CFP Class 2 - Module 2 - Investment Risk & Return
Weighted-Average Beta
The computation of a weighted beta for a portfolio is similar to the computation of a weighted return for a portfolio. Just substitute the betas for the returns.
Assume that the Small-Cap Mutual Fund has a standard deviation of 17.2, that the S&P 500 index has a standard deviation of 14.7, and that the covariance between the fund and the index is +41. What is the correlation coefficient between the fund and the market?
+41 divided by 17.2 times 14.7 = .16
Stock F Expected return 9% Standard Deviation 7% Stock G Expected return 14% Standard Deviation 16% Expected return of a portfolio that has 40% invested in Stock F and 60% invested in Stock G?
.09 Input 40 E+, .14 Input 60 E+ Shift 6 = 12% expected return.
What is the correlation coefficient between the Thor Fund and the S&P 500 Index if the coefficient of determination is .49?
.49 shift the "-" key = .7
Weighted Beta Coefficient Example: $50,000 in a Growth Mutual Fund, 1.3 beta, 15.0% return, 30,000 investment in Stock MNY .7 beta, 10.0% return
1.3 Input 50,000 E+ .7 Input 30,000 E+ SHIFT xw,b (6 key) Weighted beta = 1.075
Assume that RMC stock has had the following series of ten annual returns, signified by "rn", with an average return of 7.20%.
10 E + 12 E + 7 E + 13 E + 9 +/-, E + 9 E + 8 +/-, E + 11 E + 15 E + 12 E + SHIFT, x, y (7 key) for mean return 7.20% SHIFT, Sx, Sy (8 key) for standard deviation 8.5609
The covariance between Twin Pines Inc. and the S&P 500 is 95. The standard deviation of Twin Pines is 13, and the market's standard deviation is 12. What is their correlation coefficient?
13 x 12 = 156. 95 divided by 156 = .6090.
Libra Inc. has a mean return of 11%, and a standard deviation of 9. Assuming the returns are normally distributed, what is the probability that the stock will have a return greater than 20%?
16%. One standard deviation which will be evenly distributed accounts for 68% of the returns and would range from +2% to +20%. (11-9=2 and 11+9=20). Half of the 68% of returns would be above the mean return, and half below. So 34% of the returns would be greater than the 11% mean return, and would fall between 11% and 20%. If we know 50% of the returns will be greater than 11%, and we know that 34% of the returns will fall between 11% and 20%, we then know that 16% (what remains of the 50%, 50 - 34) of the returns will be greater than 20%.
PTV Inc. and SLK Corp. He owns 100 shares of PTV Inc. with a current market value of $5,250, and 150 shares of SLK Corp., with a current market value of $1,750. Returns of 20% and 14%, respectively, on his investments. Weighted-average expected return on portfolio?
20 INPUT 5250 E+ 14 INPUT 1750 E+ SHIFT xw,b (6 key) Rate of return = 18.50%
Correlation Coefficient Formula Example: Covariance of 276.48, and standard deviations of two different funds of 18 and 24:
276.48 divided by 18 times 24 = .64
Product of 4.5 squared
4.5, SHIFT, x squared
Benefits of Holding Bonds
60-month rolling correlation between U.S. large company stocks and long-term government bonds shows benefit of diversifying between stocks and bonds has increased in recent years. Long term gov Bonds provided diversification from U.S. stocks when needed most.
What is the required return for the following securities? The risk-free rate is 4%, and the market return is 8%. Betas for five securities are: .9, 1.1, 2, 1.4, 1.
7.6, 8.4, 12, 9.6, 8.
Beta of 1.2
A beta of 1 means that the asset is just as volatile as the market, so if the market moves up 10%, so should the individual asset. If the beta is 1.2, this means that the asset has 120% of the volatility of the benchmark (10% × 1.20). .
Standard deviation
A measure of total risk, it is a statistical measure of the degree to which an investment's returns are expected to vary from a mean return. It is the basic measure of an investment's risk.
Standard deviation
A measure of total risk. Statistical measure of the degree to which an investment's returns are expected to vary from a mean return. It is the basic measure of an investment's risk.
Negatively Skewed Distribution
A negatively skewed distribution has a large hump to the right and a long tail to the left. If there were more down years than up years, you would then have a negative skewness
Endogenous Risk Impact
A problem arises when investors do not act rationally, especially as a group. This causes endogenous risk, which is the risk that investors will behave in the same manner and literally cause panic in the marketplace. Does greatest damage to an investor.
Correlation Coefficient
A standardized version of covariance.
Coefficient of Variation
A statistical measure of the relative dispersion of data points around the mean. One of the methods for computing a risk-adjusted return for a security. If know both the standard deviation and mean return for two or more securities can calculate the coefficient of variation for each.
Five Systemic Risks - Exchange Rate Risks
An appreciating home-country currency, compared to a foreign currency, will cause an investment in a foreign security denominated in the foreign currency to be worth less, in dollar terms, than what that investment would have been worth if the currency rates were stable.
Coefficient of Variation
An investment's risk per unit of expected return.
Expected Return
An investor's estimate of a return, given the economic and market prospects for an investment. The sum of the expected dividend yield and the capital gain.
Mean return is in the middle of the symmetrically distributed bell curve
And the area under the curve is always equal to one. This means that 50% of the returns are higher than the mean return, and 50% of them are lower.
Unsystematic risk can be diversified away by holding approximately how many large cap securities?
Approximately 10 to 15 securities are required to diversify away unsystematic risk.
As Correlation Falls Beta Drops in Reliability
As correlation falls, so does beta reliability.
Risk vs. Correlation
As the correlation falls, so will the risk as measured by standard deviation.
Weighted-Average Return of the Two Assets Formula Asset A Expected Return 15%, Standard Deviation 10, Weight .4. Asset B Expected Return 25%, Standard Deviation 20, Weight .6
Asset A Expected Return 15%, Standard Deviation 10, Weight .4 Asset B Expected Return 25%, Standard Deviation 20, Weight .6 .40 × 15% + .60 × 25% = 21%. Calculating the standard deviation of a portfolio is not this easy
(.5)2 (16)2 (.5)2 (32)2 2(.5)(16)(.5)(.32)(.35) Square Root of 1st number + 2nd number + 3rd number
(.50) (16) + (.50) (32) + 2(.50)(.50)(16)(32)(0.35) = 20.24
Second Method for Calculating Beta - Used in this Course
Beta = standard deviation of individual security (Si) divided by standard deviation of the market (Sm) times the correlation coefficient between the individual asset and the market (Rim).
United Airlines fights discount carriers by lowering its fares across the country.
Business risk
Portfolio Diversification and Risk
By diversifying a portfolio, an investor is able to lower the total risk of the portfolio by eliminating the unsystematic or unique or diversifiable risk; only systematic or nondiversifiable risk is left.
Unsystematic Diversifiable Risk - Call Dates & Premium
Call dates indicate the first and successive dates on which bonds may be called, and call premium is the amount above par value to be paid if the issue is called.
Unsystematic Diversifiable Risk - Call Risk & Interest Rates
Call risk increases when interest rates decline. When investors receive their principal from called bonds, they find that they are able to reinvest the funds only at a lower rate thereby encountering reinvestment risk.
Determination of "the market" or an appropriate benchmark
Can be difficult. Most often, investors use the S&P 500 index as a proxy for the market.
Unsystematic Risk is Diversifiable Portion of Total Risk
Can be managed, and eliminated, by increasing the number of securities in a portfolio, but only if those additional securities are not highly correlated with the existing portfolio.
Systematic Risks Include: PRIME - Market Risk
Caused by investor reaction to factors independent of a particular security. Affects all securities. Comes from a significant market move in response to some new negative economic, political, or earnings information, consensus of market overvaluation, etc.
Stock F Expected return 9% Standard Deviation 7% Stock G Expected return 14% Standard Deviation 16% The covariance between stocks F and G is -22. 40% invested in Stock F and 60% invested in Stock G. What are the correlation coefficient and coefficient of determination of Stock F and Stock G?
Correlation Coefficient = 7 x 16 = 112. - 22 divided by 112 = .- 20. Coefficient of Determination = -.2 times -.2 = .04. or Shift + key.
Correlation Coefficient Table
Correlation coefficient of one asset or one asset class can be compared to only one other asset or asset class at a time. Therefore, investors typically use a correlation coefficient table.
Unsystematic Diversifiable Risk - Business Risk
Degree of uncertainty associated with a company's earnings and its ability to pay dividends or interest to investors. This uncertainty is directly related to the company's management, product line, marketing ability, and other factors specific to that company.
Unsystematic Diversifiable Risk - Liquidity & Marketability Risks
Degree of uncertainty associated with the time it takes to sell an investment with a minimum of capital loss from the current market price. Marketability risk is the risk that there is not an active market for an investment.
Limit to Diversification Benefits
Depends on the trade-off between the additional benefit to be gained from further diversification and the extra transaction costs involved in adding more assets to a portfolio. The diversification process should stop as soon as the extra benefit equals the extra cost.
Quantitative Analysis Modeling and Simulation (aka Multiple Scenario Analysis) - Sensitivity Analysis
Determine impact of unexpected, low probability outcomes. Includes many more than three scenarios. Also used in an optimization program to determine impact of changes in return or risk assumptions on an asset allocation.
Kurtosis
Distribution is more peaked leptokurtic or less peaked platykurtic than a normal distribution. Distribution has a greater percentage of small deviations or clustering around mean return and greater percentage of large deviations from mean return.
Two Uses of Correlation Coefficient & Coefficient of Determination or R-squared - Diversification vs. Reliable Beta Number
Diversification: Lower number so less correlation between an asset and a benchmark or other asset. Lower correlation = more diversification. Opposite is true to determine if beta is a reliable number and whether it can be used or not. R squared of .7 or higher preferable.
Risk Premium Component of Required Rate of Return
Does not reflect unsystematic risk. The risk premium is the extra return investors expect to earn from an investment for their assuming its systematic or nondiversifiable risk.
Risk drops with a mix of 70% stocks and 30% bonds made possible by the low correlations between the stocks and bonds (see table page 47)
Down to a standard deviation of 14.1 compared with 19.8 with all stocks. Then look at the return, which goes from 10.2% down to just 9.2%. So return has declined 9.8%, whereas the risk, as measured by standard deviation, has fallen 28.8%.
Constructing a Well-Diversified Portfolio - Two Factors
Eliminating unsystematic risk and minimizing the total risk standard deviation of the portfolio by adding assets that are not highly correlated with each other.
Constructing a systematically developed portfolio requires paying attention to which two factors?
Eliminating unsystematic risk. Minimizing total risk, which is measured by standard deviation.
The value of your investment in Sony has risen 20% during the past year; during the same period, the yen has weakened against the dollar, causing the total return on the Sony investment to be only 10%.
Exchange rate risk
Capital Market Expectations (CME).
Expectations regarding what the markets will do in the future.
Exxon Mobil decides to issue bonds instead of stock to finance a new tanker fleet.
Financial risk
Calculating Standard Deviation of the Portfolio Good Starting Point for any Problem
Find the weighted average of the standard deviations of the two assets to find the maximum standard deviation possible if the two assets were perfectly positively correlated. If they are not perfectly positively correlated then the standard deviation of the two assets together must be less, and the lower the correlation of the assets, the lower the standard deviation of the portfolio.
Ideal R2
For testing purposes you are looking for an R2 of 70% or greater, the threshold used by Morningstar. Beta is most reliable when the R2 is 100 an index fund will have an R2 of 100 with the index it is mirroring, and the reliability declines as R2 declines. 70% just one cutoff for reliability.
Beta of .7 or Higher
For testing purposes, you want to see an R-squared of 70 or higher which would be a correlation coefficient of .84 or higher to have a high enough correlation, and thus enough systematic risk, in order to use beta, or to use any of the formulas that use beta.
Systematic Risks Include: PRIME - Exchange Rate Risk Example Continued
For the U.S. investor, that interest payment decreased $0.10 due to exchange rate risk. If dollar had weakened, the interest payment would have increased in U.S. dollar terms since the interest payments are initially in appreciated British pounds. A weakening U.S. dollar is advantageous for U.S. investors owning foreign securities.
Correlation Coefficients Range
From +1 meaning the asset and the market have a perfectly positive relationship), through 0 meaning that the asset and the market have no relationship, to negative 1 meaning that the asset and the market have a perfectly negative relationship.
Fund A Mean Return 10%, Standard Deviation 10% Fund Z Mean Return 10%, Standard Deviation 20% Solve for Risk or Coefficient of Variation
Fund A: 10 divided by 10 = 1.0. Fund Z: 20 divided by 10 = 2.0. They both offer same return, but Fund Z has twice the risk as measured by standard deviation 20 versus 10.
Covariance & Diversification
Gets to the crux of what diversification is all about; if we want to diversify we need to add an asset that does not behave in the same way as the asset we already own.
Kurtosis Continued
Graph for a distribution that has a lot of returns clustered around the mean many small surprises, but some extremely large positive or negative returns few large surprises, has a higher peak and fatter tails than a normal distribution.
Correlation Coefficient
Has boundaries that range from -1.0 to +1.0. This process is called "normalizing the values." Can easily calculate the correlation coefficient between two assets once we know the covariance.
Standard Deviation - Bumpiness of the Ride
How far from mean or average return a security's returns are likely to vary. More variability, the bumpier the ride, and the more risk there is. Risk-averse investors would of course want a smoother ride, meaning less variability.
Coefficient of Determination (R2) and Systematic and Unsystematic Risk
How much of the risk is explained by the benchmark. Comparing Titanic Fund to S&P 500. Coefficient of determination is .70, 70% of the price movement of the Fund due to S&P 500 or systematic risk, other 30% not explained by the S&P 500 Index unsystematic risk.
Covariance and its importance in constructing a well diversified portfolio.
How much two investments are related to each other. How much they move together or apart. Add an asset that does not behave in the same manner as the asset we already have. The lower the covariance, the lower the correlation between the two assets.
Agressive vs. Defensive Stocks
If a stock has a beta of 1.5, the price on the stock would be expected to change by 6% when the market changes by 4%. Agressive. Stocks with high beta coefficients are referred to as "aggressive," while stocks with low beta coefficients are referred to as "defensive."
$50,000 in a fund with three-year average return of 7%, and a standard deviation of 16. Another $50,000 in a fund with a three-year average return of 11%, and a standard deviation of 32. The correlation coefficient between the two is .35. What is the standard deviation of the portfolio?
If the two funds were perfectly correlated +1 with each other we could just use 24, the weighted average of the two standard deviations. Since these two funds are not perfectly correlated we know that the standard deviation is going to be less than 24.
Standard distributions are equally distributed
In our example, half of returns higher than mean return of 7.2%, and half lower. The standard distributions equally distributed, meaning with one standard deviation, 34% of the returns higher than the mean return, and 34% lower totaling 68%, the range of one standard deviation.
Interest Rate Risk Impact on Bonds
Increases in the general level of interest rates causes bonds prices of fixed-income securities to decrease. The longer the bond's maturity, or the lower the coupon rate on the security, the greater the change in the bond's price in response to changing interest rates.
When can use weighted averages for beta returns; and standard deviation?
Input E+ Shift 6. You can use weighted averages for beta and for returns; but can only use a weighted average for standard deviation if the two assets have +1.0 correlation. If not, must use standard deviation of a portfolio formula.
The Fed has decided to increase short-term interest rates to fight increased inflation.
Interest rate risk
Systematic Risks Include: PRIME - Interest Rate Risk
Interest rate risk is the negative effect on the prices of fixed income securities caused by increases in the general level of interest rates. Inverse of reinvestment risk.
Significance of Correlation Coefficient to the accurate interpretation of the meaning of Beta?
Intuitively, an investor might conclude that a stock with a standard deviation that is twice the standard deviation of the market would have a beta of 2.0. The computations show that to be the case only when the stock in question is highly correlated with the market. Even with a low positive correlation (+.25), the beta is only .5. Many investors will conclude that a stock with a beta of .5 is half as variable as the market; what this computation should actually tell them is that they cannot use beta to judge the volatility of the stock, but instead must refer to the stock's standard deviation to measure the volatility of the stock. Beta cannot be accepted blindly without knowing how the stock in question is correlated with the market index against which its beta is calculated.
Normal Distribution (Bell Curve)
Investment returns, when examined over a large number of years, normally distributed around mean return. Each side of the distribution is equal and that the area under the curve equals 100%. Standard deviation is a critical element here.
Systematic Risk Nondiversifiable Component of Total Risk
It represents the variability in all risky assets attributed to macroeconomic variables like unanticipated changes in the GNP growth rate, industrial production, inflation, interest rates, and the money supply. No way for any investment to escape the impact.
Beta of .8
Just 80% of the volatility of the benchmark. The individual asset should move up approximately 8% (10% × .80). Aggressive investors will be comfortable with higher betas over 1, whereas risk averse investors will prefer lower betas, below 1.
The correlation coefficient formula is not provided to you on the course exam or on the CFP Board exam sheet, so you will need to memorize it:
Just remember to rearrange the covariance formula and isolate the correlation coefficient term "R" COV divided by (standard deviation of the market times standard deviation of the asset).
Minimization of Total Risk
Key to minimizing the standard deviation of a portfolio of securities is covariance. Long-term (and shorter-term) correlation coefficient statistics are helpful in the initial design of a portfolio.
Four Primary Asset Classes
Large cap, small cap, foreign, and emerging market. Each successive asset class has a lower correlation coefficient with large-cap stocks, ensuring that the portfolio standard deviation would be reduced by including an equal weighting of each of the four asset classes.
Positively Skewed Distribution
Large hump to left and a long tail to right. Generally, investment returns are positively skewed, with larger positive returns and fewer negative returns. Small levels of negative skewness, large levels of positive skewness preferred.
Most Common Equity Asset Classes Include:
Large, Mid, Small & Micro-cap value, Large, Mid, Small & Micro-cap growth, Developed international Markets & Emerging Markets. For most investors, having a fund in four to seven of the asset classes generally will suffice.
Unsystematic Diversifiable Risk - Financial Risk Continued
Larger the proportion of debt to equity, the greater the degree of financial leverage, and the greater the financial risk. Debt financing creates legal obligations to make timely payments. These fixed-payment obligations must be met prior to distributing any earnings to company owners.
Leptokurtic More Peaked
Lepto is derived from the Greek word for slender, and a good way to remember that leptokurtic is more peaked is to think of the way it "leaps" near the mean.
What is the approximate price movement of the following assets, given the following betas and a market return of +15%? Betas 1.4, .8, -.1, .5, 2.
Multiply the beta times the market change. 21, 12, -1.5, 7.5, 30.
An investor has an index mutual fund, and adds an actively managed fund that has an R2 of .94 with the index fund. Has the investor effectively diversified? Why or why not?
No. For diversification purposes, the lower the R2 the better. 94% of the price movement of the actively managed fund is explained by the index mutual fund the investor already holds.The actively managed fund will essentially mirror the performance of the index fund.
Normal Distribution vs. Lognormal Distribution.
Normal distribution always symmetric. Lognormal distribution always asymmetric. Normal used for investment returns, as returns can be positive or negative. For asset prices use lognormal distribution as can never go below zero.
Memorize Coefficient of Variation Formula
Not provided on exam. Simply standard deviation divided by the mean return (S/M), and the lower the number the better.
Required Return or Capital Asset Pricing Model (CAPM) formula.
Once beta has been calculated for a security, the required return for can be determined. This is the total return income plus capital gain that an investor should expect to obtain from a security given that security's level of risk. Risk level is measured by the security's beta.
Covariance, Correlation Coefficient & Coefficient of Determination
Once have calculated covariance, can then calculate the correlation coefficient. Once have the correlation coefficient can then calculate the coefficient of determination (R2). All three are related, and all three deal with correlation.
What does semivariance measure?
Only the returns that fall below the average, and is primarily used by portfolio managers. This measure recognizes that investors are concerned less about upside potential and more about downside risk.
Platykurtic Flatter
Platy is derived from the Greek word for broad, and if it helps you, think of platykurtic as being flat like a plate.
Systematic Risks Include: PRIME
Purchasing power or inflation risk, Reinvestment risk, Interest rate risk, Market risk, Exchange rate risk.
Interest rates have declined over the past year.
Reinvestment risk
Realized Return
The actual total return income plus capital gain earned on an investment.
As the correlation coefficient falls, so does the standard deviation
The lower the correlation between two assets, the lower the risk will be as measured by standard deviation. In one example the result of 16.0 for a +1.0 correlation is twice the standard deviation of 8.0 for a -1.0 correlation.
Systematic Risk
The nondiversifiable component of total risk. Mathematically expressed as beta for a security. There are five major types of systematic risk including purchasing power risk, reinvestment risk, interest rate risk, market risk, and exchange rate risk.
Unsystematic Risk is Diversifiable Portion of Total Risk also Known as Unique Risk
Unique to each asset and is not related to marketwide events. Can be diversified away because the unique variability of any asset can be offset by the unique variability of other assets in the portfolio.
Two-Asset Portfolio Standard Deviation Formula
σp = square root of Wi2 σi2 + Wj2 σj2 + 2WiWj COVij In this formula, W represents the percentage weight of each asset in the portfolio, expressed as a decimal e.g., .40); σ represents the standard deviation of each asset, expressed as a whole number e.g., 27; and COV represents the covariance of assets i and j, expressed as a whole number e.g., +57.
The overvalued stock market has finally fallen 15% over the past four months.
Market risk
Calculate the beta if the standard deviation of the security is 30 and the standard deviation of the market is 15 with the following correlation coefficients: 1, .25, .0, -.25 and -1.
(1) 30/15 × 1.0 = 2.0 (2) 30/15 × +0.25 = +.5 (3) 30/15 × .0 = .0 (4) 30/15 × -.25 = -.5 (5) 30/15 × -1.0 = -2.0
What is the coefficient of determination if the correlation coefficient between the Thor Fund and the S&P 500 Index is .70?
.70, SHIFT, x2 (the "+" key), answer is .49. This means that 49% of the risk is systematic risk, explained by the S&P 500 benchmark, and the other 51% of the risk is unsystematic risk, explained by other factors.
Stock F Expected return 9% Standard Deviation 7% Stock G Expected return 14% Standard Deviation 16% The covariance between stocks F and G is -22. 40% invested in Stock F and 60% invested in Stock G. What is the standard deviation?
1. (.4)2 (7)2 = 7.84 2. (.6)2 (16)2 = 92.6 + 7.84 = 100. 3. (.4)(.6)(-22) = - 5.28 x 2 = 10.56 4. 10.56 + 100 = 89.44 square root = 9.5
Standard Deviation of the Portfolio When the Correlation Coefficient is .95 (pages 55-56) Asset A Expected Return 15%, Standard Deviation 10, Weight .4 Asset B Expected Return 25%, Standard Deviation 20, Weight .6
1. 1st asset: .40 squared = .16. 10 squared = 100. .16 times 100 = 16 2. 2nd asset: .60 squared = .36. 20 squared = 400. .36 times 400 = 144 3. Weightings and covariance: 2 (.40)(.60)(10)(20)(.95) = 91.2 4. 16+144+91.2 = 251.2 Square root of 251.2 = 15.85 standard deviation of the portfolio
$75,000 investment portfolio with $40,000 invested in TRO stock fund, $20,000 invested in PES bonds, and $15,000 invested in HXQ REIT. The individual betas for each security were 1.2, .9, and .8, respectively. What is the weighted-average beta for the portfolio?
1.2 INPUT 40 E+ .9 INPUT 20 E+ .8 INPUT 15 E+ SHIFT xw,b 6 key. Beta = 1.040. Can also use 40,000 etc. Same result.
Diversification Rules of Thumb
10 to 15 large-cap securities in different industries. For mutual funds 4 to 7 funds in different asset classes optimize diversification benefits. For Mid- to small-cap stocks, a greater number of securities is needed. Allocate among 25 to 30 in different industries.
$75,000 investment portfolio with $40,000 invested in TRO stock fund, $20,000 invested in PES bonds, and $15,000 invested in HXQ REIT. The rates of return earned by each security were 12%, 9%, and 8%, respectively. Weighted-average rate of return on the portfolio?
12 INPUT 40 E+ 9 INPUT 20 E+ 8 INPUT 15 E+ SHIFT xw,b (6 key) Rate of return = 10.40%
Current market value of $175,000, 50 NLR convertible bonds with a current market value of $48,500, and 1,000 shares of MPT stock with a current market value of $72,500. Expects returns of 14%, 21%, and 9%, respectively, on her investments. Weighted-average expected return?
14 INPUT 175,000 E+ 21 INPUT 48,500 E+ 9 INPUT 72,500 E+ SHIFT xw,b (6 key) Rate of return = 13.92%
The standard deviation of the market is 14, and the standard deviation of United Enterprises Inc. is 22. The correlation coefficient between the two is .85. What is their covariance?
14 x 22 x .85 = 261.80
Square root of 20.25
20.25, SHIFT, divided by X key
Standard deviation of Mountain Enterprises is 22, the standard deviation of the market index is 16, and that the correlation coefficient between Mountain Enterprises and the market is .66. What is the beta of Mountain Enterprises?
22 divided by 16 = 1.3750 times .66 = .9075. May be a good diversifier to add to a portfolio because its lower correlation should help reduce the standard deviation of the portfolio. This lower correlation may also result in a lower beta for the portfolio.
Low Positive Correlation Coefficient Example
A better combination of funds would be one with a correlation coefficient of +.88 with the S&P 500 and a second with a correlation coefficient of +.35 with the S&P 500. Less related.
Beta
A measure of a security's volatility with respect to an index against which the stock is measured.
Beta
A measure of systematic risk, it is a statistical measure of a stock's volatility as measured against a stock index.
Term used for the difference between the market return and the risk-free rate (Rm - Rf)?
Market risk premium. The excess return earned above the risk-free rate to compensate the investor for taking extra risk by investing in the market. There is much discussion and disagreement over what the future market risk premium will be.
Semivariance
Alternative measure of risk that considers only returns lower than expectations mean return, zero, or a benchmark return. This measure recognizes that investors are concerned less about the risk of overperformance and more about the risk of underperformance.
Coefficient of Determination or R-squared - Measure of Systemic Risk
Another way to look at R2 is as a measure of systematic risk as explained by the benchmark, with the balance being unsystematic risk explained by other factors.
12 Unsystematic Risk Types
Business, call, credit or downgrading of its debt, default, event, financial, meaning greater debt equals greater financial risk, marketability or investment lacks an active trading market, liquidity or selling with a minimum capital loss, investment manager, tax, political & country risk.
Correlations or Diversification Will be Different Depending on Time Frame
Long-term correlations may be misleading if correlations have changed recently. Because correlations are not constant and change over time, investors and advisers need to be aware of the possibility of significant changes in correlation.
Most Common Choice for Risk Free Asset
Use rate for the three-month Treasury bill. This is also what the CFP Board favors.
Beta and R squared
Low R squared is generally better as means lower systemic risk as explained by the benchmark but Beta is also a measure of systematic risk so when using beta the higher the coefficient of determination (R2), the better, 70 plus reliable.
The Pueblo Fund has a standard deviation of 18, and the Durango Fund has a standard deviation of 24. The correlation coefficient between the two is .64, what is their covariance?
COVij = (.64)(18)(24) COVij = 276.48 Note that covariance is not commonly used to describe the relationship between the price movements of two securities. Instead, correlation coefficient is used.
Covariance Formula: multiply the correlation coefficient between the two assets by the standard deviation of each of the assets to arrive at their covariance.
COVij = ρij σi σj ij is the correlation coefficient between the first asset represented by the letter "i" and second asset represented by the letter "j". p is the Greek letter "rho," typically translated into our letter "R."
Correlation Coefficient Formula
COVij divided by σi times σj = Rij The correlation coefficient formula is not provided on the course exam or on the CFP Board exam sheet, so need to memorize it.
Fund A Mean Return 10%, Standard Deviation 7% Fund B Mean Return 20%, Standard Deviation 11% Solve for Risk or Coefficient of Variation
CV = σ/x Fund A 7 divided by 10 = .7. Fund B 11/20 = .55. Fund A has taken more risk than Fund B to achieve its return. Fund A's risk is 70% of its return, whereas Fund B's risk is only 55% of its return.
Coefficient of Variation Formula
CV = σ/x. CV represents the coefficient of variation; σ represents the standard deviation of an asset; and x represents the mean return of the asset. The higher the number, the more risk per unit of return, and the lower the number, the less risk per unit of return.
Calculating the Correlation Coefficient
Calculate the correlation coefficient R if given the coefficient of determination R squared, or vice versa. May be given one in a problem and need the other to arrive at an answer. If given the correlation coefficient, square it to arrive at the coefficient of determination.
Coefficient of Determination (R2) and Beta
Lower the coefficient of determination and thus the correlation coefficient the better for diversification. However, beta is a measure of systematic risk. Coefficient of determination also a measure of systematic risk, so when using beta the higher the coefficient of determination the better.
R squared of At or Near 100%
Likely an index fund as it matches the performance of the benchmark. You should not see an R squared that is close to 100% for an actively managed fund that has materially higher operating expenses, but is not materially different in portfolio structure than the benchmark index.
The Stargazer Fund has a mean (average) return of 10%, and a standard deviation of 15%. Assuming the returns are normally distributed, what range of returns would you expect 68% of the time? 95% of the time? 99% of the time?
Mean return changes but standard deviation stays the same: 68%: -5% to +25% (10 - 15 = -5, and 10 + 15 = 25) 95%: -20% to +40% (-5 - 15 = -20, and 25 + 15 = 40) 99%: -35% to +55% (-20 - 15= -35 and 40 + 15 = 55)
Positive Skewness
Mean return will be to left of normal bell curve. Good for investors because more returns are positive than negative. In "real world" about two positive years for every negative year, and this would be reflected by a positive skewness.
Correlations tend to increase in down markets
Meaning there will be less diversification when it is needed most. That is why it is important to look beyond equities, and introduce other asset classes such as cash, bonds corporate, government, and international, and commodities.
Beta of a Stock
Measures riskiness and volatility in comparison to the market in general. Beta of 1 has approximately the same risk and volatility as the market as a whole. Higher than 1 riskier. Lower than 1 less risky. Weighted average beta of a portfolio overall portfolio risk.
Covariance
Measures the extent to which two variables such as two stocks are related to each other, or how the price movements of one of the securities are related to the price movements of a second security.
Kurtosis Common with Equity Returns
Most equity returns have this characteristic. The shorter the measuring period e.g., monthly rather than yearly the larger the kurtosis.
Probability of Acheiving a Specific Rate of Return
On positive side one standard deviation is at 15.8% (7.2% mean return plus one standard deviation of 8.6%), then the probability of achieving a return of greater than 15.8% is 16%, which is the remaining area under the curve to the right of one positive standard deviation.
Unsystematic Diversifiable Risk - Call Risk
Possibility that a debt security will be called in by its issuer prior to maturity. Feature of most municipal and corporate bond issues, making it possible for the issuers to pay off existing high-coupon bond issues with new ones that have lower coupon rates. No callable U.S. Treasury securities.
Unsystematic Diversifiable Risk - Event Risk
Possibility that an investment, most often bonds, will be adversely affected by an unanticipated and damaging event. Can also affect stocks, such as BP's oil leak led to 50% drop in stock price.
Views on Risk
Predominant view is loss of principal or variability of returns. Other views are price volatility, not achieving an expected rate of return or, simply, losing money, nonmatching of cash flows investment income versus fixed living expenses & uncertainty of future returns
MSCI ACWI (standard index) R Squared = 22.57. Beta = .58 Morningstar US Real Estate (best-fit index) R Squared = 97.64. Beta = .99
Problem with this beta is only 22.5% systematic risk explained by the MSCI ACWI. 77.5% is unsystematic risk. Better indicator of volatility beta of .99 when compared with the Morningstar U.S. Real Estate index. Close to 98% systematic risk and 2% unsystematic risk
Inflation is expected to rise over the next year.
Purchasing power risk
Standard deviation of Stock A 15.2 Stock B 10.6 Stock C 16.1 Stock D 23.5. Covariance A/B +74 A/C +16 A/D -46. Compute the correlation coefficient (R) and the coefficient of determination (R2) for stocks A/B, A/C, and A/D.
R = COV divided by S1 times S2 = correlation coefficient. Then R squared = coefficient of determination. A/B: 15.2 times 10.6 = 161.2. 74 divided by 161.12 = .469 = R. Squared = 21.2% R squared. A/C = .065 or .07 squared = .49%. A/D = .129 or .12 squared = 1.69
The Coefficient of Determination
R squared. Shift + key. Explains how much variability of one factor can be caused by its relationship to another factor. Its the square of the correlation coefficient, also known as "R," which allows it to display the degree of linear correlation between two variables.
Square the correlation coefficient .66, get an R2 of .44, or 44%. This tells us that the base against which Mountain Enterprises' beta is measured (the market index) may not be appropriate if we want to predict its return for next year.
R2 tells us that only 44% (systematic risk) of the price movement of Mountain Enterprises is explained by the benchmark we are comparing it to, and the other 56% is unsystematic risk not explained by the benchmark.
Coefficient of Variation Benefit
Rather than selecting the security that has the greatest absolute return or the lowest absolute risk over the period measured, can use the coefficient of variation to determine which security exhibited the least risk per unit of return.
Systematic Risks Include: PRIME - Reinvestment Risk
Reinvesting cash flows at a lower rate than was being earned. If interest rates rise, bond prices fall, but periodic interest payments reinvested at higher rates. But if interest rates decline, bond prices will rise, but periodic interest payments can be reinvested only at a lower rate.
The coefficient of variation formula is not provided to you on the course exam or on the CFP Board exam sheet, so you will need to memorize it:
Remember that it is simply the standard deviation divided by the mean return (S/M), and the lower the number the better.
Weighted Averages for Beta, Returns and Standard Deviations.
Remember that you can use weighted averages for beta and for returns; however, you can only use a weighted average for standard deviation if the two assets are perfectly positively correlated +1.0 correlation. If not, have to use Standard Deviation for a Portfolio Formula.
Required Return & Beta
Required return is whatever the risk-free rate is plus the market risk premium. Then adjust required return based on the amount of risk we are taking, as measured by beta. As beta increases required return increases, and as beta decreases, required return decreases.
Risk Drops with 50% stocks, 50% bonds (see table page 47)
Return declines 17.7% from 10.2% to 8.4%, yet the standard deviation declines 43.9% from 19.8% to 11.1%.
Endogenous Risk
Risk from shocks that are generated and amplified within the financial system. 2008 financial crisis, where an increase in volatility led to traders reducing their position. This selling increased volatility and in turn led to traders at other firms selling, and a chain reaction ensued.
Determine the range within which the security's returns can be expected to fall 68% of the time or one standard deviation
Round up the standard deviation from 8.5609 to 8.6. Then take mean return of 7.2%, and add and subtract the standard deviation of 8.6%. This tells us that 68% of the time, the returns for RMC stock can be expected to be between -1.4% (7.2% - 8.6%) and +15.8% (7.2% + 8.6%).
Calculate the correlation coefficient, given the following coefficient of determinations between an asset and a benchmark. Coefficient of Determination (R2) .98 .86 .70 .50 .04
Shift - Key .9899 .9274 .8367 .7071 .20
Volatility of Correlations
Should be aware that an investment may be made with the expectation that it will continue to have a low correlation with current holdings, only to find it is of minimal help if the correlation changes substantially over the next year or so.
Expected Return for the Portfolio
Simply the weighted-average return of the two assets.
Two Criteria for Risk Free Rate Security
Some controversy exists over the appropriate security to use for the risk-free rate. Must meet two criteria. First, must not have any risk of default. Second, there cannot be any reinvestment risk. Tend to choose short-term U.S. Treasury securities as the risk free asset.
Unsystematic Diversifiable Risk - Financial Risk
Some use only the owners' money or equity. Others use borrowed money debt. Usually, a firm's capital structure is some combination of the two. Risk associated with the degree to which debt used to finance operations. An entity with no debt has no financial risk.
Five Systemic Risks - Reinvestment Risk
Sometimes called "reinvestment rate risk." Falling interest rates will cause the cash flow from an investment to fall when the principal or interest payments of that investment are reinvested at lower rates.
Coefficient of Determination or R-squared
Square of the correlation coefficient. Indicates percentage of one asset's movement that can be explained by movement of a second asset. Generally, the second asset is a market index or benchmark like the S&P 500, and the first asset is an individual stock or mutual fund.
However, if an investor wants to predict next year's return for Mountain Enterprises without regard to its role in a portfolio, another beta may have to be computed
Square the correlation coefficient of .66 = R2 of .44. R2 tells us that only 44% systematic risk of the price movement of Mountain Enterprises is explained by the benchmark and the other 56% is unsystematic risk not explained by the benchmark.
Standard Deviation of the Portfolio When the Correlation Coefficient is 1.0 (pages 54-55) Asset A Expected Return 15%, Standard Deviation 10, Weight .4 Asset B Expected Return 25%, Standard Deviation 20, Weight .6
Standard Deviation of the Portfolio equals square root of (.40) (10) + (.60) (20) + 2(.40)(.60)(10)(20)(1.0). 1. 1st asset: .40 squared = .16. 10 squared = 100. .16 times 100 = 16 2. 2nd asset: .60 squared = .36. 20 squared = 400. .36 times 400 = 144 3. Weightings and covariance: 2 (.40)(.60)(10)(20)(1.0) = 96 4. 16+144+96 = 256. Square root of 256 = 16 standard deviation of the portfolio
Coefficient of Variation Formula
Standard deviation divided by the mean return.
Computing Standard Deviation - First Step
Start with a series of returns. Assume 36 monthly returns. Add them all up and divide by 36 to obtain the average return for that 36-month period. Now can compute the standard deviation for that security's returns.
Liquidity & Marketability Example
Stocks listed on the New York Stock Exchange are very liquid; that is, a round lot can be sold with a small bid/ask spread. Active market of buyers and sellers or high marketability. Each listed stock can usually be sold quickly at a price close to its last posted sale value.
Correlations Inconsistent in Rising and Falling Markets
Studies show that correlations tend to decrease become lower in rising markets, and increase become higher in falling markets, which is the opposite effect an investor would desire.
Total Risk
Systematic risk and unsystematic risk comprise total risk, which is measured by standard deviation which measures variability. Systematic risk, as we will see later on, is measured by beta which measures volatility.
Determine range where returns can be expected to be within two standard deviations or 95% of the time.
Take the -1.4% and subtract 8.6% to arrive at -10.0. On the positive side, we take the 15.8% and add 8.6% to it to arrive at 24.4%. This means that 95% of the time two standard deviations the range of returns will fall within -10.0 to +24.4%.
Coefficient of Variation Calculation
Take the standard deviation and divide by the mean return. If standard deviation 12.73, mean return 9.57%, coefficient of variation = 1.33. Lowest coefficient of variation numbers mean better combination of risk and return.
Investor finds two funds that have correlation coefficients with the S&P 500 index of +.21 and +.42, respectively.
Tells her both funds react independently of the S&P 500 index, but doesn't show if both funds are closely correlated with each other or if they act independently of each other. In this case, a correlation coefficient of the two funds with each other would have to be computed.
Recency
Tendency to place more weight on recent events than on past events. When market going up, investors project going to keep going up. When going down investors don't invest, or they sell, because they project that the market is just going to keep going down.
Scorpio Inc. has a mean return of 19%, and a standard deviation of 25. What is the probability that the stock will have a return greater than 19% if the returns are normally distributed?
The answer is 50%. In a normally distributed yield curve, half of the returns will be greater than the mean return, and half of the returns will be less than the mean return.
Arithmetic vs Geometric Mean
The arithmetic mean is the sum of the numbers, divided by the quantity of the numbers. In general, you can only take the geometric mean of positive numbers. The geometric mean, by definition, is the nth root of the product of the n units in a data.
Standard Deviation
The degree to which an investment's returns can be expected to vary from its mean return.
Unsystematic Risk
The diversifiable component of total risk. The major types of unsystematic risk are business risk, financial risk, and country risk.
Assuming that the correlation of a stock and the market is high, what would beta coefficients of .5, 1.0 mean to someone investing in the stock?
The larger the beta coefficient is, the more volatile the security's historic price is relative to the market. 1.0 stock movement expected to be identical to market movement. .5 means less volatile than the market. If market changes by 4%, stock expected to change by 2%.
Coefficient of Determination
The proportion of the total variation in returns of a security that is explained by the variation in returns of another security or of a benchmark index.
Weighted-Average Return
The return on a portfolio of securities is the sum of the individual assets' returns, each weighted by that asset's proportion of the total market value of the portfolio.
Required Return
The return required to induce an investor to invest in an asset, given that asset's level of risk. An asset's expected return should exceed its required return.
Five Systemic Risks - Interest Rate Risk
The risk that a change in interest rates, especially rising rates, will cause the market value of a fixed-income security to fall. Note: These first three risks, purchasing power, reinvestment, and interest rate, are primary risks associated with bond investing.
Five Systemic Risks - Market Risk
The risk that changes in the overall market prices will cause changes in the market value of a specific security. Market risk is associated with all securities, especially equity securities, and it cannot be eliminated by diversification.
Five Systemic Risks - Purchasing Power
The risk that future inflation will cause the purchasing power of cash flow from an investment to fall.
Single Point Approach vs Monte Carlo Simulation
The single-point approach uses a rate of return and inflation rate, and applies this in a linear fashion, assuming the same return and inflation rate, year after year. Market doesn't work like this and Monte Carlo attempts to address possibility of various returns over time.
Morningstar assumes that a mutual fund with an R squared that exceeds 70% is highly correlated with the market and thus is not an effective diversifier.
The square root of 70% is approximately .84. Therefore, using the Morningstar criteria, we can assume that a mutual fund with a correlation coefficient higher than +.84 is not a good fund to add to a portfolio that already holds a fund indexed to the S&P 500 index.
Correlation Coefficient
The statistical measure of the strength of the relationship of the returns of two assets. It is a standardized version of covariance.
Covariance
The tendency of the returns of two assets to move, over time, in the same or different directions.
Covariance
The tendency of two assets to move together or apart.
Beta Coefficient
The total risk of a security or a portfolio, the sum of unsystematic and systematic risk, is measured by standard deviation or its variability. Systematic risk of a security is measured by beta or volatility. Once little unsystematic risk in the portfolio can use beta to measure risk.
Total Risk
The uncertainty that an investment will deliver its expected return; measured by a security's standard deviation.
Total Risk
The uncertainty that an investment will deliver its expected return— mathematically expressed as standard deviation for a security. Total risk consists of the sum of unsystematic risk and systematic risk.
Unsystematic Diversifiable Risk - Credit and Default Risks
The weaker the firm's financial condition, the greater the chance of the firm defaulting on financial obligations. The degree of a company's default risk is reflected in its credit rating. Credit rating drop can cause the market price of that debt to drop significantly.
Liquidity can have Different Meanings Depending on Context
The words "liquid," "illiquid," or "liquidity" can refer to respect to safety of principal, or accessibility of funds, or minimum capital loss on conversion into cash, or small price concessions incurred in selling.
Covariance and the Correlation Coefficient.
There is a standardized form of covariance that is easier to work with and understand, since it confines the correlation between -1 and +1, and this is called the correlation coefficient.
Coefficient of Determination Example: Correlation coefficient of a small-cap fund with the S&P 500 index .16. If square that number, (.16) squared, we get .0256, or 2.56%.
This tells an investor that only 2% to 3% of the price movements of the small-cap fund can be explained by the movements of the S&P 500 index. The remaining 97% to 98% of the fund's price movements are due to other factors like fund manager's stock picks.
Returns can be expected to be within three standard deviations 99% of the time
To calculate, we would again add and subtract the standard deviation: -10% - 8.6% = -18.6%, and 24.4% + 8.6% = 33%. So three standard deviations fall within the range of -18.6% to +33%.
Leptokurtic Distribution
Type of kurtosis where statistical value is positive. There are high peaks around mean return and thick tails on both sides. Generally good for investors as asset has a relatively low amount of variance with most returns close to the mean.
Platykurtic Distribution
Type of kurtosis with negative statistical value. Flatter peak around mean when compared to normal distribution. Wide dispersion of returns, high variance among those returns. Higher than usual probability for extreme price movements.
Systematic Risks Include: PRIME - Exchange Rate Risk Example
U.S. investor owns a British bond that pays an interest payment of one pound, and six months ago one pound equaled $1.80. Assume that now the U.S. dollar has strengthened because it takes less U.S. currency to buy one pound so that it takes only $1.70 to equal one pound.
Calculate the standard deviation and mean return for the following individual security: years 1-4: +8, +10, +12, +14
Use the E+ key to enter your returns, for negative returns enter the number first, then change the sign ("+/-"). Set for 1 P/YR. SHIFT, 8 for standard deviation 2,58, SHIFT, 7 for mean return 11
Things to remember about the Two-Asset Portfolio Standard Deviation - whole numbers vs decimals, shift + key
W is percentage weight of each asset, decimal (e.g., .40); o standard deviation, whole number (e.g., 27); and COV is covariance of assets i and j, whole number (e.g., +57). Shift + key for ()2, use weighted average instead of correlation 1+
High Positive Correlation Coefficient Example
When constructing an investment portfolio using mutual funds, one should not purchase a fund with a +.88 correlation coefficient with the S&P 500 and another with a +.931 correlation coefficient with the S&P 500. These two funds are far too related.
Diversification & Assets with Low Positive Correlation Coefficients
When constructing portfolios, it is not necessary and it is virtually impossible to have negative correlation coefficients among all assets and the market. Assets with low positive correlation coefficients diversify a portfolio quite well.
Correlation Coefficients Range Examples - Negative Relationship
With a perfectly negative relationship or negative 1, the asset can be expected to fall by 10% when the market rises by 10%. This would have to be some kind of derivative security. The further you go from + 1 the greater the diversification.
Correlation Coefficients Range Examples - Positive or No Relationship
With a perfectly positive relationship +1, when market rises 10%, the asset rises 10%; when it falls 10%, the asset falls 10%. With a correlation coefficient of 0 no relationship, the asset could rise, fall, or do nothing when the market rises.
a. Coefficient of variation and b. Beta
a. An investment's risk per unit of expected return. b. Measure of systematic risk, it is a statistical measure of a stock's volatility as measured against a stock index.
a. Covariance and b. Correlation Coefficient
a. Tendency of the returns of two assets to move, over time, in the same or different directions. b. The statistical measure of the strength of the relationship of two assets returns. Standardized version of covariance.
Assume that the risk-free rate is the three-month Treasury bill rate of 3.5%, that the market's expected return is 8%, and that the beta of ABC stock is 1.1. What return would an investor require to induce him or her to invest in ABC?
ri = rf + (rm - rf)Bi 8 - 3.5 = 4.5 times 1.1 = 4.95 plus 3.5 = 8.45% Note that we multiplied beta times the market's risk premium (rm - rf) or 4.5 and then added the risk-free rate to that number. Can also use decimals.
CAPM or Required Return Formula
ri = rf + (rm - rf)Bi ri = required return of the individual asset rf = the risk-free rate generally a Treasury security, usually T-bills plus rm = the return of the market such as the S&P 500 Bi = beta of the individual asset
Market Risk Premium
rm - rf = the difference between the return of the market and the risk-free rate. The additional return investors expect that rewards them for the extra risk they are taking by investing in the market
