Lesson 2 of Investment Planning: Portfolio Theory Flashcards
Standard Deviation!
-
Standard Deviation is
- a measure of risk and variable of returns.
- The higher the standard deviation, the higher the riskiness of the investment.
- SD in simple terms, measures how much something flip-flops around on average.
- SD can be used to determine the total risk of an undiversified portfolio.
- For CFP EXAM be prepared to:
- Use SD to determine the profitability of returns.
- Calculate the SD.
SD to Calculate a Probability of Returns
The graph below illustrates a normal distribution with probabilities between +/- 1, 2, and 3 standard deviations away from the average.
EXAM TIP
Memorize the 68, 95 & 99 depending if the returns are +/- 1, 2, or 3 SD sway from the average.
Example of SD to Calculate Profitability of Returns
Calculating SD with a Financial Calculator
- The first thing you do is click 2nd and then click 7 for (data)
- Second you click 2nd and then click CE/C for (CLR Work).
- The type of inputs into the calculator. Make sure X01,2,3… is what the information is being put into. Click the down arrow to navigate between the X0’s. Click enter after input is in each time you input a number in, and then click the down arrow.
- Y01 should be left as is. Should be 1. Skip to the next X0.
- When all numbers are in click 2nd and then click 8 for (stat).
- Scroll down until you see 1-V. When you find that, press enter.
- After you press the down arrow and you should see all the stats associated.
- SD = Sx
X with the line above is the mean.
https://www.youtube.com/watch?v=l1jo6CpYATA
Calculating SD with a Financial Calculator
Exam Tip
It is possible that a CFP Exam Question regarding SD could simply be “ Which of the following assets is most risky?” They are really asking you to calculate SD and select the asset with the HIGHEST standard deviation.
2nd way to calculate standard deviation
- The examiners may provide you with the expected returns for a stock or fund with the likelihood (probability that those returns will be realized.
- The calculation is simply the sum of all expected returns multiplied by their respective probabilities
- Expected Return = Σ (Return x Probability)
Example of 2nd way to calculate standard deviation
Shantele has been looking at mutual funds and her advisors tells her about a great fund with the following expected returns and probability of obtaining each return.
Expected Return:
10%
15%
18%
Probability of Returns:
30%
60%
10%
Answer:
The total expected return is 13.8%
= (10% x 30%) + (15% x 60%) + (18% + 10%)
Coefficient of Variation!
- The coefficient of variation is useful in
- determining which investment has more relative risk
- when investments have different average returns.
- Useful when comparing two assets with different average returns.
- The coefficient of variation tells us the
- probability of actually experiencing a return close to the average return.
- The higher the coefficient of variation the more risky
- an investment per unit of return.
The formula for the Coefficient of Variation
Not given on exam
CV = Standard Deviation ÷ Average Return
Example of CV
Example 1:
Which one is more risky?
Stock A: SD = 12% and Average Return = 8%
Stock B: SD = 10% and Average Return = 8%
The coefficient of variation is not necessary to determine which investment is more risky because their average returns are the same. Stock A is more risky because it has the same return, but a higher SD.
Example 2:
Alternatively, if Fred had the following investment opportunities, which one has the highest risk per unit of return earned?
Stock A: SD = 12% and Average Return = 10%
Stock B: SD =8% and Average Return = 5%
Stock A: CV = 0.12 ÷ 0.10 = 1.2
Stock B: CV = 0.08 ÷ 0.05 = 1.6
Therefore, Stock B has more risk per unit of return compared to A. You cannot assume that because Stock A has a higher standard deviation that is has a higher adjusted return.
Could ask which has a higher risk adjusted return which would be lower CV.
Distribution of Returns!
- Normal Distribution
- Lognormal Distributions
- Skewness
Normal Distribution
- A normal distribution is appropriate if an investor is considering a range of investment returns, as was covered above.
Logonormal Distribution
- A lognormal distribution is not a normal distribution.
- A lognormal distribution is appropriate if an investor is considering a dollar amount or portfolio value at a point in time.
For example:
- If an investor invests $1 into the market 60 years ago, it would be worth $60 today. With a lognormal distribition you are looking for a trend line or ending dollar amount.
Skewness
- Skewness refers to a normal distribution curve shifted to the left or right of the mean return.
- Commodity returns tend to be skewed.
- A distribution is positively skewed when
- its tail is more pronounced on the right side than it is on the left.
- Since the distribution is positive, the assumption is that its value is positive.
- As such, most of the values end up being left of the mean.
- This means that the most extreme values are on the right side.
- In statistics, a negatively skewed (also known as left-skewed) distribution is a type of distribution in which
- more values are concentrated on the right side (tail) of the distribution graph while the left tail of the distribution graph is longer.
Kurtosis
- Kurtosis refers to variation of returns. If there is little variation in returns, the distribution will have a high peak.
- Treasuries have little variation of returns, have a high peak, and, therefore, have a positive kurtosis.
- If returns are widely dispersed, the peak of the curve will be low and have a negative kurtosis.
- Leptokurtic = High peak and fat tails (higher chance of extreme events)
- Platykurtic = Low peak and thin tails (lower chance of extreme events)
Exam Tip
- Leptokurtic = High peak and fat tails (higher chance of extreme events)
- Platykurtic = Low peak and thin tails (lower chance of extreme events)
**OTHER: **
- In statistical terms, “fatter tails” refer to the relatively higher probability of extreme or outlier events in a distribution.
- Tails in a distribution represent the regions of values that are far from the mean (average).
- A distribution with fatter tails has a higher likelihood of observing values at a greater distance from the mean compared to a distribution with thinner tails.
Exam Question
Conrad has noticed that the stock she purchased tends to have a very high tight distribution around the mean but there seems to be a high probability of “outliers” (multi-deviation returns). This is most indicative of what type of curve?
a) Positive skewness
b) Leptokurtosis
c) Normal
d) Lognormal
Answer: B
The leptokurtic distribution reflects the tendency of observations to fall closely around the mean creating a peaked distribution at the mean with thicker tails. If historical returns indicate leptokurtosis then there is much more reserved variation in periodic returns but a higher probability of large multi-sigma deviations (i.ie “fat tails”)
Mean-Variance Optimization
Mean-variance optimization is the process of
- adding risky securities to a portfolio but
- keeping the expected returns the same.
- It’s finding the balance of combining asset classes that
- provide the lowest variance as measured by standard deviation.
Monte Carlo Simulation
- Monte Carlo Simulation is a spreadsheet simulation that gives a probabilistic distribution of events occurring.
- For example, what is the probability of running out of money in retirement with a client who has a withdrawal rate of 3%, 4%, or 5%.
- Monte Carlo simulation then adjusts assumptions and returns the probability of an event occurring depending upon the assumption.
- Allows for “what if” scenarios and sensitivity analysis if variables such as inflation or savings rate change.
Exam Tip
It’s not likely that the above concepts will be tested. Know the characteristics.
If there is a question it will most likely be to adjust the assumptions.
Covariance!
Covariance is the measure of
- two securities combined and their interactive risk.
- In other words, how price movements between two securities are related to each other.
Covariance is a measure of relative risk.
- If the correlation is known, or a given, covariance is calculated as
- the deviation of investment ‘A’ times the deviation of investment ‘B’ times the correlation of investment ‘A” to investment ‘B’, thus:
Covariance Formula
Exam Tip
You may need to calculate COV if you are given the correlation coefficient and need to calculate the standard deviation of a two-asset portfolio. This is a provided formula on the CFP board formula sheet.
Correlation/ Correlation Coefficient
- Correlation and the covariance measure movement of one security relative to that of another.
- Covariance and correlation coefficient are both relative measures.
- Correlation ranges from +1 to -1 and provides the investor with insight as to the strength and direction two assets move relative to each other.
- A correlation of +1 denotes that two assets are perfectly positively correlated.
- A correlation of 0 denotes that assets are completely uncorrelated.
- A correlation of -1 denotes a perfectly negative correlation.
- Diversification benefits (risk is reduced) begin anytime correlation is less than 1.
This is not a provides formula.
Correlation/ Correlation Coefficient Formula
Exam Question
When combining asset classes, an investor begins to receive diversification benefits when correlation is?
a) Equals -1
b) Less than 1
c) Less than 0
d) Less than or equal to 1
e) Equals 0
Answer: B
- When the correlation coefficient is less than 1, an investor begins to receive diversification benefits. In other words, the variability of returns is reduced.
- The most diversification benefits are received when the correlation is equal to -1, but diversification begins when correlation is less than 1.
Exam Tip
You won’t have to calculate correlation using the formula, but you need a thorough understanding of the concepts related to correlation.
Beta!
- The beta coefficient is a measure of an individual security’s volatility relative to that of the market.
- Beta is used to measure the volatility of a diversified portfolio.
- It measures the systematic risk dependent on the volatility of the security relative to that of the market.
- The beta of the market is 1.
- A stock with a beta of 1 will be expected to mirror the market in terms of direction, return, and fluctuation.
- A stock beta higher than 1 means the stock fluctuates more than the market and greater risk is associated with that particular security.
- A stock beta of less than 1 indicates that the security fluctuates less relative to market movements.
- The greater the beta coefficient of a given security, the greater the systematic risk associated with that particular security.
- Beta is also a measure of systematic risk or market risk,
- whereas standard deviation is a measure of total risk.
Beta is the slope of the line that represents a security’s return when plotted relative to market returns. The slope (or Beta) is determined below:
Beta Formula
Slope of line represents security’s return plotted relative to market return
(GIVEN on EXAM)
2nd way to calculate BETA
Beta may also be calculated by
- dividing the security risk premium by
- the market risk premium.
Example of Calculating Beta
If a fund has a return of 20% and the market has a return of 10%, the beta would be 20÷10 = 2
If a fund has a return of 5% and the market has a return of 10%, the beta would be 5÷10 = 0.5
Coefficient of Determination or R-Sqaured (r^2)!
- R-squared is a measure of how much return is due to the market or what percentage of a security’s return is due to the market.
- Calculate r-squared (r^2) by squaring the correlation coefficient.
- R-squared also provides the investor insight into how well diversified a portfolio is, because the higher the r-squared, the higher percentage of return from the market (systematic risk) and less from unsystematic risk.
- R-squared also tells the investor if Beta is an appropriate measure of risk.
- if r-squared is greater than or equal to 0.70, then Beta is an appropriate measure of total risk.
- If r-squared is less than 0.70, then Beta is not an appropriate measure of total risk and standard deviation should be used to measure total risk.
Example
If Mutual Fund XYZ has a correlation coefficient of 0.80 then r^2 is 0.64, which means 64% of fund XYZ’s return is due to the market.
Exam Question
Mutual fund XYZ has a 5-year return of 12%, with a standard deviation of 15%. Fund XYZ has a Beta of 1.4, with a correlation of 0.90 to the S&P 500. What percent of return from fund XYZ is due to the S&P 500?
a) 90%
b) 81%
c) 19%
d) 10%
Answer: B
Correlation = 0.90, therefore, r-sqaured = 0.81.
81% of the return is due to the market and 19% is due to unsystematic risk.
Exam Question
Portfolio Risk
- The risk of a portfolio can be measured through determination of the interactivity of the SD and covariance of securities in the portfolio.
- The process also utilizes the weight of both securities involved, the standard deviations of the respectice securities, and the correlation coefficient of the two securities.
- This formula is also known as Portfolio Deviation Formula or Standard Deviation of a Two Asset Portfolio.
Example of Calculating Portfolio Risk
Systematic Risk
OR (MNEMONIC Risks)
- Systematic Risk is the lowest level of risk one could expect in a fully diversified portfolio.
- It is inherent in the “system” as a result of the unknown element existing in securities that have no guarantees.
- THERE IS NO ESCAPING THE RISK. THE RISK WILL ALWAYS BE THERE
Subtopics:
- Nondiversifiable Risk
- Market Risk
- Economy-based Risk
Unsystematic Risk
- Is the risk that exists in a specific firm or investment that can be eliminated through diversification.
- Through ownership of a number of different securities or investments,
- the investor can eliminate the risk and insulate their investments.
Subtopics:
- Diversifiable Risk
- Unique Risk
- Company-specific risk
Nondiversifiable Risk & Diversifiable Risk
Nondiversifiable Risk:
- Risk that cannot be eliminated by having a large portfolio of many assets.
Diversifable Risk:
- an investor can only mitigate against unsystematic risk through diversification. An investor uses diversification to manage risk by investing in a variety of assets.
Systematic Risk!
- Purchasing Power Risk*
- Reinvestment Rate Risk*
- Interest Rate Risk*
- Market Risk
- Exchange Rate Risk
- *Most likely to be tested
- PRIME*
Can’t get away from them. Always gonna be there.
Purchasing Power Risk*
- Is the risk that (1) inflation will erode the amount of goods and services that can be purchased, and
- (2) a dollar today cannot purchase the same amount of goods and services tomorrow or the day after.
Purchasing power risk affects both equities and bonds.
Reinvestment Rate Risk*
-
Reinvestment rate risk is the risk
- that an investor will not be able to reinvest at the same rate of return that is currently being received.
Reinvestment rate risk mostly impacts bonds.
Interest Rate Risk*
- Interest rate risk is the risk that changes in interest rates will
- impact the price of both equities and bonds.
- There is an inverse relationship between interest rates and both equities and bonds.
Market Risk
- Market risk impacts all securities in the short term
- because the short-term ups and downs of the market
- tend to take all securities in the same direction.
Exchange Rate Risk
- Exchange rate risk is the risk that a change in exchange rates will impact the price of international securities.
Unsystematic Risks!
- Accounting Risk
- Business Risk**
- Country Risk**
- Default Risk**
- Executive Risk
- Finanial Risk**
- Government/Regulation Risk**
- **Most likely to be tested
ABCDEFG
Accounting Risk
- Accounting risk is the risk associated with an audit firm being too closely tied to the management of a company.
For example: Arthur Andersen and Enron
Business Risk**
- Is the inherent risk a company faces by operating in a particular industry.
For example,
- Halliburton faces much different risks in the oil industry than Microsoft does, which is primarily selling intellectual property and protecting copyrights.
Country Risk
- Country Risk is the risk a company faces by doing business in a particular country.
For example, Halliburton faces unique risks doing business in Iraq.
Default Risk**
- Default risk is the risk of a company defaulting on their debt payments.
- Default risk can be thought of as the likelihood of a firm being able to satisfy its debt obligations on time.
Executive Risk or Management Risk
- Executive is the risk associated with the moral and ethical character of the management running the company.
Financial Risk**
- Financial risk is the amount of leverage deployed by the firm.
- Financial leverage is the ratio of debt versus equity the firm has deployed, or the financial structure.
- The higher the percentage of debt deployed by the firm, the more risky.
Government/Regulation Risk
- Government or regulation risk is the risk that tariffs or restrictions may be placed on an industry or firm that may impact the firm’s ability to effectively compete in an industry.
Exam Question
Stock Index funds are exchange-traded funds that track market indices and are subject to which of the following risks?
a) Financial Risk
b) Business risk
c) Systematic risk
d) Unsystematic risk
e) Diversifiable risk
Answer: C
Stock index funds and ETFs that track market indices are subject to systematic risk or market risk since they attempt to achieve market-type returns.
All others are examples of unsystematic or diversifiable risk.
Exam Question
Which of the following are non-diversifiable risks?
- Business Risk
- Management Risk
- Company or industry risk
- Market risk
- Interest rate risk
- Purchasing power risk
a) 4,5, and 6
b) 1,2, and 3
c) 5,6, and 2
d) 1,3 and 4
e) 1,4, and 6
Answer: A
Nondiverifiable risks are systematic risks. Recall the mnemonic or systematic risk: They are prime risks.
Modern Portfolio Theory!
- Modern Portfolio Theory
- Efficient Frontier
- Indifference Curves
- Efficient Portfolio
- Optimal Portfolio
Modern Portfolio Theory Definition
- The acceptance by an investor of a given level of risk
- while maximizing expected return objectives.
Efficient Frontier
- The curve illustrates the best possible returns that could be expected from all possible portfolios.
Indifference Curves
- Constructed using selections made based on this highest level of return given an acceptable level of risk.
Efficient Portfolio
- Occurs when an investor’s indifference curve is tangent to the efficient fronter.
Optimal Portfolio
The one selected from all efficient portfolios.
- Investors seek the highest return attainable at any level of risk.
- Investors want the lowest level of risk at any level of return.
- The assumption is also made that investors are risk averse.
Efficient Frontier!
- To determine the efficient frontier, simply compare portfolios based on their risk-return relationships.
- If it is above efficient fronter it is considered unattainable.
- The below is considered inefficient.
- An investor cannot achieve a portfolio that has a higher return for each level of risk.
Exam Question
Optimal Portfolio
- An indifference curve represents how much return an investor needs to take on risk.
- If an investor is risk averse, this investor will have a very steep indifference curve. This means the investor requires significantly more return to take on just a little more risk.
- Alternatively, if an investor is risk-seeking, they will have a relatively flat indifference curve. This means the investor will NOT require a significant amount of return - to take on more risk.
- The point at which the investor’s indifference curve is tangent to the efficient frontier represents that investor’s optimal portfolio. AKA. tangency portfolio is another name.
Capital Market Line!
- The Capital Market Line (CML) is the macro aspect of the Capital Asset Pricing Model (CAPM). It specifies the relationship between risk and returns on all possible portfolios.
- The CML becomes the new efficient frontier, mixing in the risk-free asset with a diversified portfolio.
- A portfolio’s return should be on the CML.
- Inefficient portfolios are below the CML.
- The CML is NOT used to evaluate the performance of a single security.
CML EXAM TIP
What is the measure of risk the CML uses?
Standard Deviation
CML Formula
(NO LONGER ON THE FORMULA SHEET)
More CML
- Before CML touches efficient frontier: Lends a portion of the uninvested risk-free rate.
- At the optimal portfolio: investor is fully invested in the portfolio. Does not lend at the risk-free rate or borrow at the rate.
- Right of the Optimal portfolio: Investor is said to have borrowed at the risk-free rate to full invest in all capital.
Capital Asset Pricing Model!
- The Capital Asset Pricing Model (CAPM) calculates the relationship of the risk and return of an individual security using Beta (b) as its measure of risk.
- The CAPM formula is often referred to as the Security Market Line (SML) equation because its inputs and results are used to construct the SML.
- The difference between the (rm - rf) is considered the market risk premium, that is how much an investor should be compensated to take on a market portfolio versus a risk-free asset.
Capital Asset Pricing Model Formula
Exam Tip CAPM
- The CAPM formula is on the formula sheet and you will need to use it.
- You may be asked to calculate an expected return or required rate of return using the CAPM.
- Also, you may be given the market risk premium rather than the return of the market.
Be sure to remember that the market risk premium is (rm-rp).
Exam Question
If the risk-free rate of return is 3%, the beta of a security is 1.5, and the market risk premium is 9%, what is the expected return?
a) 13.5%
b) 12.5%
c) 16.5%
d) 12.0%
e) 13.0%
Answer: C
ri = rf +( rm - rf ) Bi
=0.03 + (0.09)*1.5
=16.5%
Security Market Line
- The relationship between risk and return as defined by the CAPM and graphically plotted results in the Security Market Line (SML).
- Both the CAPM and SML assume an investor should earn a rate of return at least equal to the risk-free rate of return.
- The SML intercepts the y-axis at the risk-free rate of return.
The SML uses betas as its measure of risk, whereas the CML uses Standard Deviation as its measure of risk.
- If a portfolio provides a return above the SML, it would be considered undervalued and should be purchased.
- If a portfolio provides a return below the SML, it would be considered undervalued and should not be purchased.
- The SML may also be used with individual securities.
- Security is selling, need to figure out if you should purchase or not purchase.
Exam question
What is the intersection on the y-axis of the CML/SML?
a) Risk-free rate of return
b) Market Portfolio
c) Undervalued asset
d) Overvalued asset
e) Indeterminable
Answer A
The starting point on the CML/SML is a risk-free rate of return.
Portfolio Performance Measures
- Information Ratio
- Treynor Index
- Sharpe Ratio
- Jensen Model or Jensen Alpha
Information Ratio
- A relative risk-adjusted performance measure.
- Measures the excess return and the consistency provided by a fund manager, relative to a benchmark.
- The higher the excess return (or Information Ratio) the better.
- Excess return can be positive or negative depending on the fund’s performance relative to its benchmark.
Information Ratio Formula
Formula given
Treynor Index
- The Treynor index uses the beta of a portfolio as its denominator and the difference between the portfolio return and the risk-free rate return as the numerator.
The Treynor Index is
- A risk-adjusted performance measure. It’s also a “relative” risk-adjusted performance indicator, meaning one Treynor ratio needs to be compared to another Treynor ratio to provide meaning.
- A measure of how much return was achieved for each unit of risk. The higher the Treynor ratio, the better because that means more return was provided for each unit.
- It measures the reward achieved relative to the level of systematic risk (as defined by beta).
- Accomplished by standardizing portfolio return for volatility.
- Treynor justifies the use of the model on the assumption that in a well-diversified portfolio, the unsystematic risk is already close to zero.
- Treynor Index doesn’t indicate whether a portfolio manager has outperformed or underperformed the market.
Treynor Index Formula
Formula given
Exam Question
John is considering the two mutual funds below but is uncertain which performed better over the past year. The growth fund has a beta of 1.2 relative to the market. The risk-free rate of return was 3%. Which fund would you recommend based on the Treynor Ratio?
Growth Fund: Actual Return 12%
Index Fund: Actual Return of 9%
a) Growth Fund
b) Index Fund
c) Growth and Index Fund have the same risk-adjusted returns.
d) None of the above
Answer: A
Growth fund has a higher Treynor ratio
Growth Fund Treynor: (0.12 - 0.03) ÷ 1.2 = 0.075
Index Fund Treynor: (0.09 - 0.03) ÷ 1 = 0.06
**Note the Index Fund Beta is 1 because it tracks the market.
Sharpe Index
- Sharpe provides a measure of portfolio using a risk-adjusted measure that standardized returns for their variability.
- The model measures reward to total variability, or total risk, using the following formulas.
The Sharpe Index Is:
- A risk-adjusted performance measure.
- It’s also a “relative” risk-adjusted performance indicator,
- meaning one Sharpe ratio needs to be compared to another Sharpe ratio to provide meaning.
- A measure of how much return was achieved for each unit of risk.
- The higher the Sharpe Ratio, the better because that means more return was provided for each unit of risk.
- Sharpe Index measures risk premiums of that portfolio relative to the total amount of risk in the portfolio.
The formula does not measure a portfolio’s manager’s performance against that of the market.
Sharpe Ratio Formula
Formula given
Both Sharpe and Treynor
- They both provide very similar results in the performance measurement of portfolio managers.
- In a well-diversified portfolio, the results are frequently identical.
- In the case of poor diversification,
- the results of a comparison of these two portfolio evaluation indicators can be significantly different in the rankings of the performance results in managers,
Exam Question
Craig is evaluating two sector mutual funds that are not well-diversified. How would you recommend that Craig evaluate the funds on a risk-adjusted return basis?
a) Calculate the Treynor ratio and select the fund with the highest Treynor.
b) Calculate the Sharpe ratio and select the fund with the highest Sharpe.
c) Calculate the Treynor ratio and select the fund with the lowest Treynor.
d) Calculate the Sharpe ratio and select the fund with the lowest Sharpe.
Answer: B
Because sector funds are not well-diversified the most appropriate risk measure to use is standard deviation.
Recall the standard deviation is used for nondiversified portfolios and Beta is used for well-diversified portfolios.
When evaluating both Sharpe and Treynor ratios, always select the fund that provides the highest Sharpe or Treynor ratio. The higher the Sharpe or Treynor, the more return for each unit of risk.
Jensen Model or Jensen’s Alpha
- The Jensen Index or Jensen’s Alpha is significantly different from Sharpe and Treynor in that the Jensen’s Alpha is capable of distinguishing a manager’s performance relative to that of the market and determining differences between realized or actual returns and required returns as specified by CAPM.
- Treynor and Sharpe are calculations for providing a measure and ranking of relative performance.
- Jensen’s model attempts to construct a measure of absolute performance on a risk-adjusted basis.
- An absolute performance measures simply means that looking at Jensen’s Alpha tells you something.
- A positive Alpha indicates that the fund manager provided more return that was expected for the risk undertaken.
- A negative Alpha indicates that the fund manager provided less return that was expected for the risk that was undertaken.
- An Alpha of zero indicates that the fund manager provided a return equal to the return that was expected for the risk that was undertaken.
- A portfolio manager’s performance is judged relative to the Capital Asset Pricing Model.
- The alpha is indicative of the level of a manager’s performance.
- The higher the alpha, the better the performance.
- Negative alphas indicate managers who have underperformed the market on a risk-adjusted basis.
Jensen Model or Jensen’s Alpha Formula
Formula given
Exam Question
hristos is evaluating two mutual funds to purchase. Which fund would you recommend?
Fund A: Standard Deviation = 12%, R-squared = 0.92, Alpha = 2.0, Sharpe = 1.2
Fund B: Standard Deviation = 13%, R-squared = 0.90, Alpha = 1.8, Sharpe = 1.5
a) Fund A because it has a higher Alpha.
b) Fund A because it has a lower standard deviation.
с) Fund B because it has a higher Sharpe.
d) Fund B because it has a higher standard deviation.
Answer: A
Since both portfolios are well-diversified, as indicated by r-squared, then evaluate the funds based on the risk-adjusted performance indicator that uses Beta. Since Alpha uses Beta as its measure of risk, the fund with the higher Alpha should be selected
Note: If the r-squared was less than 0.70, the funds would not be well-diversified, so standard deviation should be the risk measure. Only Sharpe uses standard deviation as its risk measure.
Then fund B would have been appropriate.
Exam Question
Donna’s mutual fund returned 19% last year, with a beta of 2. The risk-free rate of return was 3%, the market return was 8%. The standard deviation is 18%. What would you tell Donna regarding the performance of her mutual fund?
a) The standard deviation was too high; therefore, Donna was undercompensated for the risk of her fund
b) The Sharpe ratio is 1, which means Donna earned an adequate risk-adjusted return.
c) The Sharpe ratio is 1, which means Donna earned a return less than was required on a risk-adjusted basis
d) The market outperformed the mutual fund on a risk-adjusted basis.
e) The alpha is 6%, which means the fund manager returned a higher rate of return than was expected on a risk-adjusted basis.
Answer: E
Standard deviation is a measure of volatility and variability. Standard deviation by itself is not a risk-adjusted performance indicator. For Sharpe to be meaningful, one Sharpe ratio needs to be compared to another Sharpe ratio. The Alpha ratio is calculated below:
Alpha = 0.19 - [0.03 + (0.08 - 0.03)*21
0.19 - 0.13
0.06 or 6%
Exam Question
Prince Albert is reviewing his investment statement with you, his CFP® practitioner. You two are going through his investment performance and he points out that the All American Total Market fund beat the market by 2%. Having seen this you point out that this investment, you recommended of course, has an alpha of 2%. Under what circumstances is this comment true?
a) Always, as it meets the definition of alpha.
b) Only if the risk-free rate of return is zero.
c) Only if the beta of the fund is one.
d) Only if the correlation is positive.
Answer: C
The calculation for alpha is the actual return of the fund less the expected return on the fund.
The only time the alpha will equal the amount the fund beat the market is if the beta is one.
Let’s just make up some numbers. Return on the portfolio is 8%, return on the market is 6%, the risk-free return is 2%, and beta is 1.
Return on Portfolio - Return of Market
=.08 - .06 = .02 which is the alpha
Op = 0.08 - [0.02 + (0.06 - 0.02)1]
0.08 - 0.06
0.02 ог 2%
Sharpe, Treynor and Alpha Application
-
The question becomes “Which risk-adjusted performance measurement is appropriate to use and when?”
- Both Treynor and Alpha use Beta as the measure of risk; therefore,
- Treynor and Alpha are appropriate risk-adjusted performance indicators when considering a diversified portfolio.
- A portfolio is considered diversified when r-squared
- is greater than or equal to 0.70.
- If r-squared is greater than or equal to 0.70 then Beta is a reliable measure of total risk; therefore,
- Treynor and Alpha are appropriate risk-adjusted performance measures.
A portfolio is considered not well-diversified when r-squared is less than 0.70.
- If r-squared is less than 0.70, then Beta is not an appropriate measure of total risk;
- therefore, standard deviation is an appropriate risk measurement.
- If standard deviation is an appropriate measure of risk, then using Sharpe as a risk-adjusted measurement is appropriate because Sharpe uses standard deviation as its measure of risk,
Exam question
If mutual fund ABC has a correlation of 0.80 to the S&P 500, which of the following risk-adjusted performance measures would be appropriate to measure the performance of fund ABC?
a)Treynor.
b) Jensen.
c) Sharpe.
d) Treynor and Sharpe.
e) Treynor and Jensen.
Answer: C
Because the correlation is 0.80, the r-squared is 0.64. Only 64% of the return for fund ABC is due to the S&P 500, Therefore, 36% is due to unsystematic risk. Since Beta only measures market (systematic risk), too much return is due to unsystematic risk; therefore, Beta is not appropri-ate. Rule out Treynor and Jensen since both use Beta. Sharpe uses standard deviation, which measures total risk market (systematic) and unsystematic.
Summary of Performance Measures
Sharpe and Treynor are relative performance measures.
- Relative means that you must compare one Sharpe or one Treynor to another.
- The higher the Sharpe or Treynor, the more return for each unit of risk.
- Always select the higher Sharpe or Treynor ratio.
When determining which fund performed better on a risk-adjusted basis,
- always rank the Sharpe or Treynor ratios, then select the highest.
Alpha is an absolute performance measure.
- A positive Alpha is good;
- a negative Alpha is bad.
Select which risk-adjusted performance measures to use based on r-squared.
- If r-squared is ≥ 0.70,
- use Treynor and Alpha because both use Beta and a r-squared ≥ 0.70 indicates the portfolio is well-diversified and Beta is an appropriate measure of total risk.
- If r-squared is < 0.70,
- then the portfolio is not well-diversified so use Sharpe because it uses standard deviation as its measure of risk.
- Standard deviation is appropriate for portfolios that are not well- diversified.
If r-squared is too low, they are using wrong benchmark. Therefore IR doesn’t hold as true measure.
Comparison of Performance Indicators
Exam Tip
If the exam doesn’t give you r-sqaured, then select Sharpe!
Variability vs Volatility
Variability is measured using variation.
- It is a measure of how far a return varies from what is expected, as in the mean average.
Volatility, is measured, by beta measures the relative relationship between a benchmark (in this case the market) and a return of some investment relative to that market.
- How much mcuh more or less volatile is it than the market if market volatility is 1?
Exam Question
