). Portfolio B is positively skewed (long tail on the right side of the distribution).

The coefficient of variation is useful in determining the relative degree of variability of different data sets if those data sets have different: means or different units of measurement. means, but not different units of measurement. units of measurement, but not different means.

A is correct. The coefficient of variation is a relative measure of risk (dispersion) and is useful for both data sets that have different means and for data sets that do not have the same unit of measurement.

The table below gives the annual total returns on the MSCI Germany Index from 1993 to 2002. The returns are in the local currency. Use the information in this table to answer Questions 5-10. MSCI Germany Index Total Returns, 1993-2002 Year Return (%) 1993 46.21 1994 −6.18 1995 8.04 1996 22.87 1997 45.90 1998 20.32 1999 41.20 2000 −9.53 2001 −17.75 2002 −43.06 Source: Ibbotson EnCorr AnalyzerTM. To describe the distribution of observations, perform the following: Create a frequency distribution with five equally spaced classes (round up at the second decimal place in computing the width of class intervals). Calculate the cumulative frequency of the data. Calculate the relative frequency and cumulative relative frequency of the data. State whether the frequency distribution is symmetric or asymmetric. If the distribution is asymmetric, characterize the nature of the asymmetry.

A frequency distribution is a tabular display of data summarized into a relatively small number of equally sized intervals. In this example, we want five equally sized intervals. To make the frequency distribution table, we take the following seven steps. Sort the data in ascending order. −43.06% −17.75% −9.53% −6.18% 8.04% 20.32% 22.87% 41.20% 45.90% 46.21% Calculate the range. Recall that the range formula is Range = Maximum value − Minimum value. In this case, the range is 46.21 − (−43.06) = 89.27. Decide on the number of intervals in the frequency distribution, k. This number was specified as k = 5 in the statement of the problem. Determine the interval width as Range/k = 89.27/5 = 17.854 or 17.86, rounding up at the second decimal place as instructed in the statement of the problem. Note that if we rounded down to 17.85, the final class would terminate at 46.19 and not capture the largest return, 46.21. Determine the intervals by successively adding the interval width to the minimum value, to determine the ending points of intervals, stopping after we reach an interval that includes the maximum value. −43.06 + 17.86 = −25.20 −25.20 + 17.86 = −7.34 −7.34 + 17.86 = 10.52 10.52 + 17.86 = 28.38 28.38 + 17.86 = 46.24 Thus the intervals are from −43.06 up to (but not including) −25.20, −25.20 to −7.34, −7.34 to 10.52, 10.52 to 28.38, and 28.38 to (including) 46.24. Count the number of observations falling in each interval. The count is one observation in −43.06 to −25.20, two observations in −25.20 to −7.34, two observations in −7.34 to 10.52, two observations in 10.52 to 28.38, and three observations in 28.38 to 46.24. Construct a table of the intervals listed from smallest to largest that shows the number of observations falling in each interval. The heading of the last column may be "frequency" or "absolute frequency." Interval Frequency A −43.06 ≤ observation < −25.20 1 B −25.20 ≤ observation < −7.34 2 C −7.34 ≤ observation < 10.52 2 D 10.52 ≤ observation < 28.38 2 E 28.38 ≤ observation ≤ 46.24 3 We find the cumulative frequencies by adding the absolute frequencies as we move from the first interval to the last interval. Interval Absolute Frequency Calculations for Cumulative Frequency Cumulative Frequency A −43.06 ≤ observation < −25.20 1 1 B −25.20 ≤ observation < −7.34 2 1 + 2 = 3 C −7.34 ≤ observation < 10.52 2 3 + 2 = 5 D 10.52 ≤ observation < 28.38 2 5 + 2 = 7 E 28.38 ≤ observation ≤ 46.24 3 7 + 3 = 10 The cumulative frequency is a running total of the absolute frequency, and the cumulative frequency for the last interval equals the total number of observations, 10. The relative frequency of an interval is the frequency of the interval divided by the total number of observations. The cumulative relative frequency sums the relative frequencies. The cumulative relative frequency for the last interval is 100 percent. Intervals Absolute Frequency Relative Frequency (%) Calculations for Cumulative Frequency (%) Cumulative Relative Frequency (%) A −43.06 ≤ observation < −25.20 1 1/10 = 10 10 B −25.20 ≤ observation < −7.34 2 2/10 = 20 10 + 20 = 30 C −7.34 ≤ observation < 10.52 2 2/10 = 20 30 + 20 = 50 D 10.52 ≤ observation < 28.38 2 2/10 = 20 50 + 20 = 70 E 28.38 ≤ observation ≤ 46.24 3 3/10 = 30 70 + 30 = 100 10 The last interval in the frequency distribution contains 30 percent of the observations, whereas the first interval contains 10 percent of the observations. The middle three intervals each contain 20 percent of the observations. The last two intervals thus contain 50 percent of the observations. The distribution is asymmetric. With most observations somewhat concentrated to the right but with one extreme negative observation (in the first interval), we conclude that the distribution is negatively skewed.

The table below gives the annual total returns on the MSCI Germany Index from 1993 to 2002. The returns are in the local currency. Use the information in this table to answer Questions 5-10. MSCI Germany Index Total Returns, 1993-2002 Year Return (%) 1993 46.21 1994 −6.18 1995 8.04 1996 22.87 1997 45.90 1998 20.32 1999 41.20 2000 −9.53 2001 −17.75 2002 −43.06 Source: Ibbotson EnCorr AnalyzerTM. To describe the degree to which the distribution may depart from normality, perform the following: Calculate the skewness. Explain the finding for skewness in terms of the location of the median and mean returns. Calculate excess kurtosis. Contrast the distribution of annual returns on the MSCI Germany Index to a normal distribution model for returns.

According to Equation 17, sample skewness SK is SK=[n(n−1)(n−2)]∑i=1n(Ri−R¯¯¯)3s3 The sample size, n, is 10. We previously calculated R¯¯¯=10.8 and deviations from the mean (see the tables in the solution to Problems 9B and 9C). We also calculated s = 29.9474 (showing four decimal places) in the solution to 9D. Thus s3 = 26,858.2289. Using these results, we calculate the sum of the cubed deviations from the mean as follows: ∑i=110(Ri−R¯¯¯)3=∑i=110(Ri−10.8)3=35.413+(−16.98)3+(−2.76)3+12.073+ 35.13+9.523+30.43+(−20.33)3+(−28.55)3+(−53.86)3=44,399.4694−4,895.6804−21.0246+1,758.4167+ 43,243.551+862.8014+28,094.464−8,402.5699− 23,271.1764−156,242.4525=−74,474.2012 So finally we have SK=10(9)(8) −74,474.201226,858.2289=−0.39 In the sample period, the returns on the MSCI Germany Index were slightly negatively skewed. For a negatively skewed distribution, the median is greater than the arithmetic mean. In our sample, the median return of 14.18 percent is greater than the mean return of 10.80. According to Equation 18, sample excess kurtosis, KE, is KE=⎧⎩⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪[n(n+1)(n−1)(n−2)(n−3)]∑i=1n(Ri−R¯¯¯)4s4⎫⎭⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪−3(n−1)2(n−2)(n−3) The sample size, n, is 10. We previously calculated R¯¯¯ = 10.8 and deviations from the mean (see the tables in the solution to Problems 9B and 9C). We also calculated s = 29.9474 (showing four decimal places) in the answer to 9D. Thus s4 = 804,334.1230. Using these results, we calculate the sum of the deviations from the mean raised to the fourth power as follows: ∑i=110(Ri−R¯¯¯)4=∑i=110(Ri−10.8)4=35.414+(−16.98)4+(−2.76)4+12.074+ 35.14+9.524+30.44+(−20.33)4+(−28.55)4+(−53.86)4=1,572,185.212+83,128.6531+58.0278+21,224.0901+ 1,517,848.64+8,213.8694+854,071.7056+170,824.2468+ 664,392.0855+8,415,218.489=13,307,165.02 Thus we have {[10(11)(9)(8)(7)]13,307,165.02804,334.123}−3(9)2(8)(7)=3.6109−4.3393=−0.73 In the sample period, the returns on the MSCI Germany Index were slightly platykurtic. This means that there were fewer observations in the tails of the distribution than we would expect based on a normal distribution model for returns. In contrast to a normal distribution, the distribution of returns on the MSCI Germany Index is somewhat asymmetric in direction of negative skew and is somewhat platykurtic (less peaked).

If the observations in a data set have different values, is the geometric mean for that data set less than that data set's: harmonic mean? arithmetic mean? A No No B No Yes C Yes No

B is correct. Unless all the values of the observations in a data set have the same value, the harmonic mean is less than the corresponding geometric mean, which in turn is less than the corresponding arithmetic mean. In other words, regarding means, typically harmonic mean < geometric mean < arithmetic mean.

An analyst gathered the following information about a portfolio's performance over the past ten years: Mean annual return 11.8% Standard deviation of annual returns 15.7% Portfolio beta 1.2 If the mean return on the risk-free asset over the same period was 5.0%, the coefficient of variation and Sharpe ratio, respectively, for the portfolio are closest to: Coefficient of variation Sharpe ratio A 0.75 0.43 B 1.33 0.36 C 1.33 0.43

C is correct. The coefficient of variation measures total risk per unit of return or standard deviation/mean return, or 15.7/11.8 = 1.33. The Sharpe ratio is excess return per unit of risk or excess return/standard deviation. The mean excess return is 11.8% − 5.0% = 6.8%, so the Sharpe ratio is 6.8/15.7 = 0.43.

The following table repeats the annual total returns on the MSCI Germany Index previously given and also gives the annual total returns on the JP Morgan Germany five- to seven-year government bond index (JPM 5-7 Year GBI, for short). During the period given in the table, the International Monetary Fund Germany Money Market Index (IMF Germany MMI, for short) had a mean annual total return of 4.33 percent. Use that information and the information in the table to answer Questions 12-14. Year MSCI Germany Index (%) JPM Germany 5-7 Year GBI (%) 1993 46.21 15.74 1994 −6.18 −3.40 1995 8.04 18.30 1996 22.87 8.35 1997 45.90 6.65 1998 20.32 12.45 1999 41.20 −2.19 2000 −9.53 7.44 2001 −17.75 5.55 2002 −43.06 10.27 Calculate the coefficient of variation for: the 60/40 equity/bond portfolio described in Problem 12. the MSCI Germany Index. the JPM Germany 5-7 Year GBI. Contrast the risk of the 60/40 equity/bond portfolio, the MSCI Germany Index, and the JPM Germany 5-7 Year GBI, as measured by the coefficient of variation.

For the 60/40 equity/bond portfolio, the mean return (as computed in Problem 12) was 9.65 percent. We can compute the sample standard deviation of returns as s = 18.31 percent using Equation 14. The coefficient of variation for the 60/40 portfolio was CV=s/R¯¯¯=18.31/9.65=1.90 . For the MSCI Germany Index, CV=s/R¯¯¯=29.95/10.80=2.77 . For the JPM Germany 5-7 Year GBI, CV=s/R¯¯¯=6.94/7.92=0.88 . The coefficient of variation is a measure of relative dispersion. For returns, it measures the amount of risk per unit of mean return. The MSCI Germany Index portfolio, the JPM Germany GBI, and the 60/40 equity/bond portfolio, were respectively most risky, least risky, and intermediate in risk, based on their values of CV. Portfolio CV Risk MSCI Germany Index 2.77 Highest 60/40 Equity/bond portfolio 1.90 JPM Germany GBI 0.88 Lowest

The table below gives the annual total returns on the MSCI Germany Index from 1993 to 2002. The returns are in the local currency. Use the information in this table to answer Questions 5-10. MSCI Germany Index Total Returns, 1993-2002 Year Return (%) 1993 46.21 1994 −6.18 1995 8.04 1996 22.87 1997 45.90 1998 20.32 1999 41.20 2000 −9.53 2001 −17.75 2002 −43.06 Source: Ibbotson EnCorr AnalyzerTM. To describe the dispersion of the distribution, perform the following: Calculate the range. Calculate the mean absolute deviation (MAD). Calculate the sample variance. Calculate the sample standard deviation. Calculate the semivariance. Calculate the semideviation.

In the solution to Problem 5A, we calculated the range as Maximum value − Minimum value = 46.21 − (−43.06) = 89.27. The mean absolute deviation is defined in Equation 10 as MAD=∑i=110∣∣Ri−R¯¯¯∣∣10 To find the MAD for this example, we take the following four steps: Calculate the arithmetic mean of the original values. Subtract the arithmetic mean from each value. Take the absolute value of each deviation from the mean. Sum the absolute values of the deviations and divide by the total number of observations. The mean absolute deviation in this case is 24.50 percent. The following table summarizes the calculation. Original Data Ri Deviation from Mean Ri−R¯¯¯ Absolute Value of Deviation from Mean ∣∣Ri−R¯¯¯∣∣ 46.21 35.41 35.41 −6.18 −16.98 16.98 8.04 −2.76 2.76 22.87 12.07 12.07 45.90 35.10 35.10 20.32 9.52 9.52 41.20 30.40 30.40 −9.53 −20.33 20.33 −17.75 −28.55 28.55 −43.06 −53.86 53.86 R¯¯¯=10.8 ∑i=110∣∣Ri−R¯¯¯∣∣=244.98 MAD=∑i=110∣∣Ri−R¯¯¯∣∣10=24.50 Variance is defined as the mean of the squared deviations around the mean. We find the sample variance with Equation 13: s2=∑i=110(Ri−R¯¯¯)210−1 To calculate the variance, we take the following four steps: Calculate the arithmetic mean of the original values. Subtract the arithmetic mean from each value. Square each deviation from the mean. Sum the squared deviations and divide by the total number of observations minus 1. The variance, in this case, is 896.844711. The following table summarizes the calculation. Original Data Ri Deviation from Mean Ri−R¯¯¯ Squared Value of Deviation from Mean (Ri−R¯¯¯)2 46.21 35.41 1253.8681 −6.18 −16.98 288.3204 8.04 −2.76 7.6176 22.87 12.07 145.6849 45.90 35.10 1232.0100 20.32 9.52 90.6304 41.20 30.40 924.1600 −9.53 −20.33 413.3089 −17.75 −28.55 815.1025 −43.06 −53.86 2900.8996 R¯¯¯=10.8 ∑i=110(Ri−R¯¯¯)2=8,071.6024 s2=∑i=110(Ri−R¯¯¯)29=896.844711 Recall that the units of variance are the squared units of the underlying variable, so to be precise we say that the sample variance is 896.844711 percent squared. To have a result in the original units of measurement, we calculate the standard deviation. The standard deviation is the positive square root of the variance. We calculate the standard deviation with Equation 14: s=896.844711−−−−−−−−−√=29.9474or29.95% The standard deviation is thus 29.95 percent. Semivariance is the average squared deviation below the mean. Five observations (−43.06, −17.75, −9.53, −6.18, and 8.04) lie below the mean return of 10.8. We compute the sum of the squared deviations from the mean as (−43.06 − 10.8)2 + (−17.75 − 10.8)2 + (−9.53 − 10.8)2 + (−6.18 − 10.8)2 + (8.04 − 10.8)2 = 4,425.2490. Semivariance equals 4,425.2490/(10 − 1) = 491.6943. Semideviation equals 491.6943−−−−−−−√=22.174or22.17% .

The table below gives statistics relating to a hypothetical five-year record of two portfolios. Mean Monthly Return (%) Standard Deviation (%) Skewness Excess Kurtosis Portfolio A 1.6792 5.3086 −0.1395 −0.0187 Portfolio B 1.8375 5.9047 0.4934 −0.8525 Based only on the information in the above table, perform the following: Contrast the distributions of returns of Portfolios A and B. Evaluate the relative attractiveness of Portfolios A and B.

Portfolio B's returns are centered to the right of Portfolio A's, as indicated by mean return. Portfolio B's distribution has somewhat more dispersion than A's. Both return distributions are asymmetric but in different ways. The return distribution for Portfolio A is slightly negatively skewed. Portfolio B's distribution is moderately positively skewed. Portfolio A's return distribution is mesokurtic, and Portfolio B's return distribution is slightly platykurtic. We cannot know which portfolio particular investors would prefer without knowing their exact preferences for risk and return. Portfolio B has a higher mean return and moderately positive skewness, but it also has more risk as measured by standard deviation of return.

The table below gives statistics relating to a hypothetical 10-year record of two portfolios. Mean Annual Return (%) Standard Deviation of Return (%) Skewness Portfolio A 8.3 19.5 -1.9 Portfolio B 8.3 18.0 3.0 Based only on the information in the above table, perform the following: Contrast the distributions of returns of Portfolios A and B. Evaluate the relative attractiveness of Portfolios A and B.

With identical means, the two return distributions are similarly centered. Portfolio B's distribution has somewhat less dispersion, as measured by standard deviation. Both return distributions are asymmetric but in different ways. The return distribution for Portfolio A is negatively skewed; Portfolio B's distribution is positively skewed. Most investors would prefer the return distribution of Portfolio B, which has the same mean return as Portfolio A but less risk as measured by standard deviation of return. Furthermore, Portfolio B's returns are positively skewed, indicating a higher frequency of very large positive returns relative to Portfolio A. In contrast, Portfolio A's returns are negatively skewed.

Is a return distribution characterized by frequent small losses and a few large gains best described as having: negative skew? a mean that is greater than the median? A No No B No Yes C Yes No

B is correct. A distribution with frequent small losses and a few large gains has positive skew (long tail on the right side) and the mean is greater than the median.

An analyst gathered the following information: Portfolio Mean Return (%) Standard Deviation of Returns (%) 1 9.8 19.9 2 10.5 20.3 3 13.3 33.9 If the risk-free rate of return is 3.0 percent, the portfolio that had the best risk-adjusted performance based on the Sharpe ratio is: Portfolio 1. Portfolio 2. Portfolio 3.

B is correct. The Sharpe ratio is the mean excess return (mean return less risk-free rate of 3.0 percent) divided by the standard deviation of the portfolio. It is highest for portfolio 2 with a Sharpe ratio of 7.5/20.3 = 0.3695. For portfolio 1, the Sharpe ratio is 6.8/19.9 = 0.3417 and for portfolio 3 the Sharpe ratio is 10.3/33.9 = 0.3038.

The following table repeats the annual total returns on the MSCI Germany Index previously given and also gives the annual total returns on the JP Morgan Germany five- to seven-year government bond index (JPM 5-7 Year GBI, for short). During the period given in the table, the International Monetary Fund Germany Money Market Index (IMF Germany MMI, for short) had a mean annual total return of 4.33 percent. Use that information and the information in the table to answer Questions 12-14. Year MSCI Germany Index (%) JPM Germany 5-7 Year GBI (%) 1993 46.21 15.74 1994 −6.18 −3.40 1995 8.04 18.30 1996 22.87 8.35 1997 45.90 6.65 1998 20.32 12.45 1999 41.20 −2.19 2000 −9.53 7.44 2001 −17.75 5.55 2002 −43.06 10.27 Using the IMF Germany MMI as a proxy for the risk-free return, calculate the Sharpe ratio for: the 60/40 equity/bond portfolio described in Problem 12. the MSCI Germany Index. the JPM Germany 5-7 Year GBI. Contrast the risk-adjusted performance of the 60/40 equity/bond portfolio, the MSCI Germany Index, and the JPM Germany 5-7 Year GBI, as measured by the Sharpe ratio.

For the 60/40 equity/bond portfolio, we earlier computed a mean return and standard deviation of return of 9.65 percent and 18.31, respectively. The statement of the problem gave the mean annual return on the proxy for the risk-free rate, the IMF Germany MMI, as 4.33 percent. We compute the Sharpe ratio as Sh=R¯¯¯p−R¯¯¯Fsp=9.65−4.3318.31=0.29 For the MSCI Germany Index, Sh=R¯¯¯p−R¯¯¯Fsp=10.80−4.3329.95=0.22 For the JPM Germany 5-7 Year GBI, Sh=R¯¯¯p−R¯¯¯Fsp=7.92−4.336.94=0.52 The Sharpe ratio measures excess return per unit of risk as measured by standard deviation. Because we are comparing positive Sharpe ratios, a larger Sharpe ratio reflects better risk-adjusted performance. During the period, the JPM Germany GBI had the best risk-adjusted performance and the MSCI Germany Index had the worst risk-adjusted performance, as measured by the Sharpe ratio. The 60/40 equity/bond portfolio was intermediate in risk-adjusted performance. Portfolio Sharpe Ratio Performance JPM Germany GBI 0.52 Best 60/40 Equity/bond portfolio 0.29 MSCI Germany Index 0.22 Worst

The table below gives the annual total returns on the MSCI Germany Index from 1993 to 2002. The returns are in the local currency. Use the information in this table to answer Questions 5-10. MSCI Germany Index Total Returns, 1993-2002 Year Return (%) 1993 46.21 1994 −6.18 1995 8.04 1996 22.87 1997 45.90 1998 20.32 1999 41.20 2000 −9.53 2001 −17.75 2002 −43.06 Source: Ibbotson EnCorr AnalyzerTM. To describe the values at which certain returns fall, calculate the 30th percentile.

Recall the formula for the location of the percentile (Equation 8): Ly=(n+1)y100 where Ly is the location or position of the yth percentile, Py, and y is the percentage point at which we want to divide the distribution. If we apply the percentile location formula, we find L30=(10+1)30100=3.3 , which is not a whole number. To find the 30th percentile from the sorted data, we interpolate by taking the value in the third position, −9.53, and adding 30 percent of the difference between the items in fourth and third position. The estimate of the 30th percentile is P30 ≈ −9.53 + 0.3[−6.18 − (−9.53)] = −9.53 + 0.3(3.35) = −9.53 + 1.005 = −8.53. Therefore, the 30th percentile is −8.53. By definition, the 30th percentile is that value at or below which 30 percent of the observations lie. In this problem, 3 observations out of 10 lie below −8.53.

State the type of scale used to measure the following sets of data. Sales in euros. The investment style of mutual funds. An analyst's rating of a stock as underweight, market weight, or overweight, referring to the analyst's suggested weighting of the stock in a portfolio. A measure of the risk of portfolios on a scale of whole numbers from 1 (very conservative) to 5 (very risky) where the difference between 1 and 2 represents the same increment in risk as the difference between 4 and 5.

Sales in euros are measured on a ratio scale. Mutual fund investment styles are measured on a nominal scale. We can count the number of funds following a particular style, but whatever classification scheme we use, we cannot order styles into "greater than" or "less than" relationships. The ratings are measured on an ordinal scale. An analyst's rating of a stock as underweight, market weight, or overweight orders the rated securities in terms of levels of expected investment performance. The risk measurements are measured on an interval scale because not only do the measurements involve a ranking, but differences between adjacent values represent equal differences in risk. Because the measurement scale does not have a true zero, they are not measured on a ratio scale.

Explain the relationship among arithmetic mean return, geometric mean return, and variability of returns. Contrast the use of the arithmetic mean return to the geometric mean return of an investment from the perspective of an investor concerned with the investment's terminal value. Contrast the use of the arithmetic mean return to the geometric mean return of an investment from the perspective of an investor concerned with the investment's average one-year return.

So long as a return series has any variability, the geometric mean return must be less than the arithmetic mean return. As one illustration of this relationship, in the solution to Problem 6A, we computed the arithmetic mean annual return on the MSCI Germany Index as 10.80 percent. In the solution to Problem 7, we computed the geometric mean annual return as 6.7 percent. In general, the difference between the geometric and arithmetic means increases with the variability of the period-by-period observations. The geometric mean return is more meaningful than the arithmetic mean return for an investor concerned with the terminal value of an investment. The geometric mean return is the compound rate of growth, so it directly relates to the terminal value of an investment. By contrast, a higher arithmetic mean return does not necessarily imply a higher terminal value for an investment. The arithmetic mean return is more meaningful than the geometric mean return for an investor concerned with the average one-period performance of an investment. The arithmetic mean return is a direct representation of the average one-period return. In contrast, the geometric mean return, as a compound rate of growth, aims to summarize what a return series means for the growth rate of an investment over many periods.

Identify each of the following groups as a population or a sample. If the group is a sample, identify the population to which the sample is related. The S&P MidCap 400 Index viewed as representing US stocks with market capitalization falling within a stated range. UK shares that traded on 11 August 2003 and that also closed above £100/share as of the close of the London Stock Exchange on that day. Marsh & McLennan Companies, Inc. (NYSE: MMC) and AON Corporation (NYSE: AOC). This group is part of Standard & Poor's Insurance Brokers Index. The set of 31 estimates for Microsoft EPS for fiscal year 2003, as of the 4 June 2003 date of a First Call/Thomson Financial report.

The S&P MidCap 400 Index represents a sample of all US stocks in the mid-cap or medium capitalization range. The related population is "all US mid-cap stocks." The statement tells us to enumerate all members of a group and is sufficiently precise to allow us to do that. The statement defines a population. The two companies constitute a sample of US insurance brokers. The related population is "US insurance brokers." The statement defines a population. The 31 estimates for Microsoft EPS are the population of publicly available US analyst estimates of Microsoft's FY2003 EPS, as of the report's date.

Suppose a client asks you for a valuation analysis on the eight-stock US common stock portfolio given in the table below. The stocks are equally weighted in the portfolio. You are evaluating the portfolio using three price multiples. The trailing 12 months (TTM) price-to-earnings ratio (P/E) is current price divided by diluted EPS over the past four quarters.1 The TTM price-to-sales ratio (P/S) is current price divided by sales per share over the last four quarters. The price-to-book ratio (P/B) is the current price divided by book value per share as given in the most recent quarterly statement. The data in the table are as of 12 September 2003. Client Portfolio Common Stock TTM P/E TTM P/S P/B Abercrombie & Fitch (NYSE: AFN) 13.67 1.66 3.43 Albemarle Corporation (NYSE: ALB) 14.43 1.13 1.96 Avon Products, Inc. (NYSE: AVP) 28.06 2.45 382.72 Berkshire Hathaway (NYSE: BRK.A) 18.46 2.39 1.65 Everest Re Group Ltd (NYSE: RE) 11.91 1.34 1.30 FPL Group, Inc. (NYSE: FPL) 15.80 1.04 1.70 Johnson Controls, Inc. (NYSE: JCI) 14.24 0.40 2.13 Tenneco Automotive, Inc. (NYSE: TEN) 6.44 0.07 41.31 Source: www.multexinvestor.com. Based only on the information in the above table, calculate the following for the portfolio: Arithmetic mean P/E. Median P/E. Arithmetic mean P/S. Median P/S. Arithmetic mean P/B. Median P/B. Based on your answers to Parts A, B, and C, characterize the appropriateness of using the following valuation measures: Mean and median P/E. Mean and median P/S. Mean and median P/B.

The arithmetic mean P/E is (13.67 + 14.43 + 28.06 + 18.46 + 11.91 + 15.80 + 14.24 + 6.44)/8 = 15.38. Because the portfolio has an even number of stocks (eight), the median P/E is the mean of the P/Es in the n/2 = 8/2 = 4th and (n + 2)/2 = 10/2 = 5th positions in the data sorted in ascending order. (These are the middle two P/Es.) The fourth position P/E is 14.24, and the fifth position P/E is 14.43. The median P/E is (14.24 + 14.43)/2 = 14.34. The arithmetic mean P/S is (1.66 + 1.13 + 2.45 + 2.39 + 1.34 + 1.04 + 0.40 + 0.07)/8 = 1.31. The median P/S is the mean of the P/Ss in the fourth and fifth positions in the data sorted in ascending order. The fourth position P/S is 1.13, and the fifth position P/S is 1.34. The median P/S is (1.13 + 1.34)/2 = 1.24. The arithmetic mean P/B is (3.43 + 1.96 + 382.72 + 1.65 + 1.30 + 1.70 + 2.13 + 41.31)/8 = 54.53. The median P/B is the mean of the P/Bs in the fourth and fifth positions in the data sorted in ascending order. The fourth position P/B is 1.96, and the fifth position P/B is 2.13. The median P/B is (1.96 + 2.13)/2 = 2.05. The distribution of P/Es is not characterized by outliers (extreme values) and the mean P/E and median P/E at 15.38 and 14.34, respectively, are similar in magnitude. Both the mean P/E and the median P/E are appropriate measures of central tendency. Because the mean P/E uses all the information in the sample and is mathematically simpler than the median, however, we might give it preference. Both the mean P/S and the median P/S are appropriate measures of central tendency. The mean P/S and median P/S at 1.31 and 1.24, respectively, are similar in magnitude. The P/S of 0.07 for Tenneco Automotive, Inc. is very small, yet it has only a moderate influence on the mean. As price is bounded from below at zero and sales are non-negative, the lowest possible P/S is 0. By contrast, there is no upper limit in theory on any price ratio. It is extremely high rather than extremely low P/Ss that would be the greater concern in using an arithmetic mean P/S. Note, too, that the P/E of about 6.4 for Tenneco, the lowest P/E observation, is not inconsistent with the P/S of 0.07 as long as the P/S is a valid observation (rather than a recording error). The median P/B, but not the mean P/B, is an appropriate measure of central tendency. The mean P/B of 54.53 is unduly influenced by the extreme P/Bs of roughly 383 for Avon Products and roughly 41 for Tenneco Automotive. The case of Tenneco is interesting. The P/E and the P/S in particular appear to indicate that the stock is cheap in terms of the earnings and sales that a dollar buys; the P/B appears to indicate the reverse. Because book value is an accounting number subject to such decisions as write-downs, we might investigate whether book value per share for Tenneco and Avon reflects such actions.

The table below gives the deviations of a hypothetical portfolio's annual total returns (gross of fees) from its benchmark's annual returns, for a 12-year period ending in 2003. Portfolio's Deviations from Benchmark Return, 1992-2003 (%) 1992 −7.14 1993 1.62 1994 2.48 1995 −2.59 1996 9.37 1997 −0.55 1998 −0.89 1999 −9.19 2000 −5.11 2001 −0.49 2002 6.84 2003 3.04 Calculate the frequency, cumulative frequency, relative frequency, and cumulative relative frequency for the portfolio's deviations from benchmark return, given the set of intervals in the table that follows. Return Interval Frequency Cumulative Frequency Relative Frequency (%) Cumulative Relative Frequency (%) −9.19 ≤ A < −4.55 −4.55 ≤ B < 0.09 0.09 ≤ C < 4.73 4.73 ≤ D ≤ 9.37 Construct a histogram using the data. Identify the modal interval of the grouped data.

The entries in the table are as follows. Return Interval Frequency Cumulative Frequency Relative Frequency (%) Cumulative Relative Frequency (%) −9.19% ≤ A < −4.55% 3 3 25.00 25.00 −4.55% ≤ B < 0.09% 4 7 33.33 58.33 0.09% ≤ C < 4.73% 3 10 25.00 83.33 4.73% ≤ D ≤ 9.37% 2 12 16.67 100.00 The frequency column provides the count of the observations falling in each return interval. The cumulative frequency adds up (cumulates) the frequencies. For example, the cumulative frequency of 7 for Interval B is the sum of the frequency of 3 for Interval A and the frequency of 4 for Interval B. The cumulative frequency for the last interval, D, equals the total number of observations, 12. The relative frequency column gives the frequency of an interval as a percentage of the total number of observations. The relative frequency for Interval B, for example, is 4/12 = 33.33 percent. The cumulative relative frequency column cumulates the relative frequencies. After reaching the last interval, the cumulative relative frequency should be 100 percent, ignoring rounding errors. The histogram for these data is shown below. From the frequency distribution in Part A or the histogram in Part B, we can see that Interval B (−4.55 to 0.09) has the most members, 4. Interval B is thus the modal interval.

The following table repeats the annual total returns on the MSCI Germany Index previously given and also gives the annual total returns on the JP Morgan Germany five- to seven-year government bond index (JPM 5-7 Year GBI, for short). During the period given in the table, the International Monetary Fund Germany Money Market Index (IMF Germany MMI, for short) had a mean annual total return of 4.33 percent. Use that information and the information in the table to answer Questions 12-14. Year MSCI Germany Index (%) JPM Germany 5-7 Year GBI (%) 1993 46.21 15.74 1994 −6.18 −3.40 1995 8.04 18.30 1996 22.87 8.35 1997 45.90 6.65 1998 20.32 12.45 1999 41.20 −2.19 2000 −9.53 7.44 2001 −17.75 5.55 2002 −43.06 10.27 Calculate the annual returns and the mean annual return on a portfolio 60 percent invested in the MSCI Germany Index and 40 percent invested in the JPM Germany GBI.

The following table shows the calculation of the portfolio's annual returns, and the mean annual return. Year Weighted Mean Calculation Portfolio Return (%) 1993 0.60(46.21) + 0.40(15.74) = 34.02 1994 0.60(−6.18) + 0.40(−3.40) = −5.07 1995 0.60(8.04) + 0.40(18.30) = 12.14 1996 0.60(22.87) + 0.40(8.35) = 17.06 1997 0.60(45.90) + 0.40(6.65) = 30.20 1998 0.60(20.32) + 0.40(12.45) = 17.17 1999 0.60(41.20) + 0.40(−2.19) = 23.84 2000 0.60(−9.53) + 0.40(7.44) = −2.74 2001 0.60(−17.75) + 0.40(5.55) = −8.43 2002 0.60(−43.06) + 0.40(10.27) = −21.73 Sum = 96.46 Mean Annual Return = 9.65 Note: The sum of the portfolio returns carried without rounding is 96.48.

The table below gives the annual total returns on the MSCI Germany Index from 1993 to 2002. The returns are in the local currency. Use the information in this table to answer Questions 5-10. MSCI Germany Index Total Returns, 1993-2002 Year Return (%) 1993 46.21 1994 −6.18 1995 8.04 1996 22.87 1997 45.90 1998 20.32 1999 41.20 2000 −9.53 2001 −17.75 2002 −43.06 Source: Ibbotson EnCorr AnalyzerTM. To describe the compound rate of growth of the MSCI Germany Index, calculate the geometric mean return.

The geometric mean requires that all the numbers be greater than or equal to 0. To ensure that the returns satisfy this requirement, after converting the returns to decimal form we add 1 to each return. We then use Equation 6 for the geometric mean return, RG: RG=[∏t=110(1+Rt)](1/10)−1 which can also be written as RG=(1+R1)(1+R2)...(1+R10)−−−−−−−−−−−−−−−−−−−−−−−√10−1 To find the geometric mean in this example, we take the following five steps: Divide each figure in the table by 100 to put the returns into decimal representation. Add 1 to each return to obtain the terms 1 + Rt. Return (%) Return in Decimal Form 1 + Return 46.21 0.4621 1.4621 -6.18 -0.0618 0.9382 8.04 0.0804 1.0804 22.87 0.2287 1.2287 45.90 0.4590 1.4590 20.32 0.2032 1.2032 41.20 0.4120 1.4120 -9.53 -0.0953 0.9047 -17.75 -0.1775 0.8225 -43.06 -0.4306 0.5694 Multiply together all the numbers in the third column to get 1.9124. Take the 10th root of 1.9124 to get 1.9124−−−−−√10=1.0670 . On most calculators, we evaluate 1.9124−−−−−√10 using the yx key. Enter 1.9124 with the yx key. Next, enter 1/10 = 0.10. Then press the = key to get 1.0670. Subtract 1 to get 0.0670, or 6.70 percent a year. The geometric mean return is 6.70 percent. This result means that the compound annual rate of growth of the MSCI Germany Index was 6.7 percent annually during the 1993-2002 period. Note that this value is much less than the arithmetic mean of 10.80 percent that we calculated in the solution to Problem 6A.

Portfolio's Deviations from Benchmark Return, 1992-2003 (%) 1992 −7.14 1993 1.62 1994 2.48 1995 −2.59 1996 9.37 1997 −0.55 1998 −0.89 1999 −9.19 2000 −5.11 2001 −0.49 2002 6.84 2003 3.04 Tracking risk (also called tracking error) is the standard deviation of the deviation of a portfolio's gross-of-fees total returns from benchmark return. Calculate the tracking risk of the portfolio, stated in percent (give the answer to two decimal places).

To calculate tracking risk of the portfolio, we use the sample standard deviation of the data in the table. We use the expression for sample standard deviation rather than population standard deviation because we are estimating the portfolio's tracking risk based on a sample. We use Equation 14, s=∑i=1n(Xi−X¯¯¯)2n−1−−−−−−−−−−−−⎷ where Xi is the ith entry in the table of return deviations. The calculation in detail is as follows: X¯¯¯ = (−7.14 + 1.62 + 2.48 − 2.59 + 9.37 − 0.55 − 0.89 − 9.19 − 5.11 − 0.49 + 6.84 + 3.04)/12 = −2.61/12 = −0.2175 percent. Having established that the mean deviation from the benchmark was X¯¯¯ = −0.2175, we calculate the squared deviations from the mean as follows: [−7.14−(−0.2175)]2=(−6.9225)2=47.921006[1.62−(−0.2175)]2=(1.8375)2=3.376406[2.48−(−0.2175)]2=(2.6975)2=7.276506[−2.59−(−0.2175)]2=(−2.3725)2=5.628756[9.37−(−0.2175)]2=(9.5875)2=91.920156[−0.55−(−0.2175)]2=(−0.3325)2=0.110556[−0.89−(−0.2175)]2=(−0.6725)2=0.452256[−9.19−(−0.2175)]2=(−8.9725)2=80.505756[−5.11−(−0.2175)]2=(−4.8925)2=23.936556[−0.49−(−0.2175)]2=(−0.2725)2=0.074256[6.84−(−0.2175)]2=(7.0575)2=49.808306[3.04−(−0.2175)]2=(3.2575)2=10.611306 The sum of the squared deviations from the mean is ∑i=112(Xi−X¯¯¯)2=321.621825 Note that the sum is given with full precision. You may also get 321.621822, adding the terms rounded at the sixth decimal place. The solution is not affected. Divide the sum of the squared deviations from the mean by n − 1: 321.621825/(12 − 1) = 321.621825/11 = 29.238347 Take the square root: s=29.238347−−−−−−−−√=5.41 percent. Thus the portfolio's tracking risk was 5.41 percent a year.

At the UXI Foundation, portfolio managers are normally kept on only if their annual rate of return meets or exceeds the mean annual return for portfolio managers of a similar investment style. Recently, the UXI Foundation has also been considering two other evaluation criteria: the median annual return of funds with the same investment style, and two-thirds of the return performance of the top fund with the same investment style. The table below gives the returns for nine funds with the same investment style as the UXI Foundation. Fund Return (%) 1 17.8 2 21.0 3 38.0 4 19.2 5 2.5 6 24.3 7 18.7 8 16.9 9 12.6 With the above distribution of fund performance, which of the three evaluation criteria is the most difficult to achieve?

To determine which evaluation criterion is the most difficult to achieve, we need to i) calculate the mean return of the nine funds, ii) calculate the median return of the nine funds, iii) calculate two-thirds of the return of the best-performing fund, and iv) compare the results. Calculate the mean return of the nine funds. Find the sum of the values in the table and divide by 9. X¯¯¯=(17.8+21.0+38.0+19.2+2.5+24.3+18.7+16.9+12.6)/9=171/9=19.0 Calculate the median return of the nine funds. The first step is to sort the returns from largest to smallest. Return (%) Ranking 38.0 9 24.3 8 21.0 7 19.2 6 18.7 5 17.8 4 16.9 3 12.6 2 2.5 1 The median is the middle item, which occupies the (n + 1)/2 = 5th position in this odd-numbered sample. We conclude that the median is 18.7. Calculate two-thirds of the return of the best-performing fund. The top return is 38.0; therefore, two-thirds of the top return is (2/3)38.0 = 25.33. The following table summarizes what we have learned about these funds. Criterion 1 Criterion 2 Criterion 3 19.0 18.7 25.3 Criterion 3, two-thirds of the return on the top fund, is the most difficult to meet. In analyzing this problem, note that Criterion 3 is very sensitive to the value of the maximum observation. For example, if we were to subtract 10 from the maximum (to make it 28) and add 10 to the minimum (to make it 12.5), the mean and median would be unchanged. Criterion 3 would fall to two-thirds of 28, or 18.67. In this case, the mean, at 19.0, would be the most difficult criterion to achieve.

The table below gives the annual total returns on the MSCI Germany Index from 1993 to 2002. The returns are in the local currency. Use the information in this table to answer Questions 5-10. MSCI Germany Index Total Returns, 1993-2002 Year Return (%) 1993 46.21 1994 −6.18 1995 8.04 1996 22.87 1997 45.90 1998 20.32 1999 41.20 2000 −9.53 2001 −17.75 2002 −43.06 Source: Ibbotson EnCorr AnalyzerTM. To describe the central tendency of the distribution, perform the following: Calculate the sample mean return. Calculate the median return. Identify the modal interval (or intervals) of the grouped returns.

We calculate the sample mean by finding the sum of the 10 values in the table and then dividing by 10. According to Equation 3, the sample mean return, R¯¯¯ , is R¯¯¯=∑i=110Ri10 using a common notation for returns, Ri. Thus R¯¯¯ = (46.21 - 6.18 + 8.04 + 22.87 + 45.90 + 20.32 + 41.20 - 9.53 - 17.75 - 43.06)/10 = 108.02/10 = 10.802 or 10.80 percent. The median is defined as the value of the middle item of a group that has been sorted into ascending or descending order. In a sample of n items, where n is an odd number, the median is the value of the item in the sorted data set occupying the (n + 1)/2 position. When the data set has an even number of observations (as in this example), the median is the mean of the values of the items occupying the n/2 and (n + 2)/2 positions (the two middle items). With n = 10, these are the fifth and sixth positions. To find the median, the first step is to rank the data. We previously sorted the returns in ascending order in Solution 5A(i). Returns (%) Position −43.06 1 −17.75 2 −9.53 3 −6.18 4 8.04 5 20.32 6 22.87 7 41.20 8 45.90 9 46.21 10 The value of the item in the fifth position is 8.04, and the value of the item in the sixth position is 20.32. The median return is then (8.04 + 20.32)/2 = 28.36/2 = 14.18 percent. The modal return interval, as noted in Solution 5A, is Interval E, running from 28.38 percent to 46.24 percent.

The table below gives statistics relating to a hypothetical three-year record of two portfolios. Mean Monthly Return (%) Standard Deviation (%) Skewness Excess Kurtosis Portfolio A 1.1994 5.5461 −2.2603 6.2584 Portfolio B 1.1994 6.4011 −2.2603 8.0497 Based only on the information in the above table, perform the following: Contrast the distributions of returns of Portfolios A and B. Evaluate the relative attractiveness of Portfolios A and B.

With identical means, the two return distributions are similarly centered. Portfolio B's distribution has somewhat more dispersion, as measured by standard deviation. Both return distributions are negatively skewed to the same degree. Both portfolios have very large excess kurtosis, indicating much more frequent returns at the extremes, both positive and negative, than for a normal distribution. With identical mean returns and skewness, the comparison reduces to risk. Portfolio B is riskier as measured by standard deviation. Furthermore, risk-averse investors might view Portfolio B's more frequent extreme returns (both negative and positive), as indicated by greater kurtosis, as an additional risk element. Consequently, Portfolio A has the better risk-reward profile.