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Chapter 11: Estimating Betas and the SML
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In this chapter
Testing the Security Market Line (SML)
Computing betas ()
Importance of the “optimality” of the market portfolio M
CAPM as prescriptive & descriptive
Prescriptive
Tells investor how to pick optimal portfolio
Descriptive
SML describes the relation between risk and return
Relationship is linear (see Props. 3, 4, 5 on following slides)
Descriptive
Large, diversified portfolios regressed on a benchmark have very high R2s – critical for the understanding of investment performance.
Also Important…
For 10 stocks, compute their alpha (), beta (), R2
For a portfolio of the 10 stocks, compute its alpha, beta, R2
Results: where xi is portfolio weight of stock i
From above, diversified portfolio is highly dependent on the benchmark index than average dependence of the individual assets
Example: Stock prices
As shown on the next slide
Regress each asset’s returns against the SP500 returns
Use functions T-intercept and T-slope to test significance of ai and bi
None of the ai are statistically significant
All of the bi are statistically significant
Average R2 = 42% (low, typical for this kind of exercise)
Recall: T-statistic is significant if absolute value > 2
Stock returns regressed against SP500
As shown on the next slide
Create a portfolio of the 10 stocks
In this portfolio, stocks are equally weighted
where xi is weight of asset i
Compute portfolio returns as
Regress portfolio returns against SP500, computing aportfolio, bportfolio, R2portfolio
Compare aportfolio , bportfolio and R2portfolio to the average asset values
Diversified portfolio is highly dependent on the benchmark index than average dependence of the individual assets
Looking more closely at the previous slide
Portfolio alpha = -0.0049 = weighted avg of the individual stock alphas
Portfolio beta = 1.1338 = weighted avg of the individual stock betas
Portfolio R2 = 0.8762, much higher than the weighted average of the individual stock R2s (0.4156).
Stock portfolios are highly dependent on the index!
A stock portfolio has much less idiosyncratic risk than the individual stocks in the portfolio
In a portfolio context, Beta means something!
Prescriptive: If you have a risk-free rate rf, Prop. 1 of Chp. 9 identifies the optimal portfolio
Proposition 1 (Merton 1973): All envelope portfolios solve the following equation:
where S is the variance-covariance matrix and
c is an arbitrary constant
Let c = rf , and solve for the optimal portfolio
Descriptive: Props. 3-5 from Chp. 9
Propositions 3 (Black, 1972): If portfolio y is envelope, then all other portfolios are related to y through the following linear relationship:
Descriptive: Prop. 4 is the converse of Prop. 3
Proposition 4 (Black): If the linear relationship below holds, then portfolio y is an envelope portfolio.
Descriptive: Props. 3,4 if there’s a risk-free asset
Proposition 5: If there exists a risk-free asset with return rf , then the standard security market line (SML) relationship holds:
Testing the SML: the data
Use a data set of prices
Can download price data from various data sources including Yahoo
Derive returns from prices
Testing the SML: The two-pass regression
First-pass regression
Choose a proxy M for the market portfolio
In exercise following: M = S&P500
Compute bi for each asset i
You can use the Excel function Slope
In exercise following we also compute ai and the R2, though this is not necessary
Second-pass regression
Regress average asset return on their beta:
(Long) example: 6 stocks & S&P500
Using prices, calculate returns
Run regression
Second-pass regression
Poor results: if the SP500 was an optimal portfolio for the 6 stocks, then we would have g0 = rf, g1 = E(rM) – rf , R2 = 100% .
This is guaranteed by Propositions 3-5.
Conclusion: S&P500 is not an envelope portfolio in the set of the 6 stocks.
Experiment: Find an envelope portfolio for the 6 stocks and use c = rf = 0.25%
Technology: Prop 1 gives us the formula for the envelope portfolio
Optimal portfolio
We set rf = 0.25% and use Proposition 1 to derive an optimal portfolio.
Assume optimal portfolio is the market; run 1st & 2nd pass regressions
If we substitute an optimal portfolio for the S&P500, then the 2nd pass regression is perfect!
This is guaranteed by Propositions 3-5
ABCDEFGHIJKL
QualcomBoeingKelloggFord
SachsMicrosoft
Technologies
BrandsChevron
DateQCOMBAKFHPQGSMSFTUTXYUMCVXSP500
2-Nov-0941.1146.6945.068.2444.68159.2125.7859.8431.7265.9591.49
1-Dec-0942.2648.2345.69.2646.99158.4426.7261.7731.4565.0793.27
4-Jan-1035.853.9946.6410.0442.94139.5624.7160.0630.9560.9589.91
1-Feb-1033.6656.6645.0210.8846.33147.0525.2561.4930.561.6992.69
1-Mar-1038.565.1346.1211.6448.56160.4825.865.9334.6764.798.26
1-Apr-1035.4964.9747.4312.0647.48136.5626.967.1338.5769.4999.81
3-May-1032.857.9146.4510.8742.03136.0222.8360.7137.2363.6291.83
1-Jun-1030.2956.6143.739.3439.61123.7720.3658.4835.558.4487.01
10 STOCKS AND THE SP500: 5 years of monthly data
ABCDEFGHIJKLM
QualcomBoeingKelloggFord
SachsMicrosoft
Technologies
BrandsChevron
QCOMBAKFHPQGSMSFTUTXYUMCVX
Alpha-0.00390.00410.0001-0.0069-0.0206-0.0159-0.0024-0.00270.0041-0.0046<-- =INTERCEPT(K15:K75,$L$15:$L$75)
Beta1.14051.05840.50381.43941.57191.59631.00411.12170.78941.1124<-- =SLOPE(K15:K75,$L$15:$L$75)
R-squared0.42460.42080.18970.37460.33850.50710.39350.66750.24910.5907<-- =RSQ(K15:K75,$L$15:$L$75)
T-alpha-0.59110.65950.0164-0.7354-1.8636-2.0084-0.3779-0.68340.5935-0.9904<-- =tintercept(K15:K75,$L$15:$L$75)
T-beta6.59786.54723.71625.94505.49477.79106.187510.88334.42379.2280<-- =tslope(K15:K75,$L$15:$L$75)
QualcomBoeingKelloggFord
SachsMicrosoft
Technologies
BrandsChevron
DateQCOMBAKFHPQGSMSFTUTXYUMCVXSP500
1-Dec-092.76%3.25%1.19%11.67%5.04%-0.48%3.58%3.17%-0.85%-1.34%1.93%
4-Jan-10-16.59%11.28%2.26%8.09%-9.01%-12.69%-7.82%-2.81%-1.60%-6.54%-3.67%
1-Feb-10-6.16%4.83%-3.54%8.03%7.60%5.23%2.16%2.35%-1.46%1.21%3.05%
1-Mar-1013.43%13.93%2.41%6.75%4.70%8.74%2.15%6.97%12.81%4.76%5.84%
1-Apr-10-8.14%-0.25%2.80%3.54%-2.25%-16.14%4.18%1.80%10.66%7.14%1.57%
10 STOCKS AND THE SP500
Regressing returns on the SP500
DateBAKFHPQGSMSFTSP500
2-Nov-0946.6945.068.2444.68159.2125.7891.49
1-Dec-0948.2345.609.2646.99158.4426.7293.27
4-Jan-1053.9946.6410.0442.94139.5624.7189.91
1-Feb-1056.6645.0210.8846.33147.0525.2592.69
1-Mar-1065.1346.1211.6448.56160.4825.8098.26
1-Apr-1064.9747.4312.0647.48136.5626.9099.81
3-May-1057.9146.4510.8742.03136.0222.8391.83
1-Jun-1056.6143.739.3439.61123.7720.3687.01
1-Jul-1061.4843.5111.8342.13142.2022.8493.10
2-Aug-1055.4943.5410.4635.19129.4320.8888.88
6 STOCKS AND THE SP500
5 YEARS OF MONTHLY DATA
First-pass
regression
BAKFHPQGSMSFT
Average return1.68%0.61%1.04%-0.18%0.32%0.97%<-- =AVERAGE(G9:G69)
Alpha0.00410.0001-0.0069-0.0206-0.0159-0.0024<-- =INTERCEPT(G9:G69,$H$9:$H$69)
Beta1.05840.50381.43941.57191.59631.0041<-- =SLOPE(G9:G69,$H$9:$H$69)
R-squared0.42080.18970.37460.33850.50710.3935<-- =RSQ(G9:G69,$H$9:$H$69)
DateBAKFHPQGSMSFTSP500
1-Dec-093.25%1.19%11.67%5.04%-0.48%3.58%1.93%<-- =LN('Stock data'!H4/'Stock data'!H3)
4-Jan-1011.28%2.26%8.09%-9.01%-12.69%-7.82%-3.67%<-- =LN('Stock data'!H5/'Stock data'!H4)
1-Feb-104.83%-3.54%8.03%7.60%5.23%2.16%3.05%
1-Mar-1013.93%2.41%6.75%4.70%8.74%2.15%5.84%
1-Apr-10-0.25%2.80%3.54%-2.25%-16.14%4.18%1.57%
3-May-10-11.50%-2.09%-10.39%-12.19%-0.40%-16.41%-8.33%
6 STOCKS AND THE SP500
5 YEARS OF RETURNS, FIRST-PASS REGRESSION
BAKFHPQGSMSFT
Average return1.68%0.61%1.04%-0.18%0.32%0.97%
Alpha0.00410.0001-0.0069-0.0206-0.0159-0.0024
Beta1.05840.50381.43941.57191.59631.0041
R-squared0.42080.18970.37460.33850.50710.3935
Second-pass regression
0.0144<-- =INTERCEPT(B4:G4,B6:G6)
-0.0058<-- =SLOPE(B4:G4,B6:G6)
0.1491<-- =RSQ(B4:G4,B6:G6)
SECOND-PASS REGRESSION: average
BAKFHPQGSMSFT
BA0.00360.00090.00240.00240.00160.00141.68%
K0.00090.00180.00100.00060.00060.00040.61%
F0.00240.00100.00760.00360.00420.00221.04%
HPQ0.00240.00060.00360.01000.00390.0022-0.18%
GS0.00160.00060.00420.00390.00690.00240.32%
MSFT0.00140.00040.00220.00220.00240.00350.97%
<-- {=varcovar(B23:G83)}
Constant, r
HPQ-0.4115
MSFT0.3908
Variance-covariance matrix
<-- {=MMULT(MINVERSE(B4:G9),I4:I9-
B12)/SUM(MMULT(MINVERSE(B4:G9),I4:I9-
6 STOCKS, FINDING AN OPTIMAL PORTFOLIO
ABCDEFGHIJK
First-pass
regression
BAKFHPQGSMSFT
Average return1.68%0.61%1.04%-0.18%0.32%0.97%
Alpha0.00080.00210.00150.00300.00240.0016<-- =INTERCEPT(G16:G76,$H$16:$H$76)
Beta0.69160.17510.3804-0.20630.03510.3463
R-squared0.65810.08390.09590.02140.00090.1714
Second-pass regression
0.0025<-- =INTERCEPT(B4:G4,B6:G6)
0.0207<-- =SLOPE(B4:G4,B6:G6)
1.0000<-- =RSQ(B4:G4,B6:G6)
DateBAKFHPQGSMSFT
1-Dec-093.25%1.19%11.67%5.04%-0.48%3.58%3.91%
4-Jan-1011.28%2.26%8.09%-9.01%-12.69%-7.82%15.81%
1-Feb-104.83%-3.54%8.03%7.60%5.23%2.16%2.59%
1-Mar-1013.93%2.41%6.75%4.70%8.74%2.15%13.11%
1-Apr-10-0.25%2.80%3.54%-2.25%-16.14%4.18%5.47%
3-May-10-11.50%-2.09%-10.39%-12.19%-0.40%-16.41%-14.70%
1-Jun-10-2.27%-6.03%-15.17%-5.93%-9.44%-11.45%-4.26%
6 STOCKS AND THE OPTIMAL PORTFOLIO
RUNNING THE FIRST AND SECOND PASS REGRESSION
{=MMULT(B16:G16,'Optimal
portfolio'!$B$14:$B$19)}
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