Return and Risk The Capital Asset Pricing Model (CAPM)

Important Security Characteristics μ, which ____ σ2 and σ, which _____ Covariance and Correlation, which are _____

Statistics Review: Covariance Covariance: measures how two variables move in relation to one another Positive: The two variables move up together or down together Ex: Height and Weight Negative: When one moves up the other moves down Ex: Sleep and Coffee consumption

Calculation Cov (X,Y) = σXY = Σ pi * (Xi – μx) * (Yi – μY) σXY = p1*(X1 – μX)*(Y1 – μY)+p2*(X2 – μX)*(Y2 – μY)+….+pN*(XN – μX)*(YN – μY) Note that σXX = Σ pi * (Xi – μx) * (Xi – μX) Which implies?

Covariance Strength If the covariance of asset 1 and 2 is -1,000, and between asset 1 and 3 is 500. Which asset is more closely related to 1?

Correlation Coefficient Correlation Coefficient: Measures the strength of the covariance relationship The correlation coefficient is a standardization of covariance ρ12 = σ12 / (σ1 * σ2)

Correlation Coefficient Formula ρ12 = σ12 / (σ1 * σ2) ρ 12 – σ12 - σ1 - σ2 -

Possible Correlation Coefficients ρ12 has to be between -1 and 1 +1 implies that the assets are perfectly positively correlated 0 implies that the assets are not related -1 implies that the assets are perfectly negatively correlated

Comparing Strength When determining the strength of a correlation all we care about is the absolute value of the correlation coefficient If ρ13 is -0.8, and ρ23 is 0.5, which asset is more correlated with 3?

Correlation Coefficient Example σ13 is -1,000; σ23 is 500 σ1 is 10; σ2 is 1,000; σ3 is 250 Is 1 or 2 more strongly correlated with 3?

Portfolio Risk So far we’ve been examining the risk of individual assets, but what happens when we combine individual assets into a portfolio?

Portfolio Illustration

Diversification Reduces risk by combining assets that are unlikely to all move in the same direction, without sacrificing expected return Intuition: “Don’t put all your eggs in one basket” Stocks don’t move in exactly the same way. On a given day, while Boeing may yield positive 1% return, Microsoft may have gone down 1%. So, if you had invested a dollar each in Boeing and Microsoft, you would not have lost any money.

Two Types of Risk Diversifiable/Unsystematic/Unique risk Diversifiable risk affects individual or small groups of firms (industries) Ex: Lawsuits, Strikes Non-Diversifiable/Systematic/Market risk Affects all firms, economy wide risks Ex: Business Cycle, Inflations Shocks Which does σ measure?

How Diversification Works Reduces/eliminates unsystematic risk. Why can’t we diversify away systematic risk?

The Wonders of Diversifying

Portfolio Variances Formulas

Portfolio Covariance Matrix Stock 1 Stock 2 Stock 3 Stock N Stock 1 Var (1,1) Cov (2,1) Cov (3,1) Cov (N,1) Stock 2 Cov (1,2) Var (2,2) Cov (3,2) Cov (N,2) Cov (1,3) Var (3,3) Stock 3 Cov (2,3) Cov (N,3) Stock N Cov (1,N) Cov (2,N) Cov (3,N) Var (N,N)

Portfolio Variance Example Two stocks A and B have expected returns of 10% and 20%. In the past, A and B have had std dev of 15% and 25%, respectively, with a correlation coefficient of 0.2. You decide to invest 30% in A and the rest in B. Calculate the portfolio return and portfolio risk. Has diversification been of any use? Explain.

Calculations wA = 30% wB = A = 10% B = 20% A = 15% B = 25% AB = 0.2 Portfolio Return = Portfolio Variance = Portfolio Standard Deviation = Weighted Average Standard Deviation

Remarks on Diversification Diversification reduces the p from 22% to 18.9% What happens if AB = 1? What happens as AB approaches -1?

Which Stock do you Prefer? Stock A :  = 10%;  = 2% Stock B :  = 10%;  = 3% Stock C :  = 12%;  = 2%

Fundamental Premise of Portfolio Theory Rational investors prefer the highest expected return at the lowest possible risk How can investors lower risk without sacrificing return?

Possible two asset portfolios What are the possible portfolios we can create using only stocks and bonds The correlation coefficient is -0.99 Stock: std dev = 14.3% expected return = 11.0% Bond: std dev = 8.2% expected return = 7.0%

Possible Portfolios Some portfolios are better: why? which ones? 100% stocks 100% bonds

Efficient Portfolio Efficient Portfolios Can these same risk and return pairing be achieved with a single stock?

Including More Assets In the real world there are more than two assets The efficient frontier is the outer most envelope of possible portfolios, given the universe of available assets Form every possible portfolio of the various assets and the efficient frontier are the portfolios on the edge

Including More Assets Possible Portfolios return P

Including a Risk Free Asset How will the ability to lend and borrow at the risk free rate affect our possible risk-return combinations?

Risk Free and Risky Assets return P Start with our risky assets

Which Efficient Portfolio? Under certain assumptions everyone holds the MARKET PORTFOLIO, M This portfolio contains every asset in the market, according to their market weights We have simplified investing down to two assets

Why the market portfolio It offers the greatest return per unit of risk SHARPE RATIO measure the risk return trade off: (ri - rf) / I, The price of risk A higher Sharpe Ratio implies we receive more return per unit of risk

M & the Risk free asset return 100% stocks M, Market Portfolio rf 100% bonds  Investors choose to combine the risk-free asset and M, to create their portfolio

Why only two assets? Holding any risky asset other than M offers a lower risk adjusted return The investor is bearing more risk than necessary The investor is accepting too low a return Rf is how we adjust for our risk tolerance

Investing on the CML: In Two Easy Steps Find M Determine your level of risk aversion Risk aversion determines location on the CML The more risk averse the investor the safer the portfolio Closer to the risk free

Where on the CML to invest? Why do people move along the CML? More risk averse investors will hold more? More risk tolerant investors will hold more? How do we move along the CML?

How do we Lend & Borrow? When we lend $, we give $ & receive interest We do the same when we buy T-Bills So buying T-Bills is the same as lending $ When we borrow $, we get $ and pay interest We do the same when we short T-Bills So shorting T-Bills is the same as borrowing $

Borrowing & Lending $ Who lends $? Who borrows $?

Where are we borrowing & lending 100% bonds 100% stocks rf return  M, Market Portfolio

Risk compensation In a diversified portfolio, risk depends exclusively on the underlying securities exposure to systematic risk Market will not compensate investors for unsystematic risk Why? What risk will the market compensate investors for?

CAPM Market will only compensate an investor for systematic risk, which is measured by BETA (β) β – measures the sensitivity of the stock return to the market return (ex S&P 500) E(Ri) = Rf + βi (RM - Rf)

CAPM Assumptions Investors all have the same expectations Investors are risk averse and utility-maximizing Investors only care about mean and variance Expected return and risk Perfect markets No taxes No transactions costs Unlimited borrowing and lending at the risk-free rate

β Formulas βi = i,m / m2 βi = (ρim i )/ m

βi = (ρim i )/ m Interpretation I – Measures asset ‘i’ total risk ρim – Measures the proportion of i’s total risk that is systematic m – Measures the total market risk Which is??? ρimi– Measures the systematic risk of asset ‘i’ So βi:

Market βm βm = m,m / m2 What is the β of the risk free asset?

Notes on β β – tells us how sensitive a stock is to market movements “Average Stock” has a β of 1 Stocks with β > 1 amplify market movements Stocks with 0 < β < 1 reduce market movements Stocks with negative β?

Portfolio β The weighted average of the component stocks’ β Example: You invested 40% of your money in asset A, βA is 1.5 and the balance in asset B, βA is 0.5. What is the portfolio beta?

CAPM and β CAPM states that expected returns are proportional to an investment’s systematic risk A stock’s expected risk premium varies in proportion to it’s β E(Ri) = Rf + βi (RM - Rf) Security Market Line (SML) is the graphical representation of this relation

Security Market Line What is the slope of the SML?

Putting Stocks in the SML In equilibrium, all stocks should lie on the SML. This means that all stocks are correctly priced A stock under the SML is: Under or Over priced A stock above the SML is: Under or Over priced

Dis-Equilibrium SML E(ri) SML C B A β

CAPM Formula CAPM: E(Ri) = Rf + βi (RM - Rf) Market Risk Premium: (RM - Rf) Asset i’s Risk Premium: βi (RM - Rf)

Example Rf = 5% Historical average risk premium is 8.4% β = 1.5, return = β = 1, return = β = 0.5, return = β = 0, return = β = -1, return =

Example 1 The stock market moves up by 10%. Assume stock A has a beta of 1.5, stock B has a beta of 0.5 and stock C has a beta of -0.5. Predict A, B, and C’s response to the market?

Example 2 βA is 1.5, and A is 20% βB is 2, and B is 15%; Which stock has a higher expected return?

Why We Care Basic investment rule Explains the risk return trade off Big rewards are accompanied by large risks Explains the risk return trade off