Chapter 9 Risk and Return Ms. Faith Moono Simwami Mo.simwami@gmail.com

After studying Risk and Return, you should be able to: Understand the relationship between risk and return. Define risk and return and show how to measure them by calculating expected return, standard deviation, and coefficient of variation. Distinguish between avoidable (unsystematic) risk and unavoidable (systematic) risk and explain how proper diversification can eliminate one of these risks. Understand Portfolio Selection and Diversification.

Defining Return Income received on an investment plus any change in market price, usually expressed as a percent of the beginning market price of the investment. Dt + (Pt – Pt - 1 ) R = Pt - 1 Return Example The stock price for Stock A was $10 per share 1 year ago. The stock is currently trading at $9.50 per share and shareholders just received a $1 dividend. What return was earned over the past year? $1.00 + ($9.50 – $10.00 ) = 5% R = $10.00

Defining Risk Risk refers to the chance that some unfavorable event will happen Risk is the variability of returns from those that are expected. Risk is the chance of harm or loss; danger. We know that various asset classes have yielded very different returns in the past:. Risk can be measured by looking at the variability in the probability distribution with reference being made to the mean value or average (expected return). The variability or spread, which is the risk is measured using the standard deviation. The smaller the standard deviation the lesser the risk and the larger the standard deviation the higher the risk. Before we can measure the standard deviation we need to know the expected return or mean. Investment risk is the probability that actual returns may deviate from expected returns The chance that actual returns may be lower than expected return gives rise to investment risk. Higher the probability of actual returns being less than expected, higher will be investment risk

Determining Expected Return The expected return is the return on a risky asset expected in the future. Meaning summation of the product of the respective probabilities and the returns, Expected Return (ER) = ER= (R1)(P1)+ (R2)(P2)+ (R3)(P3)+ + + Pi(Ri) ER = S ( Ri )( Pi ) ER is the expected return for the asset, Ri is the return for the ith possibility, Pi is the probability of that return occurring, S The sum of the total number of possibilities.

Determining Standard Deviation (Risk Measure) Standard Deviation, s, is a statistical measure of the variability of a distribution around its mean. It is the square root of variance. Risk can be measured by looking at the variability in the probability distribution with reference being made to the mean value or average (expected return).The variability or spread, which is the risk is measured using the standard deviation. The smaller the standard deviation the lesser the risk and the larger the standard deviation the higher the risk. Before we can measure the standard deviation we need to know the expected return or mean. Pi is the probability of that return occurring, ER is the expected return for the asset, ri is the return for the ith possibility, S The sum of the total number of possibilities.

How to determine the Expected Return for Stock A State of Economy Example .The figure below shows the probability distribution for different states of an economy with probability assigned and the returns under each state of the economy. State of Economy Probability Returns of Stock A Returns of Stock B Boom 0.3 25% 19% Normal 0.4 15% 12% Recession -10% 8% Total Probability 1   Calculate the expected rate of return for Stock A and Stock B? How to determine the Expected Return for Stock A State of Economy Probability Returns of Stock A ER Boom 0.3 25% 7.5% Normal 0.4 15% 6% Recession -10% -3% Total Probability 1.0 10.5%

Example 2.How to Determine the Standard Deviation for Stock A State of Economy Pi Ri (Ri)(Pi) (Ri - ER)2(Pi) Boom 0.3 25% 7.5% 63.075 Normal 0.4 15% 6% 8.100 Recession -10% -3% 125.075 Total 1.0 10.5% 196.25 How to determine the Coefficient of Variation CV This means that for this stock A an investor takes on 1.33units of risk for every extra 1.0 unit of return earned. This is useful especially where standard deviations are equal or close and an investor needs to choose between two or more options for investment. CV = 1.33

Calculate the expected rate of return and standard deviation for Stock for Stock B?

What is the Coefficient of Variation for Stock B? CV = 0.335 Which stock has a higher investment risk? Explain why? Stock A has a higher investment Risk. This means that for this stock A an investor takes on 1.33units and stock B an investor takes on 0.335 units of risk for every extra 1.0 unit of return earned. Also the Standard deviation for stock A is 14 and for Stock B is only 4.32 This observation is useful especially where standard deviations are equal or close and an investor needs to choose between two or more options for investment.

ERp=WA x ERA+WB x ERB Portfolio Selection and Diversification Most investors tend to hold more than one investment. They hold what is known as a portfolio. A portfolio is a group of assets or investment held by an investor. The reason for holding a portfolio is to diversify the risk or reduce the level of exposure to risk associated with holding a single investment. Determining the risk in a portfolio requires knowing the portfolio expected return and the correlation or covariance of the returns among the securities that make the portfolio. How to determine the Expected return for a portfolio ERp=WA x ERA+WB x ERB Example 3.2 Let say you intend to hold a portfolio in which 50% of the funds are invested in A and the other half in B. The ER for A and B respectively are 10.5% and 12.9%.Let us assume that the calculated standard deviation for A and B are 14% and 4.32%.Find portfolio expected return. ERp= 0.5(10.5)+0.5(12.9) = 11.7%

Measuring Risk for a Portfolio A number of factors affect the risk associated with holding a portfolio; one of them is the way returns from investment move in relation to each other. Returns either move positively or negatively in relation to each other. This is known as the correlation (covariance). The covariance is measure as follows: COV (Ra ,Rb ) =∑Pi(Ra-ERa )(Rb- ERb) Example State of Economy Probability Returns of Stock A Returns of Stock B Boom 0.3 25% 19% Normal 0.4 15% 12% Recession -10% 8% Total Probability 1 ER = 10.5% ER = 12.9% COV (Ra,Rb) =∑Pi(Ra-ERa)(Rb-ERb) =0.3(25% -10.5%)(19% - 12.9%)+0.4(15%-10.5%)(12%-12.9%)+0.3(-10%-10.5%)(8%-12.9%) = 0.3(14.5)(6.1)+0.4(4.5)(-0.9)+0.3(-20.5)(-4.9) = 55.05% or 0.5505 This means that the relationship between the returns of A and B is positive. This will increase the risk for a portfolio.

To obtain the portfolio risk we need to compute the portfolio standard deviation using the formula below: Where; W is percentage weights for amount invested in A and B σ is the standard deviation for A and B COV (Ra, Rb) is the covariance for A and B The portfolio standard deviation = 0.0049+0.00046656+0.27525 = 0.5297 The correlation lies between +1 and -1. When it is 0 it means there is no relation between the returns movement.

CORRELATION AND INTERFERENCE (RISK REDUCTION) “Correlation” is the word given to the extent to which assets move together, this is measured with statistical formulae. Correlations can range from -1 (perfectly negatively correlated) through to +1 (perfectly positively correlated). If asset B tends to move in the opposite direction to asset A then these two assets are said to have “negative correlation”, and they can be highly effective at cancelling out each other’s volatility. If the assets both trend upwards over the longer term a combination of them will have a return equal to the average of the two assets’ returns but with substantially reduced volatility. If the securities are perfectly positively correlated, there is no reduction in risk from forming a portfolio. When the correlation between returns is less than +1.0, there are risk reduction benefits. Diversification can reduce the risk of the portfolio below the weighted average of the total risk of the individual securities. The maximum risk reduction is achieved when the returns on two securities move exactly opposite each other so their correlation is -1.0. Negatively correlated assets cancel the greatest amount of each other’s volatility.

INTERFERENCE AND CORRELATION Diagram 1 shows Positive Covariance or Correlation Coefficient Diagram 2 shows Negative Covariance or Correlation Coefficient Negative correlation between returns from two or more investment reduces the overall risk for a portfolio, hence it is important to understand how return move when selecting investments to add to a portfolio. Negatively correlated assets cancel the greatest amount of each other’s volatility. This means that the relationship between the returns of A and B is positive. This will increase the risk for a portfolio.

Rebalancing a portfolio Rebalancing a portfolio is the process of adjusting a portfolio to bring it back to its original asset allocation (original mix.) It involves periodically buying or selling assets in your portfolio to maintain your original desired level of asset allocation. Since assets perform differently at different times, the portfolio is likely to drift from your desired asset allocation. . Failure to rebalance means that a portfolio can change risk profile over time and may no longer be appropriate. You need to readjust your portfolio, to restore its original balance. If your investment goal hasn't changed, your portfolio's mix shouldn't, either. Because of market forces, however, it does. For example, say your original target asset allocation was 50% stocks and 50% bonds. If your stocks performed well during the period, it could have increased the stock weighting of your portfolio to 70%. You may then decide to sell some of your stocks and buy bonds to get it back to your original target allocation of 50/50. A simple rule of portfolio construction If you have two assets with roughly equal expected returns, putting 50% into each is a way to hedge one’s bets (and spread the risk) without compromising expected return. The lower the correlation of those assets, the more the risk will be reduced while not reducing expected returns at all. Actually, this holds true with a greater number of investments as well. For example, if you have five equally attractive assets you could invest one fifth in each.

Capital Asset Pricing Model (CAPM) CAPM is a model that describes the relationship between risk and expected (required) return; in this model, a security’s expected (required) return is the risk-free rate plus a premium based on the systematic risk of the security. The CAPM suggests that investors demand compensation for risks that they are exposed to and these returns are built into the decision-making process to invest or not. Investors demand compensation for expected inflation, a real rate of return over and above expected inflation compensation over and above the risk-free rate of return for any additional risk undertaken. We will make the case that investors don’t need compensation for all of the risk of an investment because some of that risk can be diversified away. Investors require compensation for risk they can’t diversify away! Security Market Line The SML essentially graphs the results from the capital asset pricing model (CAPM) formula. The x-axis represents the risk (beta), and the y-axis represents the expected return. The market risk premium is determined from the slope of the SML. The security market line is a useful tool in determining whether an asset being considered for a portfolio offers a reasonable expected return for risk.

Systematic Risk (Beta) Illustration of the Security Market Line Formula in CAPM β is the beta of stock (measures systematic risk of stock ) R is the required rate of return for a stock ERM is the expected return for the market portfolio. Rf is the risk-free rate of return, The return that a rational investor should demand is therefore based on market rates and the beta risk of the investment. To find this, you solve for the required return in the CAPM: R = Rf + (ERM – Rf) b % ERM Market Premium for risk Risk Premium Required Return Rf Real Return Premium for expected inflation Risk-free Return bM = 1.0 Systematic Risk (Beta)

Systematic Risk (non-diversifiable risk) is the variability of return on stocks or portfolios associated with changes in return on the market as a whole. Systematic risk arises due to the changes in local and global macroeconomic parameters which include economic policy decisions made by governments, decisions of central banks that affect the lending interest rates and even waves of economic recession. These are some of the systematic risk examples. They arise due to the inherent dynamic nature of an economy and the flow of resources around the world. Examples are wars, rising inflation ,Financial crisis 2008 Unsystematic Risk (Diversifiable Risk) is the variability of return on stocks or portfolios not explained by general market movements. It is avoidable through diversification.Some of the unsystematic risk examples :are labour strikes, drop in sales of a company or any other problem which arises due to human level error in judgment at the managerial level, that affects your stock or securities investment. equipment failure for that one company, management competence or management incompetence for that particular firm, a jet carrying the senior management team of the firm crashes. Obviously, diversifiable risk is that unique factor that influences only the one firm.

Determination of the Required Rate of Return (Using the CAPM formula) Example 1 Determination of the Required Rate of Return (Using the CAPM formula) Lisa Miller at Basket Wonders is attempting to determine the rate of return required by their stock investors. Lisa is using a 6% Rf and a long-term market expected rate of return of 10%. A stock analyst following the firm has calculated that the firm beta is 1.2. What is the required rate of return on the stock of Basket Wonders? R = Rf + (ERM – Rf) β RBW = 6% + (10% – 6%) 1.2 RBW = 10.8% The required rate of return exceeds the market rate of return as BW’s beta exceeds the market beta (1.0). Using the CAPM it is possible to determine the required return on a given stock if the beta is known and the risk premium is also known.

Determination of the Required Rate of Return ( Using the CAPM formula) Example 2. A stock has a beta of 1.2 and the risk premium for this stock is 6%.Given that T bills offer 8% risk free return. What required rate of return will investors demand on this stock? Determination of the Required Rate of Return ( Using the CAPM formula) R = Rf + (ERM – Rf)b = 8% + (6%)(1.2) = 15.2% Risk premium When the risk is adjusted the return responds likewise. Suppose the beta was increased to 2.0. R = 8% + (6%(2.0) = 20% The same would happen if the beta was reduced to 0.5. R = 8% + (6%)(0.5)= 11%