Chapter 6the Meaning And Measurement Of Risk And Return 2017 Pearson ✓ Solved

Chapter 6 The Meaning and Measurement of Risk and Return © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› Learning Objectives Define and measure the expected rate of return of an individual investment. Define and measure the riskiness of an individual investment. Compare the historical relationship between risk and rates of return in the capital markets. © 2017 Pearson Education, Inc.

All rights reserved. 6-‹#› Learning Objectives Explain how diversifying investments affects the riskiness and expected rate of return of a portfolio or combination of assets. Explain the relationship between an investor’s required rate of return on an investment and the riskiness of the investment. © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› EXPECTED RETURN DEFINED AND MEASURED © 2017 Pearson Education, Inc.

All rights reserved. 6-‹#› Holding-Period Return Historical or holding-period or realized rate of return Holding-period return = payoff during the “holding†period. Holding period could be any unit of time such as one day, few weeks or few years. © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› Holding-Period Return You bought 1 share of Google for 4.05 on April 17 and sold it one week later for 5.06.

Assuming no dividends were paid, your dollar gain was: 565.06 – 524.05 = .01 © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› Holding-Period Rate of Return Holding-period rate of return Google rate of return: 41.01/524.05 = .0783 or 7.83% © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› Expected Return Expected Cash Flows and Expected Rate of Return The expected benefits or returns an investment generates come in the form of cash flows.

Cash flows are used to measure returns (not accounting profits). © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› Expected Return The expected cash flow is the weighted average of the possible cash flows outcomes such that the weights are the probabilities of the occurrence of the various states of the economy. Expected Cash flow (X) = ΣPbi*CFi Where Pbi = probabilities of outcome i CFi = cash flows in outcome i © 2017 Pearson Education, Inc. All rights reserved.

6-‹#› © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› Expected Cash Flow Equation Equation 6-3 © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› Expected Cash Flow Expected Cash flow = ΣPbi*CFi = 0.2*1000 + 0.3*1200 + 0.5*1400 = ,260 on ,000 investment © 2017 Pearson Education, Inc.

All rights reserved. 6-‹#› Expected Rate of Return We can also determine the % expected return on ,000 investment. Expected Return is the weighted average of all the possible returns, weighted by the probability that each return will occur. Expected Return (%) = ΣPbi*ri where Pbi = probabilities of outcome i ri = expected % return in outcome i © 2017 Pearson Education, Inc. All rights reserved.

6-‹#› Expected Rate of Return Equation 6-4 © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› Expected Rate of Return Expected Return (%) = ΣPbi*ri where Pi = probabilities of outcome i ki = expected % return in outcome I = 0.2(10%) + 0.3(12%) + 0.5(14%) = 12.6% © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› RISK DEFINED AND MEASURED © 2017 Pearson Education, Inc.

All rights reserved. 6-‹#› Risk Three important questions: What is risk? How do we measure risk? Will diversification reduce the risk of portfolio? © 2017 Pearson Education, Inc. All rights reserved.

6-‹#› Risk Defined Risk refers to potential variability in future cash flows. The wider the range of possible future events that can occur, the greater the risk. Thus, the returns on common stock are more risky than returns from investing in a savings account in a bank. © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› Risk Measured Consider two investment options: Invest in Treasury bond that offers a 2% annual return.

Invest in stock of a local publishing company with an expected return of 14% based on the payoffs (given on next slide). © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› Probability of Payoffs Stock Treasury Bond 100% chance 2% © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› Expected Rate of Return Treasury bond = 1*2% = 2% Stock = 0.1*-10 + 0.2*5% + 0.4*15% + 0.2*25% + 0.1*30% = 14% © 2017 Pearson Education, Inc.

All rights reserved. 6-‹#› © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› Figure 6-1 Treasury Bond versus Stock We observe from Figure 6-1 that the stock of the publishing company is more risky but it also offers the potential of a higher payoff. © 2017 Pearson Education, Inc. All rights reserved.

6-‹#› Standard Deviation (S.D.) Standard deviation (S.D.) is one way to measure risk. It measures the volatility or riskiness of portfolio returns. S.D. = square root of the weighted average squared deviation of each possible return from the expected return. © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› © 2017 Pearson Education, Inc.

All rights reserved. 6-‹#› © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› Comments on S.D. There is a 66.67% probability that the actual returns will fall between 2.86% and 25.14% (= 14% ± 11.14%).

So actual returns are far from certain! Risk is relative; to judge whether 11.14% is high or low risk, we need to compare the S.D. of this stock to the S.D. of other investment alternatives. To get the full picture, we need to consider not only the S.D. but also the expected return. The choice of a particular investment depends on the investor’s attitude toward risk. © 2017 Pearson Education, Inc. All rights reserved.

6-‹#› RATES OF RETURN: THE INVESTOR’S EXPERIENCE © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› Rates of Return: The Investor’s Experience (1926–2014) Figure 6-2 shows: A.

The direct relationship between risk and return B. Only common stocks provide a reasonable hedge against inflation. The study also observed that between 1926 and 2014, large stocks had negative returns in 22 of 86 years, while Treasury bills generated negative returns in only 1 year. © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› RISK AND DIVERSIFICATION © 2017 Pearson Education, Inc.

All rights reserved. 6-‹#› Portfolio Portfolio refers to combining several assets. Examples of portfolio: Investing in multiple financial assets (stocks – 00, bonds – 00, T-bills – 00) Investing in multiple items from a single market (example: investing in 30 different stocks) © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› Risk and Diversification Total risk of portfolio is due to two types of risk: Systematic (or market risk) is risk that affects all firms (ex.: tax rate changes, war) Unsystematic (or company-unique risk) is risk that affects only a specific firm (ex.: labor strikes, CEO change) Only unsystematic risk can be reduced or eliminated through effective diversification. (Figure 6-3) © 2017 Pearson Education, Inc.

All rights reserved. 6-‹#› Total Risk & Unsystematic Risk Decline as Securities Are Added © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› Correlation and Risk Reduction The main motive for holding multiple assets or creating a portfolio of stocks (called diversification) is to reduce the overall risk exposure. The degree of reduction depends on the correlation among the assets.

If two stocks are perfectly positively correlated, diversification has no effect on risk. If two stocks are perfectly negatively correlated, the portfolio is perfectly diversified. © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› Correlation and Risk Reduction Thus, while building a portfolio, we should pick securities/assets that have negative or low-positive correlation to realize diversification benefits. © 2017 Pearson Education, Inc. All rights reserved.

6-‹#› Market Risk or Systematic Risk Measuring Market Risk: eBay vs. S&P 500 Table 6-3 and Figure 6-4 display the monthly returns for eBay and S&P 500 for the 12 months ending May 2015. © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› © 2017 Pearson Education, Inc. All rights reserved.

6-‹#› ‹#› © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› Measuring Market Risk Equation 6-6 © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› Equations 6-7 and 6-8 © 2017 Pearson Education, Inc.

All rights reserved. 6-‹#› From Figure 6-4 and Table 6-3 Average monthly return: eBay= 1.39% S&P 500 = 0.87% Risk was higher for eBay with standard deviation of 4.65% versus 2.57% for S&P 500. There is a moderate positive relationship in the movement of returns between eBay and S&P 500 (in 7 of the 12 months) (see Figure 6-5). © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› © 2017 Pearson Education, Inc.

All rights reserved. 6-‹#› Characteristic Line and Beta The relationship between eBay and S&P 500 is captured in Figure 6-5. Characteristic line is the “line of best fit†for all the stock returns relative to returns of S&P 500. The slope of the characteristic line (= 0.782) measures the average relationship between a stock’s returns and those of the S&P 500 Index Returns. This slope (called beta) is a measure of the firm’s market risk; i.e., eBay’s returns are 0.782 times as volatile on average as those of the overall market. © 2017 Pearson Education, Inc.

All rights reserved. 6-‹#› © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› Interpreting Beta Beta is the risk that remains for a company even after we have diversified our portfolio. A stock with a Beta of 0 has no systematic risk A stock with a Beta of 1 has systematic risk equal to the “typical†stock in the marketplace A stock with a Beta exceeding 1 has systematic risk greater than the “typical†stock Most stocks have betas between 0.60 and 1.60.

Note, the value of beta is highly dependent on the methodology and data used. © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› Portfolio Beta Portfolio beta indicates the percentage change on average of the portfolio for every 1 percent change in the general market. βportfolio = Σ wj*βj Where wj = % invested in stock j βi = Beta of stock j © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› Equation 6-10 © 2017 Pearson Education, Inc.

All rights reserved. 6-‹#› © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› Risk and Diversification Demonstrated The market rewards diversification. Through effective diversification, we can lower risk without sacrificing expected returns and we can increase expected returns without having to assume more risk. © 2017 Pearson Education, Inc.

All rights reserved. 6-‹#› Asset Allocation Asset allocation refers to diversifying among different kinds of asset types (such as treasury bills, corporate bonds, common stocks). Asset allocation decision has to be made today – the payoff in the future will depend on the mix chosen before, which cannot be changed. Hence asset allocation decision is considered the “most important decision†while managing an investment portfolio. © 2017 Pearson Education, Inc. All rights reserved.

6-‹#› © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› Asset Allocation Matters! We observe the following from Figure 6-8. Direct relationship between risk and return: As we move from an all-stock portfolio to a mix of stocks and bonds to an all-bond portfolio, both risk and return decline.

Holding period matters: As we increase the holding period, risk declines. © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› Asset Allocation Summary There has never been a time when investors lost money if they held an all-stock portfolio—the most risky portfolio—for 10 years. The market rewards the patient investor. © 2017 Pearson Education, Inc. All rights reserved.

6-‹#› THE INVESTOR’S REQUIRED RATE OF RETURN © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› The Investor’s Required Rate of Return Investor’s required rate of return is the minimum rate of return necessary to attract an investor to purchase or hold a security. This definition considers the opportunity cost of funds, i.e., the foregone return on the next best investment. © 2017 Pearson Education, Inc. All rights reserved.

6-‹#› The Investor’s Required Rate of Return Equation 6-11 © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› Risk-Free Rate This is the required rate of return or discount rate for risk-free investments. Risk-free rate is typically measured by the U.S. Treasury bill rate. © 2017 Pearson Education, Inc.

All rights reserved. 6-‹#› Risk Premium The risk premium is the additional return we must expect to receive for assuming risk. As the level of risk increases, we will demand additional expected returns. © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› Capital Asset Pricing Model (CAPM) CAPM equation equates the expected rate of return on a stock to the risk-free rate plus a risk premium for the systematic risk.

CAPM provides for an intuitive approach for thinking about the return that an investor should require on an investment, given the asset’s systematic or market risk. © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› Measuring the Required Rate of Return Equation 6-12 © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› Capital Asset Pricing Model If the required rate of return for the market portfolio rm is 10%, and the rf is 3%, the risk premium for the market would be 7%.

This 7% risk premium would apply to any security having systematic (nondiversifiable) risk equivalent to the general market, or beta of 1. In the same market, a security with beta of 2 would provide a risk premium of 14%. © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› CAPM Equation 6-13 CAPM suggests that beta is a factor in determining the required returns. © 2017 Pearson Education, Inc. All rights reserved.

6-‹#› CAPM Example Market risk = 10% Risk-free rate = 3% Required return = 3% + beta * (10% - 3%) If beta = 0 Required return = 3% If beta = 1 Required return = 10% If beta = 2 Required return = 17% © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› The Security Market Line (SML) SML is a graphic representation of the CAPM, where the line shows the appropriate required rate of return for a given stock’s systematic risk. © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› © 2017 Pearson Education, Inc.

All rights reserved. 6-‹#› Key Terms Asset allocation Beta Capital asset pricing model (CAPM) Characteristic line Expected rate of return Historical or realized rate of return Holding-period return Portfolio beta Required rate of return Risk Risk-free rate of return Risk premium Security market line Standard deviation Systematic risk Unsystematic risk © 2017 Pearson Education, Inc. All rights reserved. 6-‹#› Module 6 Critical Thinking Assignment: Measurement of Risk and Return in Investing *Complete the problems in an Excel spreadsheet. Be sure to show your work to receive credit, no hard keys.

Problem 6-1: Standard Deviation Given the following probabilities and returns, for MKS, Inc., find the standard deviation. DATA Probability Return 0.% 0.% 0.% 0.% Problem 6- 2 Holding-Period Period Return From the price data below, compute the holding-period returns for periods 2 through 4. DATA TIME STOCK PRICE (SAR) Problem 6-3: Holding-Period Gain SAR and Return Suppose you purchased 16 shares of DCI stock for 24.22 SAR per share on May 1, 2016. On September 1 of the same year, you sold 12 shares of the stock for 25.68 SAR per share. Calculate the holding-period dollar gain for the shares you sold, assuming no dividend was distributed, and calculate the holding-period rate of return.

Problem 6-4: Capital Asset Pricing Model Using the CAPM, estimate the appropriate required rate of return for the three stocks listed here, given that the risk-free rate is 5 percent and the expected return for the market is 12 percent. Stock Beta A .75 B .90 C 1.40 Problem 6-5: Security Market Line A. Determine the expected return and beta for the following portfolio: DATA Stock % of Portfolio Beta Expected Return % 1.% % 0.% % 1.% Problem 6-6: Required Rate of Return using CAPM a. Compute an appropriate rate of return for ABC common stock, which has a beta of 1.2. The risk-free rate is 2 percent, and the market portfolio has an expected return of 11 percent. b.

Why is the rate you computed an appropriate rate? Problem 6-7: Expected Return, Standard Deviation Below are the historical prices for Citigroup and the S&P500 Index. Calculate the average monthly returns and the standard deviation for each. DATA Month Adidas S&P 500 Stock Prices Index May-. June-.

July-. August-. September-. October-. November-.

December-. January-. February-. March-. April-. May-.

Paper for above instructions

The Meaning and Measurement of Risk and Return in Investing

Introduction

The concepts of risk and return are fundamental to investing, influencing investor behavior and portfolio management strategies. This essay elaborates on the definitions and measurements of both risk and return, highlights the relationship between them, and discusses implications for investment decisions utilizing methods such as the Capital Asset Pricing Model (CAPM). The relationship between historical risk-return trade-offs also provides evidence for risk-based investment strategies.

Expected Return

Definition: The expected return is the anticipated return on an investment based on potential future cash flows, weighted according to their probabilities (Pearson, 2017).
Measurement: The expected return can be calculated using the formula:
\[ \text{Expected Return} (\%) = \sum P_{i} \cdot r_{i} \]
where \( P_{i} \) is the probability of each outcome and \( r_{i} \) is the return of that outcome. For example, if an investment has a 20% chance of returning 10%, a 30% chance of returning 12%, and a 50% chance of returning 14%, the expected return would be:
\[ 0.2 \cdot 10 + 0.3 \cdot 12 + 0.5 \cdot 14 = 12.6\% \]
This technique of weighing potential outcomes is crucial for assessing investment choices (Bodie, Kane, & Marcus, 2014).

Risk

Definition: Risk refers to the potential variability in returns, implying uncertainty about future cash flows from an investment. Essentially, more uncertain investments are considered riskier. (Brealey, Myers, & Marcus, 2023).
Measurement: One common measure of risk is the standard deviation, which quantifies the dispersion of returns around the expected return. It is calculated as follows:
\[ \sigma = \sqrt{\sum P_{i} (r_{i} - E[r])^2} \]
High standard deviation indicates high volatility, and thus higher risk, while low standard deviation suggests stable returns (Markowitz, 1952).
For instance, if you were to analyze the stock of eBay, with returns exhibiting a standard deviation of 4.65%, this indicates a more volatile investment compared to an index like the S&P 500, which may have a standard deviation of 2.57%, indicating lower risk (Pearson, 2017).

The Relationship Between Risk and Return

The historical risk-return relationship demonstrates that higher risk is generally associated with higher expected returns. For instance, equities over the long term have historically outperformed fixed income securities, but they have also exhibited much greater variability in returns (Fama & French, 2015). The Capital Asset Pricing Model (CAPM) formalizes this relationship by quantifying the expected return of an asset based on its systematic risk (represented by beta) compared to the market.
The CAPM equation is:
\[ E(R) = R_f + \beta \cdot (E(R_m) - R_f) \]
where:
- \( E(R) \) is the expected return on the asset,
- \( R_f \) is the risk-free rate,
- \( E(R_m) \) is the expected market return,
- \( \beta \) measures the asset’s volatility relative to the market (Sharpe, 1964).
For example, if a stock has a beta of 1.2, with a risk-free rate of 2% and an expected market return of 11%, the required return can be computed as follows:
\[ E(R) = 2\% + 1.2 \cdot (11\% - 2\%) = 10.8\% \]
This computation illustrates how investors adjust their return expectations according to perceived risks.

Diversification

Diversification is a risk management strategy that mixes various investments within a portfolio to minimize risks. When combined properly, assets that do not move synchronously (e.g., stocks and bonds) can lower overall portfolio risk without a proportional decrease in expected returns (Elton & Gruber, 1997).

Systematic and Unsystematic Risk

Investors encounter two principal types of risks:
1. Systematic Risk: This is the inherent risk that affects the entire market, such as economic downturns or geopolitical crises (Ross, 1976). It cannot be eliminated through diversification.
2. Unsystematic Risk: This risk is specific to a company or industry and can be mitigated through diversification. Examples include management changes or product recalls.
Thus, while developing a diversified portfolio, investors aim to maintain low levels of unsystematic risk while still exposing their investments to potentially greater systematic risk due to its possible higher returns (Lintner, 1965).

Conclusion

Effective risk and return measurement is crucial for making informed investment decisions. The expected return and risk assessments, which include standard deviation and the CAPM, empower investors to align their portfolio with their financial goals and risk tolerances. The relationship between historical data and expected outcomes reinforces the prudent approach of diversification as a means to balance risk and return, highlighting the significance of strategic asset allocation.

References

1. Bodie, Z., Kane, A., & Marcus, A. J. (2014). Investments (10th ed.). McGraw-Hill Education.
2. Brealey, R. A., Myers, S. C., & Marcus, A. J. (2023). Principles of Corporate Finance (13th ed.). McGraw-Hill Education.
3. Elton, E. J., & Gruber, M. J. (1997). Modern Portfolio Theory and Investment Analysis (6th ed.). Wiley.
4. Fama, E. F., & French, K. R. (2015). A five-factor asset pricing model. Journal of Financial Economics, 116(1), 1-22.
5. Lintner, J. (1965). The valuation of risky assets and the selection of risky investments in stock portfolios and capital budgets. Review of Economics and Statistics, 47(1), 13-37.
6. Markowitz, H. M. (1952). Portfolio selection. The Journal of Finance, 7(1), 77-91.
7. Pearson Education. (2017). Chapter 6: The Meaning and Measurement of Risk and Return. Pearson.
8. Ross, S. A. (1976). The arbitrage theory of capital asset pricing. Journal of Economic Theory, 13(3), 341-360.
9. Sharpe, W. F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk. The Journal of Finance, 19(3), 425-442.
10. Sharpe, W. F., Alexander, G. J., & Bailey, J. V. (1999). Investments (6th ed.). Prentice Hall.