Jensen's Alpha
motivated by e-mail from Ron Mc.

There's this thing called Jensen's alpha, designed to measure how well a mutual fund manager does ... compared to "the market".

>The market?
Yeah, maybe the Wilshire 5000, maybe the DOW, maybe the S&P500, maybe ...
>And does Jensen's thing work?
Define "work".

Anyway, let's take a look at Jensen's alpha (invented my Michael Jensen in the 1970s):
  • We collect the returns for some stock (or mutual fund) over the past n months (weeks? years?)
  • We do the same for some market index, say the S&P500.
  • Then, using these returns, we generate a scatter plot and identify the slope of the "best line" fit: that's called beta.
  • Then we calculate the CAPM according to the magic formula:
    CAPM Return = Rf + beta (Rmkt - Rf)
    where Rf = risk-free return and Rmkt = (average or "expected") market return
  • CAPM (presumably) gives the "expected" stock return ... unless that stock didn't behave like the market as a whole.

If the stock behaved "like the market", then we'd expect beta = 1, meaning a 1% change in the market return would be reflected in a 1% change in the stock return.
After all, the slope of that "best line fit" is (change in GE return) / (change in mkt return).
In Figure 1, since beta is significantly larger than 1, it seems that market changes are amplified in GE returns.
Indeed, for the situation in Fig. 1 we have:
beta = 1.470, Rmkt = 0.117% and (let's assume) Rf = 0.01% so CAPM return = 0.01+ 1.470 (0.117 - 0.01) = 0.167%

Figure 1 (using daily returns over one year)
>And if the stock did NOT behave like the market?
Aah, that's where Jensen comes in.
Now we take a look at the (average or "expected") stock return to see if'n it did much better (or worse) than would be the case if it just followed market machinations.
To do that, we ...

>You calculate how much better the stock did, right?
Jensen's alpha = Rstock - [ Rf + beta (Rmkt - Rf)]
where Rstock = (average or "expected") stock return
CAPM = 0.167% says that we'd expect the (daily) returns of GE stock to average 0.167% if it just followed the market.
However, if we calculate the actual (average) GE return over the past year, we get 0.066%.

>So GE underperformed the market, right?
That's how it'd be interpreted. In fact, we'd get J-alpha = Rstock - CAPM = 0.066 - 0.167 = - 0.101% and that ain't good.

>Uh ... but I can see that from the chart in Fig. 2 so why that ritual to calculate J-alpha?
You asking me? I just write about these things. I don't have to understand 'em.
Our conclusion about how well (or poorly) a stock or fund did (compared to "the market"), will depend upon beta.
In our GE vs S&P example, I've used daily returns over the past year.
One normally uses monthly returns ... maybe over several years.
However, beta depends upon whether you take end-of-month returns or mid-month returns ... or whatever.

For example, check out Beta here, Beta there.

Figure 2 : performance for year 2009
It's worthwhikle noting that beta is also a measure of correlation and volatilities of the two stocks involved.
If GE, for example, had a high correlation with the market (which is usually does) and its volatility is significantly larger than the market volatility, then beta will be large.
That's because:
    Beta[x,y] = r SD[y] / SD[x]
    where r is the (Pearson) correlation and SD[y]/SD[x] is the ratio of Standard Deviations.

Another thing: we look at historical returns. That tells us what happened in the past, but does it tell us anything about what'll happen in the future?
Further, the derivation of the magic formula for the Capital Asset Pricing Model (CAPM) involves the standard deviation of returns as the measure of risk.
I really dislike that association. Besides ...

>Do you have a spreadsheet?
Of course. It looks like this:

Click the picture to download

Note the following:

  1. Three years worth of data is downloaded from Yahoo and monthly returns are calculated.
  2. The values of beta and CAPM and the mean stock return Rstock are calculated.
    In the above example (GE vs S&P): beta = 1.688, CAPM = -1.07%, annual risk-free = 4% (so monthly rate = 0.33%) and mean GE return = -1.66%
  3. Then Jensen's alpha is calculated. (In the above example: J-alpha = - 0.59%)
Note: If you'd like to consider only deviations from the Rf (the risk-free return), stick a y into cell O5.
Then, the risk-free return is subtracted from all returns and beta is calculated as the slope of the modified regression line.

Note that, if J-alpha = 0%, then (Rstock - Rf) / (Rmkt - Rf) = beta.

Note, too, that the spreadsheet uses the S&P500 as the "market".
If you calculate the S&P500 vs the S&P500, you get (surprise!) J-alpha = 0%.
If you calculate the DOW vs the S&P500, you get J-alpha = 0.07%.

I might point out the Jensen's "alpha" ain't the same as the intercept of the regression line which is also called alpha ?!#$@!*?

>So what do you conclude?
GE didn't do good over the past three years ... compared to the S&P500.
However, note something interesting: monthly GE stock returns amplified changes in market returns, 'cause beta = 1.688 was pretty large.
In other words, if an S&P return changed by X% then we'd expect GE return (for the same period) to change by 1.688X%.
That's the interpretation of the regression line: that "best line fit" in the right graph.

>So you'd expect that to hold in the future?
Me? Of course not.
In fact, if you change the End Date for the data download (in cell B3) to a few years earlier your conclusion may be quite different.
For example, GE did worser than the S&P over the period 2004-2006, yet J-alpha was positive.
So don't think that positive J-alpha means the stock done good, compared to the market.

>So what do you think?
Me? Think?

I should mention that, at this particular time and this particular "definition" of J-alpha (i.e. using monthly returns over three years),
our friend GE has the worsest J-alpha on the DOW.

Did I mention that (somewhere on the spreadsheet) you can calculate J-alpha for a gaggle of stocks?

P.S. Here's a list of J-alphas for a couple of hundred ETFs ... compliments of Ron McEwan:
1AAXJ2.09%iShares MSCI All Country Asia ex Japan Index Fund
2ACWI0.33%iShares MSCI ACWI (All Country World Index) Index Fund
3ACWX0.41%iShares MSCI ACWI ex US Index Fund
4AFK0.97%Market Vectors-Africa Index ETF
5AIA1.87%iShares S&P Asia 50
7BJK0.34%Market Vectors Gaming ETF
8BKF3.14%iShares MSCI BRIC Index Fund
9CGW0.52%Claymore S&P Global Water Index ETF
10CRO-0.03%Claymore/Zacks Country Rotation ETF
11CUT0.67%Claymore/Beacon Global Timber Index ETF
13DBN1.02%WisdomTree International Basic Materials Sector Fund
14DBR-0.50%WisdomTree International Health Care Sector Fund
15DBT-0.31%WisdomTree International Technology Sector Fund
16DBU-1.70%WisdomTree International Utilities Sector Fund
17DDI0.54%WisdomTree International Industrial Sector Fund
18DEB-0.12%WisdomTree Europe Total Dividend Fund
19DEM1.70%WisdomTree Emerging Markets High-Yielding Fund
20DEW0.04%WisdomTree Europe High-Yielding Equity Fund
21DFE1.62%WisdomTree Europe SmallCap Dividend Fund
22DFJ-1.40%WisdomTree Japan SmallCap Fund
23DGG-0.73%WisdomTree International Communications Sector Fund
24DGS3.01%WisdomTree Emerging Market SmallCap Fund
25DGT0.03%SPDR DJ Global Titans ETF
26DIM0.31%WisdomTree International MidCap Dividend Fund
27DKA0.89%WisdomTree International Energy Sector Funds
28DLS0.63%WisdomTree International SmallCap Fund
29DND1.48%WisdomTree Pacific ex-Japan Total Dividend Fund
30DNH2.25%WisdomTree Pacific ex-Japan High-Yielding Equity Fund
31DNL-0.73%WisdomTree Japan High-Yielding Equity Fund
32DOL-0.10%WisdomTree International LargeCap Dividend Fund
33DOO0.31%WisdomTree International Dividend Top 100 Fund
34DPC0.49%WisdomTree International Consumer Cyclical Sector Fund
35DPN0.92%WisdomTree International Consumer Non-Cyclical Sector Fund
36DRF0.20%WisdomTree International Financial Sector Fund
37DRW0.56%WisdomTree International Real Estate Fund
38DTH0.28%WisdomTree DEFA High-Yielding Equity Fund
39DWM-0.03%WisdomTree DEFA Fund
40DWX1.55%SPDR S&P International Dividend ETF
41DXJ-1.56%WisdomTree Japan Total Dividend Fund
42ECH4.85%iShares MSCI Chile Index Fund
43EEB3.31%Claymore/BNY Mellon BRIC ETF
44EEM2.17%iShares MSCI Emerging Index Fund
45EEN0.35%Claymore/BNY Mellon EW Euro-Pacific LDRs ETF
46EEV-7.35%ProShares UltraShort MSCI Emerging Markets
47EFA-0.09%iShares MSCI EAFE Index Fund
48EFG-0.03%iShares MSCI Growth Index Fund
49EFU-1.54%ProShares UltraShort MSCI EAFE
50EFV-0.16%iShares MSCI Value Index Fund
51EIS3.90%iShares MSCI Israel Capped Investable Market Index Fund
52ENY1.77%Claymore/SWM Canadian Energy Income Index ETF
53EPI3.28%WisdomTree India Earnings Fund
54EPP1.78%iShares MSCI Pacific Ex-Japan Index Fund
55EUM-3.55%ProShares Short MSCI Emerging Markets
56EWA1.99%iShares MSCI Australia Index Fund
57EWC1.57%iShares MSCI Canada Index Fund
58EWD1.71%iShares MSCI Sweden Index Fund
59EWG-0.91%iShares MSCI Germany Index Fund
60EWH2.08%iShares MSCI Hong Kong Index Fund
61EWI-0.75%iShares MSCI Italy Index Fund
62EWJ-1.44%iShares MSCI Japan Index Fund
63EWK1.65%iShares MSCI Belgium Index Fund
64EWL-0.34%iShares MSCI Switzerland Index Fund
65EWM2.24%iShares MSCI Malaysia Index Fund
66EWN0.87%iShares MSCI Netherlands Index Fund
67EWO1.56%iShares MSCI Austria Index Fund
68EWP0.15%iShares MSCI Spain Index Fund
69EWQ-0.17%iShares MSCI France Index Fund
70EWS2.13%iShares MSCI Singapore Index Fund
71EWT2.39%iShares MSCI Taiwan Index Fund
72EWU0.71%iShares MSCI United Kingdom Index Fund
73EWV1.60%ProShares UltraShort MSCI Japan
74EWW1.45%iShares MSCI Mexico Index Fund
75EWX3.27%SPDR S&P Emerging Markets Small Cap ETF
76EWY1.77%iShares MSCI South Korea Index Fund
77EWZ4.99%iShares MSCI Brazil Index Fund
78EXB0.35%Claymore/Beacon Global Exchanges Brokers & Assets Managers Index ETF
79EXI-0.35%iShares S&P Global Industrials Sector Index Fund
80EZA1.16%iShares MSCI South Africa Index Fund
81EZU-0.33%iShares MSCI EMU Index Fund
82FAN-0.27%First Trust ISE Global Wind Energy ETF
83FCHI1.45%iShares FTSE China (HK Listed) Index Fund
84FDD0.99%First Trust Dow Jones STOXX Select Dividend 30 Index Fund
87FFR-0.27%First Trust FTSE EPRA/NAREIT Global Real Estate Index Fund
88FGD1.40%First Trust DJ Global Select Dividend
89FLM-0.60%First Trust ISE Global Engineering and Construction Index Fund
90FRN1.90%Claymore/BNY Mellon Frontier Markets ETF
91FXI1.36%iShares FTSE/Xinhua China 25 Index Fund
92FXP-6.63%ProShares Ultrashort FTSE/Xinhua China
93GAF1.56%SPDR S&P Middle East & Africa ETF
94GCC0.56%GreenHaven Continuous Commodity Index Fund
95GEX-1.96%Market Vectors Global Alternative Energy ETF
96GII-0.97%SPDR FTSE/Macquarie GI 100 ETF
97GMF2.22%SPDR S&P Emerging Asia Pacific ETF
98GML4.14%SPDR S&P Emerging Latin America ETF
99GMM2.33%SPDR S&P Emerging Markets ETF
100GRI0.07%Cohen & Steers Global Realty Majors ETF
101GULF-1.02%WisdomTree Middle East Dividend ETF
102GUR2.61%SPDR S&P Emerging Europe ETF
103GWL-0.08%SPDR S&P World ex-US ETF
104GWX0.63%SPDR S&P International SmallCap ETF
105GXC2.02%SPDR S&P China ETF
106HAO3.53%Claymore/AlphaShares China Small Cap Index ETF
107HAP1.33%Market Vectors Hard Assets Producers ETF
108HGI1.40%Claymore/Zacks International Multi-Asset Income Index ETF
109ICLN-1.99%iShares S&P Global Clean Energy Index Fund
110IDV1.73%iShares Dow Jones EPAC Select Dividend
111IEV-0.02%iShares S&P Europe 350 Index Fund
112IFAS0.36%iShares FTSE EPRA/NAREIT Asia Index Fund
113IFEU0.19%iShares FTSE EPRA/NAREIT Europe Index Fund
114IFGL0.60%iShares FTSE EPRA/NAREIT Global Real Estate ex-U.S. Index Fund
115IFSM0.90%iShares FTSE Developed Small Cap ex-North America Index Fund
116IGF-0.55%iShares S&P Global Infrastructure
117ILF3.59%iShares Latin America 40 Index Fund
118IOO-0.15%iShares S&P Global 100 Index Fund
119IPD2.61%SPDR S&P International Consumer Discretionary Sector ETF
120IPF0.14%SPDR S&P International Financial Sector ETF
121IPK0.93%SPDR S&P International Technology Sector ETF
122IPN1.35%SPDR S&P International Industrial Sector ETF
123IPS0.33%SPDR S&P International Consumer Staples Sector ETF
124IPU-1.27%SPDR S&P International Utilities Sector ETF
125IPW1.09%SPDR S&P International Energy Sector ETF
126IRV1.63%SPDR S&P International Materials Sector ETF
127IRY0.04%SPDR S&P International Health Care Sector ETF
128IST-0.61%SPDR S&P International Telecommunications Sector ETF
129ITF-1.51%iShares S&P/TOPIX 150 Index Fund
130IXC0.35%iShares S&P Global Energy Index Fund
131IXG-0.55%iShares S&P Global Financial Index Fund
132IXJ-0.17%iShares S&P Global Healthcare Index Fund
133IXN1.88%iShares S&P Global Technology Index Fund
134IXP-0.46%iShares S&P Global Telecommunications Index Fund
135JPP-1.62%SPDR Russell/Nomura PRIME Japan ETF
136JSC-1.35%SPDR Russell/Nomura Small Cap Japan
137JXI-1.10%iShares S&P Global Utilities Sector Index Fund
138KOL5.16%Market Vectors Coal ETF
139KWT-1.72%Market Vectors Solar Energy ETF
140KXI0.21%iShares S&P Global Consumer Staples Sector Index Fund
141LVL1.96%Claymore/S&P Global Dividend Opportunities Index ETF
142MDD0.33%SPDR S&P International Mid Cap ETF
143MES-1.45%Market Vectors Gulf States Index ETF
144MOO2.54%Market Vectors-Agribusiness ETF
145MXI1.75%iShares S&P Global Materials Sector Index Fund
146NLR-0.76%Market Vectors-Nuclear Energy ETF
147NUCL0.96%iShares S&P Global Nuclear Energy Index Fund
148ONEQ1.42%Fidelity Nasdaq Composite Index Tracking Stock
149PAF2.12%PowerShares FTSE RAFI Asia Pacific ex-Japan Portfolio
150PAGG2.24%PowerShares Global Agriculture Portfolio
151PAO-0.11%PowerShares Autonomic Balanced Growth NFA Global Asset Portfolio
152PBD-0.04%PowerShares Global Clean Energy Portfolio
153PBP0.57%PowerShares S&P 500 BuyWrite Portfolio
154PBTQ-0.41%PowerShares Global Biotech Portfolio
155PCA-0.17%PowerShares Autonomic Balanced NFA Global Asset Portfolio
156PDN1.52%PowerShares FTSE RAFI Developed Markets ex-U.S. Small-Mid Portfolio
157PEF0.78%PowerShares FTSE RAFI Europe Portfolio
158PFA0.40%PowerShares Dynamic Developed International Opportunities Portfolio
159PFF-0.08%iShares S&P US Preferred Stock Fund
160PGF-0.17%PowerShares Financial Preferred
161PGJ1.99%PowerShares Golden Dragon Halter USX China Portfolio
162PIE1.40%PowerShares DWA Emerging Market Technical Leaders? Portfolio
163PIN2.64%PowerShares India Portfolio
164PIO0.71%PowerShares Global Water Portfolio
165PIZ0.49%PowerShares DWA Developed Market Technical Leaders Portfolio
166PJO-1.74%PowerShares FTSE RAFI Japan Portfolio
167PKN0.03%PowerShares Global Nuclear Portfolio
168PKOL4.92%PowerShares Global Coal Portfolio
169PMNA-0.99%PowerShares MENA Frontier Countries Portfolio
170PQBW2.29%PowerShares NASDAQ-100 BuyWrite Portfolio
171PSAU1.91%PowerShares Global Gold and Precious Metals Portfolio
172PSP-0.49%PowerShares Listed Private Equity
173PSTL1.89%PowerShares Global Steel Portfolio
174PTO0.05%PowerShares Autonomic Growth NFA Global Asset Portfolio
175PTRP1.36%PowerShares Global Progressive Transportation Portfolio
176PWND0.35%PowerShares Global Wind Energy Portfolio
177PXF0.14%PowerShares FTSE RAFI Developed Markets ex-U.S. Portfolio
178PXH2.01%PowerShares FTSE RAFI Emerging Markets Portfolio
179PXR2.99%PowerShares Global Emerging Markets Infrastructure Portfolio
180PZD0.41%PowerShares Cleantech Portfolio
181ROB1.12%Claymore/Robb Report Global Luxury Index ETF
182RSX5.64%Market Vectors TR Russia ETF
183RWO-0.20%SPDR DJ Wilshire Global Real Estate ETF
184RWX0.17%SPDR DJ Wilshire Intl Real Estate
185RXI0.72%iShares S&P Global Consumer Discretionary Sector Index Fund
186SCJ-1.45%iShares MSCI Japan Sm Cap
187SCZ0.89%iShares MSCI EAFE Small Cap
188SEA0.19%Claymore/Delta Global Shipping Index ETF
189TAN-1.43%Claymore/MAC Global Solar Energy Index ETF
190TAO2.42%Claymore/AlphaShares China Real Estate ETF
191TBH-2.59%HOLDRS TeleBras
192THD3.55%iShares MSCI Thailand Investable Market Index Fund
193TOK0.42%iShares MSCI Kokusai
194TUR3.77%iShares MSCI Turkey Investable Market Index Fund
195VEA-0.09%Vanguard Europe Pacific
196VEU0.51%Vanguard FTSE All World Ex US ETF
197VGK0.21%Vanguard European ETF
198VPL-0.46%Vanguard Pacific ETF
199VT0.37%Vanguard Total World Stock Index
200VWO2.74%Vanguard Emerging Markets ETF
201WOOD-0.05%iShares S&P Global Timber & Forestry Index Fund
202WPS0.17%iShares S&P World ex-US Property