Risk and Return & The Capital Asset Pricing Model (CAPM) MBAC 6060 Chapter 11 Risk and Return & The Capital Asset Pricing Model (CAPM)

Chapter Overview Last chapter we looked at risk and return for CATEGORIES of stocks (large and small) This chapter we look risk and return in more detail Risk and return calculations for individual stocks Risk and return calculations for a portfolio of stocks

Chapter Outline Calculate returns and risk (variance and standard deviation) for individual stocks Calculate returns and risk for portfolios Consider Expected Returns and Realized Returns The difference between them is called the Surprise Announcements by companies, the government… Two components of Risk “Systematic” and “Unsystematic” How Diversification lowers Portfolio Risk Measuring Systematic Risk with Beta The Security Market Line (SML)

Risk and Return Calculations Two ways to deal with expected values Depending on how the data is presented We’ll consider height First Way: We have a large sample of data: Get everybody’s height (measure or ask?) Calculate the mean and s: The MEAN height is the EXPECTED height

Risk and Return Calculations (Part 2) Second Way: Set categories or ranges of height Calculate mean for each category Calculate (or assign) probabilities for each category Calculate expectation: E(H) = 66.3 inches Category (i) Prob (pi) Inches (Hi) piHi Shortest 30% 61 18.3 Middle 40% 66 26.4 Tallest 72 21.6 100% 66.3

Why might the second way better? Because you can change category probabilities based on new information or expectations Intramural basketball meeting, So reset the probabilities to 10%, 10% and 80% E(H) = 70.3 inches Category (i) Prob (pi) Inches (Hi) piHi Shortest 10% 61 6.1 Middle 66 6.6 Tallest 80% 72 57.6 100% 70.3

Benefit to this method: You can adjust your calculation to produce an expected height that is more inline with your understanding of the situation If you know something about the next group in the room, you can adjust your expectation calculation We’ll see how this relates to stocks soon…

Risk-Return Calculation Notation: Single Stock Notation Return Calculations for a Single Stock E(Ri) = Expected Return of some stock i Risk Calculations for a Single Stock si2 (and si) = Variance (and SD) of some Stock i Portfolio Notation Return Calculations for a Portfolio E(RP) = Expected Return of a Portfolio of stocks Risk Calculations for a Portfolio sP2 (and sP) = Variance (and SD) of a Portfolio

Some More Notation: ps = Probability of some “state obtaining” S = The Number of possible states (big S) s = One of the states (little s) R = Return RA = Return of stock A (Distinguish between stocks if we are considering more than one) Rs = Return given some state of the economy Rs,A = Return given some state of the economy for stock A E(R) = Expected Return E(RA) = Expected Return of stock A E(RP) = Expected Return of a portfolio

Even More Notation : sA,B = rA,B sA sB Var(R) = s2 = Variance of the returns of a stock Var(RA) = sA2 = Variance of the returns of stock A Var(RP) = sP2 = Variance of the returns of a portfolio SD(R) = s = Standard Deviation of the returns SD(RA) = sA = Standard Deviation of the returns of stock A SD(RP) = sP = Standard Deviation of the returns of a portfolio Covar(A,B) = sA,B = Covariance between returns of stocks 1 and 2 Corr(A,B) = rA,B = Correlation between the returns of stocks 1 and 2 rA,B = sA,B/(s1 s2) sA,B = rA,B sA sB

11.1, 11.2 Return Calculations for Single Assets Expected return for a stock We will now call categories “states” This is a statistical term used to describe outcomes The state of the weather: Sunny, Cloudy, Rainy The state of the economy: Boom, Normal, Recession Calculate an expected (or mean) return for each state (Rs) This will usually be given Calculate (or assign) a probability for each state (ps) Again, will usually be given Calculate the expected return E(R) = Sum of psRs E(R) is similar to the Arithmetic Mean which we thought of as the expected return

Return Calculations for a Single Asset Calculate the Expected return for a stock State Probabilities: pBoom = 30%, pNormal = 50%, pRecession = 20% Returns Given States: RBoom = 15%, RNormal = 10%, RRecession = 2% E(R) = 9.90% Economy (s) Prob (ps) Return (Rs) psRs Boom 30% 15% 0.045 Normal 50% 10% 0.050 Recession 20% 2% 0.004 100% 0.099

Risk Calculations (s2 and s) for a Single Asset Var(R) = s2 = 0.002029 SD(R) = s = 0.0450 = 4.50% Economy (s) ps Rs E(R) ps[Rs- E(R)]2 Boom 30% 15.00% 9.90% 0.000780 Normal 50% 10.00% 0.000001 Recession 20% 2.00% 0.001248 100% 0.002029

Two New Stocks: Supertech and Slowpoke: E(R) and s for Supertech: E(R) = 17.50% s = (0.066875)½ = 25.86% Economy (s) ps Rs psRs E(R) ps[Rs- E(R)]2 Depression 25% -20% -5.0% 17.5% 0.035156 Recession 10% 2.5% 0.001406 Normal 30% 7.5% 0.003906 Boom 50% 12.5% 0.026406 100% 0.066875

E(R) and s for Slowpoke: Economy (s) ps Rs psRs E(R) ps[Rs- E(R)]2 Depression 25% 5% 1.3% 5.5% 0.000006 Recession 20% 5.0% 0.005256 Normal -12% -3.0% 0.007656 Boom 9% 2.3% 0.000306 100% 0.013225

Recap:   E(R) σ Supertech 17.50% 25.86% Slowpoke 5.50% 11.50%

11.3 Now Lets Look at a Portfolio: A Portfolio is defined by its components… The risk of each component The return of each component The relationship between the components And the Weight of each component Weights are the portion of the portfolio invested in each asset Weights must sum to one So lets look at a 60-40 portfolio for Supertech (Stock A) and Slowpoke (Stock B) XA = 0.60 and XB = 0.40

Return for a Portfolio of Stocks Expected Return of a Portfolio E(RP): E(RP) = XAE(RA) + XBE(RB) E(RP) = (0.60)(17.50%) + (0.40)(5.50%) = 12.70%

Risk of a Portfolio of Stocks Data for Portfolio and Portfolio Stocks: XA = 60% and σA = 28.86% XB = 40% and σB = 11.50% sP2 = XA2 sA2 + XB2 sB2 + 2XAsAXBsBA,B rA,B is the CORRELATION between the returns The CORRELATION defines the relationship between A and B

Calculating Correlation: First Calculate the Covariance (sA,B): Covar(A,B) = pDep[RA,Dep – E(RA)][RB,Dep – E(RB)] + pRec[RA,Rec – E(RA)] [RB,Rec – E(RB)] + pNorm[RA,Norm – E(RA)] [RB,Norm – E(RB)] + pBoom[RA,Boom – E(RA)] [RB,Boom – E(RB)]

Covariance (sA,B): Covar(A,B) = -0.004875 Covar(A,B) = pDep[RA,Dep – E(RA)][RB,Dep – E(RB)] + pRec[RA,Rec – E(RA)] [RB,Rec – E(RB)] + pNorm[RA,Norm – E(RA)] [RB,Norm – E(RB)] + pBoom[RA,Boom – E(RA)] [RB,Boom – E(RB)] Covar(A,B) = -0.004875 RA - E(RA) RB - E(RB) ps RA RB RA - 17.50% RB - 5.50% Product Dep 25% -20% 5% -37.5% -0.5% 0.000469 Rec 10% 20% -7.5% 14.5% -0.002719 Norm 30% -12% 12.5% -17.5% -0.005469 Boom 50% 9% 32.5% 3.5% 0.002844 -0.004875

rA,B = sA,B/(sA sB) = -0.004875/[(0.2586)(0.1150)] = -0.1639 The Correlation (r) is the Standardized Covariance: Standardize by dividing by the product of the standard deviations rA,B = sA,B/(sA sB) = -0.004875/[(0.2586)(0.1150)] = -0.1639 How do we interpret Correlation? What does r = 1.00 mean? What does r = 0.00 mean? What does r = -1.00 mean? What does r = -0.1639 mean?

Correlation:

sP2 = XA2 sA2 + XB2 sB2 + 2XAsAXBsBA,B sP = [sP2]½ Portfolio Risk is calculated using this formula: sP2 = XA2 sA2 + XB2 sB2 + 2XAsAXBsBA,B sP = [sP2]½ Standard Deviation is the square-root of the Variance Think about this formula: What is the largest possible value for rA,B? What is the smallest possible value for rA,B? rA,B must be between 1 and -1

sP2 = XA2 sA2 + XB2 sB2 + 2XAsAXBsBA,B Risk Calculation for the 60-40 Portfolio Plug all the values into Portfolio Variance formula: sP2 = XA2 sA2 + XB2 sB2 + 2XAsAXBsBA,B sP2 = (.6)2(.2586)2 + (.4)2(.1150)2 + 2(.6)(.2586)(.4)(.1150)(-.1639) sP2 = 0.023851 sP = (sP2)½ = (0.023851)½ = 0.1544 = 15.44%

Recap: Consider E(RP) = 12.70%: Consider σP = 15.44% E(R) σ Supertech 17.50% – 5.50% = 12% 60% x 12% = 7.2% 5.50% + 7.2% = 12.70% = E(RP) Consider σP = 15.44% 25.86 – 11.50% = 14.36% 60% x 14.36% = 8.62% 11.50% + 8.62% = 20.12% > 15.44% = σP   E(R) σ Supertech 17.50% 25.86% Slowpoke 5.50% 11.50% 60-40 Portfolio 12.70% 15.44%

sP2 = XA2 sA2 + XB2 sB2 + 2XAsAXBsBA,B Portfolio Risk Equation Again: sP2 = XA2 sA2 + XB2 sB2 + 2XAsAXBsBA,B We can see how the portfolio’s risk changes when the relationship between the stocks (r) changes This leads to a FUNDAMENTAL RESULT IN FINANCE!

sP2 = XA2 sA2 + XB2 sB2 + 2XAsAXBsB(1) Diversification and Portfolio Risk: What if rA,B = 1? What does it say about the stocks if their correlation is 1? The two stocks’ returns move together No Diversification! So lets look at the risk of a portfolio made up of two undiversified stocks How do we know they are undiversified? Because rA,B = 1 So substitute 1 for rA,B in the portfolio risk calculation: sP2 = XA2 sA2 + XB2 sB2 + 2XAsAXBsB(1) And now do some algebra 

Diversification and Portfolio Risk: Portfolio standard deviation formula: sP = [sP2]½ = [XA2 sA2 + XB2 sB2 + 2XAsAXBsBrA,B]½ Set rA,B = 1 (no diversification) sP = [XA2 sA2 + XB2 sB2 + 2XAsAXBsB1]½ Rewrite in a way that allows us to simplify: sP = [(XAsA)2 + (XBsB)2 + 2(XAsA)(XBsB)]½ Let a = XAsA and b = XBsB: [a2 + b2 + 2ab]½ [(a + b)2]½ (a + b) sP = XAsA + XBsB (but only if rA,B = 1)

Diversification and Portfolio Risk: If rA,B = 1 then sP = XAsA + XBsB So Portfolio Risk equals: Weight of A x Risk of A + Weight of B x Risk of B This is the weighted average risk What if the stocks’ correlation is 1.00? sP = [XA2 sA2 + XB2 sB2 + 2XAsAXBsBrA,B ]½ = [(.6)2(.2586)2 + (.4)2(.115)2 + 2(.6)(.2586)(.4)(.115)(1)] ½ sP = 20.12% Same as the weighted average risk: sP = XA sA + XB sB = (.6)(.2586) + (.4)(.115)

Diversification and Portfolio Risk: So if rA,B = 1 then sP = XAsA + XBsB The weighted average risk When is this true? ONLY when rA,B = 1 What is the most rA,B can be? The MAX for rA,B is 1 So what if rA,B < 1? Then sP < XAsA + XBsB Portfolio risk is less than the weighted average risk! If rA,B < 1 then the portfolio is diversified And risk is lower!

Diversification lowers portfolio risk!!! Diversification and Portfolio Risk: This is a fundamental result in finance Diversification lowers portfolio risk!!! sP = [XA2 sA2 + XB2 sB2 + 2XAsAXBsBrA,B]½ Not diversified? Then rA,B = 1 and sP = XA sA + XB sB Diversified? Then rA,B < 1 and sP < XA sA + XB sB What if the stocks correlation is 0.20 (or 20%)? sP = [XA2 sA2 + XB2 sB2 + 2XAsAXBsBrA,B ]½ = [(.6)2(.2586)2 + (.4)2(.115)2 + (.6)(.2586)(.4)(.115)(0.2)]½ sP = 17.04% If correlation is 20%, portfolio risk is 17.04% (less than 20.12%)

Recap: rA,B = 1.00 then sP = 20.02% rA,B = 0.20 then sP = 17.04% Portfolio risks for different correlations: rA,B = 1.00 then sP = 20.02% rA,B = 0.20 then sP = 17.04% rA,B = -0.1639 then sP = 15.44% rA,B = -1.00 then sP = 10.92% Note that in each case we are looking at a 60-40 portfolio So the weights are not changing, only the degree of diversification This is hypothetical. The degree of diversification is a function of the stocks and does not change

Portfolio Risk and Return Recap: A Portfolio’s Expected Return equals: E(RP) = XAE(RA) + XBE(RB) This is a function of: The weights of the stocks: XA and XB The expected returns of the stocks: E(RA) and E(RB) A Portfolio’s Risk equals: sP = [XA2 sA2 + XB2 sB2 + 2XAsAXBsBrA,B]½ The risks of the individual stocks: sA and sB The diversification of the two stocks: rA,B

Portfolio Risk and Return Recap: E(RP) = XAE(RA) + XBE(RB) sP = [XA2 sA2 + XB2 sB2 + 2XAsAXBsBrA,B]½ The returns, risks and correlation of the assets are given We only get to choose the weights E(RA) = 17.50% E(RB) = 5.50% sA = 25.86% sAB = 11.50% rA,B = -0.1639 So look at the possible portfolios for A and B Portfolios defined by weights Go to Two Asset Portfolio Spreadsheet 

The Opportunity Set can’t get worse with the addition of more assets… 11.5 The Efficient Set for Many Securities What if there are more than two assets? return Individual Assets P The Opportunity Set can’t get worse with the addition of more assets…

The Efficient Set for Many Securities return Efficient Frontier minimum variance portfolio Individual Assets P The section of the opportunity set above the minimum variance portfolio is the Efficient Frontier.

11.6 Components of Risk and Diversification Think about the return of a stock as having two components Two for now (It will be three components in a minute) The Expected Return: E(R) The Unexpected return: U

Expected Returns and Risk Components The Realized Return (we’ll call this R) equals: R = E(R) + U For any period, the unexpected (U) return can be either positive or negative Over time, by definition, the average of the unexpected component is zero Positive and negative variations cancel each other out Think about Expected Height: Realized Height = H = E(H) + U Over time, the average of the unexpected portion of height (the deviations from the mean) will equal zero

Announcements and News Announcements and news contain: An Expected component A Surprise component The Surprise affects a stock’s price And therefore the stock’s return Prices move only when an announcement is different from the expectation If the announced earnings equals the expected earnings, prices don’t move Where do expectations come from? Analysts, Economists, … Earnings Example: Go to: finance.yahoo.com Enter a ticker symbol (AAPL) Click Analysts Estimates

Think About Surprises: The Unexpected part (U) Realized Return: R = E(R) + U Two components to the unexpected part (U): Things that affect the whole economy (the “System” or “Market”) And therefore affect ALL stocks GDP, inflation, energy prices affect all stocks Things that affect only one stock (Unsystematic or Unique) Corporate earnings, losses, a law suit, personnel changes… So we can breakup U (Unexpected) into 2 parts Or breakup R into three parts: R = E(R) + Market Surprises + Unique Surprises R = E(R) + m + e

U has two components: The Expected Return: E(R) Recap: The Realized Return of a stock (R) equals: The Expected Return: E(R) The Unexpected Return: U U has two components: Systematic (or “Market” part): m Unsystematic (or “Unique” or “Idiosyncratic” part): e

Think about the Unique Part (e): Things that affect one company will not affect another (unique) By definition, the e from one company will be uncorrelated with the e from another company When one e is positive, the other e may be positive, negative or zero So if we look at a whole bunch of uncorrelated e’s… The sum of the e’s will be ZERO!

Recap of an extremely important point: A stock’s return can be thought of as having 3 parts: The Expected part: E(R) The Unexpected part that affects all stocks: m The Unexpected part that affects only that stock: e So the Realized Return: R = E(R) + m + e Now think about RISK Risk is Return Volatility Why will the realized return, R, be different from the expected return, E(R)? m and e So volatility of m and e are a stock’s risk So now think of risk as the amount the realized return varies from the expected return

Back to the Height Example What will be the realized height of the next person to walk in the room? Realized Height will be expected height plus person’s “variation” Realized H = E(H) + e Ignore the m part for now. It doesn’t really fit this analogy But if we look at the average realized height of the next 30 people: Each person’s e will cancel out So the average realized height (H) will be close to the E(H) Holding 30 stocks (diversification) does the same thing Because the 30 e’s cancel each other out… The average realized return of the 30 stocks (the portfolio’s return) will be closer to the expected return (Plus the m part) RP = E(RP) + mP

Back to Stocks: Called Systematic Risk or Market Risk A stock’s risk is the volatility of the 2 unexpected parts: Volatility of the market: the m part Volatility of the individual stock: e part The Volatility of the m part: Called Systematic Risk or Market Risk We can’t eliminate the m part But we can hold stocks with potentially larger or smaller market volatility A potentially larger or smaller m We’ll get to this in a minute…

Second part is the Volatility of the e part: Called Unsystematic Risk or Unique Risk or Idiosyncratic Risk or Non-Market Risk We can Eliminate the e part through diversification: Total Risk = Systematic Risk + Unsystematic Risk Diversification eliminates the unsystematic portion… Total Risk = Systematic Risk

How many stock’s do you are needed to eliminate unique risk? Now let’s look at the Systematic Risk or Market Risk or the m part 

Systematic Risk and Beta (b) There is a reward for bearing risk (higher Return) But there is NOT a reward for bearing unnecessary risk The expected return on a risky asset depends only on the asset’s systematic (or market) risk You do not get compensated for unsystematic risk Since it can be diversified away Why are you not compensated? If you don’t diversify and others do They will pay more for a stock Since they don’t need as much compensation as you!

Volatility and Return for 500 Individual Stocks 1926–2004 Portfolios (Red Squares) have significantly lower risk than individual stocks Large stocks (Yellow Triangles) have less risk than Small stocks (Orange Dots) The Small Stock Portfolio has more return and more market risk than the Large Stock Portfolio

Systematic Risk and Beta (b) How do we measure systematic risk? We use the Beta (b) coefficient to measure systematic risk What does Beta tell us? b = 1 means the same systematic risk as the overall market b < 1 means the asset has less systematic risk than the overall market b > 1 means the asset has more systematic risk than the overall market

Measuring Systematic Risk (Beta) Some Beta’s from Google Finance (2011-03-25): Beta Measures the “Market Risk” Name Symbol Beta Altria MO 0.45 Procter and Gamble PG 0.52 3M MMM 0.82 Target TGT 0.95 Apple AAPL 1.37 Level 3 LVLT 1.41 US Steel X 2.43

Systematic Risk and Return The Greater the Beta (the Systematic aka Market Risk measure) The greater the variation in realized return Given some unexpected market news In other words, the greater the risk The Greater the Risk, the Greater the Return: If a small unexpected downturn in the economy causes a company’s earnings to drop dramatically… Then you won’t pay a lot for the stock If the stock’s Price Drops, the Expected Return Increases The Greater the Expected Return for some stock X The Greater the Risk Premium for that stock Recall the Expected Risk Premium for stock X: E(RX) – Rf

[E(RX) – Rf]/bX Systematic Risk and Return Now look at the risk premium (the return) given the risk: Divide the Risk Premium for stock X: E(RX) – Rf By the Market Risk Measure for stock X: bX [E(RX) – Rf]/bX This is the Risk Premium per Unit of Market Risk In an Efficient Market Equilibrium, this ratio will be Equal for all stocks Why? Let’s see what happens if the ratio isn’t equal 

Systematic Risk and Return Let’s see what happens if the ratio is not equal: E(RX) = 17% bX = 0.8 E(RY) = 22% bY = 1.2 Rf = 5% We expect E(RY) > E(RX) since Y has more market risk But is it enough greater?

Sell Y until the price drops enough to make Y’s return high enough Systematic Risk and Return [E(RX) – Rf]/bX = (0.17 – 0.05)/0.8 = 0.15 [E(RY) – Rf]/bY = (0.22 – 0.05)/1.2 = 0.125 People holding Y are not receiving enough extra return to compensate them for Y’s risk What will they do? Sell Y until the price drops enough to make Y’s return high enough Or Buy X (since it is such a good deal) until its return is low enough

E(RX) – Rf = bX[E(RM) – Rf] Systematic Risk and Return Back to comparing X’s and Y’s risk-premium and risk: Compare to the Market’s Risk-Premium and risk: If [E(RY) – Rf]/bY must be equal for all stocks Then it must be equal to the aggregate market portfolio: [E(RX) – Rf]/bX = [E(RY) – Rf]/bY = [E(RM) – Rf]/bM Look at just [E(RX) – Rf]/bX = [E(RM) – Rf]/bM: What is bM? This is the “market risk” of the market So bM = 1 [E(RX) – Rf]/bX = [E(RM) – Rf]/1 Solve for stock X’s expected risk premium: [E(RX) - Rf]: E(RX) – Rf = bX[E(RM) – Rf]

E(RX) = Rf + bX[E(RM) – Rf] Systematic Risk and Return Stock X’s expected risk premium: E(RX) – Rf Equals: The stocks market risk measure: bX Multiplied by the expected market risk premium:[E(RM) – Rf] E(RX) – Rf = bX[E(RM) – Rf] Now solve for stock X’s expected return, E(RX): E(RX) = Rf + bX[E(RM) – Rf] This is the Capital Asset Pricing Model (CAPM)

Systematic Risk and Return - The CAPM According to the CAPM, the expected return of a stock equals: E(RX) = Rf + bX[E(RM) – Rf] The risk-free rate (Rf) Plus the stock’s market risk measure (b) Multiplied by the expected market risk premium [E(RM) – Rf] What is the expected return for AAPL next year? The one year risk-free rate = Rf = 0.30% AAPL’s Market Risk Measure = bAAPL = 1.37 Expected Market Risk Premium = [E(RM) – Rf] = 7.9% (See chapter 10, Table 10.3, page 317 or Chapter 10, Slide 24) E(RAAPL) = Rf + bAAPL[E(RM) – Rf] = 0.0030 + 1.37[0.079] = 11.123%

E(RX) = Rf + bX[E(RM) – Rf] Systematic Risk and Return - The CAPM According to the CAPM, the expected return of a stock equals: E(RX) = Rf + bX[E(RM) – Rf] The risk-free rate (Rf) Plus the stock’s market risk measure (b) Multiplied by the expected market risk premium [E(RM) – Rf]

CAPM Examples: E(Ri) = Rf + bi[E(RM) – Rf] The beta for P&G is 0.52 and the beta for US Steel is 2.43 The one-year risk-free rate is 0.30% The expected market risk premium is 7.90% Calculate the expected return of P&G and US Steel: E(Ri) = Rf + bi[E(RM) – Rf] E(RP&G) = Rf + bP&G[E(RM) – Rf] = 0.0030 + 0.52[0.079] = 4.408% E(RUS Steel) = Rf + bUS Steel[E(RM) – Rf] = 0.0030 + 2.43[0.079] = 19.497% Note that the only difference between the calculations for E(RP&G) and E(RUS Steel) is the market risk measure (beta) A larger beta means larger expected return.

Systematic Risk and Return - The CAPM Recap: P&G and Us Steel: E(RPG) = 4.408% E(RUS Steel) = 19.497% So why not just buy US Steel since it’s E(R) is bigger? Because in one year, the Realized Return (R) will equal E(R) + m m is unexpected return due to unexpected market returns The “market risk”

Recall from Chapter 10: sM = 20.60% Market Risk: m US Steel has a potentially really big m Measured by its b = 2.43 RUS Steel is expected to be 19.497% But it could be much higher or lower: If the market risk premium is 2% higher than expected [E(RM) – Rf] = 9.9% (instead of 7.9%) Then US Steel’s m will be 2.43(2.00%) = 4.86% If the market risk premium is 3% lower than expected [E(RM) – Rf] = 4.9% (instead of 7.9%) Then US Steel’s m will be 2.43(-3.00%) = -7.29% Recall from Chapter 10: sM = 20.60%

Five-Year Returns for US Steel (X) and P&G (PG) USX is at $55.02, down from $58.56 (down 6.05% over the last 5 years) PG is at $60.88, up from $51.55 (up 18.10% over the last 5 years) S&P500 is at 1313.80, up from 1302.95 (up 0.83% over the last 5 years)

Total Risk vs. Market Risk Recall: Total Risk = Systematic Risk + Unsystematic Risk But diversification eliminates the unsystematic portion… Total Risk = Systematic Risk + 0 (if you diversify) Total Risk is measured by the standard deviation (s) Systematic Risk (or Market Risk) is measured by beta (b) Since Unsystematic risk can be eliminated without cost You are not compensated for this portion of total risk Why?

Main Result of the CAPM: E(RX) = Rf + bX[E(RM) – Rf] E(RX) - Rf = bX[E(RM) – Rf] Risk-Premium of X = Something[Risk-Premium of Market] The return you expect to get from holding X in excess of the risk-free rate (E(RX) - Rf) equals: The return you expect to get from holding all stocks in excess of the risk-free rate (E(RM) – Rf) Times something (bX) The Something is the portion of the Total Market Risk If you hold more market risk, you get more return Less market risk, you get less return And THAT IS IT! That is all you get!

Total Risk vs. Market Risk (Continued 1) Consider Stocks 1 and 2: Which Stock has the greater: Total Risk? Systematic risk? Unsystematic risk? Expected risk premium? Expected return? σ Β Stock 1 40% 0.50 Stock 2 20% 1.50

Total Risk vs. Market Risk (Continued 2) Again Consider Stocks 1 and 2: Calculate the Expected Risk Premiums and Expected Returns Assume Rf = 1.00% and [E(RM) – Rf] = 8.5% Expected Risk Premiums: [E(R1) - Rf] = b1[E(RM) – Rf] = 0.5[8.5%] = 4.25% [E(R2) - Rf] = b1[E(RM) – Rf] = 1.5[8.5%] = 12.75% Expected Returns: E(R1) = Rf + b1[E(RM) – Rf] = 1.00% + 0.5[8.5%] = 5.25% E(R1) = Rf + b1[E(RM) – Rf] = 1.00% + 1.5[8.5%] = 13.75% So Stock 1 has greater total risk but lower expected return Why? σ β Stock 1 40% 0.50 Stock 2 20% 1.50

Calculating Beta Beta is the slope-coefficient of OLS Regression The slope-coefficient of the trend line Also Calculated as: 𝛽= 𝐶𝑜𝑣 𝑅 𝑖 , 𝑅 𝑀 𝑉𝑎𝑟 𝑅 𝑀 Go to AAPL Beta 2011-03 spreadsheet 

11.9 The Security Market Line The SML is a line depicting the CAPM So Back to the CAPM: For some stock (or portfolio): E(R) = Rf + b[E(RM) – Rf] For any stock, plug in a b, and get an E(R) We can “plot” this relationship on a graph with b on the horizontal axis E(R) on the vertical axis Think about the equation of the line: E(R) = Rf + b[E(RM) – Rf] = 0.30% + b[7.9%] The intercept is Rf = 0.30% The slope is [E(RM) – Rf] = 7.9% y = intercept + x(slope)

The Security Market Line Where is the point for the “market”? bM = 1 E(RM) – Rf = 7.90% E(RM) = Rf + bM[E(RM) – Rf] = 0.30% + 1[7.90%] = 8.20% Same as y = intercept + x(slope) = 0.30% + 1(7.90%) = 8.20% What about something with zero market risk (b = 0)? b = 0 [E(RM) – Rf] = 7.9% Rf = Rf + 0[E(RM) – Rf] = 0.30% + 0[7.90%] = 0.30% This is the “y intercept” So zero market risk earns the risk-free rate! So what does the graph look like? 

The Security Market Line

The Security Market Line The Theory: In equilibrium, all stocks are on this line If a stock’s E(R) is above the line: Then Return is too high given the risk People buy it until the price goes up and the return goes down If a stock’s E(R) is below the line: Return is to low given the risk People sell it until the price goes down and the return goes up See Slides 54 through 56

Portfolio Beta Portfolio beta is calculated the same way as the portfolio’s expected return: E(RP) = X1R1 + X2R2 + … + XNRN bP = X1b1 + X2b2 + … + XNbN