Risk and Return in General: Theory and Evidence. Abstract

Transcription

1 Rsk and Return n General: Theory and vdence Abstract mprcally, standard, ntutve measures of rsk lke volatlty and beta do not generate a postve correlaton wth average returns n most asset classes. It s possble that rsk, however defned, s not postvely related to return as an equlbrum n asset markets. Ths paper presents a survey of data across 20 dfferent asset classes, and presents a model hghlghtng the assumptons consstent wth no rsk premum. The key s that when agents are concerned about relatve wealth, rsk takng s then devatng from the consensus or market portfolo. In ths envronment, all rsk becomes lke dosyncratc rsk n the standard model, avodable so unprced. (JL D01, D81, G11, G12) rc Falkensten Mnneapols, MN, USA efalken@gmal.com June

2 To the degree rsk s not dversfable, someone must hold t, and standard utlty theory suggests those who do hold t should be compensated va a rsk premum relatve to rsk-free alternatves. Yet t s strkng that a frst approxmaton to rsk va volatlty or beta aganst the market return generates no postve rsk premums. Consder that assets such as houses have characterstcs that requre compensaton, such as crme or bad schools, and these factors are abstract n a sense, yet ther effects emnently measurable and consstent wth ntuton (Black 1999; Tta, Petras and Greenbaum 2006). Rsk, meanwhle, has devolved nto the fnancal equvalence of dark matter, evdent solely by ts effects. As asset prcng models have ncreased n complexty from the smple one-factor CAPM, to stochastc dscount factor(s) so general, they place almost no restrctons on fnancal data (Campbell (2000)). xplanng asset returns va rsk s often more calbraton than predcton, as when the rsk premums are functons of atheoretcally observed rsk factors (see Da and Sngleton 2002; Fama and French 1992). In my book Fndng Alpha, (Falkensten (2009)), I outlne the emprcal anomales to the standard rsk-reward theory n asset markets, and provdes motvaton for my hypothess that we use relatve status, as opposed to absolute wealth, n utlty functon. The book also addresses the refnements needed to explan why there appears a rsk premum n some areas, such as the AAA to BBB bond return spread, or the short end of the yeld curve, and argues ths s prmarly a cash premum, a rsk-premum from the super-safe to safe spectrum of rsk. Further, there appears a negatve rsk premum n many hghly rsky areas, such as for the most volatle stocks. Ths paper outlnes the central theme of that book, documentng the scope of the absence of a rsk-return fndng, and showng that a relatve status utlty functon s consstent wth ths as an equlbrum. In secton 1, I gve an overvew of data relevant to the proposton rsk s at best uncorrelated wth return. In Secton 2 I present dfferent models that capture the same dea, 2

3 showng that a relatve utlty functon generates a zero rsk premum, and also hghlghts how heterogeneous agents can generate a negatve rsk return relaton. I. The Scope of the mprcal Falure As the current extensons to the CAPM va Arbtrage Prcng Theory or Stochastc Dscount Factors has no consensus on what that rsk factors actually are, ths creates a problem n crtczng the general proposton that no rsk measure s postvely correlated wth expected returns. Current asset prcng theory s really a framework, as opposed to a theory, and so any sngle bt of evdence, such as the low return to hghly volatle stocks, n solaton, merely suggest a flawed rsk proxy (e.g., volatlty). Yet fundamentally, the theory of rsk premums s based on the dea that we do not lke thngs that covary wth our wealth, broadly defned, because they ncrease our net wealth volatlty, broadly defned. It seems reasonable to presume, therefore, that prced rsk s somehow postvely correlated wth volatlty, because volatlty s postvely correlated wth CAPM betas. Intutvely, the ultmate prced rsk factor(s) should also be correlated wth betas and volatlty. Thus, the most damnng evdence s the scope of the volatlty-return correlaton falure across many asset classes. Ths evdence has never been presented as an argument for the falure of the conventonal theory because as ths theory cuts across several asset classes, each wth ther own measurement and stochastc characterstcs, a sngle statstcal test s dffcult f not mpossble. But the welter of data s broad, and examples of a volatlty-return relaton are the anomales, rather than the observaton that rsky securtes have lower returns n a partcular asset class. 3

4 A. Total Volatlty and Cross-Sectonal Returns My dssertaton was ttled Mutual Funds, Idosyncratc Varance, and Asset Returns. Its central focus was the negatve relaton between dosyncratc rsk s negatvely related to returns (Falkensten (1994)). Ang, Hodrck, Xng and Zhang (2006) documented the relatonshp between dosyncratc varance and returns, cross sectonally, and followed ths up wth another paper documentng t nternatonally (Ang, Hodrck, Xng, and Zhang (2009)). Fgure 1 shows the average excess returns for quntles sorted by dosyncratc volatlty. xcess returns, whch are lke the alpha n a market model that ncludes factors, n ths case, the 4-factor Fama-French factors. You take the total returns, and subtract the factor returns, because the excess return s that whch s unexplaned by the Fama-French factors. As these factors are generally postvely correlated to varance, that merely ncreases the alpha, because total volatlty and beta are postvely correlated, meanng hghly volatle stocks have a hgher beta and hgher expected return. The excess returns show a strong negatve relaton to dosyncratc volatlty. The followng year, they documented ths effect n 23 developed country markets and leave ths fndng as a global puzzle. 4

5 Fgure 1. Annual xcess Returns to G-7 qutes Sorted by Idosyncratc Volatlty Low Hgh Idosyncratc Volatlty From Ang, Hodrck, Xng and Zhang 2007, Table 7. Returns are annualzed B. Beta and Returns It s well known that a CAPM beta, n the context of sze, s unrelated to returns (Fama and French (1992)). Yet the connecton between beta and returns s actually much worse than that when put nto tangble nvestable portfolos. Table I shows the returns to varous portfolos grouped by ther betas. Ths used an nvestable unverse, whch was assumed to have a lower bound at the twenteth percentle of the NYS lsted frms at any tme. As Nasdaq and AMX frms are generally smaller, ths gets rd of about half of the stocks actually lsted currently, but t s more realstc n that t corresponds currently to about a $500MM market captalzaton cutoff. Most nsttutonal nvestors are wary of gong much below ths because t gets dffcult to 5

6 put on large postons, and usng the percentle we can account for the upward drft n the average market captalzaton of ths perod. As the equty market has become broader over tme, gong back to 1962 ths rule only exclude the bottom 20% of stocks wth at least 36 monthly returns, whle currently t excludes about 50% of all companes. No TFs, RITs, closed-end funds, etc. All common stocks. Data were taken from 1962 through March Only monthly data were used pror to 2000, but subsequent to that ths was combned wth daly return data, so that the beta estmates are better n the more recent perod (pror months for the monthly data, pror year for daly data). Table I Statstcs to Portfolos Sorted by Betas, 2/1962-3/2009 Betas are constructed from monthly data, the pror months as avalable. Beta-Low and Beta-Hgh are the extremum 100 low and hgh beta stocks. Beta 0.5, 1.0, and 1.5 are from the 100 stocks nearest these numbers. Portfolos are formed every 6 months. Daly data s used for betas after Beta- Beta- qual Low Beta-0.5 Beta-1.0 Beta-1.5 Hgh S&P500 Weghted AnnRet Arth 11.20% 11.50% 12.60% 11.80% 10.90% 10.30% 12.11% AnnRet Geo 10.90% 11.40% 11.60% 8.70% 5.20% 9.50% 10.99% AnnStdev 13.10% 11.60% 17.40% 26.20% 33.90% 15.00% 17.74% Beta Sharpe

7 We see that average arthmetc annual returns are hghest for the Beta-1.0 portfolo. Ths portfolo contans only those stocks were the prospectve beta forecast s closest to a beta of 1.0, thus trmmng off the hgher and lower beta equtes. The beta 0.5, and beta 1.5, smlarly targeted those numbers. Over ths perod, we see the average beta actually experence by these portfolos was closer to one, reflectng some of the neradcable mean reverson n beta. Nonetheless, they were relatvely close to ther beta targets. The hgh and low beta portfolos, meanwhle, were merely the 100 most extreme projected betas, and these betas vared consderably over the 47 year perod. The nterestng thng s that returns are not ncreasng n terms of beta, and ths suggests several obvous nvestng strateges for those who do care about beta. Ths becomes even clearer when one uses geometrc returns as opposed to arthmetc returns, because arthmetc returns are for someone nvestng a fxed amount, each month, whereas the geometrc return s for the buyand-hold nvestor, the latter beng more relevant to the longer term nvestor. These geometrc returns really fall off for the hgher beta portfolos. Because ths sample selected from about 2000 stocks currently, t tended to have a more equal-weghted bas than the S&P500, whch s value weghted. The equal weghted column on the far rght of Table I shows the equal weghtng of stocks wthn the 20th percentle of NYS market cap cut-off. As smaller stocks strongly outperformed larger stocks n ths perod, the average return among these equtes s hgher than for the S&P500 by about 150 bass ponts annually. Yet the Beta 1.0 portfolo generates a good 61 bass pont lft relatve to ths effect, hghlghtng that the Beta 1.0 portfolo, by excludng the low-returnng, hgh beta stocks, domnates the ndex wth the same beta. D. Call Optons. 7

8 Theoretcally, beta or any covarance wth the elusve rsk factor measures the how much of rsk, and so f rsk s prced, optons wth hgher strke prces (e, more out-of-themoney), have hgher beta(s) per dollar nvested, whch mples hgher average return. Therefore far out-of-the-money call optons should offer extremely hgh expected returns as a percent of ther prce. As underlyng stocks always, n practce, have postve CAPM betas, all calls wll have postve betas that exceed the beta of the underlyng stock, and call betas wll ncrease n the strke prce as the calls get further out-of-the-money. Hence, all calls wll have postve expected returns and the expected returns wll be larger for greater strke prces, because the betas, as a functon of the call prce, ncrease as you go out of the money. Coval and Shumway (2001) prove that expected uropean call returns must be postve and ncreasng n the strke prce provded only that (1) nvestor utlty functons are ncreasng and concave and (2) stock returns are postvely correlated wth aggregate wealth. 8

9 Fgure 2. Monthly Reruns for Call Optons Ranked by ther Senstvty to Stock Prce (Omega) Low Hgh Call Opton Delta to Stock (Omega) Sophe N (2007). Table 5 Sophe N (2007) looked at data from 1996 through 2005, and found that the hghest outof-the-money calls, wth one month to expraton, have average returns of 37%, over a month! Fgure 2 above shows that f you bucket call optons nto groups based on ther deltas, you fnd that call optons, ndeed, are ndeed hghly levered stock postons. Lower deltas mean the call opton s less senstve, n dollar terms, to a stock movng, but more senstve, n percentage terms. Thus, an at-the-money call opton wth a delta of 0.5 move 0.5 dollars for every 1 dollar move n the underlyng, whle an out-of-the-money opton may have a delta of On a percentage bass, snce the at the money opton has a prce of around 5, whle the out-of-the- 9

10 money opton a prce of 0.25, the percent change n prce for the low delta opton s much greater. The key to remember s that the average stock has a beta of 1.0, these betas range from 4 to 15 gvng one 4 to 15 tmes the juce of the daly return. An opton's beta s the beta of the stock, tmes the 'omega', whch s a measure of the percent return n the opton prce gven a 1% change n the stock prce. If the omega on a Ford call opton s calculated to be 1.6, then for every 1% change n the prce of Ford the prce of the call opton wll rse by 1.6%. Not only s the average return negatve for call optons, these returns get worse the more mplctly levered, the more rsky, the optons become, n contrast to what the weak assumpton descrbed by Coval and Shumway. Returns are negatvely correlated wth the betas. Investors bascally are overpayng for lottery tckets when they buy optons, and just lke the lottery, the average payout s worse they more rsk one takes. If there s a rsk premum n equtes, t certanly s not amplfed n optons n any way, because you lose money, on average, buyng leverage market postons va call optons. Prvate Investments ntrepreneural nvestment, such as n small propretorshps (S-corps and prvate LLCs) s a hghly undversfed nvestment for most entrepreneurs. The reasons are straghtforward, n that when one person has a sgnfcant effect on the busness through hs effort and competence, t s natural that he should have the most skn n the game. Ths s a classc ssue of moral hazard because a busness manager, who has sgnfcant upsde and, wthout ownershp, no downsde, s motvated to take wld rsks on the theory of heads I wn, tals the banker loses. However, f the manager s the majorty owner, hs falure should affect hs net wealth too. About 75 percent of all prvate equty s owned by households for whom t consttutes at least half of ther total net worth. Furthermore, households wth entrepreneural equty nvest on 10

11 average more than 70 percent of ther prvate holdngs n a sngle prvate company n whch they have an actve management nterest. Despte ths dramatc lack of dversfcaton, prvate equty returns are on average no hgher than the market return on all publcly traded equty. Fgure 3 shows the basc results of Moskowtz and Vssng-Jorgensen (2002), that over an 8 year perod, f anythng returns to prvate busness, be t partnershps, propretorshps, S corps, C corps, and two entrely dfferent sets of data, there s no demonstrable premum. Gven an nvestor can nvest n a dversfed, and lqud equty portfolo, t s puzzlng why households wllngly nvest substantal amounts n an asset wth an equvalent return, but much hgher volatlty, ncludng a postve correlaton wth the market. Fgure 3. Annual Returns to Small Busness Investments Annual Returns, CRSP-Value Weghted CRSP-Small Cap Propretorshp & Partnershp S & C Corps From Table 4, Moskowtz, Tobas J, and Annette Vssng-Jorgensen "The Returns to ntrepreneural Investment: A Prvate quty Premum Puzzle?,"92(4), pp

12 The forced nondversfcaton of a prvate equty nvestment, from a pure portfolo perspectve, mples a requste hgher return. Usng standard utlty models to calbrate the hurdle rate that would make a household ndfferent between nvestng n a portfolo of a sngle prvate frm, a publc equty ndex, and T-blls, or a portfolo of just the publc equty ndex and T-blls, researchers estmate that prvate equty rsk generates a hurdle rate of about 10 percent hgher the publc equty return (Heaton and Lucas (2000)). You should receve a huge premum for the large dosyncratc rsk you are takng, rsk that unlke dosyncratc rsk n the market, s mpossble to dversfy away. ntreprenuers appear to be takng extra rsk, for no extra return. F. Leverage and Returns. The Mller-Modglan theorem states that regardless of the debt and equty proportons, the value of the frm s the same. As a frm ncreases ts leverage usng more debt, ts equty concentrates the varable returns of the busness on a smaller and smaller base, makng them both rsker: the equty s beta and volatlty wll ncrease, the debt wll have a hgher chance of defaultng. The mplcaton s that hghly levered frms should have lower rated debt (junk), and more volatle equty, but because debt has a lower return than equty, the net, total return to all a company s securtes (debt and equty) s a constant. 12

13 Fgure 4. Annual Return to Portfolos sorted by Market Leverage (Debt/Market Cap) Adjusted for Book/Market and Sze xposure 8% 4% 0% Penman, Rchardson, and Tuna, 2007, Table 1 In fgure 4 above, we see that leverage s, anomalously, clearly negatvely correlated wth returns. These researchers held constant sze and book/market. Hgher leverage mples lower returns for equty even though ths should ncrease rsk of that equty, and thus should ncrease returns (Haugen and Baker (1996), Penman, Rchardson, and Tuna (2007)). Further, there have been no papers lnkng how leverage s postvely related to expected returns, even though ths result would have been consstent wth a Nobel-prze wnnng theory. mprcally supportng Nobel-wnnng theory for the frst tme s worthy of a publcaton n a top journal, and for 13

14 academcs, ths s ther number one prorty. The absence of a postve fndng n ths context s perhaps more powerful than the handful of negatve results. G. Mutual Funds The orgnal tests of the CAPM were on mutual fund returns, hopng to show that mutual fund performance would be explaned by the new rsk factor (Sharpe (1965), Treynor and Mazuy (1966). Subsequent work found no relatonshp between a stock s return and ts beta, and t was suffcently unnterestng that n more recent work, the relaton between beta and returns s addressed only as an asde n Malkel (1995), who notes that a fund manager s beta s uncorrelated wth hs fund s average return. As wth leverage studes, the absence of any volatlty or beta correlaton wth mutual fund returns s most relevant here, because t hghlghts an absence of confrmaton n an area examned snce the very begnnngs of asset prcng theory. Absense of evdence s evdence of absence from a Bayesan perspectve, not proof, but suggestve especally when you know there has been a systematc, thoughtful search for such evdence. Carhart (1997) presented a well documented study of mutual funds s most well known for ntroducng momentum as a factor, akn to Fama and French s value and sze factors, but t was also notable for the manfest rrelevance of beta n hs analyss. Good or bad, a mutual fund s beta was never an ssue n explanng the results, and so that paper hardly mentoned the null results, and nstead centered on the mportance of momentum as a factor that explaned the postve one-year persstence. Later studes of mutual funds by Wermers (2000) does not even address beta. H. Currences 14

15 Uncovered Interest Rate Party s a theory that connects current to future spot rates. Ths theory states that you have two ways of nvestng, whch should be equal. Frst, you can nvest n your home country at the rskless rate. So f the US nterest rate s 5%, you can make a 5% return n one year, n USD. Alternatvely, you can buy, say, yen, nvest at the yen nterest rate (each currency has a dfferent rsk-free rate), and then convert back to USD when your rskless securty matures. For ths to be equal, you need somethng lke r USD r % change n yen yen Where r usd s the US nterest rate, etc. So, f you make 5% n USD, an Amercan nvestor should receve that same return n yen, va the nterest rate n yen, plus the expected apprecaton/deprecaton n the yen aganst the dollor. If the nterest rate n yen s 1%, ths means one expects the yen to apprecate by 4% aganst the dollar. When the foregn nterest rate s hgher than the US nterest rate, rsk-neutral nvestors should expect the foregn currency to deprecate aganst the dollar by the dfference between the two nterest rates. Ths way, nvestors are ndfferent between borrowng at home and lendng abroad, or the converse. Ths s known as the uncovered nterest rate party condton, and t s volated n the data except n the case of very hgh nflaton currences. In practce hgher foregn nterest rates predct that foregn currences apprecate aganst the dollar, makng nvestng n hgher nterest rate countres wnwn: you get apprecaton on your currency, and hgher rskless nterest rates whle n that currency. Now the rates of expected return va the two nvestment paths can dffer accordng to rsk, of course. So one can magne, lookng at the yen, or the dollar, or varous uropean currences n the 1970 s, etc., tryng to te each to some measure of a home currency s rsk factor: consumpton, the stock market. 15

16 Lke hgh returns to low volatlty stocks, t s dffcult, but not theoretcally mpossble, to make sense of ths. Hodrck (1987) wrote a comprehensve techncal overvew of the theory and evdence of currency markets n Hodrck looked at CAPM models, latent varable models, condtonal varance models, models that use expendtures on durables, or nondurables and servces, Kalman flters. None outperformed the spot rate as a predctor of future currency prces. Hodrck leaves off wth the dea that smple models may not work well. He summed up hs fndngs n ths paragraph: "We have found a rch set of emprcal results... We do not yet have a model of expected returns that fts the data. Internatonal fnance s no worse off n ths respect than more tradtonal areas of fnance." For the next 20 years, and many hedge funds specalzed n the carry trade, whch was as smple as t was successful: lend captal to hgh nterest rate currences, enjoy the hgh rskless rates and currency apprecaton on the spot rate; borrow captal at the low nterest rate currency, and make money on the deprecaton of ths debt over tme. In 2008 these strateges suffered sgnfcantly, but the net effect s stll there s no clear relaton between rsk and return n currences. Brunnermeer, Nagel and Pedersen (2008) noted that Overall, we argue that our fndngs call for new theoretcal macroeconomc models n whch rsk prema are affected by fundng and lqudty constrants, not just shocks to productvty, output, or the utlty functon. By our fndngs they mean, the carry trade contnued to work 30 years after beng dentfed by Farber and Fama (1979), and t has contnued as a puzzle because no reasonable rsk factor can explan t. 16

17 quty Premum I. World Country Returns rb, Harvey and Vskanta (1995) look at data from between developed and nondeveloped countres. As the arthmetc returns are much hgher than the geometrc returns, he hghlghts those monthly returns for dsplayng a rsk premum based on the volatlty, but f you look at the geometrc returns, they are about the same as the developed country data (13.5% for developng, 12.3% for developed). Usng a broader set of data from 1989 through 2000, Bansal and Dahlqust (2002) report approxmately smlar arthmetc returns (15.8% vs. 16.1%), but then usng annualzed geometrc returns, the return for the developed countres was 13.8% vs. 6.7% for the developng countres. Thus t s nterestng that among country equty returns, there s no clear rsk premum. Dmson, March, and Staunton (2006) fnd that the USA had about the same average return relatve to short term debt from compared to 17 other developed countres worldwde, about 5%. Fgure 5 shows the volatlty and returns for 17 countres over the perod. Fgure 5 quty Premums vs. Volatlty for 17 Country Returns, Standard Devaton From Dmson, Marsh, and Staunton (2006). 17

18 It s strange that there s not a pattern among ther returns n terms of volatlty, because ntutvely, those countres where the stock market ndex s especally volatle, would have a hgher rsk premum, as foregners would not be able to nvest suffcently due to tax and nsttutonal reasons, and elmnate the rsk premum for ths dosyncratc rsk. One can look for global rsk factors that explan ths, and the usual ones (eg, a world stock market ndex) do not work. J. Corporate Bonds The conventonal corporate bond puzzle s that spreads are too hgh. The most conspcuous bond ndex captures US Baa and Aaa bond yelds gong back to 1919, whch generates enough data to make t the corporate spread measure, especally when lookng at correlatons wth busness cycles. Yet Baa bonds are stll nvestment grade, and have only a 4.7% 10 year cumulatve default rate after ntal ratng. As the recovery rate on defaulted bonds s around 50%, ths annualzes to a mere 0.23% annualzed loss rate. Snce the spread between Baa and Aaa bonds has averaged around 1.2% snce 1919, ths generates an approxmate 0.97% annualzed excess return compared to the rskless Aaa yeld, creatng the puzzle that spreads are too hgh for the rsk ncurred. Altman and Bana (2004) and Kozhemekn (2007) note there s no premum to Hgh Yeld portfolos relatve to nvestment grade portfolos, a set of bonds wth a 3.84% average annual default rate from Further, Altman and Stoneberg (2006) note that a bankrupt bond portfolo underperforms nvestment grade bonds. Both Hgh Yeld and Bankrupt bonds have more volatlty and cyclcalty than nvestment grade bonds, and do ther worst when returns are most valued, n bad tmes. Junk bonds are ntutvely rsky. Data from the Merrll 18

19 Lynch Hgh Yeld ndex show an 6.77% annualzed return relatve to the 7.18% return of ther nvestment grade ndex from 1987 through December Indces are really an overstatement because how such ndces have a systematc bas when portrayng llqud or unaudted asset classes. For a long tme, average junk bond s bd-ask spreads were 10 ponts wde, and ths transacton cost s mplct n the fact that for a set of mutual Hgh Yeld mutual funds that currently exst (e, obvous survvorshp bas), ther total annualzed return from , was 3.44%, whereas the Merrll Hgh Yeld Index rose 6.77% (see fgure 6 below). Investment Grade funds underperformed ther ndex by much less, n that the Index return was 7.18% compared to fund returns of 6.48%. The smple dea s that llqud assets have hgher transacton costs, hgher ' management fees', that cause actual performance to be lower than that of an ndex. Thus the closng prces of llqud assets, such as n an ndex, wll be a based proxy for real returns f based on merely closng prces. If you take a couple percent off the hgh yeld ndex due to prce mpact, commssons, bd-ask spread, there s a negatve rsk premum to these rsky assets. 19

20 Fgure 6 Total Returns to Corporate Bonds, % 7% 6% 5% 4% 3% 2% Merrll Lynch Indces Closed nd Funds 1% 0% Hgh Yeld Investment Grade Data are from the Merrll Hgh Yeld Master II and Merrll BBB-AA Index (H0A0 and C0C0, respectvely). Closed end fund data are from Bloomberg. Therefore, the excess corporate rsk premum puzzle pertans to one porton of the rsk spectrum the dfference between a 0.03% (AAA) and a 0.3% (BBB) annualzed default rate, a dstncton wthout a dfference to many nvestors. When one goes from a 0.3% to a 15% default rate, as one does when you go from BBB to C rated bonds, there s no return premum at all. Gven reasonable expectatons of transacton costs, and the actual dfference between the hgh yeld ndces and actual hgh yeld returns, t seem probable people extend nto hgher credt rsk wth a lower average return. It s dffcult to see how the lttle rsk s prced, the bg one not, f rsk s to have any consstent meanng. If the corporate spread s a functon of rsk at one end, why s t not at the other, more ntutve end? K. Yeld Curve 20

21 Ann Ret StDev he general shape of the yeld curve s as follows. It rses untl about 3 years, then flattens out. Whle ths may be decevng because bond prces have postve convexty, n practce ths effect on US Treasury returns, and yelds, s bascally the same n the post 1954 perod. Thus, I took data on US government bond yeld snce 1958 through 2008 from the Fed s H.15 report, and constructed a set of annualzed returns based on a buy-and-hold strategy. ach monthly return subtracted the Fed Funds rate, and ncluded both the monthly prce apprecaton plus the coupon yeld. These data are presented n fgure 6 below. Fgure 6 Total Return and Standard Devaton to Long Treasury Bond Postons, % 10% 0.4% 8% 6% 0.0% 4% 2% -0.4% 6mo 1yr 3yr 5yr 10yr 20yr 0% Ann xcess Ret (over Fed Funds) AnnStDev As we can see, the returns to a funded poston n bonds, an excess return, s very close to zero over ths perod. Indeed, buyng 6-month T-blls funded at the Fed Funds rate was a money loser, and for the 1-year bond, about a zero return. Returns are ncreasng over maturty, yet, the 21

22 ncrease n yeld from 5 years to 10 years s mnscule, from 0.55% to 0.65%, and t actually declne to 0.60% for the 20 year bonds. 30 year bond data starts n 1977, and has a 2 year gap around 2003, and the data ovelappng wth the 20 year suggest a slghtly (10 bass pont) lower annualzed return from 20 to 30 years. It s far to say the returns to yeld curve extensons past fve years are bascally zero. The prce volatlty, meanwhle, ncreases consstently as maturty ncreases, and thus the Sharpe rato, the rato of the return on the bond mnus a rsk free rate, falls as the maturty ncreases beyond 5 years. The rsk premum from 3-months to 3-5 years does not extrapolate. L. Futures Futures are dervatve securtes, blateral agreements, one sde to buy, the other to sell, at a future date, a spot commodty at a prespecfed prce. Futures returns are not drven by lower expected spot prces because such prces are reflected n a low current futures prce (Black, (1976)). What return can an nvestor n futures expect to earn f he does not beneft from expected spot prce movements, and s unable to outsmart the market? The dfference between the current futures prce and the expected future spot prce. Assume the current futures prce s below the current spot prce. Usually, ths mples the expected spot prce s above the futures prce (we don t truly observe the actual expected futures prce, but ths s generally true). On average, gong long the futures makes money when t s below the current spot prce because the futures prce rses to the eventual spot prce. At maturty, whle the spot prce may have fallen, the futures prce has rsen too. Ths s called normal backwardzaton because f you put the futures prces out lke a bond yeld curve (yelds up, prces down), the more dstant futures prces are below the current prce. 22

23 Fgure 7 Forward Curves n August Gold n Contango, Copper n Normal Backwardzaton Gold Copper Fgure 7 shows the term structure of futures for Gold and Copper n August Copper s n backwardzaton, whle Gold s n contango, a fun name for the opposte. Hstorcally, gold s always n contango, meanng, f you are long gold futures, you lose money on average as t rolls to maturty. Other commodtes flop around, sometmes flat, sometmes n normal backwardzaton, and sometmes n contango. Harvey and rb (2006) fnd that copper, heatng ol, and lve cattle were on average n backwardzaton, whle corn, wheat, slver, gold and coffee were n contango, on average. The expected roll returns (called because the futures prces rolls to the current spot prce over tme), based on the current relaton of the futures to the spot, are uncorrelated wth the promnent rsk factors for equtes (e, the market, value, and sze factors) or for corporate bonds 23

24 (e, the Baa-Aaa yeld spread). Changes n nflaton adversely affects the roll returns from normal backwardzaton, whle adversely affectng the roll returns for contango (see Gorton, and Rouwenhorst (2005)). There are predctable returns n futures returns, prmarly due to the roll, whch s foreseen n the current relaton of the futures prce to the spot prce. But what drves ths, from a rsk perspectve, s a mystery. M. Dstress Rsk arly on n the sze effect, researchers were at a loss to fgure out what knd of rsk that frm sze, outsde of beta, captured. The obvous rsk, resdual rsk from these very small stocks, was dversfable, and so not rsk. Fama and French suggested that both the value premum and the small stock premum were related to some sort of dstress factor, that s, value stocks, whose prce was beaten down by pessmsts, and small stocks, whch had less access to captal markets, probably had more rsk of defaultng, f the economy faltered. It may not show up n correlatons or covarances, but that s merely because such rsks are very epsodc lke the rsk of a heart attack: The frst symptom of a heart attack s a heart attack. Dchev (1998) had documented ths, but ths fndng could be dsmssed because he presumably had a poor default model (he used the Altman s (1968) model that was based on only 33 defaultng companes). Then several others documented a smlar result, and fnally Campbell, Hlscher, and Szlagy (2005) fnd the dstress factor can hardly explan the sze and book or market factors; n fact, t merely creates another anomaly because the returns are sgnfcantly n the wrong drecton. Dstressed frms have much hgher volatlty, market betas, and loadngs on value and small cap rsk factors than stocks wth a low rsk of falure; furthermore, they have much worse performance n recessons. These patterns hold n all sze 24

25 quntles but are partcularly strong n smaller stocks. Dstress was not a rsk factor that generated a return premum, as suggested by theory, but rather a symptom of a hgh default rate, hgh bond and equty volatlty, hgh bond and equty beta, and low equty return. Whle I was at Moody s n 2000, I was able to use ther database of ratngs back to 1975 and fnd that the rate of return lned up almost perfectly wth the ratng, wth AAA havng the hghest return, C the lowest. Updatng that data usng S&P ratngs, and used the ratng n June, to form a portfolo over the followng 12 months, a very straghtforward strategy. I use 1987 as the startng pont here because pror to ths junk bonds, those wth ratng below nvestment grade, were small n number, as there was a structural shft n the junk market n the late 1980 s when these nstruments started to have good prcng data (ths does not change the results anyway). 25

26 Fgure 8 Annual Stock Returns to Portfolos Formed by Ratng Portfolos were formed every June. Frms delstng wthn the 12 months were then reallocated to the remander of the portfolo 15% 12% 9% 6% 3% 0% -3% -6% -9% -12% AAA AA A BBB BB B C The returns are pretty flat untl you get to the sgnature junk bonds, the Bs, and then t falls precptously, and the Cs are actually negatve. Thus, the equty returns to frms wth low fnancal strength are low, and ther debt does not seem to compensate ether. Hgh rsk, from a fnancal dstress perspectve, appears negatvely related to returns for Agency-rated companes. When consdered n conjuncton wth the flat to lower returns for B-rated debt, ths suggests a negatve rsk premum for rsky company s assets, broadly defned. 26

27 Gross Return N. Move Producton Art DeVany (2003) found that between G-rated moves generated lower volatlty and approxmately the same returns as R-rated moves, though there was a clear preference towards R-rated moves (over 1000 R-rated moves and only 60 G-rated ones). But moves has a strong pareto dstrbuton, where the mean s much hgher than the medan or mode. It seems studo executves are generally bettng on the next Ttanc, because the very hghest grossng moves are R rated. Fgure 9 Move Gross Return and Volatlty by Ratng 2015 moves from G PG PG Volatlty R Devany and Walls (1997) 27

28 O. Sports Book Bets on hgh probablty low payoff gambles have hgh expected value and low probablty hgh payoff gambles have low expected value. For example, a 1-10 horse havng more than a 90% chance of wnnng has an expected value of about $1.03 (for every $1 bet), whereas a horse has an expected value of about 14 cents per dollar nvested (Hausch, Lo, and Zemba (1994). Ths bas has appeared across many years and across all szes of race track bettng pools. The effect of these bases are that for a gven fxed amount of money bet, the expected return vares wth the odds level. The favorte long-shot bas s monotone across odds and the drop n expected value s especally large for the lower probablty horses. For bets on extreme favortes, there s an postve expected return. For all other bets, the expected return s negatve. Zemba, and Hausch (1984) document the favorte long-shot bas s monotone across odds and the drop n expected value s especally large for the lowest probablty horses (worse than 50-1). P. Lotteres The annual per capta lottery expendture n the US s about $170, and the rate of return s about -47% per dollar played. It clearly presents a challenge to the dea that people are lookng at these games on a rsk-return contnuum, and motvated the earlest nflected utlty functons (Fredman and Savage (1948). These nvestments clearly cater to what s commonly called those seekng rsk, or postve skew. There are two prmary characterstcs of lotteres. Frst, poor people play them more, n both relatve and absolute terms, than wealthy people. Bhattacharyya and Garret (2006) fnd that poorer households spend consderably more than wealther households on lotteres. 28

29 Garrett and Sobel (2004) fnd the popularty (sales) of lotteres found that average payout (expected return), but the sze of the top prze was hghly sgnfcant. In other words, the $100 mllon super lotto has the most sales even though the probablty of wnnng s so small t bascally s outsde the realm of ntuton (1 n 195 mllon for the popular Powerball). People who buy lottery tckets seem to prefer those lotteres that offer the worst odds, but the greatest payout. Gamblng seems to be totally outsde the assumptons of rsk averson, and s the common motvator for Prospect Theory applcatons. Q. Total Volatlty and Aggregate Volatlty and Returns The most basc rsk models assume that the expected return on an asset s proportonal to the expected nondversfable varance of the asset: the hgher the varance, the hgher the expected return (see Merton (1980)). Modern models tend to focus on some abstract thng we don t lke, lke declnes n consumpton, wealth, or output, but those bad states are generally concdent wth hgher volatlty, as volatlty ncreases when the economy s dong poorly. In general, researchers document a null relatonshp between volatlty and future returns, some fnd a negatve relatonshp (Campbell (1987), Whtelaw (1994), Nelson (1991). Sharpe and Amromn (2005) used survey data, and found nvestor expectatons were totally nconsstent wth standard models. They found that when nvestors have a more favorable assessment of macroeconomc condtons, they tend to expect hgher returns. Second, they found that the expectaton of more favorable economc condtons has a strong negatve effect on expected stock market volatlty. A good example of ths knd of thnkng s a Gallup poll put out by Pane Weber (Graham, Benjamn and Jason Zweg. 2003). In 1998, at the begnnng of the stock market boom, they surveyed an expected return of 13% from nvestors. After back to back 20%+ returns, when the Nasdaq doubled, nvestors rased ther expectatons to 18% n 29

30 February of 2000, rght before the peak. Two years later, after a 50% correcton, and a 50% rse n the VIX (a measure of expected volatltes), they antcpated only a 7% expected return. So from a Sharpe rato perspectve, when nvestors expect a hgh numerator, they expected a low denomnator. They expect good tmes to be hgh returns and low volatlty, and bad tmes to be low returns and hgh volatlty. R. Intal Publc Offerngs An ntal publc offerng has a great deal of uncertanty, especally for an economst wshng to apply a factor senstvty to t, because there s no hstorcal tme seres for the equty. One usually apples a factor based on ts characterstcs, such as sze, book/market, and perhaps ndustry. But wthout a track record, these assgnments are hghly uncertan n the Keynesan/Knghtan sense. One would expect, gven uncertanty averson, for these stocks to have postve returns to compensate for ths uncertanty. xamnng IPOs from , Rtter and Welch (2002) fnds the geometrc average return for these IPOs fve years after ssuance are 5.1% below sze and book-to-market matched frms. People who buy IPOs pay a premum, perhaps on the hope of buyng the next Yahoo! or Google. S. Analyst Dsagreement Dfferences of opnon should proxy for parameter uncertanty, a perhaps better estmate of rsk (see Hansen and Sargent (2001)). Usng analysts earnngs forecasts as a proxy for dfferences of opnon among nvestors, Karl Dether et al (2002) fnd the quntle of stocks wth the greatest opnon dspersons underperformed a portfolo of otherwse smlar stocks. 30

31 ach month, they take stocks and sort them nto fve groups based on sze (market cap), and then wthn these groups, sort agan nto quntle based on analyst forecast dsperson, as measured by the rato of the standard devaton of analyst current fscal year annual earnngs per share forecasts to the absolute value of the mean forecast. They fnd that the stocks wth the hgher dsperson n analyst s earnngs forecasts earn sgnfcantly lower returns than otherwse smlar stocks. Specfcally, the hghest dsperson group had a 9.5% annual return defct over the perod. T. Tradng Volume Another metrc of dsagreement s the amount of tradng volume n a stock, normalzed by ts stock volume. On average, f dsagreement leads to more transactons as optmsts and pessmsts take sdes, we should see a hgher return to these stocks. In the US, snce 1997, I created an ndex of stocks n the top 1500 that had the hghest tradng volume/shares outstandng, each sx months. The annualzed geometrc return for these hgh volume stocks was 1.6%, vs 9.7% for the low volume stocks. The hgh volume stocks tend to be hghly correlated wth hgher beta portfolo returns, and low volume wth low beta stocks. If one s educaton was unaware of utlty functons one would have to look at these data and say that volatlty s nversely correlated wth returns. The only clear area that a rsk premum appears s n the BBB-AAA spread, the short end of the yeld curve, and the equty rsk premum. They are exceptons to the rule. Further, n Falkensten (2009), I note that after adjustments for transacton costs, peso problems, taxes, and several other reasonable adjustments, the average nvestor s equty premum s probably zero. The theory that metrcs of rsk such as volatlty or covarance s postvely correlated wth average returns fals 31

32 spectacularly when appled to volatlty, moves, beta, developng country equtes, aggregate volatlty and aggregate returns, gamblng, lotteres, optons, fnancal leverage, fnancal dstress, currences, mutual funds, small busnesses, analyst forecast dsperson, IPOs, and futures. These are not mnor lacunae, but the heart of the rsk-return theory. As a frst approxmaton, volatlty should be generally postvely correlated wth returns f prced rsk s to have any meanng n an asset prcng theory. II. How a Relatve Utlty Functon Generates Zero Rsk Premum The purpose here s to present a model where there s no rsk premum n equlbrum. The models presented are smple, but an even smpler example s useful n seeng the drver of ths the result. There are two assets, X and Y, and two states of nature, 1 and 2. An nvestor faced wth asset X or Y can see the followng returns: Table II Payoffs to Assets X and Y n States 1 and 2 Total Return Avg. Relatve Return X Y X Y State State

33 As shown n Table II, Y s conventonally consdered rsker, wth a 40 pont range n payoffs versus a 20 pont range for X. Yet on a relatve bass, each asset generates dentcal rsk. In State 1, X s a +5 out performer; n State 2, X s a -5 underperformer, and vce versa for asset Y. In relatve return space, the hgher absolute volatlty asset s not rsker; the reader can check ths for any example n whch the two assets have the same mean absolute payout over the states (.e., the average for asset X and asset Y s the same) The rsk n low volatlty assets s ts losng ground durng good tmes. If X and Y are the only two assets n the economy, equvalent relatve rsk can be acheved by takng on an undversfed bet on X or Y, whch s dentcal to takng a poston on not-y and not-x. The postons, from a relatve standpont, are mrror mages. Buyng the market, n ths case allocatng half of each, meanwhle, generates zero rsk. verythng really flows from ths smple nsght. Implctly the equlbrum and arbtrage derves from the fact that when relatve portfolo wealth s the argument n the utlty functon, systematc volatlty s symmetrc, as the complement to any portfolo subset wll necessarly have dentcal though opposte sgned relatve return. Thus gans from trade can always be made f there s a rsk premum. A. The Arbtrage Model Assume an economy wth rsky assets that are a functon of a market factor r m. For any nvestor who chooses an asset wth a specfc beta, returns are generated va the factor model r r ( 1) m 33

34 Where s a constant for an asset wth the specfc beta, and m ~ m, m 2 r N. We wll assume no dosyncratc rsk from assets, because the gst of ths approach s wthout loss of generalty. The return on the rsk-free asset s the constantr f. The market prce of all assets, rsky or rsk-free, are assumed equal to 1, so we are solvng for a rf, and m such that ths s an equlbrum. That s, the prces are set to one, but the returns are free parameters. The market return n ths model s the benchmark to whch nvestors compare themselves, just as mutual fund managers typcally try to outperform ther benchmark. Ther objectve s to maxmze ther out performance, subject to mnmzng ts varance. Defne r relatve performance of nvestor to the market return out, whch s the out r r r ( 2) m Here r s the return on the nvestor s portfolo wth ts partcular factor loadng, and r s the return on the market. Investors all have the smple objectve of maxmzng r m mnmzng a proporton of ts varance, as n out whle Max r out 2 a 2 ( 3) Where 2 Var r rm. Substtutng equaton (1) nto (2) generates out r 1r ( 4) m Snce r m s the only random varable, the varance of outperformance s just ( 5) 1 m 34

35 quaton (5) mples that the beta bet s bascally rsky to the extent t devates from the average, n ether drecton. We can replcate the relevant rsk of a stock wth a beta of, 2, va a portfolo consstng of unts of the market portfolo, and borrowng rsk-free asset (cost s unts of the, same as for the stock, as all assets have a prce of 1 by assumpton). Arbtrage then mples that these have the same expected returns, so r r r ( 6) m 1 f m The LHS of equaton (6) s the market portfolo levered tmes by borrowng (1- ) n the rsk-free asset n fnancng, whle the RHS s the unlevered asset portfolo va equaton (1). They have the same factor exposure, and cost the same, so they should have the same return n equlbrum. Thus equaton (6) mples 1 r ( 7) f Ths allows us to replace the wth r n equaton (4) and leads to the factor 1 f model r 1 r 1r ( 8) out f m 2 If the degree of rsk of relevance to nvestors s ther out performance,, the expected return for assets wth k should be the same as those wth 2 k, because they have the m m 2 same rsk n ths envronment: k k The rsk of a 2 asset s dentcal n magntude to a 0 asset, so the expected returns must be the same out out 2 r k r k ( 9) 35

36 Usng equaton (9) on the equvalence of 2-k and k beta assets, and applyng the expectatons operator, we have 1 k r k 1 r 1 2 k r 2 k 1 r ( 10) f m f m The LHS of equaton (10) s the expected return on the expected return on the 2 k asset. Solvng for r we get m k asset, and the RHS s the rm r ( 11) f quatons (1), (7) and (11) mply r r r r f m r m ~ N r, f m f ( 12) Thus no arbtrage, n the sense thngs equvalent n rsk are prced the same (as rsk s defned here), generates the tradtonal CAPM wth the sgnfcant dfference that the expected market return s equvalent to the rsk-free rate. Just as the equlbrum model n pror secton mples, the expected return on all assets s the same, becauser r 0 m. f In contrast, a tradtonal arbtrage model would take from the arbtrage equaton (7), and, combned wth the market model equaton (1), generate the standard factor model ( 13) r r r r f m f But now the maxmzaton functon reflects the fact that the nvestor only cares about absolute volatlty, not volatlty relatve to some benchmark. Max 2 r a 2 2 m ( 14) 36

37 Substtutng equaton (13) for r, the frst order condton on equaton (14) generates the famlar equaton rm rf ( 15) a2 m So nvestor s optmal wll be equal to the rsk premum over the rsk averson coeffcent tmes the market varance. Assumng a representatve nvestor, conventonal parameters for ths approach of 6% for the rsk premum, 3 for a rsk coeffcent, and 15% for market volatlty, ths mples an equlbrum beta choce of 1, consstent wth an equlbrum where the representatve nvestor holds the market basket. But f, as argued n Falkensten (2009), the market premum s n effect zero for the average nvestor, the choce wll be zero, whch s not an equlbrum because on average the market beta s 1 by defnton and n postve net supply. In the tradtonal approach, a postve market premum s necessary for nvestors to hold the market n equlbrum, whereas n a relatve rsk model, combnng equatons (3) and (8), we get Max 1 2 rf 1r a m 2 1 ( 16) m 2 rm rf 1 ( 17) a2 m Here the optmal choce of s 1 only f the rsk premum s zero (.e., [r m ]=r f ), because rsk s uncompensated va arbtrage, and rsk can be avoded n ths model by choosng a beta of 1. A postve rsk premum would nduce a desred optmal beta greater than 1, whch would then not be an equlbrum. 37

38 Whle ths s a smple model, t has at ts essence no more smplcty than what generates tradtonal rsk premums. The only dfference s whether one puts relatve as opposed to absolute wealth n the utlty functon. Both the absolute and relatve rsk approach generate the famlar factor prcng model, but n the relatve rsk approach the rsk premum s zero n equlbrum, whereas n the absolute rsk approach the rsk premum must be postve. B. qulbrum wth Heterogeneous Investors There exsts a two-perod economy wth two dentcal ndvduals, and -. There are two types of assets; one s a rsk-free bond that pays off R f wth certanty n perod 1. There also exsts an equty wth a return of R, where 2 R ~ N, ( 18) Total wealth for the ndvdual n perod 0 s gven by hs portfolo of assets. 0 R f w R ( 19) F and f represent the holdngs for nvestor on the rsky and rsk-free asset, respectvely. ach ndvdual s endowed wth k unts of wealth, so the budget constrant s w0 k ( 20) Agent s utlty functon s drven by hs wealth relatve to the other agent (there s no consumpton) n an exponental utlty functon wth a rsk averson coeffcent a exp U w w a w w ( 21) As the argument n equaton (21) s normally dstrbuted, the ndvdual therefore maxmzes the followng functon 38