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.

>Huh?
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?
Yes.
 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 ...

Of course. It looks like this:

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:
 1 AAXJ 2.09% iShares MSCI All Country Asia ex Japan Index Fund 2 ACWI 0.33% iShares MSCI ACWI (All Country World Index) Index Fund 3 ACWX 0.41% iShares MSCI ACWI ex US Index Fund 4 AFK 0.97% Market Vectors-Africa Index ETF 5 AIA 1.87% iShares S&P Asia 50 6 BIK 2.99% SPDR S&P BRIC 40 7 BJK 0.34% Market Vectors Gaming ETF 8 BKF 3.14% iShares MSCI BRIC Index Fund 9 CGW 0.52% Claymore S&P Global Water Index ETF 10 CRO -0.03% Claymore/Zacks Country Rotation ETF 11 CUT 0.67% Claymore/Beacon Global Timber Index ETF 12 CWI 0.46% SPDR MSCI ACWI ex-US ETF 13 DBN 1.02% WisdomTree International Basic Materials Sector Fund 14 DBR -0.50% WisdomTree International Health Care Sector Fund 15 DBT -0.31% WisdomTree International Technology Sector Fund 16 DBU -1.70% WisdomTree International Utilities Sector Fund 17 DDI 0.54% WisdomTree International Industrial Sector Fund 18 DEB -0.12% WisdomTree Europe Total Dividend Fund 19 DEM 1.70% WisdomTree Emerging Markets High-Yielding Fund 20 DEW 0.04% WisdomTree Europe High-Yielding Equity Fund 21 DFE 1.62% WisdomTree Europe SmallCap Dividend Fund 22 DFJ -1.40% WisdomTree Japan SmallCap Fund 23 DGG -0.73% WisdomTree International Communications Sector Fund 24 DGS 3.01% WisdomTree Emerging Market SmallCap Fund 25 DGT 0.03% SPDR DJ Global Titans ETF 26 DIM 0.31% WisdomTree International MidCap Dividend Fund 27 DKA 0.89% WisdomTree International Energy Sector Funds 28 DLS 0.63% WisdomTree International SmallCap Fund 29 DND 1.48% WisdomTree Pacific ex-Japan Total Dividend Fund 30 DNH 2.25% WisdomTree Pacific ex-Japan High-Yielding Equity Fund 31 DNL -0.73% WisdomTree Japan High-Yielding Equity Fund 32 DOL -0.10% WisdomTree International LargeCap Dividend Fund 33 DOO 0.31% WisdomTree International Dividend Top 100 Fund 34 DPC 0.49% WisdomTree International Consumer Cyclical Sector Fund 35 DPN 0.92% WisdomTree International Consumer Non-Cyclical Sector Fund 36 DRF 0.20% WisdomTree International Financial Sector Fund 37 DRW 0.56% WisdomTree International Real Estate Fund 38 DTH 0.28% WisdomTree DEFA High-Yielding Equity Fund 39 DWM -0.03% WisdomTree DEFA Fund 40 DWX 1.55% SPDR S&P International Dividend ETF 41 DXJ -1.56% WisdomTree Japan Total Dividend Fund 42 ECH 4.85% iShares MSCI Chile Index Fund 43 EEB 3.31% Claymore/BNY Mellon BRIC ETF 44 EEM 2.17% iShares MSCI Emerging Index Fund 45 EEN 0.35% Claymore/BNY Mellon EW Euro-Pacific LDRs ETF 46 EEV -7.35% ProShares UltraShort MSCI Emerging Markets 47 EFA -0.09% iShares MSCI EAFE Index Fund 48 EFG -0.03% iShares MSCI Growth Index Fund 49 EFU -1.54% ProShares UltraShort MSCI EAFE 50 EFV -0.16% iShares MSCI Value Index Fund 51 EIS 3.90% iShares MSCI Israel Capped Investable Market Index Fund 52 ENY 1.77% Claymore/SWM Canadian Energy Income Index ETF 53 EPI 3.28% WisdomTree India Earnings Fund 54 EPP 1.78% iShares MSCI Pacific Ex-Japan Index Fund 55 EUM -3.55% ProShares Short MSCI Emerging Markets 56 EWA 1.99% iShares MSCI Australia Index Fund 57 EWC 1.57% iShares MSCI Canada Index Fund 58 EWD 1.71% iShares MSCI Sweden Index Fund 59 EWG -0.91% iShares MSCI Germany Index Fund 60 EWH 2.08% iShares MSCI Hong Kong Index Fund 61 EWI -0.75% iShares MSCI Italy Index Fund 62 EWJ -1.44% iShares MSCI Japan Index Fund 63 EWK 1.65% iShares MSCI Belgium Index Fund 64 EWL -0.34% iShares MSCI Switzerland Index Fund 65 EWM 2.24% iShares MSCI Malaysia Index Fund 66 EWN 0.87% iShares MSCI Netherlands Index Fund 67 EWO 1.56% iShares MSCI Austria Index Fund 68 EWP 0.15% iShares MSCI Spain Index Fund 69 EWQ -0.17% iShares MSCI France Index Fund 70 EWS 2.13% iShares MSCI Singapore Index Fund 71 EWT 2.39% iShares MSCI Taiwan Index Fund 72 EWU 0.71% iShares MSCI United Kingdom Index Fund 73 EWV 1.60% ProShares UltraShort MSCI Japan 74 EWW 1.45% iShares MSCI Mexico Index Fund 75 EWX 3.27% SPDR S&P Emerging Markets Small Cap ETF 76 EWY 1.77% iShares MSCI South Korea Index Fund 77 EWZ 4.99% iShares MSCI Brazil Index Fund 78 EXB 0.35% Claymore/Beacon Global Exchanges Brokers & Assets Managers Index ETF 79 EXI -0.35% iShares S&P Global Industrials Sector Index Fund 80 EZA 1.16% iShares MSCI South Africa Index Fund 81 EZU -0.33% iShares MSCI EMU Index Fund 82 FAN -0.27% First Trust ISE Global Wind Energy ETF 83 FCHI 1.45% iShares FTSE China (HK Listed) Index Fund 84 FDD 0.99% First Trust Dow Jones STOXX Select Dividend 30 Index Fund 85 FEU -0.04% SPDR DJ STOXX 50 ETF 86 FEZ -0.46% SPDR DJ EURO STOXX 50 ETF 87 FFR -0.27% First Trust FTSE EPRA/NAREIT Global Real Estate Index Fund 88 FGD 1.40% First Trust DJ Global Select Dividend 89 FLM -0.60% First Trust ISE Global Engineering and Construction Index Fund 90 FRN 1.90% Claymore/BNY Mellon Frontier Markets ETF 91 FXI 1.36% iShares FTSE/Xinhua China 25 Index Fund 92 FXP -6.63% ProShares Ultrashort FTSE/Xinhua China 93 GAF 1.56% SPDR S&P Middle East & Africa ETF 94 GCC 0.56% GreenHaven Continuous Commodity Index Fund 95 GEX -1.96% Market Vectors Global Alternative Energy ETF 96 GII -0.97% SPDR FTSE/Macquarie GI 100 ETF 97 GMF 2.22% SPDR S&P Emerging Asia Pacific ETF 98 GML 4.14% SPDR S&P Emerging Latin America ETF 99 GMM 2.33% SPDR S&P Emerging Markets ETF 100 GRI 0.07% Cohen & Steers Global Realty Majors ETF 101 GULF -1.02% WisdomTree Middle East Dividend ETF 102 GUR 2.61% SPDR S&P Emerging Europe ETF 103 GWL -0.08% SPDR S&P World ex-US ETF 104 GWX 0.63% SPDR S&P International SmallCap ETF 105 GXC 2.02% SPDR S&P China ETF 106 HAO 3.53% Claymore/AlphaShares China Small Cap Index ETF 107 HAP 1.33% Market Vectors Hard Assets Producers ETF 108 HGI 1.40% Claymore/Zacks International Multi-Asset Income Index ETF 109 ICLN -1.99% iShares S&P Global Clean Energy Index Fund 110 IDV 1.73% iShares Dow Jones EPAC Select Dividend 111 IEV -0.02% iShares S&P Europe 350 Index Fund 112 IFAS 0.36% iShares FTSE EPRA/NAREIT Asia Index Fund 113 IFEU 0.19% iShares FTSE EPRA/NAREIT Europe Index Fund 114 IFGL 0.60% iShares FTSE EPRA/NAREIT Global Real Estate ex-U.S. Index Fund 115 IFSM 0.90% iShares FTSE Developed Small Cap ex-North America Index Fund 116 IGF -0.55% iShares S&P Global Infrastructure 117 ILF 3.59% iShares Latin America 40 Index Fund 118 IOO -0.15% iShares S&P Global 100 Index Fund 119 IPD 2.61% SPDR S&P International Consumer Discretionary Sector ETF 120 IPF 0.14% SPDR S&P International Financial Sector ETF 121 IPK 0.93% SPDR S&P International Technology Sector ETF 122 IPN 1.35% SPDR S&P International Industrial Sector ETF 123 IPS 0.33% SPDR S&P International Consumer Staples Sector ETF 124 IPU -1.27% SPDR S&P International Utilities Sector ETF 125 IPW 1.09% SPDR S&P International Energy Sector ETF 126 IRV 1.63% SPDR S&P International Materials Sector ETF 127 IRY 0.04% SPDR S&P International Health Care Sector ETF 128 IST -0.61% SPDR S&P International Telecommunications Sector ETF 129 ITF -1.51% iShares S&P/TOPIX 150 Index Fund 130 IXC 0.35% iShares S&P Global Energy Index Fund 131 IXG -0.55% iShares S&P Global Financial Index Fund 132 IXJ -0.17% iShares S&P Global Healthcare Index Fund 133 IXN 1.88% iShares S&P Global Technology Index Fund 134 IXP -0.46% iShares S&P Global Telecommunications Index Fund 135 JPP -1.62% SPDR Russell/Nomura PRIME Japan ETF 136 JSC -1.35% SPDR Russell/Nomura Small Cap Japan 137 JXI -1.10% iShares S&P Global Utilities Sector Index Fund 138 KOL 5.16% Market Vectors Coal ETF 139 KWT -1.72% Market Vectors Solar Energy ETF 140 KXI 0.21% iShares S&P Global Consumer Staples Sector Index Fund 141 LVL 1.96% Claymore/S&P Global Dividend Opportunities Index ETF 142 MDD 0.33% SPDR S&P International Mid Cap ETF 143 MES -1.45% Market Vectors Gulf States Index ETF 144 MOO 2.54% Market Vectors-Agribusiness ETF 145 MXI 1.75% iShares S&P Global Materials Sector Index Fund 146 NLR -0.76% Market Vectors-Nuclear Energy ETF 147 NUCL 0.96% iShares S&P Global Nuclear Energy Index Fund 148 ONEQ 1.42% Fidelity Nasdaq Composite Index Tracking Stock 149 PAF 2.12% PowerShares FTSE RAFI Asia Pacific ex-Japan Portfolio 150 PAGG 2.24% PowerShares Global Agriculture Portfolio 151 PAO -0.11% PowerShares Autonomic Balanced Growth NFA Global Asset Portfolio 152 PBD -0.04% PowerShares Global Clean Energy Portfolio 153 PBP 0.57% PowerShares S&P 500 BuyWrite Portfolio 154 PBTQ -0.41% PowerShares Global Biotech Portfolio 155 PCA -0.17% PowerShares Autonomic Balanced NFA Global Asset Portfolio 156 PDN 1.52% PowerShares FTSE RAFI Developed Markets ex-U.S. Small-Mid Portfolio 157 PEF 0.78% PowerShares FTSE RAFI Europe Portfolio 158 PFA 0.40% PowerShares Dynamic Developed International Opportunities Portfolio 159 PFF -0.08% iShares S&P US Preferred Stock Fund 160 PGF -0.17% PowerShares Financial Preferred 161 PGJ 1.99% PowerShares Golden Dragon Halter USX China Portfolio 162 PIE 1.40% PowerShares DWA Emerging Market Technical Leaders? Portfolio 163 PIN 2.64% PowerShares India Portfolio 164 PIO 0.71% PowerShares Global Water Portfolio 165 PIZ 0.49% PowerShares DWA Developed Market Technical Leaders Portfolio 166 PJO -1.74% PowerShares FTSE RAFI Japan Portfolio 167 PKN 0.03% PowerShares Global Nuclear Portfolio 168 PKOL 4.92% PowerShares Global Coal Portfolio 169 PMNA -0.99% PowerShares MENA Frontier Countries Portfolio 170 PQBW 2.29% PowerShares NASDAQ-100 BuyWrite Portfolio 171 PSAU 1.91% PowerShares Global Gold and Precious Metals Portfolio 172 PSP -0.49% PowerShares Listed Private Equity 173 PSTL 1.89% PowerShares Global Steel Portfolio 174 PTO 0.05% PowerShares Autonomic Growth NFA Global Asset Portfolio 175 PTRP 1.36% PowerShares Global Progressive Transportation Portfolio 176 PWND 0.35% PowerShares Global Wind Energy Portfolio 177 PXF 0.14% PowerShares FTSE RAFI Developed Markets ex-U.S. Portfolio 178 PXH 2.01% PowerShares FTSE RAFI Emerging Markets Portfolio 179 PXR 2.99% PowerShares Global Emerging Markets Infrastructure Portfolio 180 PZD 0.41% PowerShares Cleantech Portfolio 181 ROB 1.12% Claymore/Robb Report Global Luxury Index ETF 182 RSX 5.64% Market Vectors TR Russia ETF 183 RWO -0.20% SPDR DJ Wilshire Global Real Estate ETF 184 RWX 0.17% SPDR DJ Wilshire Intl Real Estate 185 RXI 0.72% iShares S&P Global Consumer Discretionary Sector Index Fund 186 SCJ -1.45% iShares MSCI Japan Sm Cap 187 SCZ 0.89% iShares MSCI EAFE Small Cap 188 SEA 0.19% Claymore/Delta Global Shipping Index ETF 189 TAN -1.43% Claymore/MAC Global Solar Energy Index ETF 190 TAO 2.42% Claymore/AlphaShares China Real Estate ETF 191 TBH -2.59% HOLDRS TeleBras 192 THD 3.55% iShares MSCI Thailand Investable Market Index Fund 193 TOK 0.42% iShares MSCI Kokusai 194 TUR 3.77% iShares MSCI Turkey Investable Market Index Fund 195 VEA -0.09% Vanguard Europe Pacific 196 VEU 0.51% Vanguard FTSE All World Ex US ETF 197 VGK 0.21% Vanguard European ETF 198 VPL -0.46% Vanguard Pacific ETF 199 VT 0.37% Vanguard Total World Stock Index 200 VWO 2.74% Vanguard Emerging Markets ETF 201 WOOD -0.05% iShares S&P Global Timber & Forestry Index Fund 202 WPS 0.17% iShares S&P World ex-US Property