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### Risk, Return, and CAPM

### Risk, Return, and CAPM economy: boom, expansion, recession and project the returns of GM stock for the three states

Professor Dr. Rainer Stachuletz

Corporate Finance

Berlin School of Economics and Law

Fußzeile

Expected Returns

Methods used to estimate expected return

Decisions must be based on expected returns

Historical approach

Probabilistic approach

Risk-based approach

Fußzeile

Using 1900-2003 annual returns, the average risk premium for U.S. stocks relative to Treasury bills is 7.6%. Treasury bills currently offer a 2% yield to maturity

Expected return on U.S. stocks: 7.6% + 2% = 9.6%

Historical Approach for Estimating Expected ReturnsAssume that distribution of expected returns will be similar to historical distribution of returns.

Can historical approach be used to estimate the expected return of an individual stock?

Fußzeile

Historical Approach for U.S. stocks relative to Treasury bills is 7.6%. Treasury bills currently offer a 2% yield to maturityEstimating Expected Returns

Assume General Motors long-run average return is 17.0%. Treasury bills average return over same period was 4.1%

GM historical risk premium: 17.0% - 4.1% = 12.9%

GM expected return = Current Tbill rate + GM historical risk premium = 2% + 12.9% = 14.9%

May reflect GM’s past more than its future

Limitations of historical approach for individual stocks

Many stocks have a long history to forecast expected return

Fußzeile

For example, assign probabilities for possible states of economy: boom, expansion, recession and project the returns of GM stock for the three states

Outcome

Probability

GM Return

Recession

20%

-30%

Expansion

70%

15%

Boom

10%

55%

Probabilistic Approach for Estimating Expected ReturnsIdentify all possible outcomes of returns and assign a probability to each possible outcome:

GM Expected Return = 0.20(-30%) + 0.70(15%) +0.10(55%) = 10%

Fußzeile

Risk-Based Approach for economy: boom, expansion, recession and project the returns of GM stock for the three statesEstimating Expected Returns

1. Measure the risk of the asset

2. Use the risk measure to estimate the expected return

1. Measure the risk of the asset

- Systematic risks simultaneously affect many different assets
- Investors can diversify away the unsystematic risk
- Market rewards only the systematic risk: only systematic risk should be related to the expected return

How can we capture the systematic risk component of a stock’s volatility?

Fußzeile

Risk-Based Approach for economy: boom, expansion, recession and project the returns of GM stock for the three statesEstimating Expected Returns

- Collect data on a stock’s returns and returns on a market index
- Plot the points on a scatter plot graph
- Y–axis measures stock’s return
- X-axis measures market’s return

- Plot a line (using linear regression) throughthe points

Slope of line equals beta, the sensitivity of a stock’s returns relative to changes in overall market returns

Beta is a measure of systematic risk for a particular security.

Fußzeile

Sharper Image weekly returns economy: boom, expansion, recession and project the returns of GM stock for the three states

S&P 500 weekly returns

Scatter Plot for Returns on Sharper Image and S&P 500

Fußzeile

ConAgra weekly returns economy: boom, expansion, recession and project the returns of GM stock for the three states

S&P 500 weekly returns

Scatter Plot for Returns ConAgra and S&P 500

Fußzeile

Risk-Based Approach for economy: boom, expansion, recession and project the returns of GM stock for the three statesEstimating Expected Returns

Capital

Average

Market Line

Return

Individual Stock A:

rM

CAPM

R

f

Slope CML:

s2M

Risk

Fußzeile

E(R economy: boom, expansion, recession and project the returns of GM stock for the three statesP)

SML

A - Undervalued

•

•

A

Slope = E(Rm) - RF = MarketRisk Premium

RM

•

•

B

•

•

RF

B - Overvalued

•

=1.0

i

The Security Market LineFußzeile

Risk-Based Approach for economy: boom, expansion, recession and project the returns of GM stock for the three statesEstimating Expected Returns

2. Use the risk measure to estimate the expected return:

- Plot beta against expected return for two assets:
- A risk-free asset that pays 4% with certainty, with zero systematic risk and
- An “average stock”, with beta equal to 1, with an expected return of 10%.

- Draw a straight line connecting the two points.
- Investors holding a stock with beta of 0.5 or 1.5, for example, can find the expected return on the line.

Beta measures systematic risk and links the risk and expected return of an asset.

Fußzeile

Expected returns economy: boom, expansion, recession and project the returns of GM stock for the three states

•

18%

•

14%

•

ß = 1.5

•

•

10%

“average” stock

•

4%

Risk-free asset

•

•

•

•

•

•

•

•

•

•

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Beta

Risk and Expected ReturnsSecurity Market LineWhat is the expected return for stock with beta = 1.5 ?

Fußzeile

Estimating the Risk Free Rate economy: boom, expansion, recession and project the returns of GM stock for the three states

Extract of UK interest rate data from the Financial Times (17 February, 2005)

The Steps Towards the Estimation of economy: boom, expansion, recession and project the returns of GM stock for the three statesBeta Using Ordinary Least Squares Regression

The Security Market Line economy: boom, expansion, recession and project the returns of GM stock for the three states

Arbitrage Drivers and the economy: boom, expansion, recession and project the returns of GM stock for the three statesLinearity of the Security Market Line

Portfolio Expected Returns economy: boom, expansion, recession and project the returns of GM stock for the three states

How does the expected return of a portfolio relate to the expected returns of the securities in the portfolio?

Expected return of a portfolio with N securities

The portfolio expected return equals the weighted average of the portfolio assets’ expected returns

E(Rp) = w1E(R1)+ w2E(R2)+…+wnE(Rn)

- w1, w2 , … , wn : portfolio weights
- E(R1), E(R2), …, E(RN): expected returns of securities

Fußzeile

Portfolio Expected Returns economy: boom, expansion, recession and project the returns of GM stock for the three states

E (Rp) = w1 E (R1)+ w2 E (R2)+…+wn E (Rn)

E (Rp) = (0.125) (10%) + (0.25) (12%) + (0.125) (8%) + (0.5) (14%) = 12.25%

Fußzeile

Portfolio Risk economy: boom, expansion, recession and project the returns of GM stock for the three states

Portfolio risk is the weighted average of systematic risk (beta) of the portfolio constituent securities.

ß P = (0.125) (1.00) + (0.25) (1.33) + (0.125) (0.67) + (0.50) (1.67) = 1.38

But portfolio volatility is not the same as the weighted average of all portfolio security volatilities

Fußzeile

Portfolio composed of the following two assets: economy: boom, expansion, recession and project the returns of GM stock for the three states

- An asset that pays a risk-free return Rf, , and
- A market portfolio that contains some of every risky asset in the market.

Security market line: The line connecting the risk-free asset and the market portfolio

Fußzeile

The Security Market Line economy: boom, expansion, recession and project the returns of GM stock for the three states

- In equilibrium, all assets lie on this line.
- If individual stock or portfolio lies above the line:
- Expected return is too high.
- Investors bid up price until expected return falls.

- If individual stock or portfolio lies below SML:
- Expected return is too low.
- Investors sell stock driving down price until expected return rises.

- If individual stock or portfolio lies above the line:

Plots relationship between expected return and betas

Fußzeile

Efficient Markets economy: boom, expansion, recession and project the returns of GM stock for the three states

Efficient market hypothesis (EMH): in an efficient market, prices rapidly incorporate all relevant information

Financial markets much larger, more competitive, more transparent, more homogeneous than product markets

Much harder to create value through financial activities

Changes in asset price respond only to new information. This implies that asset prices move almost randomly.

Fußzeile

Efficient Markets economy: boom, expansion, recession and project the returns of GM stock for the three states

If asset prices unpredictable, then what is the use of CAPM?

CAPM gives analyst a model to measure the systematic risk of any asset.

On average, assets with high systematic risk should earn higher returns than assets with low systematic risk.

CAPM offers a way to compare risk and return on investments alternatives.

Fußzeile

- Decisions should be made based on expected returns.
- Expected returns can be estimated using historical, probabilistic, or risk approaches.
- Portfolio expected return/beta equals weighted average of the expected returns/beta of theassets in the portfolio.
- CAPM predicts that the expected return on a stock depends on the stock’s beta, the risk-free rate and the market premium.

Fußzeile

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