Learning Objectives 1. Calculate the expected rate of return and - - PDF document

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Chapter 8

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Learning Objectives

1. Calculate the expected rate of return and volatility for a portfolio of investments and describe how diversification affects the returns to a portfolio of investments. 2. Understand the concept of systematic risk for an individual investment and calculate portfolio systematic risk (beta). 3. Estimate an investor’s required rate of return using the Capital Asset Pricing Model.

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Principles Applied in This Chapter

 Principle 2: There is a Risk‐Return Tradeoff.  Principle 4: Market Prices Reflect Information.

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Portfolio Returns and Portfolio Risk

 With appropriate diversification, you can lower the risk of your portfolio without lowering the portfolio’s expected rate of return.  Those risks that can be eliminated by diversification are not rewarded in the financial marketplace.

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Calculating the Expected Return of a Portfolio

E(rportfolio) = the expected rate of return on a portfolio of n assets. Wi = the portfolio weight for asset i. E(ri ) = the expected rate of return earned by asset i. W1 × E(r1) = the contribution of asset 1 to the portfolio expected return.

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CHECKPOINT 8.1: CHECK YOURSELF

Calculating a Portfolio’s Expected Rate of Return

Evaluate the expected return for Penny’s portfolio where she places a quarter of her money in Treasury bills, half in Starbucks stock, and the remainder in Emerson Electric stock.

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Step 1: Picture the Problem

T-bills Emerson Electric Starbucks 0% 2% 4% 6% 8% 10% 12% 14%

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Step 2: Decide on a Solution Strategy

 The portfolio expected rate of return is simply a weighted average of the expected rates of return of the investments in the portfolio.  Step 3: Solve E(rportfolio) = .25 × .04 + .25 × .08 + .50 × .12 = .09 or 9%

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Step 2: Decide on a Solution Strategy

 We have to fill in the third column (Product) to calculate the weighted average.

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Portfolio E( Return) X W eight = Product Treasury bills 4 .0 % .2 5 EMR stock 8 .0 % .2 5 SBUX stock 1 2 .0 % .5 0

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Step 3: Solve

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Alternatively, we can fill out the following table from step 2 to get the same result.

Step 4: Analyze

The expected return is 9% for a portfolio composed of 25% each in treasury bills and Emerson Electric stock and 50% in Starbucks.

If we change the percentage invested in each asset, it will result in a change in the expected return for the portfolio.

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Evaluating Portfolio Risk: Portfolio Diversification

 The effect of reducing risks by including a large number

• f investments in a portfolio is called diversification.

 The diversification gains achieved will depend on the degree of correlation among the investments, measured by correlation coefficient.

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Portfolio Diversification

The correlation coefficient can range from ‐1.0 (perfect negative correlation), meaning that two variables move in perfectly opposite directions to +1.0 (perfect positive correlation).

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Diversification Lessons

1. A portfolio can be less risky than the average risk of its individual investments in the portfolio. 2. The key to reducing risk through diversification ‐ combine investments whose returns are not perfectly positively correlated.

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Calculating the Standard Deviation

• f a Portfolio’s Returns

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Figure 8‐1 Diversification and the Correlation Coefficient— Apple and Coca‐Cola

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Figure 8‐1 Diversification and the Correlation Coefficient— Apple and Coca‐Cola

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CHECKPOINT 8.2: CHECK YOURSELF

Evaluating a Portfolio’s Risk and Return

• Evaluate the expected return and standard deviation of the

portfolio of the S&P500 and the international fund where

• The correlation is assumed to be .20 and
• Sarah still places half of her money in each of the funds.

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Step 1: Picture the Problem

Sarah can visualize the expected return, standard deviation and weights as shown below.

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Step 2: Decide on a Solution Strategy

 The portfolio expected return is a simple weighted average of the expected rates of return of the two investments  The standard deviation of the portfolio can be calculated using the formula. We are given the correlation to be equal to 0.20.

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Step 3: Solve

E(rportfolio) = WS&P500 E(rS&P500) + WInt E(rInt) = .5 (12) + .5(14) = 13%

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Step 3: Solve

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SDportfolio = √ { (.52x.22)+(.52x.32)+(2x.5x.5x.20x.2x.3)} = √ {.0385} = .1962 or 19.62%

Systematic Risk and Market Portfolio

CAPM assumes that investors chose to hold the

• ptimally diversified portfolio that includes all of the

economy’s assets (referred to as the market portfolio). According to the CAPM, the relevant risk of an investment relates to how the investment contributes to the risk of this market portfolio.

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Systematic Risk and Market Portfolio

To understand how an investment contributes to the risk

• f the portfolio, we categorize the risks of the individual

investments into two categories:

 Systematic risk, and  Unsystematic risk

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Figure 8.2 Portfolio Risk and the Number of Investments in the Portfolio

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Systematic Risk and Beta

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Table 8.1 Beta Coefficients for Selected Companies

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Calculating Portfolio Beta

The portfolio beta measures the systematic risk of the portfolio.

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Calculating Portfolio Beta

Example Consider a portfolio that is comprised of four investments with betas equal to 1.50, 0.75, 1.80 and 0.60

• respectively. If you invest equal amount in each

investment, what will be the beta for the portfolio?

= .25(1.50) + .25(0.75) + .25(1.80) + .25 (0.60) = 1.16

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The Security Market Line and the CAPM

 CAPM describes how the betas relate to the expected rates of return. Investors will require a higher rate of return on investments with higher betas.

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Risk and Return for Portfolios Containing the Market and the Risk‐Free Security

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Risk and Return for Portfolios Containing the Market and the Risk‐Free Security

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The Security Market Line and the CAPM

The straight line relationship between the betas and expected returns in Figure 8‐4 is called the security market line (SML)



Its slope is often referred to as the reward to risk ratio.

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The Security Market Line and the CAPM

SML is a graphical representation of the CAPM. SML can be expressed as the following equation This is known as the CAPM pricing equation:

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CHECKPOINT 8.3: CHECK YOURSELF

Estimating the Expected Rate of Return Using the CAPM

Estimate the expected rates of return for the three utility companies, found in Table 8‐1, using the 4.5% risk‐free rate and market risk premium of 6%.

Table 8.1 Beta Coefficients for Selected Companies

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Step 2: Decide on a Solution Strategy

We can determine the required rate of return by using CAPM equation The betas for the three utilities companies (Yahoo Finance estimates) are: AEP = 0.74 DUK = 0.40 CNP = 0.82

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Step 3: Solve

AEP: E(rAEP) = 4.5% + 0.74(6) = 8.94% DUK: E(rDUK) = 4.5% + 0.40(6) = 6.9% CNP: E(rCNP) = 4.5% + 0.82(6) = 9.42%

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Step 4: Analyze

The expected rates of return on the stocks vary depending on their beta. Higher the beta, higher is the expected return.

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CHECKPOINT 8.3: CHECK YOURSELF #2

Estimating the Expected Rate of Return Using the CAPM

Estimate the expected rates of return for the three utility companies, found in Table 8‐1, using the 4.5% risk‐free rate and market return of 11%

Table 8.1 Beta Coefficients for Selected Companies

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Step 2: Decide on a Solution Strategy

We can determine the required rate of return by using CAPM equation The betas for the three utilities companies (Yahoo Finance estimates) are: AEP = 0.74 DUK = 0.40 CNP = 0.82

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Step 3: Solve

AEP: E(rAEP) = 4.5% + 0.74(11‐ 4.5) = 9.31% DUK: E(rDUK) = 4.5% + 0.40(11‐ 4.5) = 7.1% CNP: E(rCNP) = 4.5% + 0.82(6) = 9.83%

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Step 4: Analyze

The expected rates of return on the stocks vary depending on their beta. Higher the beta, higher is the expected return.

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