Section I.A Statistics and Methods

Skewness applied:

- For distributions that are non-normal (e.g., exhibit skewness, which is a measure of the distribution's asymmetry around the mean), the standard mean-variance analysis is limited, which means standard deviation is simply less meaningful. - With a negative skew, we have fewer, but more extreme outcomes to the left of the mean. Those outcomes pull the distribution and mean to the left. because the possibility of that extreme left tail event is not captured by the stat. - With a positive skew, we have fewer, but more extreme outcomes to the right of the mean. Those outcomes pull the distribution and mean to the right. For a negative skew, the standard deviation may be underestimating the risk For a positive skew, the standard deviation may be overestimating the risk.

What if excess returns are not normally distributed?

- Standard deviation is no longer a complete measure of risk - Sharpe ratio is not a complete measure of portfolio performance - Need to consider skew and kurtosis

Kurtosis

- kurtosis measures how fat the tails are on a distribution (i.e., the tendency for for extreme returns) relative to a normal distribution curve; - if kurtosis is positive (leptokurtic), a chart will show more extreme outcomes, creating a tendency for the observations around the mean to seem great (relative to those between the mean and tails) and appearing to have a higher peak; - if kurtosis is low (platykurtic), a chart will show thinner tails (i.e., fewer extreme outliers) with a tendency for a distribution that appears to have a flatter top (less peakedness) because those observations that would otherwise be concentrated around the mean are elsewhere in the distribution; - higher kurtosis suggests greater risk -than reflected in the normal distribution relied upon in the traditional mean-variance framework.

N vs. N-1

-If you are using all of the data available (i.e., a population) then use "N"; if they are only using a sample then use "N-1" when calculating standard deviation. -IWI gives some guidance on this issue at the bottom of page 92 in "Math for Investment Consultants" by stating, "It is important to recognize the different uses for n and n − 1; however, within the scope of this text the student should assume using n for the denominator unless otherwise instructed.". -That being said, when it comes to the certification exam, look for whether they actually tip you off by stating this is the entire data set or just a sampling. Most candidates report that the language in the question(s) on the cert-exam is/are clear when it comes to whether you should be using N (population) N-1 (sample). -Candidates do not report "both" answer choices (one using N and one using N-1) being presented to you. If you use N and the answer isn't there, you should be able to go back, plug in the same numbers into the formula, but use N-1 this time, and see that answer choice available - or vice versa. So, if you're not sure (and they don't tip you off), I would start with N first and then go to N-1 on the exam based on that feedback which comes straight from IWI.

Test Guidance

-Special Z-table cases (and the probability/proportion of area under a curve that each represent) such as: -z = -1.00 (i.e., 1 std dev left) covers 15.87% or about 16% which is why the area within 1 std deviation of a mean represents 68% of the outcomes -z = -2.00 covers 2.28% so +/- 2 std dev represents about 95% -z = -3.00 covers 0.13% so +/- 3 std dev represents about 99% z = -1.65 covers 5.0% (useful for VaR) -z = -2.33 covers 1.0% (useful for VaR)

Test Guidance

-We expect that you'll be asked to estimate the chance (probability) of some fund or portfolio experiencing some specified loss. -You can use the prior examples and your knowledge of standard deviation and the bell curve (without needing the z-score) to quickly estimate such a loss now. -And here are a few additional benchmarks (next slides) to keep in mind...

Standard Deviation Outcome Ranges

1SDEV = 68.26% 2SDEV=95.44% 3SDEV=99.74%

You are presented with an investment strategy with a mean return of 20% and a standard deviation of 10 percent. What is the approximate probability of a negative return if the returns are normally distributed?

2.40% A negative return would fall just outside of 2 standard deviations from the mean return of a normal distribution because the mean plus or minus 2 SD represents returns between 0% and +40%. Since returns within 2 standard deviations of the mean represent 95% of the expected outcomes, the remaining 5% of events fall outside of 0% and +40%. We're only interested in the left tail side of this problem, so we need to divide 5% by 2. This gives us 2.50%. Since the problem asks us to estimate, the best answer among these choices is 2.4%.

Detailed Content Outline

5% (6 questions)

Given the following data points: 4.8, 5.6, 7.1, 8.4, and 11.9 ... which of the following choices below accurately reflects the mean (listed first) and median (listed second) of the set?

7.56, 7.56 7.10, 7.56 7.10, 7.10 7.56, 7.10 The mean = (4.8+5.6+7.1+8.4+11.9)/5. = 7.56 The median is the central most data point. = 7.10 LAST ANSWER IS CORRECT

Mean Reversion

A theory that asserts data points or events (e.g., stock prices or returns) eventually move back toward their long-term average.

Tech Fund has a standard deviation of 14.96 and Energy Fund has a standard deviation of 12.43. What is the correlation coefficient between the two investments below? Year TECH FUND ENERGY FUND 1 36.45 14.10 2 -5.17 15.17 3 12.46 -15.7 4 9.10 2.20

A) .0694 B) .6199 C) .1290 D) .5159 Answer A First, calculate the average returns of both funds. Tech Fund = 13.21. Energy Fund = 3.94. Now take the differences between the tech fund's single and average returns and multiply by the differences between the energy fund's single and average returns. [(36.45 - 13.21) x (14.10 - 3.94)] + [(-5.17 - 13.21) x (15.17 - 3.94)] + [(12.46 - 13.21) x (-15.70 - 3.94)] + [(9.10 - 13.21) x (2.20 - 3.94)] = (236.12) + (-206.410) + (14.73) + (7.15) = 51.59/4 = 12.90.Second, calculate the correlation coefficient by plugging in the covariance you calculated into the formula: correlation coefficient = [(covarianceAB)/(standard deviationA x standard deviationB)] = 12.90/(14.96 x 12.43) = 12.90/185.95 = .0694.

Consider the following probability distribution for stocks A and B: The coefficient of correlation between A and B is: State Probability Return Stock A Return Stock A B 1 .10 10% 8% 2. .20 13% 7% 3 .20 12% 6% 4 .30 14% 9% 5 .20 15% 8%

A) 0.46. B) 1.20. C) 0.58. D) 0.60. covA,B = 0.1(10% - 13.2%)(8% - 7.7%) + 0.2(13% - 13.2%)(7% - 7.7%) + 0.2(12% - 13.2%)(6% - 7.7%) + 0.3(14% - 13.2%)(9% - 7.7%) + 0.2(15% - 13.2%)(8% - 7.7%) = 0.76; rA,B = 0.76/[(1.1)(1.5)] = 0.46.

Which would have higher standard deviations.....annual returns or monthly returns?

Annual returns have a higher standard deviation. Intuitively most would probably guess that monthly returns should have a higher standard deviation because monthly returns appear to jump around more than annual returns. But actually, standard deviations are higher over longer periods of time as more can happen (i.e., there is greater variability over time).

BETA

Another measure of investment risk is beta (β). Beta is a measure of systematic or market-related risk. It cannot be diversified away because it is the risk of being invested in the market. The beta of an investment reflects both the risk of an investment relative to the market and also reflects the correlation with the market. It is, in some sense, a measure of the response of the security to mar- ket movement. The beta of an investment may be found by a number of methods,

Which of the following statements is accurate? A) Standard deviations of individual securities are generally higher than the standard deviation of the market. B)Monthly standard deviations of stock returns are higher than annual standard deviations. C)Semi-deviation is the average of the squared variance of all values less than the average or mean. D)Variance is the square root of standard deviation.

Answer A Monthly standard deviations of stock returns are lower than annual standard deviations. Standard deviations of individual securities are generally higher than the standard deviation of the market. Standard deviation is the square root of variance. Semi-variance is the average of the squared deviations of all values less than the average or mean.

You are evaluating an emerging markets fund that has an average return of 16% and a standard deviation of 8%. If the portfolio has a beta of 1.25, what is the closest approximate probability that the fund will have a negative return for the year? A) 5.00% B) 2.50% C) 3.75% D) 1.25%

Answer B Average return of 16% - 8% - 8% = 0% (low) or 16% + 8% + 8% = 32% (high) at 2 standard deviations from average. That leaves a 5% chance of the return falling outside of that range (i.e., greater than 16% or less than zero). 5% divided by 2 (to capture the lower range) = 2.5% chance.

Question: The covariance between investment A and B is .015 and the standard deviations of investments A and B are 13.65% and 17.44% respectively. What is the correlation coefficient of investments A and B?

Answer: [(.015)/(.1365 x .1744)] = 0.63 Formula Slide 43

Question: The correlation coefficient between investment A and B is .63 and the standard deviations of investments A and B are 13.65% and 17.44% respectively. What is the covariance of investments A and B?

Answer: [(.63) x (.1365 x .1744)] = 0.015 Formula Slide 40

Variance

Calculate the variance of the following returns for Fund A in the select years: SAMPLE N-1 Year 1 = 2% Year 2 = 6% Year 3 = 18% Year 4 = 11% 1)Solve by calculating the mean first: 2+6+18+11 = 37; 37/4 = 9.25 2) Plug into the equation: [(2-9.25) 2+(6-9.25) 2+(18-9.25) 2+(11-9.25)2] = 142.75 142.75 divided by (n) 3 = 47.5833 Calculate the variance of the following returns for Fund A in its only years: Population N Year 1 = 2% Year 2 = 6% Year 3 = 18% Year 4 = 11% 1)Solve by calculating the mean first: 2+6+18+11 = 37; 37/4 = 9.25 2) Plug into the equation: [(2-9.25) 2+(6-9.25) 2+(18-9.25) 2+(11-9.25)2] = 142.75 142.75 divided by (n) 4 = 35.6875

Calculate Correlation Coefficient

Consider the following probability distribution for stocks A and B: State Probability Return on Stock A Return Stock B 1 0.10 10% 8% 2 0.20 13% 7% 3 0.20 12% 6% 4 0.30 14% 9% 5 0.20 15% 8% The coefficient of correlation between A and B is: a. 0.46 b. 0.60 c. 0.58 d. 1.20 Answer: a. 0.46 Solution: covA,B = 0.1(10% - 13.2%)(8% - 7.7%) + 0.2(13% - 13.2%)(7% - 7.7%) + 0.2(12% - 13.2%)(6% - 7.7%) + 0.3(14% - 13.2%)(9% - 7.7%) + 0.2(15% - 13.2%)(8% - 7.7%) = 0.76; rA,B = 0.76/[(1.1)(1.5)] = 0.46

Measures of Dispersion

variance standard deviation semi-variance semi-deviation coefficient of variation

Measures of Central Tendency

• Mean - also referred to as the "average" of a set of data points; it is the sum of the digits (data) divided by the number of digits (data) • Median - In a set of numbers, it is the digit located in the very middle of the number set. It is not an average of that number set. • Mode - In a given number set, mode is the number most often repeated in that set.

Normal and Non-normal Distributions

• Normal distribution is considered foundational in the development of Modern Portfolio Theory. SDEV distribution chart-68-95-99 • This is controversial as it relates to investment analysis and inputs used to build optimal portfolios (e.g., black swans).

1. Basic statistical measures

• measures of central tendency • dispersion • variability • skewness • kurtosis

5. Time series and trend analysis concepts, methods, and interpretation

• seasonality • mean reversion • multi-period forecasting • smoothing

2. Basic statistical concepts

• the normal distribution • probability • sampling from a population • significance testing

Covariance and Correlation

Covariance is a measure of how two investments (or portfolios) move together or apart. Covariance, however, may be a large number so we use the concept of correlation to compress the covariance into a range of +/− 1. Correlation is denoted by the Greek letter for R which is ρ (pronounced rho).

Monte Carlo Simulations

Defined Statistical modeling method used to approximate the probability of future outcomes through multiple trials (simulations) using random variables Advantages Can help one visualize and understand variability of future growth (and returns); offers a way to analyze risk; powerful tool for illustrating a variety of possible outcomes that could be useful in planning Disadvantages Generates a normal distribution where the most likely scenario is found in the middle of the events - this is not always realistic and can create overconfidence which may lead to developing overly aggressive, risky portfolios; these models are not built to allow for a wide range of inputs: factors, expectations, etc.; model assumes efficient markets

Random Walk

Defined: A mathematical event in which a set of events or samples follows a pattern of random (unpredictable) steps. Application: Theory that hypothesizes stock prices cannot be accurately predicted based on historical data due to the random walk theory (made popular by Princeton's Burton Malkiel).

Standard Deviation

Defined: • A measure of dispersion expressed as the square root of the variance. • Measures the amount of variability around the average or mean. • An advantage of using standard deviation (as compared to variance) is that it expresses dispersion in the same units as the original values in the sample or population. Application: • A low standard deviation indicates that data points gather close to the mean while a high standard deviation indicates that data points are spread far apart from the mean. Remember • Standard deviation (SD) is represented by Greek letter sigma (σ) • Measure of volatility (i.e., risk) • Measures amount of variation or dispersion from anaverage • **SD is considered a measure of "total risk" • The SD of a random variable is the square root of its variance • SD is used in the Capital Allocation Line (CAL) • **SD is used in the Sharpe Ratio, M2, information ratio

Semi-Variance

Defined: • Measures data that is below the mean or target value of a data set. • **Considered a better measurement of downside risk. • **Semi-variance is the average of the squared deviations of all values less than the average or mean.

Variance

Defined: • Measures how far a set of numbers is spread out. • Defined as the average squared difference between the mean and each item in the population or in the sample. • Variance is always non-negative. Application: • A high variance means that data points are very spread out. -Variance Value of zero means that all values within the set are identical

Correlation Coefficients

Defined: • The correlation coefficient indicates the degree of relationship between two variables; • The correlation coefficient always lies between -1 and +1; -1 indicates perfect negative relationship between two variables, +1 indicates perfect positive linear relationship, and 0 indicates lack of any linear relationship

Correlation Coefficients

Defined: • The correlation coefficient indicates the degree of relationship between two variables; • The correlation coefficient always lies between -1 and +1; -1 indicates perfect negative relationship between two variables, +1 indicates perfect positive linear relationship, and 0 indicates lack of any linear relationship Calculate Correlation Coefficient Question: The covariance between investment A and B is .015 and the standard deviations of investments A and B are 13.65% and 17.44% respectively. What is the correlation coefficient of investments A and B? Answer: [(.015)/(.1365 x .1744)] = 0.63

Coefficient of Variation (CV)

Defined: • The ratio of the standard deviation to the mean. • AKA: relative standard deviation. • **Formula: CV = standard deviation/mean-Memorize • Result: it shows the extent of variability in relation to mean of the population. • Assumptions: normal distribution. Application: • For comparison of data sets with different units or widely different means, one should use the CV instead of standard deviation.

Covariance

Defined: • indicates how two variables are related • a measure of the degree to which the returns of two assets move together • a positive covariance indicates that assets move together while a negative covariance indicates that assets move inversely • assets possessing a high covariance with each other do not offer much diversification

Smoothing past values with an n-period moving average

Defined• • A calculation that creates a series of averages. • Also called a rolling or moving mean. Application: • Used to smooth out short-term fluctuations or volatility. • Accentuates longer-term trends and cycles.

Probability: expected return on a portfolio

E(r)= Sum p(r)*r(s) p(s) = probability of a state r(s) = return if a state occurs s = state Q: You expect there is a 68% probability of an investment returning 7% over the next 12 months and a 32% probability of a 0% return. What is the expected return on this investment? a. 5.83% b. 6.80% c. 4.76% d. 7.00% Answer: c. 4.76% Solution: Multiply probability of each outcome by that outcome and add them together. Expected Return = (68% * 7%) + (32% * 0%) = 0.0476 + 0.0 = 0.0476 = 4.76%

The mid-cap blend fund you're tracking has an expected return of 9.7% and a standard deviation of 14.1%. The S&P 400 mid-cap index has an expected return of 8.9% and a standard deviation of 15.0%. The correlation between the two is .88. What is the covariance between the two?

Group of answer choices 0.0076 0.0086 0.0186 0.0212 Covariance = correlation of .88 multiplied by the standard deviation of the fund (.141) multiplied by the standard deviation of the index (.15) = 0.0186.

The frontier markets fund you're tracking has an expected return of 16.8% and a standard deviation of 21.7%. The S&P 500 index has an expected return of 9.7% and a standard deviation of 12.5%. The covariance between the two is .0123. What is the correlation between the two?

Group of answer choices 1.3249 0.4535 0.0092 0.5844 The Coefficient of Correlation = Covariance divided by the product of the standard deviations of the two assets = 0.0123 / (0.217 * 0.125) = 0.4535.

The following statements are correct or accurate regarding the disadvantages of Monte Carlo simulation ("MCS"). I. MCS generates a normal distribution which is not always realistic and can create overconfidence. II. MCS often leads to developing overly conservative portfolios. III. MCS models are not built to allow for a wide range of inputs (e.g., factors, expectations, etc.). IV. MCS models assumes efficient markets.

II and III only I and IV only I, III and IV only ​​​​​​​ I, II and III only I,III, AND IV ONLY IS CORRECT

If excess returns are not normally distributed _______________________. A)We should consider skewness but not kurtosis. B)We do not need to consider skewness and kurtosis. C)Sharpe Ratio is then likely a complete measure of portfolio performance. D)Standard deviation is no longer a complete measure of risk.

If excess returns are not normally distributed _______________________. A)We should consider skewness but not kurtosis. B)We do not need to consider skewness and kurtosis. C)Sharpe Ratio is then likely a complete measure of portfolio performance. D)Standard deviation is no longer a complete measure of risk. What if excess returns are not normally distributed? Standard deviation is no longer a complete measure of risk. Sharpe ratio is not a complete measure of portfolio performance. We need to consider skew and kurtosis. Answer D

The Normal Distribution

Investment management is easier when returns are normal. - Standard deviation is a good measure of risk when returns are symmetric. - If security returns are symmetric, portfolio returns will be, too. - Future scenarios can be estimated using only the mean and the standard deviation.

Kurtosis measures the peakedness of a probability distribution or normal distribution curve. If kurtosis is positive, the chart will show ____________________________ that is __________________________ and may have __________________________________________. A) less concentrated mean / more slender in distribution / fatter tails B) a more slender distribution (higher peak) / concentrated more around the mean / fatter tails C) distribution less concentrated around the mean / concentrated around the mean / a flatter peak D) flatter peak / more concentrated around the mean / slender distribution

Kurtosis defined: measures the peakedness of a probability distribution or normal distribution curve; if kurtosis is positive (leptokurtic), the chart will show a more slender distribution, i.e., higher peak, that is concentrated more around the mean and may have fatter tails; if kurtosis is low (platykurtic), a chart will show thinner tails and distribution that is less concentrated around the mean, thus a flatter peak.

M-Squared

M-squared is a return measure that adjusts for total risk from the mix of funds or managers being evaluated. The weightings of the mix are selected to match the total risk of an index (e.g., S&P 500).

Information Ratio

Non Market Risk Measure The information ratio is the average excess return of a portfolio over a benchmark divided by the standard deviation of the excess returns. This measures the ability to select securities relative to a benchmark. It captures both the size of the excess return and the ability to do so consistently. Information Ratio = excess return / tracking error

TRACKING ERROR

Non Market Risk Measure Tracking error is the annualized standard deviation of monthly excess returns. You can generally assume the tracking error is normally distributed. To calculate an annual tracking error, multiply the observed tracking error by the square root of the number of periods in one year. If you use quarterly data, the annual tracking error is found by multiplying the observed tracking error by the square root of 4.

Alpha (Jensen's Performance Index)

Positive alpha means that the manager has added value through his or her security selection and timing. Negative alphas indi- cate value subtracted. α = E(Rp) - [Rf + βp (Rm - Rf)]

Another Calculate Covariance

Question: The correlation coefficient between investment A and B is .63 and the standard deviations of investments A and B are 13.65% and 17.44% respectively. What is the covariance of investments A and B? Answer: [(.63) x (.1365 x .1744)] = 0.015

Annualized Standard Deviation Question: The monthly standard deviation of a portfolio has been calculated to be 2.17. Convert this monthly standard deviation to annualized standard deviation.

Question: The monthly standard deviation of a portfolio has been calculated to be 2.17. Convert this monthly standard deviation to annualized standard deviation. a. 4.34 b. 5.32 c. 7.52 d. 26.04 Answer: c. 7.52 Solution: Multiply 2.17 by the square root of 12. (2.17) x (3.4641) = 7.5171 =SDEV*square root of time (frequency) Monthly=12; Quarterly = 4 Semiannual=2

Application of Probability

Question: You are presented with an investment strategy with a mean return of 20% and a standard deviation of 10 percent. What is the approximate probability of a negative return if the returns are normally distributed? a. 2.00% b. 2.40% c. 4.00% d. 4.80% Answer: b. 2.40% Solution: A negative return would fall just outside of 2 standard deviations from the mean return of a normal distribution because the mean plus or minus 2 SD represents returns between 0% and +40%. Since returns within 2 standard deviations of the mean represent 95% of the expected outcomes, the remaining 5% of events fall outside of 0% and +40%. We're only interested in the left tail side of this problem, so we need to divide 5% by 2. This gives us 2.50%. Since the problem asks us to estimate, the best answer among these choices is 2.4%.

Multiple regression analysis

R2 (also known as the coefficient of determination) • gives the proportion of variation in one variable that can be explained by another variable • this metric indicates the "closeness of fit" or "accuracy of fit"• keep in mind that no causality is claimed by the coefficient of determination Applications: • the higher the number the more meaning meaningful the relationship • measurements below 0.6 would indicate a weak relationship between variables (i.e., the benchmark is not as helpful as a comparative tool) • R-squared gives us an indication of the level of diversification in a portfolio • assuming the benchmark/index is diversified, a portfolio is well diversified if it has a high R-squared relative to that index • R-squared on a portfolio also gauges the reliability of alpha as an indicator of the manager's return and beta as an indicator of risk

Multi-period Forecasting

Research that uses historical data over multiple time periods (or series) to model forecasts. Multi-period forecasting often includes the use of different models. Benefits to multi-period forecasting may include improved accuracy, consistency, and smoothing of volatility.

Skewness

Skewness defined: - describes asymmetry of data points from a normal distribution; if data points are skewed to the left it is described as negative skew and if data points are skewed to the right it is described as positive skew

Scenario of Returns

State. Prob. of State. r in State Excellent .25 0.31 Good .45 0.14 Poor .25 -0.0675 Crash .05 -0.52 E(r) = (.25)(.31) + (.45)(.14) + (.25)(-.0675) + (0.05)(-0.52) E(r) = .0976 or 9.76%

Calculate Covariance

Step 1. Find the differences between each return and the arithmetic mean for that asset. This is exactly what we did for standard deviation. Step 2. Multiply the differences of the two sets of assets, for each period. Step 3. Add those up and divide by n (if solving for covariance of a population). Question: Fund A has a standard deviation of 6.28 and Fund B has a standard deviation of 8.28. What is the covariance between the two investments below? Year FUND A FUND B 1 12.56 6.21 2 19.76 2.33 3 8.11 14.44 4 2.62 23.89 Solution: First, calculate the average returns of both funds. Fund A = 10.76. Fund B = 11.72. Now take the differences between Fund A's single and average returns and multiply by the differences between Fund B's single and average returns. [(12.56 - 10.76) x (6.21 - 11.72)] + [(19.76 - 10.76) x (2.33 - 11.72)] + [(8.11 - 10.76) x (14.44- 11.72)] + [(2.62 - 10.76) x (23.89 - 11.72)] = (-9.918) + (-84.51) + (-7.208) + (-99.0638) = -200.70. -200.40 / 4 = -50.175. Note - the question did not mention whether to calculate based on the population or a sample, so we used the population and we recommend you do the same on the exam (if neither is specified).

Sharpe Ratios

The Sharpe ratio measures the total risk of the portfolio by including standard deviation instead of only the systematic risk (i.e., beta). It does not implicitly assume that a portfolio is well diversi- fied. The foundation of the Sharpe ratio is Modern Portfolio Theory (MPT), and the idea is the higher the Sharpe ratio, the closer the portfolio is to the mean variance portfolio. This means that The Sharpe index standardizes the return in excess of the risk-free rate by the variability of the returns. Sharpe Ratio = average actual return −average riskfree ---------------------------------------------- return standard deviation of the portfolio ___. ___ Sharpe Ratio = Rp −Rf /σp

Sortino Ratio

The Sortino ratio is also based on MPT and is closely related to the Sharpe ratio except that it uses downside risk. The Sortino ratio gives the same ranking as the Sharpe ratio of portfolio performance. ____ ____ Sortino Ratio =Rp −Rf Downside σp

Treynor Ration

The Treynor ratio is similar to the Sharpe ratio, except it uses the beta of the portfolio in the denominator. This means that it is using systematic risk instead of total risk. This measure assumes the portfolio is fully diversified. Treynor Ratio = avg actual return −avg risk free return beta of the portfolio ___. ___ Treynor Ratio = Rp −Rf βp

Capital Asset Pricing Model (CAPM)

The capital asset pricing model advances the relationship between risk and return in the efficient market context, and adds the possibility of earning a risk-free return. The concept is applied at the macro level, which specifies the relationship between risk and return on a portfolio, and the micro context, which specifies this relationship on a specific asset. The macro aspect of CAPM is the development of the capital market line, where risk is measured by the portfolio's standard deviation. In effect, the capital market line becomes the efficient fron- tier. Portfolios on this line represent the best attainable combination of risk and return. Portfolio combinations range from no risk by earning the risk-free rate return to high-risk in which securi- ties are bought on margin.

Coefficient of Determination (R-squared)

The coefficient of determination gives the variation in one variable explained by another, and is an important statistic in investments. No causality is claimed by the coefficient of determination. It is the job of the investor to interpret it. R-squared gives us an indication of the level of diversification in a portfolio. A portfolio is well diversified if it has a high R-squared relative to an index. R-squared on a portfolio also gauges the reliability of alpha as an indicator of the manager's return and beta as an indicator of risk. If the R-squared is close to 1.00, alpha and beta statistics can be used to explain the return of the portfo- lio as a linear function of the market.

Semivariance and Downside Deviation

The concepts are the same as variance and standard deviation, except we use only those values below our average or target return.

SDEV and Variance

There are two common measures of volatility or risk: standard deviation and variance. These measures evaluate how much returns fluctuate from a mean or some target return. If one value is known, the other can be calculated. While expressed in a complex mathematical formula, the cal- culation of standard deviation is fairly straightforward. 1) Calculate the arithmetic mean rate of the returns. 2) Subtract this mean rate of return from each year's returns. 3) Square these differences. 4) Add the squares of the differences and find the arithmetic average of these sums. ***This value is the variance. **The square root of this value is the standard deviation. Notice that the descriptive label of the variance is the "percent squared," but standard deviation is a "percent." Standard deviation and variance measure variability, meaning that they measure BOTH upside and downside risk. SEMI VARIANCE and DOWNSIDE DEVIATION ONLY calculate downside

Which of the following statements is not accurate or true? A) Kurtosis identifies the fatness of the distribution tails. B) Skewness refers to the asymmetry of a return distribution. C) Stock returns are usually positively skewed because there is a higher frequency of positive returns compared to negative returns. D) Kurtosis measures the higher than normal probabilities for extreme returns.

Which of the following statements is not accurate or true? A) Kurtosis identifies the fatness of the distribution tails. B) Skewness refers to the asymmetry of a return distribution. C) Stock returns are usually positively skewed because there is a higher frequency of positive returns compared to negative returns. D) Kurtosis measures the higher than normal probabilities for extreme returns. Answer C Stock returns are usually negatively skewed because there is a higher frequency of negative deviations from the mean. See CIMA textbook, pages 330-331.

Which of the following statements regarding correlation coefficients are accurate? I. When two stocks have a correlation coefficient that is greater than zero but less than 1.00, it means that they have a positive correlation and there may be some benefit through diversification. II. Mutual fund A and mutual fund B have a correlation coefficient of -0.89. This indicates beneficial diversification given the low level of positive correlation or association between these funds. III. In a situation where two investments have a perfectly negative correlation coefficient of -1.00, the variance or standard deviation would be zero in a perfectly balanced portfolio allocation. IV. A portfolio of investments that exhibits a vast range of correlation coefficients from 0.20 to 0.60 would be considered highly diversified and ensure a very low risk level as measured by standard deviation.

A) I and III only B) II and III only C) II and IV only D) I and II only Answer A I. TRUE: When two stocks have a correlation coefficient that is greater than zero but less than 1.00, it means that they have a positive correlation and there may be some benefit through diversification. II. FALSE: Mutual fund A and mutual fund B have a correlation coefficient of -0.89. This indicates beneficial diversification given the low level of positive correlation or association between these funds. THESE FUNDS DO NOT HAVE A POSITIVE CORRELATION TO EACH OTHER III. TRUE: In a situation where two investments have a perfectly negative correlation coefficient of -1.00, the variance or standard deviation would be zero in a perfectly balanced portfolio allocation. The negative correlation relates to positive and negative movements from a mean, and thus can equal positive or negative returns. But the variance or standard deviation of those returns would be zero. IV. FALSE: A portfolio of investments that exhibits a vast range of correlation coefficients from 0.20 to 0.60 would be considered highly diversified and ensure a very low risk level as measured by standard deviation. THE RISK LEVELS OF THESE INVESTMENTS COULD ALL BE VERY HIGH INDIVIDUALLY; AND WHILE DIVERSIFICATION MAY BRING THE PORTFOLIO'S STANDARD DEVIATION DOWN, IT MAY STILL BE VERY HIGH AND CONSIDERED RISKY.

A theory that describes stock prices as unpredictable; and therefore, most methods of valuing and/or predicting expected stock prices are futile. A) Random Walk Theory B) Behavioral Economics Theory C) Efficient Markets Theory D) Arbitrage Pricing Theory

ANSWER A A mathematical event in which a set of events or samples follows a pattern of random (unpredictable) patterns. Application: Theory that hypothesizes stock prices cannot be accurately predicted based on historical data due to the random walk theory (made popular by Princeton's Burton Malkiel). Conclusion: most if not all methods of predicting stock prices will be ineffective.

Standard deviation is a good measure of risk when ____________________. A) skewness is significantly positive B) returns are symmetric C) returns are asymmetric D) skewness is significantly negative

ANSWER B The Normal Distribution -Investment management is easier when returns are normal. -Standard deviation is a good measure of risk when returns are symmetric. -If security returns are symmetric, portfolio returns will be, too. -Future scenarios can be estimated using only the mean and the standard deviation.

When calculating the average (and standard deviation) of a discrete probability distribution, you don't simply add the potential outcomes and divide by the number of data points. You have to apply the likelihood of each outcome. For calculating variance, you have to ____________________________________. A) add up the possible outcomes and then apply equal weights to each before taking the square root of the result and multiplying by the sample or population size. B)determine the equal weights of each outcome, discount the sum of the result by the square root of the possible high and low result and then divide by n or n-1. C) multiple the weighted average by the least sums result squared and divide by n or n-1 as appropriate. D) apply the probabilities as well to the difference of each outcome to its mean (then squared) and, as a result, do not have to divide by the number of data points n (or n-1) because you've already weighted it by the probabilities.

ANSWER D Discrete Probability Distribution Occurs when there are different probabilities related to different specific outcomes. That fact impacts the calculation of the mean, variance, covariance and correlation coefficient. When calculating the average (and standard deviation) of a discrete probability distribution, you don't want to simply add the potential outcomes and divide by the number of data points. To do that means that you are effectively weighting the data points equally; five date points, as in this case, are given equal weight of 20%, effectively. Instead, you have to apply the likelihood of each outcome. So, in this case, you do that by multiplying the probability of an outcome to that outcome and then adding them altogether. For variance, you have to apply the probabilities as well to the difference of each outcome to its mean (then squared) and, as a result, do not have to divide by the number of data points n (or n-1) because you've already weighted it by the probabilities.

Seasonality

Adjusting past performance or forecasts for demonstrated or expected impact of factors such as seasonality may be beneficial when analyzing data including specific company and industry stock returns and volatility.

Coefficient of Determination

Also known as R2 (R squared) R2 Formula: Beta2 x Standard Deviation of Market2 _______________________________________________ Standard Deviation of Portfolio2