CH 13 - Lecture notes 13 PDF

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Professor: Peter Trager...

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CH 13 Expected Returns

EXAMPLE:

E(R) = Expected Return - probability of each scenario happening (recession, boom) x expected return of that scenario - Add all answers above Expected Return for stock A = 11.25%, stock B = 12.50%

Variance and Standard Deviation • Variance and standard deviation measure the volatility of returns. • Using unequal probabilities for the entire range of possibilities. • Variance = Weighted average of squared deviations • Standard Deviation = Square root of variance

n 2

σ =

å

p i(R i - E (R )) 2

i=1

Expected return of scenario – E(R) = #^2 = # x probability = ANS -.20 -.1125 = -.3125, (-.3125)^2nd = .097656 x .25 = .0244141 .15 -.1125 = .0375, (.0375)^2nd = .001406 x .5 = .0007031 .35 - .1125 = .2375, (.2375)^2nd = .056406 x .25 = .014102 • Variance = .0244141 + .0007031 + .014106 = .03921875 • Standard Deviation is square root of variance = .198 or 19.8% • One standard deviation / 68% chance = o 11.25 – 19.8  -8.5% o 11.25 + 19.8  31.05%

• • •

.30-.1250 = .1750, (.1750)2nd = .030625  .25 X .030625 = .007656 .15-.1250 = .025, (.025)2nd = .000625  .5 X .000625 = .0003125 -.10 -.1250 = -.2250, (-.2250)2nd = .050625  .25 X .050625 = .0126563 • Variance = .007656 + .0003125 + .0126563 = .0206 • Standard Deviation is the square root of .0206 = .144 or14.4%

Expected Returns • Suppose you have predicted the following returns for stocks C and T in three possible states of the economy. What are the expected returns? State Probability C T___ 0.3 15% 25% Boom Normal 0.5 10% 20% Recession 0.2 2% 1% Stock C Expected Return: • RC = .3(15) + .5(10) + .2(2) = 9.9% Stock T Expected Return: • RT = .3(25) + .5(20) + .2(1) = 17.7% Variance & Standard Deviation • Considering the expected return for each stock from the previous slide (C=9.9%, T = 17.7%) What are the variance and standard deviation for each stock? State Probability C T___ 0.3 15% 25% Boom Normal 0.5 10% 20% Recession 0.2 2% 1% Expected Return 9.9% 17.7% Stock C: § §

2 = .3 X (0.15-0.099)2 + .5 X (0.10-0.099)2 + .2 X (0.02-0.099)2 = = .000780 + .000001 + .001248  .002029 is variance (square root ANS)   = .045 or 4.50% is standard deviation

Stock T: § §

2 = .3 X (0.25-0.177)2 + .5 X (0.20-0.177)2 + .2 X (0.01-0.177)2 = = .001599 + .000265 + .005578  .007441 is variance (square root ANS)   = .0863 or 8.63% is standard deviation

EXAMPLE 2: Consider the following information: State Probability Boom .25 Normal .50 Slowdown .15 Recession .10

ABC, Inc. Return 0.15 0.08 0.04 -0.03

What is the expected return? • E(R) = .25(0.15) + .5(0.08) + .15(0.04) + .1(-0.03) = 8.05% What is the expected variance? • .25 X (.15-0.0805)2 + .5 X (0.08-0.0805)2 + .15 X (0.04-0.0805)2 + .1 X (-0.03-0.0805)2 = 0.00267475 What is the expected standard deviation? • Square root (.00267475) = 5.17%

Portfolios • A portfolio is a collection of assets. • An asset’s risk and return are important in how they affect the risk and return of the portfolio. • The risk-return trade-off for a portfolio is measured by the portfolio expected return and standard deviation, just as with individual assets. Portfolio Expected Returns • The expected return of a portfolio is the weighted average of the expected returns of the respective assets in the portfolio. • Weights (Wj) = % of portfolio invested in each asset

m

E (RP) =

å

j =1

w jE (R j)

Example 1 Portfolio Weights

*** Given: column 1, 2, 4 *** Third column: total dollars invested ($50,200) / asset dollars invested ($15,000) = 30% *** Fifth column: asset A  (.30 x .125) = .0375 or 3.75%

Portfolio Variance • Compute the portfolio return for each state: RP = w1R1 + w2R2 + … + wmRm • Compute the expected portfolio return using the same formula as for an individual asset. • Compute the portfolio variance and standard deviation using the same formulas as for an individual asset. Portfolio  Expected Asset Return, Variance & Standard Deviation • Consider the following information on returns and probabilities: § Invest 50% of your money in Asset A & Asset B. State Probability Boom .4 Bust .6

A B 30% -5% -10% 25%

Portfolio 12.5% (.5 x .3) + (.5 x -.05) 7.5% (.5x-.1 + .5x.25)

What is the expected return, variance and standard deviation for each asset? Asset A: E(RA) = .4(.30) + .6(-.10) = 6% • Variance(A) = .4 x (.30-.06)2 + .6 x (-.10-.06)2 = .0384 • Std. Dev.(A) = square root (.0384)  19.6% Asset B: E(RB) = .4(.-05) + .6(.25) = 13% • Variance(B) = .4 x (-.05-.13)2 + .6 x (.25-.13)2 = .0216 • Std. Dev.(B) = square root (.0216)  14.7%

What is the expected return, variance and standard deviation for the portfolio?

•

•

• •

Expected return for portfolio = .4(.125) + .6(.075) = .095 • .4 & .6 from probability of return • .125 & .075 are portfolio return in market Expected return for portfolio  A = .5(.06) + B = .5(.13) = .095 • .5 bc 50% money invested in Asset A/B • .6 / .13 are expected returns from Asset A/B Variance of portfolio = .4(.125.-095)2 + .6(.075-.095)2 = .0006 Standard deviation = 2.45%

Example 2: Invest 60% in Asset A & 40% in Asset B State Growth Recession

Probability .3 .7

A 20% 4%

B 12% -2%

Portfolio 16.8% (.6x.2 + .4x.12) 1.6% (.6x.04 +.4x-.02)

What is the expected return, variance & standard deviation for each asset? • Asset A: E(Ra) = .3(.20) + .7(.04) = 8.8% • Var (A) =.3(.20-.088)2nd + .7(.04-.088)2nd  .003763 + .001613 = .005376 • Standard Deviation (A) = .0733 or 7.33% • • •

Asset B: E(Rb) = .3(.12) + .7 (-.02) = 2.2% Var (B)= .3(.12-.022)2nd + .7(-.02-.022)2nd  .002881+ .001235 = .004116 Standard Deviation (B) = .0642 or 6.42%

What is the expected return, variance and standard deviation for the portfolio? • Expected Return of portfolio  .3(.168) + .7(.016) = .0616 or 6.2% • Expected Return of portfolio  A = .6(.088) + B =.4 (.022) = .0616 or 6.2% • Variance of Portfolio: .3(.168-.062)2nd + .7(.016-.062)2nd  .003371 + .001481 = .004852 • Standard Deviation = .0696 or 6.96%

Slide 29 Security market line: Systematic vs unsystematic risk  R=e(R) Statistics Beta  measure of risk  1 is average risk Risk free rate Diversification

Expected vs. Unexpected Returns • Realized returns are generally not equal to expected returns. • There is the expected component and the unexpected component. § At any point in time, the unexpected return can be either positive or negative. § Over time, the average of the unexpected component is zero. Announcements and News • Announcements and news contain both an expected component and a surprise component. • It is the surprise component that affects a stock’s price and therefore its return. • This is obvious when we watch how stock prices move when an unexpected announcement is made, or earnings are different than anticipated. Returns • Total Return = expected return + unexpected return • Unexpected return = systematic risk + unsystematic risk • Therefore, total return can be expressed as follows: Total Return = expected return + systematic risk + unsystematic risk Systematic Risk • Risk factors that affect a majority of assets in an asset field • Also known as non-diversifiable risk or market risk • EX: corona virus, changes in GDP, inflation, interest rates, oil price Unsystematic Risk • Risk factors that affect a limited number of assets • Also known as unique risk and asset-specific risk • This risk can be diversified away by adding lots of assets • EX: labor strikes, part shortages, E.coli at chipotle, prohibition Diversification • Portfolio diversification is the investment in several different asset classes or sectors. • Diversification is not just holding a lot of assets. • •

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• •

For example, if you own 50 Internet stocks, you are not diversified. However, if you own 50 stocks that span 20 different industries, then you are diversified.

The Principle of Diversification

• • •

Diversification can substantially reduce the variability of returns without an equivalent reduction in expected returns. This reduction in risk arises because worse than expected returns from one asset are offset by better-than-expected returns from another. However, there is a minimum level of risk that cannot be diversified away and that is the systematic portion.

Diversifiable Risk • The risk that can be eliminated by combining assets into a portfolio. • Often considered the same as unsystematic, unique or asset-specific risk • If we hold only one asset, or assets in the same industry, then we are exposing ourselves to risk that we could diversify away. Total Risk • Total risk = systematic risk + unsystematic risk • The standard deviation of returns is a measure of total risk. • For well-diversified portfolios, unsystematic risk is very small. • Consequently, the total risk for a diversified portfolio is essentially equivalent to the systematic risk. Systematic Risk Principle • There is a reward for bearing risk. • There is not a reward for bearing risk unnecessarily. • The expected return on a risky asset depends only on that asset’s systematic risk since unsystematic risk can be diversified away. BETA  Measuring Systematic Risk • How do we measure systematic risk? § We use the beta coefficient. • What does beta tell us? § A beta of 1 implies the asset has the same systematic risk as the overall market. § A beta < 1 implies the asset has less systematic risk than the overall market. § A beta > 1 implies the asset has more systematic risk than the overall market. § Most stocks have betas in the range of .5 to 1.5 § Beta of the market is 1.00 § Beta of a Treasury Bill is 0 US Stock Betas • Selected Beta’s of stocks effective 4/01/21 • Company Long-Term Avg Beta • Apple 1.1 • Amazon .95 • Apple/Amazon in line with the market • Boeing 2.14

• • • • • •

• Boeing is higher so market has large effect on company Costco .62 • Costco is low, so if the markets are good or bad it won’t affect them much Disney 1.0 Ford Motor 1.34 McDonalds .79 Tesla 1.83 Walmart .39

Finding Beta on a stock

Total vs. Systematic Risk • Consider the following information: Standard Deviation Beta Security C 20% 1.25 Security K 30% 0.95 • Which security has more total risk? K  standard deviation is higher • Which security has more systematic risk? C  beta is higher • Which security should have higher expected return in positive market? C  beta is higher

Example: Portfolio Betas

Security Weight Beta C .133 1.685 KO .2 0.195 INTC .267 1.161 BP .4 1.434 • What is the portfolio beta? • .133(1.685) + .2(.195) + .267(1.161) + .4(1.434) = 1.147 Beta and the Risk Premium • Remember that the risk premium = expected return – risk-free rate. • The higher the beta, the greater the risk premium should be. • Can we define the relationship between the risk premium and beta so that we can estimate the expected return? § YES!

Reward-to-Risk Ratio: Definition and Example • The reward-to-risk ratio is the slope of the line illustrated in the previous example. § Slope = (E(RA) – Rf) / (A – 0) § Reward-to-risk ratio for previous example = (20 – 8) / (1.6 – 0) = 7.5  slope of line • What if an asset has a reward-to-risk ratio of 8? § If the reward-to-risk ratio = 8, (dot above slope line) then investors will want to buy the asset. This will drive the price up and the expected return down • What if an asset has a reward-to-risk ratio of 7? § If the reward-to-risk ratio = 7, (dot below slope line) then investors will want to sell the asset. This will drive the price down and the expected return up

Market Equilibrium

•

In equilibrium, all assets and portfolios must have the same reward-to-risk ratio, and they all must equal the reward-to-risk ratio for the market.

E (R A) - R A

f

=

E (RM - R f ) M

Security Market Line • The security market line (SML) is the representation of market equilibrium. • The slope of the SML is the reward-to-risk ratio: (E(R M) – Rf) / M • Since the beta for the market is always equal to one, the slope can be rewritten. • Slope = E(RM) – Rf = market risk premium The Capital Asset Pricing Model (CAPM) • The capital asset pricing model defines the relationship between risk and return. • E(RA) = Rf + A(E(RM) – Rf) • E(Ra) = Expected Required Return of asset • Rf = Risk-Free rate (T-Bill rate) • Ba = Beta of asset • E(Rm) = Expected Market Return • E(RM)-Rf = Market Risk Premium • If we know an asset’s systematic risk (Beta), we can use the CAPM to determine its expected return. Example - CAPM • Consider the betas for each of the assets given earlier. If the risk-free rate is 3.15% and the market risk premium is 7.5%, what is the expected return for each? • E(RA) = Rf + A(E(RM) – Rf) Security

Beta

Expected Return

Apple

1.10

3.15 + 1.10(7.5) = 11.40%

Costco

0.62

3.15 + 0.62(7.5) = 7.800%

TSLA

1.83

3.15 + 1.83(7.5) = 16.87%

Walmart

0.39

3.15 + 0.39(7.5) = 6.075%

Quick Quiz Consider an asset with a beta of 1.2, a risk-free rate of 5%, and a market return of 13%. What is the reward-to-risk ratio in equilibrium? Slope = (E(RA) – Rf) / (A – 0)  Remember equilibrium has market beta of 1 § (13-5)/1 = 8 What is the expected return on the asset? E(RA) = Rf + A(E(RM) – Rf)

The risk-free rate is 4%, and the required return on the market is 12%. § What is the required return on an asset with a beta of 1.5? • R = .04 + 1.5 × (.12 - .04) =.16 § What is the reward/risk ratio? • (.12-.04)/1 = 8%

§

What is the required return on a portfolio consisting of 40% of the asset above and the rest in an asset with an average (market) amount of systematic risk? • R = (.4 × .16) + (.6 × .12) • = .064 + .072  .136 or 13.6%

Connect ?’s A portfolio consists of 290 shares of Stock C that sells for $55 and 255 shares of Stock D that sells for $20. What is the portfolio weight of Stock C? - Weight of C = 290($55) / [290($55) + 255($20)] Weight of C = .7577 Which one of the following stocks is correctly priced if the risk-free rate of return is 3.9 percent and the market risk premium is 8.4 percent? Stock A B C D E -

Beta .77 1.55 1.36 1.33 .95

Expected Return 7.86% 12.65 17.33 11.93 11.88

E Risk free rate + beta(market risk premium) = expected return 3.9 + .95(8.4) = 11.88

You have a portfolio that is equally invested in Stock F with a beta of .99, Stock G with a beta of 1.41, and the risk-free asset. What is the beta of your portfolio? - .80 - (1/3)(.99) + (1/3)(1.41) + (1/3)(0) = .80 Based on the following information, what is the standard deviation of returns?

State of Economy Recession Normal Boom

Probability of State of Economy .22 .47 .31

Rate of Return if State Occurs −.090 .105 .215

-

10.97%

The risk-free rate is 3.3 percent and the market expected return is 12 percent. What is the expected return of a stock that has a beta of 1.18? - Risk free rate + beta(market risk premium) = expected return - Market risk premium = market return – risk free rate - 3.3 + 1.18(12-3.3) = 13.57 If the economy booms, RTF, Inc., stock is expected to return 9 percent. If the economy goes into a recessionary period, then RTF is expected to only return 4 percent. The probability of a boom is 68 percent while the probability of a recession is 32 percent. What is the variance of the returns on RTF, Inc., stock? - .000544 You have a portfolio that is invested 24 percent in Stock R, 38 percent in Stock S, and the remainder in Stock T. The beta of Stock R is .69, and the beta of Stock S is 1.24. The beta of your portfolio is 1.35. What is the beta of the Stock T? - 1.350 = .24(.69) + .38(1.24) + .38(x) - .7132 = .38(x) - 1.876 = x A stock will have a loss of 12.7 percent in a recession, a return of 11.4 percent in a normal economy, and a return of 26.1 percent in a boom. There is 24 percent probability of a recession, 45 percent probability of normal economy, and 31 percent probability of boom. What is the standard deviation of the stock's returns? - 14.31% You recently purchased a stock that is expected to earn 25 percent in a booming economy, 14 percent in a normal economy, and lose 5 percent in a recessionary economy. There is 23 percent probability of a boom, 62 percent chance of a normal economy, and 15 percent chance of a recession. What is your expected rate of return on this stock? - .23(25) + .62(14) + .15(-5) - 13.68% You have a portfolio that is 37 percent invested in Stock R, 21 percent invested in Stock S, with the remainder in Stock T. The expected return on these stocks is 9.0 percent, 10.4 percent, and 12.7 percent, respectively. What is the expected return on the portfolio? - .37(.09) + .21(.104) + .42(.127) - 10.85%...