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Michael C. Ehrhardt 04/08/2023

4/11/2010

PORTFOLIO RISK AND RETURN: THE TWO-ASSET CASE

ATTAINABLE PORTFOLIOS: THE TWO ASSET-CASE

Asset A Asset BExpected return, r hat 5% 8%

4% 10%

Chapter 24. Tool Kit for Portfolio Theory, Asset Pricing Models, and Behavioral Finance

EFFICIENT PORTFOLIOS (Section 24.1)

Suppose there are two assets, A and B. wA is the percent of the portfolio invested in asset A. Since the total percents invested in the asset must add up to 1, (1-wA) is the percent of the portfolio invested in asset B.

The expected return on the portfolio is the weighted average of the expected returns on asset A and asset B.

The standard deviation of the portfolio, sp, is not a weighted average. It is:

Standard deviation, s

Using the equations above, we can find the expected return and standard deviation of a portfolio with different percents invested in each asset.

BAABAA2B

2A

2A

2Ap )W1(W2)W1(W sssss

B

^

AA

^

Ap

^

r)w1(rwr



Correlation = 1

1.00 0.00 5.00% 4.0%0.90 0.10 5.30% 4.6%0.80 0.20 5.60% 5.2%0.70 0.30 5.90% 5.8%0.60 0.40 6.20% 6.4%0.50 0.50 6.50% 7.0%0.40 0.60 6.80% 7.6%0.30 0.70 7.10% 8.2%0.20 0.80 7.40% 8.8%0.10 0.90 7.70% 9.4%0.00 1.00 8.00% 10.0%

Correlation = 0

1.00 0.00 5.00% 4.0%0.90 0.10 5.30% 3.7%

Proportion of Portfolio in Security A

(Value of wA)

Proportion of Portfolio in Security B

(Value of 1-wB) rp sp


(Value of wA)


(Value of 1-wA) rp sp

0%

5%

10%

= +1: rAB Attainable Set of/ Risk Return Combinations

Risk, sp

Expected return

B

A



0.80 0.20 5.60% 3.8%0.70 0.30 5.90% 4.1%0.60 0.40 6.20% 4.7%0.50 0.50 6.50% 5.4%0.40 0.60 6.80% 6.2%0.30 0.70 7.10% 7.1%0.20 0.80 7.40% 8.0%0.10 0.90 7.70% 9.0%0.00 1.00 8.00% 10.0%

0%

5%

10%

= 0: /rAB Attainable Set of Risk Return Combinations

Risk, sp

Expected return

B

A



Correlation = -1

1.00 0.00 5.00% 4.0%0.90 0.10 5.30% 2.6%0.80 0.20 5.60% 1.2%0.70 0.30 5.90% 0.2%0.60 0.40 6.20% 1.6%0.50 0.50 6.50% 3.0%0.40 0.60 6.80% 4.4%0.30 0.70 7.10% 5.8%0.20 0.80 7.40% 7.2%0.10 0.90 7.70% 8.6%0.00 1.00 8.00% 10.0%


(Value of wA)



0%

5%

10%

= 0: /rAB Attainable Set of Risk Return Combinations

Risk, sp

Expected return

B

A

0% 2% 4% 6% 8% 10% 12%0%

5%

10%

= -1: /rAB Attainable Set of Risk Re- turn Combinations

Risk, sp

Expected return

B

A



Table 1: Expected Return and Standard Deviation under Various Assumptions

1.00 0.00 5.00% 4.0% 4.0% 4.0%0.75 0.25 5.75% 5.5% 3.9% 0.5%0.50 0.50 6.50% 7.0% 5.4% 3.0%0.25 0.75 7.25% 8.5% 7.6% 6.5%0.00 1.00 8.00% 10.0% 10.0% 10.0%

DateMarch 2009 8.5% 18.8% 13.9% 0.02% 8.5% 18.8% 13.9%February 2009 -11.0% -27.8% -7.6% 0.03% -11.0% -27.8% -7.6%January 2009 -8.6% -25.2% -7.5% 0.01% -8.6% -25.2% -7.5%December 2008 0.8% -3.8% 4.9% 0.00% 0.8% -3.9% 4.9%November 2008 -7.5% -12.0% -11.3% 0.02% -7.5% -12.0% -11.4%October 2008 -16.8% -23.5% -21.6% 0.06% -16.9% -23.5% -21.7%September 2008 -9.2% -8.1% -18.0% 0.09% -9.3% -8.2% -18.1%August 2008 1.2% -0.7% -0.1% 0.14% 1.1% -0.8% -0.2%July 2008 -1.0% 6.0% -3.9% 0.14% -1.1% 5.9% -4.1%June 2008 -8.6% -12.1% -8.4% 0.16% -8.8% -12.3% -8.5%May 2008 1.1% -6.1% 3.1% 0.14% 0.9% -6.2% 2.9%April 2008 4.8% -11.6% 6.5% 0.11% 4.6% -11.8% 6.4%


(Value of wA)



Case I rAB = +1.0

Case II rAB = 0.0

Case III rAB = -1.0

CALCULATING BETA COEFFICIENTS (Section 24.5)

We downloaded stock prices and dividends from http://finance.yahoo.com for General Electric using its ticker symbol, GE. We also downloaded data for the S&P 500 (^SPX) which contains most actively traded stocks, and the Fidelity Magellan mutual fund (FMAGX). We computed returns, as shown in Chapter 6. We also obtained the monthly rates on 3-month Treasury bills from the FRED II data base at the St. Louis Federal Reserve, http://research.stlouisfed.org.

rM, Market Return (S&P 500 Index) ri, GE Return

rp, Fidelity Magellan

Fund Return

rRF, Risk-Free Rate (Monthly Return on 3-Month T-Bill)

Excess market return (rM-rRF)

Excess stock return

(ri-rRF)

Excess portfolio return

(rp-rRF)

0% 2% 4% 6% 8% 10% 12%0%

5%

10%

= -1: /rAB Attainable Set of Risk Re- turn Combinations

Risk, sp

Expected return

B

A

D162

Mike Ehrhardt: We have adjusted this to reflect the large capital gains distribution in May 2006.



March 2008 -0.6% 11.7% -2.2% 0.11% -0.7% 11.6% -2.3%February 2008 -3.5% -5.4% -1.6% 0.18% -3.7% -5.6% -1.8%January 2008 -6.1% -4.6% -8.9% 0.23% -6.3% -4.9% -9.2%December 2007 -0.9% -2.4% 0.3% 0.25% -1.1% -2.6% 0.1%November 2007 -4.4% -7.0% -3.8% 0.27% -4.7% -7.2% -4.1%October 2007 1.5% -0.6% 5.3% 0.33% 1.2% -0.9% 5.0%September 2007 3.6% 7.2% 6.2% 0.32% 3.3% 6.9% 5.9%August 2007 1.3% 0.3% 1.2% 0.35% 0.9% -0.1% 0.8%July 2007 -3.2% 1.3% -1.8% 0.40% -3.6% 0.9% -2.2%June 2007 -1.8% 2.6% -0.4% 0.38% -2.2% 2.2% -0.8%May 2007 3.3% 2.0% 4.1% 0.39% 2.9% 1.6% 3.7%April 2007 4.3% 4.2% 4.8% 0.41% 3.9% 3.8% 4.4%March 2007 1.0% 1.3% 0.8% 0.41% 0.6% 0.9% 0.4%February 2007 -2.2% -2.4% -1.7% 0.42% -2.6% -2.8% -2.1%January 2007 1.4% -3.1% 2.9% 0.42% 1.0% -3.5% 2.5%December 2006 1.3% 6.3% -0.5% 0.40% 0.9% 5.9% -0.9%November 2006 1.6% 0.5% 2.6% 0.41% 1.2% 0.1% 2.2%October 2006 3.2% -0.5% 2.9% 0.41% 2.7% -1.0% 2.5%September 2006 2.5% 4.4% 1.1% 0.40% 2.1% 4.0% 0.7%August 2006 2.1% 4.2% 2.4% 0.41% 1.7% 3.8% 2.0%July 2006 0.5% -0.8% -3.1% 0.41% 0.1% -1.2% -3.5%June 2006 0.0% -3.1% -0.7% 0.40% -0.4% -3.5% -1.1%May 2006 -3.1% -1.0% -4.9% 0.39% -3.5% -1.4% -5.3%April 2006 1.2% -0.6% 1.9% 0.38% 0.8% -0.9% 1.5%March 2006 1.1% 5.8% 2.7% 0.38% 0.7% 5.4% 2.3%February 2006 0.0% 1.1% -1.5% 0.37% -0.3% 0.8% -1.9%January 2006 2.5% -6.6% 4.8% 0.35% 2.2% -6.9% 4.4%December 2005 -0.1% -1.2% 2.0% 0.32% -0.4% -1.5% 1.6%November 2005 3.5% 5.3% 3.4% 0.32% 3.2% 5.0% 3.0%October 2005 -1.8% 0.7% -1.3% 0.31% -2.1% 0.4% -1.6%September 2005 0.7% 0.8% 0.7% 0.29% 0.4% 0.5% 0.4%August 2005 -1.1% -2.6% -1.2% 0.29% -1.4% -2.9% -1.4%July 2005 3.6% -0.4% 4.0% 0.27% 3.3% -0.7% 3.8%June 2005 0.0% -4.4% 0.0% 0.25% -0.3% -4.7% -0.2%May 2005 3.0% 0.7% 3.6% 0.24% 2.8% 0.5% 3.4%April 2005 -2.0% 0.4% -2.1% 0.23% -2.2% 0.2% -2.3%Average -8.5% -22.9% -7.0% 3.3% -11.7% -26.2% -10.3%

15.9% 28.9% 21.1% 0.5% 15.7% 28.7% 20.9%Correlation with market return, r 0.76 0.94 0.44 1.00 0.75 0.93R-square 0.57 0.88 0.19 1.00 0.57 0.87Slope 1.37 1.24 0.01 0.99 1.37 1.25

Standard deviation (annual)

Using the AVERAGE function and the STDEV function, we found the average historical returns and standard deviations. (We converted these from monthly figures to annual figures. Notice that you must multiply the monthly standard deviation by the square root of 12, and not 12, to convert it to an annual basis.) These are shown in the rows above.



We also use the CORREL function to find the correlation of the market with the other assets.

GE Analysis

GE Regression Results (See columns J-N) SUMMARY OUTPUTBeta

Coefficient 1.3744 1.374 Regression Statisticst statistic 7.84 Multiple R

Probability of t stat. 0.000% R SquareLower 95% confidence interval 1.02 Adjusted RUpper 95% confidence interval 1.73 Standard Er

ObservatioIntercept

Coefficient -0.00944 -0.00944 ANOVAt statistic -1.17

Probability of t stat. 24.8% RegressionLower 95% confidence interval -0.03 ResidualUpper 95% confidence interval 0.01 Total

InterceptX Variable

Magellan Analysis

Magellan Regression Results (See columns J-N) SUMMARY OUTPUTBeta

Using the AVERAGE function and the STDEV function, we found the average historical returns and standard deviations. (We converted these from monthly figures to annual figures. Notice that you must multiply the monthly standard deviation by the square root of 12, and not 12, to convert it to an annual basis.) These are shown in the rows above.

Using the function Wizard for SLOPE, we found the slope of the regression line, which is the beta coefficient. We also use the function Wizard and the RSQ function to find the R-Squared of the regression.

Using the Chart Wizard, we plotted the GE returns on the y-axis and the market returns on the x-axis. We also used the menu Chart > Options to add a trend line, and to display the regression equation and R2 on the chart. The chart is shown below. We also used the regression feature to get more detailed data. These results are also shown below.

-30% -20% -10% 0% 10% 20% 30%

-30%

-20%

-10%

0%

10%

20%

30%

f(x) = 1.37442291807652 x − 0.00943802153613873R² = 0.571900259505067

Historic Realized Returnson the Market (%)

Historic Realized Returns

on GE (%)

-20% -10% 0% 10% 20%

-20%

-10%

0%

10%

20%

f(x) = 1.23994673597531 x + 0.0028965722635326R² = 0.876260867381232



on Magellan (%)

H237

This calculation of the beta uses the slope function

H244

The calculation of the intercept uses the intercept function.





ObservatioIntercept

Coefficient 0.00290 0.00290 ANOVAt statistic 0.92

Probability of t stat. 36.4% RegressionLower 95% confidence interval 0.00 ResidualUpper 95% confidence interval 0.01 Total

InterceptX Variable

-20% -10% 0% 10% 20%

-20%

-10%

0%

10%

20%

f(x) = 1.23994673597531 x + 0.0028965722635326R² = 0.876260867381232



on Magellan (%)



The Market Model vs. CAPM

(See columns J-N) SUMMARY OUTPUTBeta



ObservatioIntercept

Coefficient -0.00844 ANOVAt statistic -1.03

Probability of t stat. 30.7% RegressionLower 95% confidence interval -0.02 ResidualUpper 95% confidence interval 0.01 Total

InterceptX Variable

Table 24-3 Regression Results for Calculating Beta

t Statistic

We have been regressing the stock (or portfolio) returns against the market returns. However, CAPM actually states that we should regress the excess stock returns (the stock return minus the short-term risk free rate) against the excess market returns (the market return minus the short-term risk free rate). We show the graph for such a regression below. Notice that it is virtually identical to the market model regression we used earlier for GE. Since it usually doesn't change the results whether we use the market model to estimate beta instead of the CAPM model, we usually use the market model.

CAPM (excess return) Model Regression Results

Regression Coefficient

Probability of t Statistic

Lower 95% Confidence

Interval

Upper 95% Confidence

Interval

Panel a: General Electric (Market model)

-30% -20% -10% 0% 10% 20% 30%

-30%

-20%

-10%

0%

10%

20%

30%

f(x) = 1.37250955725169 x − 0.00844177561216085R² = 0.564108579100218

Excess Returnson the Market, %

Excess Returnson GE, %



Intercept -0.01 -1.17 0.25 -0.03 0.01Slope 1.37 7.84 0.00 1.02 1.73

Intercept 0.00 0.92 0.36 0.00 0.01Slope 1.24 18.05 0.00 1.10 1.38

Intercept -0.01 -1.03 0.31 -0.02 0.01Slope 1.37 7.72 0.00 1.01 1.73

Note: The market model uses unadjusted returns, the CAPM model uses returns in excess of the risk-free rate.

Peformance Measures for Magellan

Jensen's AlphaIntercept from CAPM regression

4.32% per year1.13 t statistic

26.431% Probability that the intercept is not zero

Sharpe's Reward-to-Variability RatioAverage annual return in excess of risk-free rate divided by standard deviation

Magellan -10.3% divided by 21.1%-0.49

S&P 500 -11.7% divided by 15.9%-0.74

Treynor's Reward-to-Volatility RatioAverage annual return in excess of risk-free rate divided by beta

Magellan -10.3% divided by 1.25-8.2%

S&P 500 -11.7% divided by 1.00-11.7%

Panel b: Magellan Fund (Market model)

Panel c: General Electric (CAPM: Excess returns)

B354

We calculate the intercept using the INTERCEPT function.



SUMMARY OUTPUT

Regression Statistics0.756240873997870.571900259505070.562593743407350.05522582306735

48

df SS MS F Significance F1 0.1874206998 0.1874207 61.451595 5.111E-10

46 0.1402950105 0.003049947 0.3277157103

Coefficients Standard Error t Stat P-value Lower 95%Upper 95%Lower 95.0%Upper 95.0%-0.00943802153614 0.0080662277 -1.170066 0.2480009 -0.025674 0.0067985 -0.025674 0.00679851.37442291807652 0.175329021 7.8391068 5.111E-10 1.0215039 1.7273419 1.0215039 1.7273419

SUMMARY OUTPUT



Regression Statistics0.936088066039320.876260867381230.873570886237350.02163960495883

48


46 0.0215405351 0.000468347 0.1740802176

Coefficients Standard Error t Stat P-value Lower 95%Upper 95%Lower 95.0%Upper 95.0%0.00289657226353 0.0031606588 0.9164457 0.3642129 -0.003466 0.0092586 -0.003466 0.00925861.23994673597531 0.0687006647 18.048541 1.672E-22 1.1016595 1.378234 1.1016595 1.378234



SUMMARY OUTPUT

Regression Statistics0.7510716204865

0.564108579100220.554632678645880.05532185501662

48


46 0.1407833516 0.003060547 0.322978028

Coefficients Standard Error t Stat P-value Lower 95%Upper 95%Lower 95.0%Upper 95.0%-0.00844177561216 0.0081713295 -1.033097 0.3069596 -0.02489 0.0080063 -0.02489 0.00800631.37250955725169 0.1778870241 7.7156249 7.789E-10 1.0144416 1.7305776 1.0144416 1.7305776



LINEST Results: y=mx+bRead me.Slope (m) 1.2455775652 0.0036019 Intercept (b)Std. Error of m 0.0693868854 0.0031873 Std. Error of b

0.8750833108 0.0215789 Std. Error of yF 322.24543068 46 Degrees of freedomSS Regression 0.1500536221 0.0214199 SS Residual

t-stat for slope 17.951195801 1.1300673 t-stat for interceptProb of t 2.080485E-22 0.2643061 Prob of t

R2

K349

The LINEST function returns several statistics from a regression. The results are shown in the shaded box. Using these results, we calculated the t-statistics and the probability of the t-statistic, as shown in red.

SECTION 24.1SOLUTIONS TO SELF-TEST

Stock A: expected return 10%Stock A: standard deviation 35%Stock B: expected return 15%Stock B: standard deviation 45%Correlation between A and B 0.30

% portfolio in A% portfolio in B 60%

40%

Portfolio: expected return 12.0%Portfolio: standard deviation 31.5%

Stock A has an expected return of 10 percent and a standard deviation of 35 percent. Stock B has an expected return of 15 percent and a standard deviation of 45 percent. The correlation coefficient between Stock A and B is 0.3. What are the expected return and standard deviation of a portfolio invested 60 percent in Stock A and 40 percent in Stock B?

Stock A has an expected return of 10 percent and a standard deviation of 35 percent. Stock B has an expected return of 15 percent and a standard deviation of 45 percent. The correlation coefficient between Stock A and B is 0.3. What are the expected return and standard deviation of a portfolio invested 60 percent in Stock A and 40 percent in Stock B?


Standard deviation of Park 60%Standard deviation of market 20%Correlation between Park and market 0.40

Park's beta 1.20

The standard deviation of stock returns of Park Corporation is 60 percent. The standard deviation of the market return is 20 percent. If the correlation between Park and the market is 0.40, what is Park's beta?

The standard deviation of stock returns of Park Corporation is 60 percent. The standard deviation of the market return is


Risk-free rate 5%

10%

15%

0.50

1.30

Brown's required return 20.50%

An analyst has modeled the stock of Brown Kitchen Supplies using a two-factor APT model. The risk-free rate is 5 percent, the required return on the first factor (r1) is 10 percent and the required return on the second factor (r2) is 15 percent. If bi1 = 0.5 and bi2 = 1.3, what is Brown's required return?

r1

r2

b1

b2

An analyst has modeled the stock of Brown Kitchen Supplies using a two-factor APT model. The risk-free rate is 5 ) is 10 percent and the required return on the second factor (r2) is 15


Risk-free rate 5.0%

11.0%

3.2%

4.8%

0.0%

0.70

1.20

0.70

Required return 16.40%

An analyst has modeled the stock of a company using a Fama-French three-factor model. The risk-free rate is 5 percent, the required market return is 11 percent, the risk premium for small stocks (rSMB) is 3.2 percent, and the risk premium for value stocks (rHML) is 4.8 percent. If ai = 0, bi = 0.7, ci = 1.2, and di = 0.7, what is the stock's required return?

rM

rSMB

rHML

ai

bi

ci

di

An analyst has modeled the stock of a company using a Fama-French three-factor model. The risk-free rate is 5 SMB) is 3.2 percent, and the risk

= 0.7, what is the stock's required return?

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