Chapter 3: Risk & Return

"Black Swan Events"

Low probability, high impact economic events. They are events that might cause investment returns to fall into the "tails" of our normal distribution graphic. Examples: Market bubbles, war, extreme natural disasters.

Coefficient of Variation (CV)

Measure the amount of risk per unit of return. Standard deviation divided by expected returns

Z-score

Have a mean of 0 and a standard deviation of 1.0, which is the same as a standard normal distribution. A z-score equals the number of standard deviation that a result is from the mean.

Net Present Value (NPV)

The difference between the initial cash outflow and the discounted future cash flows

Real Rate of return (Inflation adjusted return)

Adjusts the nominal return to account for the loss of purchasing power due to inflation

Beta

- A measure of SYSTEMATIC RISK that measures volatility - Beta is an indication of a security's volatility relative to the market; With the market having a beta of 1.0 -A positive beta means that the security and the market move in the same direction. A negative Beta means that the security and the market move in opposite directions. Beta higher than 1- More volatile than the market Beta above 0 but less than 1: Less volatilve Beta of 0= Uncorrelated to the market negative Beta= Reverse volatility

The Four "Cs"

1. Coefficient of Variation: Standard deviation divided by expected return (total risk per unit of return) 2. Correlation Coefficient: Ranges from -1.0 to +1.0. Maximum risk reduction occurs at perfect negative (-1.0). No risk reduction occurs at perfect positive (+1.0). 3. Covariance: A measure of how much two assets move together. 4. Coefficient of Determination: Known as R(squared). This statistic provides the percentage variation in portfolio returns that is explained by the variation in the benchmark returns.

Probability of Returns Assuming a Normal Distribution (Standard Deviation)

68%- Approximately 68% of returns fall within 1 standard deviation of the mean 95% Approximately 95% of returns fall within 2 standard deviation of the mean 99%- Approximately 98% of returns fall within 3 standard deviations of the mean.

Covariance

A Measure of how two assets move together. It combines volatility of one asset's returns with the tendency of those returns to move up or down at the same time that another asset's returns move up or down. -Key point: The more positive the covariance between two variables, the more they move together. When two assets move together are combined in a portfolio, there is little risk reduction. The more negative the covariance between the two variables, the less their movements are together, resulting in a greater diversification. Cov(ab)= Standard deviation ofA* Standard Deviation of B * Correlation between assets A and B

Platykurtic

A distribution that is less peaked (less kurtosis) than a normal distribution

Leptokurtic

A distribution that is more peaked (more kurtosis) than a normal distribution

Coefficient of variation (CV)

A measure of relative risk that is useful when comparing assets with different returns and risk. It measures the amount of risk per unit of return - The higher the CV, the greater the risk per unit of return. - An investor prefers the asset with the lower Coefficient of Variation because it is less risk, for each unit of return Formula: CV= Standard deviation/ average return

Holding period Return (HPR)

A measure of return that includes: -Capital appreciation - Current income It is the return earned over a specific period of time. HPR is most meaningful if holding period is one year.

Skewness (Positive and Negative)

A measure of the lack of symmetry for a data set With a normal distribution, the mean, median and mode are generally equal. Positively Skewed Distribution: This has a mean greater than the median (Data is shifted left) Negatively Skewed Distribution: (Data shifted to the right)

Kurtosis (Leptokurtic vs. Platykurtic) (Fat Tails vs Thin tails)

A measure of whether the data are heavy tailed or light tailed relative to a normal distribution. Leptokurtic: A distribution that is more peaked than normal (high peak and fat tails) Platykurtic: A distribution that is less peaked than Kurtosis. (Low peak and thin tails) "Fat Tails"- A higher probability of outcomes resides in the tails of a leptokurtic distribution than for a normal distribution. "Thin tails"- A lower probability of outcomes resides in the tails of a platykurtic distribution than for a normal distribution.

Regression Analysis

A statistical method for estimating the relationship among variables

Arithmetic Mean vs. Geometric Mean

Arithmetic: Calculated by summing the annual holding period returns and dividing the sum by the number of years Geometric: Calculates the average return over time assuming all earnings remain invested.

Variance

Calculated by squaring the standard deviation. Investments with high variances tend to be riskier than those with low variances.

Coefficient of Determination (r^2 or R^2)

Coefficient of Determination and Correlation Coefficient are two statistics used when measuring Correlation. is a measure of strength of the relationship between two variables. It ranges from 0 to 100% and is defined as the percent change in the dependent variable that is explained by changes in the independent variable. With a R2 of 1.0 or 100%, it means that 100% of the change in one variable is explained by changes in the other variable. Example: Consider the S&P index fund, it will have an R2 of 100%, because 100% of the change In the fund is attributable to changes in the market.

Correlation (Correlation Coefficient)

Correlation is a statistical measure of the relationship, if any between two variables. It answers the question: "how do. you determine the level of diversification within a portfolio?" The statistics used to provide the answers are the "correlation coefficient (R) and the "coefficient of determination (R^2)" Correlation Coefficient: - Correlation can be positive, negative or zero -Correlation ranges from -1 to +1 - Correlation provides insight to the strength and direction of two assets move relative to each other +1= Perfect positive relationship 0.50= Positive relationship between two variables 0= No relationship; two variables move independently -0.50= Negative relationship between variables -1.0= Perfect negative relationship

Dollar Weighted Internal Rate of Return

Measures the effect of all of the cash flows an investor controls. The objective is to determine the combined result of the timing and dollar volume of the transactions during the period, as well as the performance of the investment security. It would also be appropriate for gauging the performance of an investment manager with full discretion over an investor's account.

time weighted Internal Rate of Return

Measures the effect of any cash flows associated with the investment security (such as mutual fund or stock), but ignores the dollar volume and timing of investor driven transactions during the period. This method is more appropriate for assessing the performance of a fund manager or a security because it includes the capital appreciation and dividends earned, but it is not affected by an investor's timing decisions.

Semivariance

Only takes into consideration volatility below the mean return

What is the weighted average beta of the following portfolio? Stock L has a beta of 1.45 and constitutes 10% of the portfolio, stock M has a value of $125,000 with a beta of .93, while stock N makes up 40% of the portfolio with a beta of .65, and stock O, with a 2.2 beta has a dollar value of $175,000.

Please be certain to avoid rounding errors in arriving at the correct solution. For weighted average beta, use only the two places to the right of the decimal. Stock L = 10%; Stock M=$125,000; Stock N=40%; Stock O=$175,000; L+N=50%; therefore M+O=50% or $300,000. Thus the total portfolio value is $600,000. Stock L = $60,000/$600,000 = .10 × 1.45 = .14; Stock M = $125,000/$600,000 = .2083 × 0.93 = .19; Stock N = $240,000/600,000 = .40 × 0.65 = .26; Stock O = $175,000/$600,000 = .2917 × 2.2 = .64. Adding the results (.14+.19+.26+.64=1.23) will result in the weighted average beta of 1.23.

Semi Variance

Semivariance is similar to Standard deviation by measuring return. Except, it only takes into consideration volatility below the mean return. This is not commonly used.

Bob Conrad's investment portfolio consists of several types of stocks, bonds, and money market instruments. The portfolio has an overall standard deviation of 12%, a beta of 1.06, and a total return for the year of 11%. Bob is considering adding one of two alternative investments to his portfolio. Stock A has a standard deviation of 13%, a beta of .87, and a correlation coefficient with the portfolio of .6. Stock B has a standard deviation of 11%, a beta of .97, and a correlation coefficient of .95. Which stock should Bob consider adding to his portfolio, and why? Stock A, because it has a lower correlation coefficient. Stock A, because it has a lower beta than that of the portfolio. Stock B, because it has a lower standard deviation than that of the portfolio. Stock B, because it has a higher correlation coefficient.

Solution: The correct answer is A. In the process of adding new investments to a portfolio, the lowest correlation coefficient makes the best addition. Closest to negative one (-1) is always best.

To obtain the maximum reduction in risk, an investor should combine assets that: A. are negatively correlated. B. are uncorrelated. C. have a correlation coefficient of positive one. D. have a correlation coefficient of negative one.

Solution: The correct answer is D. Uncorrelated assets have a correlation equal to 0. Perfect negatively correlated assets will have a correlation equal to -1 and achieve the more diversification.

If fund A has a correlation coefficient of .9, then how much of its return is due to systematic (or market) risk? A. 0 B. .19 C. .64 D. .81

Solution: The correct answer is D. r = .9, r2 = .81; therefore, 81% of the return for the fund A is due to the market.

The type of risk which measures the extent to which a firm uses debt securities and other forms of debt in its capital structure to finance is known as: A. Business risk B. Systematic risk C. Default risk D. Financial risk

Solution: The correct answer is D. Financial risk has to do with the amount of leveraging or use of borrowed funds a firm utilizes to structure its investment and finance its assets.

Systematic Risk. (PRIME)

Systematic risk is NON DIVERSIFIABLE P- Purchasing Power Risk R- Reinvestment Risk I- Interest Rate Risk M- Market Risk E- Exchange Rate Risk Systematic risk are risks that are inherent in the "system."

Yield

The amount of cash the investment generates and the amount the investor receives or reinvests in a year.

Internal Rate of Return (IRR)

The earnings rate of a series of cash inflows and outflows over a period of time assuming all earnings are reinvested. It equates discounted future cash flows to the present value of an asset.

Realized taxable return

The return from the investment without considering income tax

Reinvestment Rate risk

The risk that the investor is unable to reinvest cash flows at the IRR.

Marginal Income Tax Rate

The tax rate percentage applied to incremental taxable income

Standard deviation

This is a measure of TOTAL RISK and measures variation of returns around an average; Systematic risk and unsystematic risk. - If the _______ is low, that means that the returns from an investment are fairly consistent from period to period. - The greater the _________, the greater the risk. -68-95-99 rule

Required Rate of Return

This is the rate of return an investor must earn on an investment to be fully compensated for its risk. Components include: - Real Reate -Expected inflation premium -Risk Premium

Portfolio Risk (total risk)

Total Risk= Systematic Risk + Unsystematic Risk Total return is measured by standard deviation. Systematic Risk is measured in Beta

Unsystematic Risk (Business Risk) (ABCDEFG)

Unsystematic Risk can be diversified away by combining multiple asset classes and industries in a portfolio. Unsystematic Risk Includes: - Accounting -Business -Country -Default -Financial -Government/ regulatory