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BCOR 2200 Chapter 11

BCOR 2200 Chapter 11. Risk and Return. Chapter Overview Last chapter we looked at RISK and RETURN for CATEGORIES of stocks (large and small) Now look at Risk and Return in more detail Risk and Return calculations for an individual stock

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BCOR 2200 Chapter 11

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  1. BCOR 2200Chapter 11 Risk and Return

  2. Chapter Overview • Last chapter we looked at RISK and RETURN for CATEGORIES of stocks (large and small) • Now look at Risk and Return in more detail • Risk and Return calculations for an individual stock • Risk and Return calculations for a portfolio of stocks • Specific Skills • Calculate expected returns and risk • Understand the impact of diversification • Understand the systematic risk principle • Understand the security market line • Understand the risk-return trade-off

  3. Chapter Outline • Calculate returns and risk for individual stocks • Calculate returns and risk for portfolios • Consider Expected numbers and Announced numbers • Numbers like corporate earnings, GDP, unemployment… • The difference between them Announced and Expected and is called the Surprise • Announcements by companies, the government… • Two components of Risk • Things that affect the Whole Economy • Things that affect a Single Stock • Called Systematic Riskand Unsystematic Risk • Also called MarketRiskand Unique Risk • How Diversification lowers Portfolio Risk • Measuring Systematic Riskwith the CAPM Beta • The Security Market Line (SML) • The SML and the Cost of Capital: A Preview

  4. 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

  5. Risk and Return Calculations (Part 2) Second Way: • Set categoriesor ranges of height • Calculate mean for each category • Calculate (or assign) probabilitiesfor each category • Calculate expectation: E(H) = 66.3 inches Why might this way be better? • You can change category probabilities based on new information or expectations • Intramural basketball meeting, so reset probs to 10%, 10% and 80% • New E(H) = 70.3 inches

  6. Clicker Question: • The table shows the group averages and probabilities of observing someone from each group. • Calculate the expected height for the next person to walk into the room for the general population and for an intramural basket ball meeting.

  7. Clicker Answer: • General Population: E(H) = 0.25(60) + 0.50(66) + 0.25(74) = 66.5 • Intramural Basketball: E(H) = 0.10(60) + 0.40(66) + 0.50(74) = 69.4 The Answer is E. The 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…

  8. Risk-Return Calculation Outline: 11.1 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 11.2 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

  9. Some More Notation (Part 1): • ps = Probability of some “state obtaining” • S = The Number of possible states (big S) • s = One of the states (little s) • R = Return • Ri = Return of stock i (Distinguish between stocks if we are considering more than one) • RS = Return given some state of the economy • RS,i = Return given some state of the economy for stock i • E(R) = Expected Return • E(Ri) = Expected Return of stock i • E(RP) = Expected Return of a portfolio

  10. Some More Notation (Part 2): • Var(R) = s2 = Variance of the returns of a stock • Var(R1) = s12= Variance of the returns of stock 1 • Var(RP) = sP2 = Variance of the returns of a portfolio • SD(R) = s = Standard Deviation of the returns • SD(R1) = s1= Standard Deviation of the returns of stock 1 • SD(RP) = sP = Standard Deviation of the returns of a portfolio • Covar(1,2) = s1,2= Covariance between returns of stocks 1 and 2 • Corr(1,2) = r1,2= Correlation between the returns of stocks 1 and 2 r1,2= s1,2/(s1 s2) s1,2= r1,2s1 s2

  11. 11.1 Return Calculations for a Single Stock (Part 1) • 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 stateof 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

  12. Return Calculations for a Single Stock (Part 2) • 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%

  13. Risk Calculations (s2 and s) for a Single Stock • Old way using all the historic data: • New way using state returns and state probabilities: • s2 = 0.0020029 • s = 0.0450 = 4.50%

  14. Risk and Return Examples • Two New Stocks: L and U: • Calculate the E(R) then s for each stock Table 11-2, Page 352: E(R)

  15. Risk Calculations for the Stocks: Table 11-4 page 354: s2 (For each individual stock) • Take the square-root of the Variance to get the Standard Deviation: σL = 0.2025½ = 0.45 = 45% σU= 0.0100½ = 0.10 = 10%

  16. Recap: Expected Returns Expected Return for Stock L = E(RL): • 50% chance of -20% and 50% chance of 70% • So expected return of 25% Expected Return for Stock U = E(RU): • 50% chance of 30% and 50% chance of 10% • So expected return of 20%

  17. Recap: Standard Deviations Standard Deviation for Stock L = σL: • -20% - 25% = -45% = -0.45 • 70% - 25% = 45% = 0.45 • Square and multiply by probability: 0.50(-0.45)2 + 0.50(0.45)2 = 0.2025 • Take the square root: 0.2025½ = 0.45 = 45% = σL Standard Deviation for Stock U = σU: • 10% - 20% = -10% = -0.10 • 30% - 20% = 10% = 0.10 • Square and multiply by probability: 0.50(-0.10)2 + 0.50(0.10)2 = 0.0100 • Take the square root: 0.0100½ = 0.10 = 10% = σU

  18. Clicker Question: Assume that over the next year you expect: • A 40% chance of a economic expansion (a boom) • A 60%chance of an economic contraction (a bust) • A stock will beup 20% if there is an expansion • And bedown 10% if there is a contraction. • Calculate the Expected Returnof the stock. • -15.00% • -2.00% • 2.00% • 15.00% • 40.00%

  19. Clicker Answer: E(R) = 0.4(0.20) + 0.6(-0.10) = 0.08 + -0.06 = 0.02 = 2% The Answer is C Bonus Question: • Your economic forecast changes such that you expect a 70% chance of an expansion and 30% chance of a contraction. • How would this change your expectation about the stock’s return? E(R) = 0.7(0.20) + 0.3(-0.10) = 0.14 + -0.03= 0.11 = 11%

  20. Clicker Question: Assume that over the next year you expect: • A 40% chance of a boom • A 60% chance of a bust • A stock will be up 20% if there is boom and down 10% if there is a bust • So E(R) = 2.00% • Calculate the Standard Deviation (σ) of the return. • 2.46% • 3.56% • 14.70% • 19.85% • 35.30%

  21. Clicker Answer: 40%chance of being up 20%with a 2% expectation: • (0.20 – 0.02) = 0.18 • (0.18)2 = 0.0324 • 0.40(0.0324) = 0.01296 60% chance of being down 10% with a 2% expectation: • (-0.10 – 0.02) = -0.12 • (-0.12)2 = 0.0144 • 0.60(0.0144) = 0.00864 Sum = 0.01296 + 0.00864 = 0.02160 = Variance = σ2 Square Root = (0.02160)½ = 0.1470 = 14.70% = Stdev = σ Note that the variance is NOT EQUAL to 2.16%

  22. Now Lets Look at a Portfolio: • A Portfoliois 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 simple equally-weighted portfolio for stocks L and U: • WL= 0.50 and WU= 0.50

  23. Return for a Portfolio of Stocks Expected Return of a Portfolio E(RP): • We’ll come up with a better formula later on • For now Table 11.5: E(RP)

  24. Risk for a Portfolio of Stocks Risk of a Portfolio sP: • Again a better formula later on • For now Table 11-6: sP

  25. Return Calculations For a Portfolio of stocks U and L Data for Portfolio and Portfolio Stocks: • Weight of L = WL= 50% and E(RL) = 25% • Weight of U = WU = 50% and E(RU) = 20% E(RP) = WLE(RL) + WUE(RU) E(RP) = 0.5(0.25) + 0.5(0.20) = 0.225 = 22.50%

  26. Risk Calculations For a Portfolio of stocks U and L Data for Portfolio and Portfolio Stocks: • WL= 50% and sL= 45% • WU= 50% and sU= 10% sP2 = WL2 sL2 + WU2 sU2 + 2WLsLWUsUrL,U • rL,U is the CORRELATIONbetween the returns The CORRELATIONdefines the relationship between L and U

  27. Calculating Correlation: • First Calculate the Covariance (sL,U): Covar(L,U) =pRec[RL,Rec– E(RL)][RU,Rec – E(RU)] + pBoom[RL,Boom – E(RL)] [RU,Boom – E(RU)] = 0.5[-0.20 – 0.25][0.30 – 0.20] + 0.5[0.70 – 0.25][0.10 – 0.20] • Covar(L,U) = -0.045 = sL,U • The Correlation (r) is the Standardized Covariance: • Standardize by dividing by the standard deviations rL,U = sL,U/(sLsU) = -0.045/[(0.45)(0.10)] = -1.00 (What does r= -1mean?)

  28. Portfolio Risk is calculated using this formula: Var = sP2 = W12 s12+ W22 s22+ 2W1W2 r1,2s1 s2 Stdev = sP= [sP2]½ • Stdev is the square-root of the Variance • What is the largest possible value for r1,2? • What is the smallest possible value for r1,2? • r1,2 must be between 1 and -1 • What do the values of 1, 0 and -1 mean?

  29. Return and Risk Calculations for a Portfolio of stocks U and L Now plug all the values into Portfolio Variance formula: sP2 = WL2 sL2 + WU2sU2 + 2WLWUrL,UsLsU = (.5)2(.45)2+ (.5)2(.10)2 + 2(.5)(.5)(-1)(.45)(.1) = 0.030625 sP= (sP2)½ = (0.030625)½ = 0.175 = 17.5% Compare this sP calculation to Table 11.6 

  30. Table 11-6 Again: The first method for calculating sP: • So same answer (17.5%) but second formula is better. sP= (W12 s12 + W22 s22 + 2W1W2 r1,2s1 s2)½ • Why? • Because we can see how the portfolio’s risk changes when the relationship between the stocks (r) changes • This will lead to a FUNDAMENTAL RESULT IN FINANCE!

  31. More on Portfolio Risk: • What if r1,2 = 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 r1,2 = 1 • So substitute 1 for r1,2 in the portfolio risk calculation: sP2 = W12 s12 + W22 s22 + 2W1W2 (1) s1 s2 And now do some algebra

  32. More on Portfolio Risk: • Portfolio SD is calculated using this formula: sP= [sP2]½ = [W12 s12 + W22 s22 + 2W1W2 r1,2s1 s2]½ • Set r1,2 = 1 (no diversification) sP= [W12 s12 + W22 s22 + 2W1W2 1s1 s2]½ • Rewrite in a way that allows us to simplify: sP= [(W1s1)2 + (W2s2)2 + 2(W1s1)(W2s2)]½ Let W1s1 = a and W2s2 = b: [a2 + b2 + 2ab]½ [(a + b)2]½ (a + b) sP= W1s1 + W2s2(but only if r1,2 = 1)

  33. More on Portfolio Risk: • If r1,2 = 1 then sP= W1s1 + W2s2 • So Portfolio Risk equals: (Weight of 1 x Risk of 1) + (Weight of 2 x Risk of 2) Call this the weighted average risk • What if the stocks correlation is 1.00 or 100%? sP= [W12 s12 + W22 s22 + 2W1W2 r1,2s1 s2]½ = [(.5)2(.45)2 + (.5)2(.10)2 + 2(.5)(.5)(1.00)(.45)(.1)] ½ sP= 27.5% • Same as the weighted average risk: sP= W1s1 + W2s2 = (.5)(.45) + (.5)(.10) sP= 27.5%

  34. More on Portfolio Risk: • So if r1,2 = 1 then sP= W1s1 + W2s2 • So Portfolio Risk equals: (Weight of 1 x Risk of 1) + (Weight of 2 x Risk of 2) Called the weighted average risk • WHEN is this true? ONLY when r1,2 = 1 • What is the most r1,2 can be? • The MAX for r1,2 is 1 • So what if r1,2 < 1? • sP< W1s1 + W2s2 • Portfolio risk is less than the weighted average risk! • If r1,2< 1 then the portfolio is diversified and risk is lower!

  35. More on Portfolio Risk: • This is a fundamental result in finance Diversification lowers portfolio risk!!! sP= [W12 s12 + W22 s22 + 2W1W2 r1,2s1 s2]½ • Not diversified? • Then r1,2 = 1 and sP= W1s1 + W2s2 • Diversified? • Then r1,2 < 1 and sP< W1s1 + W2s2 • What if the stocks correlation is 0.20 (or 20%)? sP= [W12 s12 + W22 s22 + 2W1W2 r1,2s1 s2]½ = [(.5)2(.45)2 + (.5)2(.10)2 + 2(.5)(.5)(.20)(.45)(.1)] ½ = 24.01% If correlation is 20%, portfolio risk is 24.01% (less than 27.50%)

  36. Recap: Portfolio risks for different correlations: r1,2 = 1.00 then sP= 27.50% r1,2 = 0.20 then sP= 24.01% r1,2 = -1.00 then sP= 17.5% • Note that in each case we are looking at an equally weighted (or 50-50) portfolio

  37. Zero-risk portfolios for stocks with r1,2 = -1: Please do this example on your own for HW: • Now assume W1 = 2/11 and W2 = 9/11 • Recalculate the portfolios risk: • sP2 = WL2 sL2 + WU2 sU2 + 2WLWU rL,UsLsU = (2/11)2(0.45)2 + (9/11)2(0.1)2 + (2/11)(9/11)(0.45)(0.1)(-1) = 0 Show this is true using the other method:: • For a Recession: RP = 2/11(-0.20) + 9/11(0.30) = 20.91% • For a Boom: RP = 2/11(0.70) + 9/11(0.10) = 20.91% • So the portfolio return is same for both states • For these weights, the portfolio has ZEROrisk

  38. Homework Example Continued… Zero-Risk Using the Other Equation: • s2 = pRec[RP,Rec – E(RP)]2 + pBoom[RP,Boom – E(RP)]2 • pRec = 0.5; pBoom = 0.5 • RP,Rec = 0.2091; RP,Boom = 0.209 • E(RP) = 0.2091 s2 = 0.5[0.2091 – 0.2091]2 + 0.50[0.2091 – 0.2091]2 = 0 Like Table 11.6:

  39. Recap: A Portfolio’s Expected Returnequals: E(RP) = W1E(R1)+ W2E(R2) This is a function of: • The weights of the stocks: W1 and W2 • The expected returns of the stocks: E(R1) and E(R2) A Portfolio’s Riskequals: sP= [W12 s12 + W22 s22 + 2W1W2r1,2s1 s2]½ This is a function of: • The weights of the stocks: W1 and W2 • The risksof the individual stocks: s1and s2 • The relationshipbetween the two stocks: r1,2

  40. Clicker Question: • Stock A has an expected return of 20% • Stock B has an expected return of 12% • Calculate the expected returns for a portfolio of the two stocks if the weight of Stock A is 10% • Calculate the expected returns for a portfolio of the two stocks if the weight of Stock A is 75% • E(R10-90) = 10.00% and E(R75-25) = 75.00% • E(R10-90) = 90.00% and E(R75-25) = 25.00% • E(R10-90) = 12.00% and E(R75-25) = 20.00% • E(R10-90) = 88.00% and E(R75-25) = 80.00% • E(R10-90) = 12.80% and E(R75-25) = 18.00%

  41. Clicker Answer: • E(RA) = 20% and E(RB) = 12% • If WA = 10% and WB = 90% E(R10-90) = (0.10)(0.20) + (0.90)(0.12) = 12.80% • If WA = 75% and WB = 25% E(R75-25) = (0.75)(0.20) + (0.25)(0.12) = 18.00% The Answer is E.

  42. Clicker Question: • sA = 30% and sB = 15% • WA = 75% and WB = 25% • Calculate sP for a portfolio of the two stocks if rA,B= 0.20 • Calculate sP for a portfolio of the two stocks if rA,B= 1.00 Recall: sP= [WA2 sA2 + WB2 sB2 + 2WAWB rA,BsAsB]½ sP= WAsA + WBsB (if rA,B = 1)

  43. Clicker Answer: • For rA,B = 0.20: sP= [WA2 sA2 + WB2 sB2 + 2WAWB rA,BsAsB]½ = [(.752)(.32) + (.252)(.152) + 2(.75)(.25)(0.2)(.3)(.15)] ½ = [0.0506 + 0.0014 + 0.0034] ½ = [0.0554] ½ sP= 0.2354 = 23.54% • For rA,B = 1.00: sP= WAsA + WBsB = (.75)(.3) + (.25)(.15) = 0.2250 + 0.0375 sP= 0.2625 = 26.25% Note that portfolio risk is lower for rA,B = 0.20

  44. 11.3 & 11.4 Expected Returns and Risk Components • 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

  45. Expected Returns and Risk Components • Total RealizedReturn (we’ll call this R) equals: • R = E(R) + U • For any period, the unexpected 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

  46. Announcements and News • Announcements and news contain two parts: • An Expected component • A Surprisecomponent • 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

  47. Think About Surprises: • The Unexpected part (U) • RealizedReturn: 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 (Unsystematicor 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

  48. 11.5 Diversification (Part 1) • 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 uncorrelatedwith the e from another company • When one eis positive, the other emay 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!

  49. Diversification (Part 2) 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 allstocks: 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 eare a stock’s risk • So now think of risk as the amount the realized return varies from the expected return

  50. Diversification (Part 3) • Back to the Height Example • What will be the realizedheight 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

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