Active Portfolio Management By Richard C. Grinold and Ronald N. Kahn

Transcription

1 Notes: Active ortfolio Maagemet y Zhipeg Ya Active ortfolio Maagemet y Richard C. Griold ad Roald N. Kah art I Foudatios... Chapter 1 Itroductio... Chapter Cosesus Expected Returs: The CAM... 3 Chapter 3 Risk... 3 Chapter 4 Exceptioal Retur, echmarks, ad Value Added... 5 Chapter 5 Residual Risk ad Retur: The Iformatio Ratio... 6 Chapter 6 The Fudametal Law of Active Maagemet... 9 art II Expected Returs ad Valuatio Chapter 7 Expected Returs ad the Arbitrage ricig Theory Chapter 8 Valuatio i Theory Chapter 9 Valuatio i ractice art III Implemetatio Chapter 10 Forecastig Chapter 11 Iformatio Aalysis Chapter 1 ortfolio Costructio Chapter 13 Trasactios Costs, Turover, ad Tradig... Chapter 14 erformace Aalysis... 4 Chapter 15 echmark Timig... 9 Chapter 16 Summary

2 Notes: Active ortfolio Maagemet y Zhipeg Ya art I Active ortfolio Maagemet y Richard C. Griold ad Roald N. Kah Foudatios Chapter 1 Itroductio I. A process for active ivestmet maagemet The process icludes researchig ideas, forecastig exceptioal returs, costructig ad implemetig portfolios, ad observig ad refiig their performace. II. Strategic overview 1. Separatig the risk forecastig problem from the retur forecastig problem.. Ivestors care about active risk ad active retur (relative to a bechmark). 3. The relative perspective will focus us o the residual compoet of retur: the retur ucorrelated with the bechmark retur. 4. The iformatio ratio is the ratio of the expected aual residual retur to the aual volatility of the residual retur. The iformatio ratio defies the opportuities available to the active maager. The larger the iformatio ratio, the larger the possibility for active maagemet. 5. Choosig ivestmet opportuities depeds o prefereces. The preferece poit toward high residual retur ad low residual risk. We capture this i a mea/variace style through residual retur mius a (quadratic) pealty o residual risk (a liear pealty o residual variace). We iterpret this as riskadjusted expected retur or value added. 6. The highest value added achievable is proportioal to the squared iformatio ratio. The iformatio ratio measures the active maagemet opportuities, ad the squared iformatio ratio idicates our ability to add value. 7. Accordig to the fudametal law of active maagemet, there are two sources of iformatio ratio: IR = IC * R - Iformatio coefficiet: a measure of our level of skill, our ability to forecast each asset s residual retur. It is the correlatio betwee the forecasts ad the evetual returs. - readth: the umber of times per year that we ca use our skill. 8. Retur, risk, bechmarks, prefereces, ad iformatio ratios costitute the foudatios of active portfolio maagemet. ut the practice of active maagemet requires somethig more: expected retur forecasts differet from the cosesus. 9. Active maagemet is forecastig. Forecastig takes raw sigals of asset returs ad turs them ito refied forecasts. This is a first step i active maagemet implemetatio. The basic isight is the rule of thumb ALHA = VOLATILITY*IC*SCORE that allows us to relate a stadardized (zero mea ad uit stadard deviatio)

3 Notes: Active ortfolio Maagemet y Zhipeg Ya score to a forecast of residual retur (a alpha). The volatility is the residual volatility. IC is the correlatio betwee the scores ad the returs. Chapter Cosesus Expected Returs: The CAM 1. The CAM is about expected returs, ot risk.. There is a tedecy for betas towards the mea. 3. Forecasts of betas based o the fudametal attributes of the compay, rather their returs over the past 60 or so moths, tur out to be much better forecasts of future beta. 4. eta allows us to separate the excess returs of ay portfolio ito two ucorrelated compoets, a market retur ad a residual retur. (o theory or assumptio are eeded to get this poit) 5. CAM states that the expected residual retur o all stocks ad ay portfolio is equal to zero. Expected excess returs will be proportioal to the portfolio s beta. 6. Uder CAM, a idividual whose portfolio differs from the market is playig a zero-sum game. The player has additioal risk ad o additioal expected retur. This logic leads to passive ivestig;, i.e., buy ad hold the market portfolio. 7. The ideas behid the CAM help the active maager avoid the risk of market timig, ad focus research o residual returs that have a cosesus expectatio of zero. 8. The CAM forecasts of expected retur will be as good as the forecasts of beta. Chapter 3 Risk I. Itroductio 1. Risk is stadard deviatio of retur. The cost of risk is proportioal to variace.. Ivestors care more about active ad residual risk tha total risk. 3. Active risk depeds primarily o the size of the active positio ad ot the size of the bechmark positio. II. Defiig risk 1. Variace will add across time if the returs i oe iterval are ucorrelated with the returs i other itervals of time. The autocorrelatio is close to zero for most asset classes. Thus, variaces will grow with the legth of the forecast horizo ad the risk will grow with the square root of the forecast horizo.. Active risk = Std (active retur) = Std(r r ) 3. Residual risk of portfolio relative to portfolio is defied by ω = σ β σ Cov( r, r ) Where, β = Var( r ) 3

4 Notes: Active ortfolio Maagemet y Zhipeg Ya 4. The cost of risk equates risk to a equivalet loss i expected retur. This cost will be associated with either active or residual risk. III. Structural Risk Models r ( t, t + 1) = β, ( t) f ( t, t + 1) u ( t) k k + k Where, r is excess retur. eta is the exposure of asset to factor k. it is kow at time t. IV. Choosig the factors 1. All factors must be a priori factors. That is, eve though the factor returs are ucertai, the factor exposures must be kow a priori, i.e., at the begiig of the period. Three types of actors:. Reposes to exteral ifluece: macro-factors. They suffer from two defects: - The respose coefficiet has to be estimated through a regressio aalysis or some similar techique. Error i variables problem. - The estimate is based o behavior over a past period of approximately five years. It may ot be a accurate descriptio of the curret situatio. These respose coefficiets ca be ostatioary. 3. Cross-sectioal comparisos These factors compare attributes of the stocks with o lik to the remaider of the ecoomy. These cross-sectioal attributes ca themselves be classified i two groups: fudametal ad market. - Fudametal attributes iclude ratios such as divided yield ad earigs yield, plus aalysts forecasts of future earigs per share. - Market attributes iclude volatility over a past period, mometum, optio implied volatility, share turover, etc. 4. Statistical factors - pricipal compoet aalysis, maximum likelihood aalysis, expectatios maximizatio aalysis, usig returs data oly; - We usually avoid statistical factors, because they are very difficult to iterpret, ad because the statistical estimatio procedure is proe to discoverig spurious correlatio. These models also caot capture factors whose exposures chage over time. 5. Three criteria: icisive, ituitive ad iterestig. - Icisive factors distiguish returs. - Ituitive factors relate to iterpretable ad recogizable dimesios of the market. - Iterestig factors explai some part of performace. 6. Typical factors: - Idustries - Risk idices: measure the differig behavior of stocks across other, oidustry dimesios, such as, volatility, mometum, size, liquidity, growth, value, earigs volatility ad fiacial leverage. - Each broad idex ca have several descriptors. E.g. volatility measures might iclude recet daily retur volatility, optio implied volatility, recet price rage, ad beta. Though typically correlated, each descriptor captures oe aspect of the 4

5 Notes: Active ortfolio Maagemet y Zhipeg Ya risk idex. We costruct risk idex exposures by weightig exposures of the descriptors withi the risk idex. 7. Quatify exposures to descriptors ad risk idices stadardize exposures! Chapter 4 Exceptioal Retur, echmarks, ad Value Added I. Itroductio 1. Exceptioal expected retur is the differece betwee our forecasts ad the cosesus.. echmark portfolios are a stadard for the active maager. 3. Active maagemet value added is expected exceptioal retur less a pealty for active variace. 4. Maagemet of total risk ad retur is distict from maagemet of active risk ad retur. 5. echmark timig decisios are distict from stock selectio decisios. II. Termiology 1. eta is the beta betwee portfolio ad bechmark.. Active positio is the differece betwee the portfolio holdigs ad the bechmark holdigs. h A = h - h 3. Active variace: T AVar = h V h = σ + σ σ = β σ Var( residualrisk) A A, + III. Compoets of Expected retur (R is the total retur o asset ) E(R ) = 1+ if + β μ + β Δf + α (4.1) - Time premium, i F the compesatio for time. - Risk premium: beta*μ, where μ is expected excess retur o the bechmark, usually a very log-ru average (50 years). - Exceptioal bechmark retur: beta* f, f is your measure of that differece betwee the expected excess retur o the bechmark i the ear future ad the log-ru expected excess retur. - Alpha: expected residual retur. - Exceptioal expected retur: beta* f + alpha: the first term measures bechmark timig; the secod compoet measures stock selectio. IV. Maagemet of total risk ad retur 1. Active maagemet starts whe the maager s forecasts differ from the cosesus.. The forecast of expected excess retur for portfolio p ca be expressed as: f = β f + α Same as (4.1) 5

6 Notes: Active ortfolio Maagemet y Zhipeg Ya Where, f is the forecast of expected excess retur for the bechmark. These forecasts will differ from cosesus forecasts to the extet that f differs from the cosesus estimate μ, ad alpha differs from zero. 3. The total retur total risk tradeoff (too aggressive) U ( ) = f λ T σ, where f is the expected excess retur ad the secod term is a pealty for risk. λ measures aversio to total risk. μ λt = σ f Δf β = = 1+ = 1+ active beta( β A ), which is the ratio of our λt σ μ forecast for bechmark exceptioal retur to the cosesus expected excess retur o the bechmark we will argue that this expected utility criterio will lead to portfolios that are typically too aggressive for istitutioal ivestmet maagers. V. Focus o value added 1. Expected utility objective high residual risks. The root cause is our evehaded treatmet of bechmark ad active risk. However, maagers are much more adverse to the risk of deviatio from the bechmark tha they are adverse to the risk of the bechmark.. A ew objective that splits risk ad retur ito three parts: - Itrisic, f λ T σ. This compoet arises from the risk ad retur of the bechmark. It is ot uder the maager s cotrol. λ is aversio to total risk. - Timig, β A Δf λt β A σ. This is the cotributio from timig the bechmark. It is govered by the maager s active beta. Risk aversio λt to the risk caused by bechmark timig. - Residual, α λr ω. This is due to the maager s residual positio. Here we have a aversio to the residual risk. - The last two parts of the objective measure the maager s ability to add value: VA = ( β A Δf λt β A σ ) + ( α λr ω ) (4.15) - The value added is a risk-adjusted expected retur that igores ay cotributio of the bechmark to risk ad expected retur. - The ew objective fuctio splits the value added ito value added by bechmark timig ad value added by stock selectio. Chapter 5 Residual Risk ad Retur: The Iformatio Ratio I. Itroductio: The iformatio ratio measures achievemet ex-post ad cootes opportuity ex-ate. Here, we are cocered about the trade off betwee residual risk ad alpha. Whe portfolio beta is equal to oe, residual risk ad active risk coicide. 6

7 Notes: Active ortfolio Maagemet y Zhipeg Ya II. The defiitio of Alpha 1. Look-forward (ex-ate), alpha is a forecast of residual retur. Lookig backward (ex-post), alpha is the average of the realized residual returs.. r () t = α + β r () t + ε () t (5.1) Where, r s are excess returs. The estimates of alpha ad beta obtaied from the regressio are the realized or historical alpha ad beta. The residual returs for portfolio are: θ() t = α + ε() t, where alpha is the average residual retur ad ε(t) is the mea zero radom compoet of residual retur. 3. Lookig forward, alpha is a forecast of residual retur. α = E( θ ) 4. Alpha has the portfolio property sice both residual returs ad expectatios have the portfolio property. α = h ( 1) α1 + h () α 5. y defiitio the bechmark portfolio will always have a residual retur equal to zero; i.e. θ = 0 with certaity. The alpha of the bechmark portfolio must be zero. Risk-free portfolio also has a zero residual retur; so the alpha for cash is always equal to zero. Thus, ay portfolio made up of mixture of the bechmark ad cash will have a zero alpha. III. Ex-post iformatio ratio: A measure of achievemet 1. A iformatio ratio is a ratio of (aualized) residual retur to (aualized) residual risk.. A realized iformatio ratio ca (ad frequetly will) be egative. 3. The ex-post iformatio ratio is related to the t-statistic oe obtais for the alpha i the regressio (equatio 5.1). If the data i the regressio cover Y years, the the iformatio ratio is approximately the alpha s t-statistic divided by the square root of Y. IV. Ex-ate iformatio ratio: A measure of opportuity 1. The iformatio ratio is the expected level of aual residual retur per uit of aual residual risk. The more precise defiitio of the iformatio ratio is the highest ratio of residual risk to residual stadard deviatio that the maager ca obtai.. Reasoable levels of ex-ate iformatio ratios ru from 0.5 to Give alpha ad portfolio residual risk, ω, the iformatio ratio for portfolio is: α IR =, (5.5) ω 4. Our persoal iformatio ratio is maximum iformatio ratio that we ca attai over all portfolios: IR= Max IR { } 7

8 Notes: Active ortfolio Maagemet y Zhipeg Ya 5. The iformatio ratio is idepedet of the maager s level of aggressiveess. ut it does deped o the time horizo. Iformatio ratio icrease with the square root of time. V. The Residual Frotier: The Maager s Opportuity Set the alpha versus residual risk (omega) tradeoffs. The residual frotier will describe the opportuities available to the active maager. The ex-ate iformatio ratio determies the maager s residual frotier. VI. The active maagemet objective 1. To Maximize the value added from residual retur where value added is measured as: VA[ ] = α λr ω (5.7) (igorig bechmark timig here) awards a credit for the expected residual retur ad a debit for residual risk.. Value added is sometimes referred to as a certaity equivalet retur. VII. refereces meet opportuities: The iformatio ratio describes the opportuities. The active maager should explore those opportuities ad choose the portfolio that maximizes value added VIII. Aggressiveess, Opportuity, ad residual risk aversio. 1. Max 5.7, subject to 5.5 the optimal level of residual risk must satisfy. * IR ω = (5.9) our desired level of residual risk will λ icrease with our opportuities ad decrease with our residual risk aversio.. It is possible to use 5.9 to determie a reasoable level of residual risk aversio. IR λ = (5.10) * ω IX. Value added: risk-adjusted residual retur * * * IR ω IR 1. Combie 5.5, 5.7 ad 5.9 VA = VA[ ω ] = = ability of the 4λR maager to add value icreases as the square of the iformatio ratio ad decreases as the maager becomes more risk averse. X. The beta = 1 frotier How do our residual risk/retur choices look i the total risk/total retur picture? The portfolios we will select (i the absece of ay bechmark timig) will lie alog the beta = 1 frotier XI. Forecast alphas directly 1. Oe way to get alpha is to start with expected returs ad the go through the procedure described i chapter 4.. Forecast alpha directly. 8

9 Notes: Active ortfolio Maagemet y Zhipeg Ya Step 1: sort the assets ito five bis: strog buy, buy, hold, sell ad strog sell. Assig them respective alphas of %, 1%, 0%, -1% ad -% Step : fid the bechmark average alpha. If it is zero, quit. Step 3: Modify the alphas by subtractig the bechmark average times the stock s beta from the origial alpha. These alphas will be bechmark-eutral. I the absece of costraits they should lead the maager to hold a portfolio with a beta of 1. More ad more elaborate variatios o this theme. For example, we could classify stocks ito ecoomic sectors ad the sort them ito strog buy, buy, hold, sell ad strog sell bis. 3. This example first, we eed ot forecast alphas with laser-lie precisio. The accuracy of a successful forecaster of alphas is apt to be fairly low. Ay procedure that keeps the process simple ad movig i the correct directio will probably compesate for losses i accuracy i the secod ad third decimal poits. Secod, although it may be difficult to forecast alphas correctly, it is ot difficult to forecast alphas directly. Chapter 6 The Fudametal Law of Active Maagemet I. The Fudamet Law 1. R: the strategy s breadth is defied as the umber of idepedet forecasts of exceptioal retur we make per year ad;. IC: the maager s iformatio coefficiet is measure of skill the correlatio of each forecast with the actual outcomes. We assume that IC is the same for all forecasts. 3. The fudametal law coects breadth ad skill to the iformatio ratio through the (approximately true) formula: IR= IC R (6.1) - The approximatio igores the beefits of reducig risk that our forecasts provide. For relatively low values of IC (below 0.1) this reductio i risk is extremely small. * IR IC R 4. y 5.9 ad 6.1 ω = = λ λ - the desired level of aggressiveess will icrease directly with the skill level ad as the square root of the breadth. The breadth allows for diversificatio amog the active bets so that overall level of aggressiveess, ω* ca icrease. The skill icreases the possibility of success; thus, we are willig to icur more risk sice the gais appear to be larger. * * IR IC R 5. y 5.11 ad 6.1 VA = VA[ ω ] = = value added by a strategy 4λR 4λR (the risk-adjusted retur) will icrease with the breadth ad with the square of the skill level. 9

10 Notes: Active ortfolio Maagemet y Zhipeg Ya 6. The fudametal law is desiged to give us isight ito active maagemet; it is t a operatioal tool. 7. A maager eeds to kow the tradeoffs betwee icreasig the breadth of the strategy, - by either coverig more stocks or shorteig the time horizos of the forecasts ad improvig skill, IC. II. Additivity - The fudametal law is additive i the square iformatio ratio. IR = R IC + R IC Ca be applied to two differet asset categories. - Ca be applied to oe asset category + market timig - We ca carry this otio to a iteratioal portfolio. The active retur of a iteratioal portfolio comes from three mai sources: active currecy positios, active allocatios across coutries, ad active allocatios withi coutry markets. - The additivity holds across maagers. I this case we have to assume that the allocatio across the maagers is the optimal. - The law s use i scalig alphas; i.e., makig sure that forecasts of exceptioal stock retur are cosistet with the mager s iformatio ratio. III. Assumptios. 1. The forecasts should be idepedet. Forecast # should ot be based o a source of iformatio that is correlated with the sources for forecast #1. - I a situatio where aalysts provide recommedatios o a firm-by-firm basis it is possible to check the level of depedece amog these forecasts by first quatifyig the recommedatios ad the regressig them agaist attributes of the firms. - Aalysts may like all the firms i a particular idustry: their stock picks are actually a sigle idustry be. - All recommeded stocks may have a high earigs yield: the aalysts have made a sigle bet o earigs to price ratios. - The same maskig of depedece ca occur over time. If you reassess your idustry bets o the basis of ew iformatio each year, while rebalacig your portfolios mothly, you should t thik that you make 1 idustry bets per year: you just make the same bet 1 times. - Suppose two sources of i have the same level of skill IC. If γ is the correlatio betwee the two iformatio sources, the the skill level of the combied sources, IC (com), will be: IC( com) = IC (1 + γ ). The law is based o the assumptio that each of the R active bets has the same level of skill - I fact, the maager will have greater skills i oe area tha aother. 3. The strogest assumptio behid the law is that the maager will accurately gauge the value of his iformatio ad build portfolios that use that iformatio i a optimal way. 10

11 Notes: Active ortfolio Maagemet y Zhipeg Ya IV. Tests: 1. It is desirable to have some faith i the law s ability to make reasoable predictios. Whe we impose istitutioal costraits limitig short sales, the realized iformatio ratios drop slightly. V. You must play ofte ad play well to wi at the ivestmet maagemet game. art II Expected Returs ad Valuatio Chapter 7 Expected Returs ad the Arbitrage ricig Theory I. Itroductio 1. The AT is a model of expected returs. - The flexibility of the AT makes it iappropriate as a model for cosesus expected returs, but a appropriate model for a maager s expected returs. - The AT is a source of iformatio to the active maager. It should be flexible. If all active maagers share the same iformatio it would be worthless.. We eed to defie a qualified model ad fid the correct set of factor forecasts. II. The easy part: fidig a qualified model 1. Amog ay group of N stocks there will be a efficiet frotier for portfolios made up out of the N risky stocks. ortfolio Q (taget portfolio) has the highest reward to risk ratio (Sharpe ratio).. A factor model s qualified, if ad oly if portfolio Q is diversified with respect to that factor model. Diversified with respect to the factor model meas that portfolio Q has miimum risk amog all portfolios with the same factor exposures as portfolio Q. 3. A frotier portfolio like Q should be highly diversified i the covetioal sese of the world. ortfolio Q will cotai all of the stocks, with o exceptioally large holdigs. We wat portfolio Q to be diversified with respect to the multiple-factor model. 4. The ARRA model was costructed to help portfolio maagers cotrol risk, ot to explai expected returs. - However, it does attempt to capture those aspects of the market that cause some groups of stocks to behave differetly tha others. - Well over 99% of the variace of highly diversified portfolios is captured by the factor compoet. 5. Ay factor model that is good at explaiig the returs of a diversified portfolio should be (early) qualified as a AT model. - The exact specificatio of the factor model may ot be importat i qualifyig a model. What is importat is that the model cotais sufficiet factors to capture movemet i the importat dimesios. III. The Hard art: Factor Forecasts 11

12 Notes: Active ortfolio Maagemet y Zhipeg Ya 1. The simplest approach to forecastig factor returs is to calculate a history of factor returs ad take their average. We are implicitly assumig a elemet of statioarity i the market. The AT does ot provide ay guaratees here. However, there is hope. Oe of the o-at reasos to focus o factors is the kowledge that the factor relatioship is stable tha the stock relatioship.. Most structure ca be helpful i developig good forecasts. AT models ca either be purely statistical or structural. The factors have some meaig i the structural model; they do t i a purely statistical model. 3. Factor forecasts are easier if there is some explicit lik betwee the factors ad our ituitio. suggests a opportuistic approach to buildig a AT model. 4. We should take advatage of our coclusio that we ca easily build qualified AT models. We should use factors that we have some ability to forecast. 5. Factor forecasts are difficult. Structure should help. IV. Applicatios: structural vs. statistical. 1. Structural model 1: give exposures, estimate factor returs: The ARRA model takes the factor exposures as give based o curret characteristics of the stocks, such as their earigs yield ad relative size. The factor returs are estimates.. Structural model : give factor returs, estimate exposure: e.g. take the factor returs as the retur o the value-weighted NYSE, gold, a govermet bod idex, ad a basket of foreig currecies. Set the exposure of each stock to the NYSE equal to 1. For the other factors, determie the past exposure of the stock to the factor returs by regressig the differece betwee the stock retur ad the NYSE retur o the returs of the other factors. 3. Structural model 3: combie structural models 1 ad : start with some primitive factor defiitios, estimate the stock s factor exposure as i structural model, the attribute returs to the factors as i structural model Statistical model 1: pricipal compoets aalysis: - Look at 50 stocks over 00 moths. Calculate the 50 by 50 matrix of realized covariace betwee these stocks over the 00 moths. - Do a pricipal compoet aalysis of the covariace matrix. - Typically, oe will fid that the first 0 compoets will explai 90% or more of the risk. Call these 0 pricipal compoet returs the factors. - The aalysis will tell us the exposures of the 50 stocks to the factors ad give us the returs o those factors over the 00 moths. - The factor returs will be ucorrelated. - We ca determie the exposures to the factors of stocks ot icluded i the origial group by regressig the returs of the ew stocks o the returs to the factors. 5. Statistical model : maximum likelihood factor aalysis: - Look at 500 stocks over 60 moths ad 10 factors. have 500*60 = returs. There would be 500*10 = 5000 exposures to estimate ad 60*10 = 600 factor returs to estimate. 1

13 Notes: Active ortfolio Maagemet y Zhipeg Ya Chapter 8 Valuatio i Theory Active maagers must believe their assessmet of value is better tha the market or cosesus assessmet. I. Risk adjusted expectatios 1. Itroduce risk adjusted discout rate, by usig CAM or AT. Ad use usually expected cash flows i the omiator. Or,. Itroduce risk-adjusted expectatios risk-eutral pricig. E * [ cf ()] t = E [ v () t cf ()] t = π (, t s ) v (, t s ) cf (, ts ), s Where v(t,s) is called value multiples, it is: - positive - with expected value oe: E[v(t,s)] = 1. - a fuctio of the retur to portfolio Q, ad proportioal to the total retur o the portfolio S, the portfolio with miimum secod momet of total retur. 3. The value multiples v(t,s) help defie a ew set of probabilities, π*(t,s) = π (t,s)*v(t,s). 4. The role of covariace: E * [ cf( t)] = E[ v( t)] + cov[ cf( t), v( t)] the covariace term will, i geeral, be egative. 5. Value multiples modify the cash flows. The value multiples v(t,s) chage the cash flows by amplifyig some, if it>1, ad reducig others, if v(t,s)<1. sice E[v(t)] = 1, they are o average ubiased. II. Market-depedet valuatio: both risk-free rate ad the value multipliers, are marketdepedet ad ot stock-depedet. E[R] = (1 + risk-free rate) Cov(v, R) E*[R] = (1 + risk-free rate) = E[v*R] expected excess retur o all stocks is determied by their covariace with v. Chapter 9 Valuatio i ractice I. Itroductio 1. The basic theory of corporate fiace provides groud rules for acceptable valuatio models.. The stadard model is Divided Discout Model (DDM). DDMs are oly as good as their growth forecasts. II. Modelig growth d + p p = i d F + r = + ξ = y, p0 p0 Where, r = the excess retur ξ= the ucertai amout of capital appreciatio. Let g = E(ξ) ad f = E(r). Suppose the expected excess retur f icludes both cosesus expected returs ad alphas: f = β f + α 13

14 Notes: Active ortfolio Maagemet y Zhipeg Ya. d i + β f + α = + g = y F p d α = ( if) + ( g f) p β (9.19) The most importat isight we must keep i mid while usig a DDM: The golde rule of DDM: g i, g out each additioal 1% of growth adds 1% to the alpha. The alphas that come out of the DDM are as good as (or as bad as) the growth estimates that go i. 3. Implied growth rates: - use 9.19, assume that the assets are fairly priced ad determie the growth rates ecessary to fairly price the assets: * d - g = if + β f p - The implied growth rate ca idetify compaies priced with urealistic growth prospects. art III Implemetatio Chapter 10 Forecastig I. Itroductio 1. Active maagemet is forecastig. The ucoditioal or aïve forecast is the cosesus expected retur. The coditioal or iformed forecast is depedet o the ifo sources. Historical averages make poor ucoditioal forecasts. 3. The refied forecast has the form volatility*ic*score. 4. Forecasts of retur have egligible effect o forecasts of risk. II. Naïve, raw, ad refied forecasts 1. The aïve forecast is the cosesus expected retur. It is the iformatioless forecast. The aïve forecast leads to the bechmark holdigs.. The raw forecast cotais the active maager s ifo i raw form: a earigs estimate, buy or sell recommedatio, etc. It is ot directly a forecast of exceptioal retur. 3. The basic forecastig formula trasforms raw forecasts ito refied forecasts. 1 Er ( g) Er ( ) Covrg (, ) Var ( g) [ g E( g)] = +, where - r = excess retur vector (N assets); g = raw forecast vector (K forecasts) E(r) = aïve (cosesus) forecast E(g) = expected forecast E(r g) = iformed expected retur, coditioal o g. 14

15 Notes: Active ortfolio Maagemet y Zhipeg Ya - Refied forecast = the chage i expected retur due to observig g: φ 1 = E( r g) E( r) = Cov( r, g) Var ( g) [ g E( g)], this is the exceptioal retur referred to i previous chapters. It ca iclude both residual retur forecasts ad bechmark timig. Ad, give a bechmark portfolio, the aïve (cosesus) forecast is: E(r) = β μ III. Refiig raw ifo: oe asset ad oe forecast 1. Assume: r = θ1+ θ θ81, where theta is biary: -1 or 1 with probability ½. They capture the ucertai compoet of the returs. Each theta has mea 0 ad variace 1.. Assume our forecast: g = + θ1+ θ + θ3+ η η13. The forecast is a combiatio of useful ad useless ifo. Thetas are bits of sigal ad η s are bits of oise. 3. IC = Corr( g, r) = Cov( r, g)/[ std( r) std( g)] =3/(9*4)= g E( g) φ = STD() r corr(, r g) STD( g) score. IV. The forecastig rule of thumb Refied forecast = Volatility * IC * Score, we call the last term as score or z- V. Refiig forecasts: oe asset ad two forecasts * * φ = STD() r ICg zg + STD() r ICg' zg', where ICs take ito accout the correlatio betwee the forecasts. - A good forecaster has a IC of 0.05, a great forecaster has a IC = 0.1, ad a world class forecaster has a IC = A IC higher tha 0. usually sigals a faulty backtest or immiet ivestigatio for isider tradig. VI. Forecastig ad risk. 1. Forecasts of returs have a egligible effect o forecasts of volatility ad correlatio. The little effect there is has othig to do with the forecast ad everythig to do with the skill of the forecaster. We ca cocetrate o the expected retur part of the problem ad ot worry about the risk part.. Let σ RIOR adσ OST be estimates of volatility without forecast ifo ad with forecast ifo. The formula relatig these is: 1/ σost = σrior 1 IC (it is derived from coditioal variace formula). Whe IC is small, havig ifo has very little effect o the volatility forecasts. 15

16 Notes: Active ortfolio Maagemet y Zhipeg Ya Chapter 11 Iformatio Aalysis VII. Itroductio 1. Iformatio aalysis begis by trasformig iformatio ito somethig cocrete: ivestmet portfolios.. Iformatio aalysis is ot cocered with the ituitio or process used to geerate stock recommedatios, oly with the recommedatios themselves. 3. Iformatio aalysis occurs i the ivestmet process before backtestig. Iformatio aalysis looks at the ufettered value of sigals. acktestig looks ot oly at iformatio cotet, but also at turover, tradability, ad trasactios costs. Iformatio aalysis is a two-step process. - Step 1 is to tur iformatio ito portfolios. - Step is to aalyze the performace of those portfolios. VIII. Iformatio ad active maagemet 1. Active maagers use iformatio to predict the future exceptioal retur o a group of stocks. The emphasis is o predictig alpha, or residual retur: beta adjusted retur relative to a bechmark.. So, whe we talk about iformatio i the cotext of active maagemet, we are really talkig about alpha predictors. Iformatio aalysis is a effort to fid the sigal-to-oise ratio. 3. We ca classify iformatio alog the followig dimesios: - rimary or processed - Judgmetal or impartial - Ordial or cardial - Historical, cotemporary, or forecast IX. Iformatio aalysis: step 1: iformatio ito portfolios. 1. As a geeral commet, the ivestmet time period should match the iformatio time period. ortfolios based quarterly iformatio iformatio which chages quarterly ad iflueces quarterly returs should be regeerated each quarter.. Here are six possibilities. Usig book-to-price ratios as a example: - rocedure 1: with buy ad sell recommedatios (rak stocks by b/p, put the top half o the buy list ad the bottom half o the sell list) we could equal (or value) weight the buy group ad the sell group. - rocedure : with scores (rak stocks ito several groups) we could build a portfolio for each score by equal (or value) weightig withi each score category. - rocedure 3: with straight alphas we could split the stocks ito two groups: oe group with higher tha average alphas ad oe with lower tha average alphas. The we ca weight the stocks i each group by how far their alpha exceeds (or lies below) the average. Oe way to geerate alphas from b/p is to assume that they are liearly related to the b/p. So we ca weight each asset i our buy ad sell list by how far its b/p lies above or below the average. This is a elaboratio of procedure 1. 16

17 Notes: Active ortfolio Maagemet y Zhipeg Ya - rocedure 4: with straight alphas we could rak the assets accordig to alpha, ad the group the assets ito quitiles ad the equal (or value) weight withi each groups. This is a elaboratio of procedure - rocedure 5: with ay umerical score we ca build a factor portfolio that bets o the predictio ad does ot make a market bet. The factor portfolio cosists of a log portfolio ad a short portfolio. The log ad short portfolios have equal value ad equal beta, but the log portfolio will have a uit bet o the predictio, relative to the short portfolio. Give these costraits, the log portfolio will tract the short portfolio as closely as possible. For b/p data, we ca build log ad short portfolios with equal value ad beta, with the log portfolio exhibitig a b/p oe stadard deviatio above that of the short portfolio, ad desiged so that the log portfolio will track the short portfolio as closely as possible. - rocedure 6: with ay umerical score we could build a factor portfolio, cosistig of a log ad a short portfolio, desiged so that the log ad short portfolios are matched o a set of pre-specified cotrol variables. For example, we could make sure the log ad short portfolios match o idustry, sector, or smallcap stock exposures. This is a more elaborate form of procedure 5, where we oly cotrolled for beta (as a measure of exposure to market risk). 3. While procedure 5 ad 6 are more elaborate, they are also more precise i isolatig the iformatio cotaied i the data. These procedures build portfolios based solely o ew iformatio i the data, cotrollig for other importat factors i the market. We recommed rocedure 5 or 6 as the best approach for aalyzig the iformatio cotaied i ay umerical scores. X. Iformatio aalysis: step : performace evaluatio 1. t-statistics, iformatio ratio, ad iformatio coefficiets Regress the excess portfolio returs agaist the excess bechmark returs: r () t = α + β r () t + ε () t. The iformatio ratio is the best sigle statistic to capture the potetial for value added from active maagemet. The t is the ratio of alpha to its stadard error. The iformatio ratio is the ratio of aual alpha to its aual risk. 3. If we observe returs over a period of T years, the iformatio ratio is approximately the t divided by the square root of the umber of years of observatios: t stat IR T - the relatioship becomes more exact as the umber of observatios icreases. - The t measures the statistical sigificace of the retur; the iformatio ratio captures the risk-reward tradeoff of the strategy ad the maager s value added. A iformatio ratio of 0.5 observed over five years may be statistically more sigificatly tha a iformatio ratio of 0.5 observed over oe year, but their value added will be equal. 17

18 Notes: Active ortfolio Maagemet y Zhipeg Ya - The distictio betwee t ad iformatio ratio arises because we defie value added based o risk over a particular horizo, i this case oe year. 4. Iformatio coefficiet: i the cotext of iformatio aalysis, it is the correlatio betwee our data ad realized alpha. XI. Advaced topics i performace aalysis: 1. ortfolio turover: give trasactio costs, turover will directly affect performace. Turover becomes importat as we move from iformatio aalysis to backtestig ad developmet of ivestable strategies.. The maximum iformatio ratio should be achieved whe the portfolio holdig period matches the iformatio horizo. We ca also ivestigate the importace of cotrollig for other variables: idustries, size, etc. We ca costruct portfolios with differet cotrols, ad aalyze the performace i each case. XII. Four guidelies ca help keep iformatio aalysis from turig ito data miig: ituitio, restrait, sesibility, ad out-of-sample testig 1. Ituitio must guide the search for iformatio before the backtest begis. Ituitio should ot be drive strictly by data. Ideally, it should arise from a geeral uderstadig of the forces goverig ivestmet returs ad the ecoomy as a whole.. Restrait should gover the backtestig process. I priciple, researchers should map out possible iformatio variatios of before the testig begis. 3. erformace should be sesible. The iformatio deservig most scrutiy is that which appears to perform too well. Oly about 10% of observed realized iformatio ratios lie above Out-of-sample testig ca serve as a quatitative check o data miig. Chapter 1 ortfolio Costructio I. Itroductio 1. Implemetatio icludes both portfolio costructio ad tradig. This chapter will take a maager s ivestmet costraits (e.g., o short sales) as give ad build the best possible portfolio subject to those limitatios. It will assume the stadard objective: maximizig active returs mius a active risk pealty.. ortfolio costructio requires several iputs: the curret portfolio, alphas, covariace estimates, trasactios costs estimates, ad a active risk aversio. Of these iputs, we ca measure oly the curret portfolio with ear certaity. II. Alphas ad portfolio costructio 1. We ca always replace a very complicated portfolio costructio procedure that * * leads to active holdigs, h A, active risk, ψ, ad a ex-ate iformatio ratio, IR, by a direct, ucostraied mea-variace optimizatio usig a modified set of alphas ad the appropriate level of risk aversio (Here we are explicitly focusig portfolio costructio o active retur ad active risk, istead of residual retur ad risk. Without bechmark timig these perspectives are idetical). The 18

19 Notes: Active ortfolio Maagemet y Zhipeg Ya IR modified alphas are: α ' = V h * ψ ad the appropriate active risk aversio is: * A λ ' A IR = ψ * III. Alpha aalysis Here are some procedures for refiig alphas that ca simplify the implemetatio procedure ad explicitly lik our refiemet i the alphas to the desired properties of the resultig portfolios: 1. echmark ad cash eutral alphas. - The first ad simplest refiemet is to make the alphas bechmark eutral. y defiitio, the bechmark portfolio has zero alpha, though the bechmark may experiece exceptioal retur. Settig the bechmark alpha to zero isures that the alphas are bechmark eutral, ad avoids bechmark timig. - We may also wat to make the alphas cash eutral; i.e., the alphas will ot lead to ay active cash positio. It is possible to make the alphas both cash ad bechmark eutral. - Table 1.1 ad 1.: the bechmark alpha is 1.6 basis poits, subtractig β α from each modified alpha the alpha of the bechmark = 0. Idex weight modified alpha beta weight*alpha beta*bechmark alpha modified alpha - beta*bechmark alpha weight*ew alpha Stock 1.8% -1.14% % 0.019% -1.16% -0.06% 4.68% 0.30% % 0.015% 0.8% 0.013% % 0.11% % 0.007% 0.10% 0.007% % -0.78% % 0.015% -0.80% % % 0.60% % 0.019% 0.58% 0.03% 6 5.5% 0.% % 0.018% 0.0% 0.011% 7 4.3% -0.65% % 0.017% -0.67% -0.09% 8 3.7% 0.14% % 0.009% 0.13% 0.005% % -0.19% % 0.007% -0.0% % % -1.10% % 0.00% -1.1% % 11.96% -0.5% % 0.014% -0.53% % 1 4.6% -0.51% % 0.010% -0.5% -0.04% % 0.01% % 0.019% -0.01% % % 0.66% % 0.018% 0.64% 0.030% % 0.14% % 0.017% 0.1% 0.006% % 0.0% % 0.017% 0.18% 0.007% % 0.91% % 0.01% 0.90% 0.083% % 0.1% % 0.015% 0.11% 0.007% % 0.44% 1 0.0% 0.016% 0.4% 0.01% % 0.35% % 0.016% 0.33% 0.014% echmark alpha= 0.016% ew bechmark alpha= 0.001%. Scale the alphas 19

20 Notes: Active ortfolio Maagemet y Zhipeg Ya - Alpha has a atural structure: Alpha = IC*volatility*score. We expect the iformatio coefficiet (IC) ad residual risk (volatility) for a set of alphas to be approximately costat, with the score havig mea zero ad stadard deviatio oe across the set. Hece, the alphas should have mea zero ad stadard deviatio, or scale, equal to IC*volatility. 3. Trim alpha outliers - Closely examie all stocks with alphas greater i magitude tha, say, three times the scale of the alphas - A detailed aalysis may show that some of these alphas deped upo questioable data ad should be igored (set to zero), while others may appear geuie. ull i these remaiig geuie alphas to three times scale i magitude. - A more extreme approach to trimmig alphas forces them ito a ormal distributio with bechmark alpha equal to zero ad the required scale factor. Such approaches are extreme because they typically utilize oly the rakig iformatio i the alphas ad igore the size of the alphas. After such a trasformatio, you must recheck bechmark eutrality ad scalig. 4. Risk factor eutral alphas. - The multiple-factor approach to portfolio aalysis separates retur alog several dimesios. A maager ca idetify each of those dimesios as either a source of risk or as a source of value added. y this defiitio, he does ot have ay ability to forecast the risk factors. He should eutralize his alphas agaist the risk factors. - The eutralized alphas will oly iclude ifo o the factors he ca forecast, plus specific asset ifo. Oce eutralized, the alphas of the risk factors will be zero. - E.g., to make alphas idustry eutral calculate the (cap weighted) alpha for each idustry. The subtract the idustry average alpha from each alpha i that idustry. IV. Trasactios costs 1. Whe we cosider oly alphas ad active risk i the portfolio costructio process, we ca offset ay problem i settig the scale of the alphas by icreasig or decreasig the active risk aversio. Fid the correct tradeoff betwee alpha ad active risk is a oe-dimesioal problem. Trasactio costs make this a two-dimesioal problem.. We must amortize the trasactios costs to compare them to the aual rate of gai from the alpha ad the aual rate of loss from the active risk. The rate of amortizatio will deped o the aticipated holdig period. The aualized trasactios cost is the roud-trip cost divided by the holdig period i years. V. ortfolio revisios 1. The returs themselves become oiser at shorter horizos. Rebalacig at very short horizos would ivolve frequet reactios to oise, ot sigal. ut the trasactios costs stay the same, whether we are reactig to sigal or oise.. We ca capture the impact of ew ifo, ad decide whether to trade, by comparig the margial cotributio to value added for stock, MCVA, to 0

21 Notes: Active ortfolio Maagemet y Zhipeg Ya the trasactios costs. The margial cotributio to value added show how value added, as measure by risk-adjusted alpha, chages as the holdig of the stock is icreased with a offsettig decrease i the cash positio. - As our holdig i stock icrease, α measures the effect o portfolio alpha. - The chage i value added also depeds upo the margial impact o active risk of addig more of stock, MCAR, which measures the rate at which active risk chages as we add more of stock. MCVA = α λ ψ MCAR A - Let C be the purchase cost ad SC the sales cost for stock. The whe SC MCVA C, we should ot make a trade. a bad aroud the alpha for each stock - λ ψ MCAR SC α C + λ ψ MCAR A A VI. Techiques for portfolio costructio: α λa ψ TC 1. Screes - Step1. Rak the stocks by alpha. - Step. Choose the first 50 stocks, say. - Step 3. Equal weight (or cap weight) the stocks. - The scree is robust it depeds solely o rakig. Wild estimates of positive or egative alphas will ot alter the result. - ut screes igore all ifo i the alphas apart from the rakigs. They do ot protect agaist biases i the alphas. If all of the utility stocks happe to be low i the alpha rakigs, the portfolio will ot iclude ay utility stocks.. Stratificatio glorified screeig. - The key is splittig the list of followed stocks ito categories. These categories are geerally exclusive. E.g. - Step 1: classify stocks ito te ecoomic sectors - Step : withi each sector, classify stocks by size: big, medium, ad small. - Step 3: withi each category (30), rak the stocks by alpha, place them ito buy, hold ad sell groups. Weight the stocks so that the portfolio s weight i each category matches the bechmark s weight i those categories. - Stratificatio igores some ifo ad does ot cosider slightly overweightig oe category ad uder-weightig aother. Ofte, little substative research uderlies the selectio of the categories, so risk cotrol is rudimetary. 3. Liear programmig space-age stratificatio - It characterizes stocks alog dimesios of risk, e.g., idustry, size, volatility, beta. - It does ot require that these dimesios distictly ad exclusively partitio the stocks. We ca characterize stocks alog all of these dimesios. The liear program will the attempt to build portfolios that are reasoably close to the bechmark portfolio i all of the dimesios used for risk cotrol. - The liear program takes all of the ifo about alpha ito accout ad cotrols risk by keepig the characteristics of the portfolio close to the characteristics of the bechmark. ut, 1

22 Notes: Active ortfolio Maagemet y Zhipeg Ya - It has difficulty producig portfolios with a pre-specified umber of stocks. Also, the risk-cotrol characteristics should ot work at cross purposes with the alphas. E.g., if the alphas tell you to shade the portfolio toward smaller stocks at some times ad toward larger stocks at other times, you should ot cotrol risk o the size dimesio. 4. quadratic programmig (Q) the ultimate i portfolio costructio - It explicitly cosiders each of the three elemets i our figure of merit: alpha, risk, ad trasactios costs. - Sice a Q is a glorified liear program, it ca iclude all the costraits ad limitatios oe fids i a liear program. - However, the Q requires a great may more iputs tha the other portfolio costructio techiques. More iputs meas more oise. VII. Summary: i the real world, alpha iputs are ofte urealistic ad biased. Covariaces ad trasactios costs are measured imperfectly. The stadard reactio is to compesate for flawed iputs by regulatig the outputs of the portfolio costructio process: placig limits o active stock positios, limitig turover, ad costraiig holdigs i certai categories of stocks to match the bechmark holdigs. Chapter 13 Trasactios Costs, Turover, ad Tradig I. Itroductio 1. Trasactios costs iclude commissios, the bid/ask spread, ad market impact. - Commissios are the charge per share paid to the broker for executig the trade. These ted to be the smallest compoet of the trasactios costs ad the easiest to measure. - The bid/ask spread is approximately the cost of tradig oe share of stock. - Market impact is the cost of tradig additioal shares of stock. It is hard to measure because it is the cost of tradig may shares relative to the cost of tradig oe share. Every trade alters the market.. A strategic questio how we ca reduce trasactios costs while preservig as much of the strategy s value added as possible. We ca attack this i two ways: reducig trasactios costs by reducig turover while retaiig as much as the value added as possible, ad reducig trasactios costs through optimal tradig. 3. Trasactios costs icrease with trade size ad the desire for quick executio, which help idetify the maager as a iformed trader, ad require icreased ivetory risk by the liquidity provider. 4. Trasactios costs are difficult to measure. 5. Trasactios costs lower value added, but you ca ofte achieve at least 75% of the value added with oly half the turover.

23 Notes: Active ortfolio Maagemet y Zhipeg Ya 6. Tradig is itself a portfolio optimizatio problem, distict form the portfolio costructio problem. Optimal tradig ca lower trasactios costs, though at the expese of additioal short-term risk. II. Market microstructure Several cosideratios determie what price the liquidity supplier will charge. 1. The liquidity supplier would like to kow why the mager is tradig. He could oly guess at the value of the maager s ifo by the volume ad urgecy of the proposed trade.. Ivetory risk: Whe the liquidity supplier trades, his goal is to hold the ivetory oly util a opposig trade comes alog. III. Aalyzig ad estimatig trasactios costs 1. The theory of market microstructure says that trasactios costs ca deped o maager style, with tradig speed maily accoutig for differeces i maager style. Maagers who trade more aggressively should experiece higher trasactios costs.. Waye Wager (1993) fids that the most aggressive ifo trader was able to realize very large short-term returs, but they were offset by very large trasactios costs. The slowest traders ofte eve experieced egative short-term returs, but with small or eve egative trasactios costs. 3. Estimatio of expected trasactios costs requires measuremet ad aalysis of past trasactios costs. The best place to start is with the maager s past record of trasactios, ad the powerful implemetatio short-fall approach to measurig the overall cost of tradig. The idea is to compare the returs to a paper portfolio to the returs to the actual portfolio. Differeces i returs to these two portfolios will arise due to commissios, the bid/ask spread, ad market impact, as well as to the opportuity costs of trades which were ever executed. E.g., some trades ever execute because the trader keeps waitig for a good price while the stock keeps movig away from him. Waye Wager has estimated that such opportuity costs ofte domiate all trasactios costs. 4. Most services do t use the implemetatio shortfall approach, because it ivolves cosiderable recordkeepig. They use more simple methods like comparig executio prices to the volume weighted average price (VWA) over the day. Such as approach measures market impact extremely crudely ad misses opportuity costs completely. 5. The most difficult approach is to directly research market tick-by-tick data. Whatever the data aalyzed, the goal is a estimate of expected trasactios costs for each stock, based o maager style, for the possible rage of trade volumes. 6. The ivetory risk model: - Give a proposed trade of size, V trade, the estimated time before a opposig trade appears to clear out the liquidity supplier s et ivetory i the stock is: V trade τ clear =, V V daily is the average daily volume i the stock. daily 3

24 Notes: Active ortfolio Maagemet y Zhipeg Ya τ clear - Ivetory risk: σivetory = σ, where σ is the stock s aual volatility Last step assumes the liquidity supplier demads a retur proportioal to this ivetory risk: Δ = c σ ivetory, where c is the risk/retur tradeoff - Combie the above three equatios together: Vtrade Trasactios costs = commissios + spread/price + ctc, where c icludes the V stock s volatility, a risk/retur tradeoff, ad the coversio from aual to daily uits. IV. Turover, trasactios costs, ad value added VA = α λ ψ, suppose the maager plas to move from portfolio I to 1. A portfolio Q.. urchase turover: TO = * Max[0, h h ],, * 3. sales turover: TO = Max[0, h h ], S,, 4. TO = mi {TOp, TOs} TO TO 5. A lower boud: VA( TO) VA +Δ I VAQ[( ) ( ) TO TO ] 6. You ca achieve at least 75% of the icremetal value added with 50% of the turover. 7. Trasactios costs: Max: VAp TC*TO p whe TC = slope of value added/turover frotier (VA over TO), optimal. 8. Implied trasactios costs: we ca fix the level of turover at the required level, TO R, ad the fid the slope, SLOE(TO R ), of the frotier at TO R. implied trasactios costs = the slope. 9. Reasoable levels of roud trip costs (%) do ot call for large amouts of turover ad that very low or high restrictios o turover correspod to urealistic levels of trasactios costs. 10. It is good ews for the portfolio maager if the trasactios costs differ. Differeces i trasactios costs further ehace our ability to discrimiate our ability to discrimiate adds value. Q daily Q Chapter 14 erformace Aalysis I. Itroductio 1. The goal of performace aalysis is to distiguish skilled from uskilled ivestmet magers. Simple cross-sectioal comparisos of returs ca distiguish wiers from losers. Time series aalysis of the returs ca start to separate skill from luck, by measurig retur ad risk. Time series aalysis of 4

25 Notes: Active ortfolio Maagemet y Zhipeg Ya returs ad portfolio holdigs ca go the farthest toward aalyzig where the maager has skill: what bets have paid off ad what bets have t. The maager s skill ex-post should lie alog dimesios promised ex-ate.. For owers of fuds, some assumptios: - skillful active maagemet is possible; - skill is a iheret quality that persists over time; - that statistically abormal returs are a measure of skill; - Skillful maagers idetified i oe period will show up as skillful i the ext period. 3. For fud maagers: performace aalysis ca be used to moitor ad improve the ivestmet process. erformace aalysis ca, ex-post, help the maager avoid two major pitfalls i implemetig a active strategy. - The first is icidetal risk: maagers may like growth stocks without beig aware that growth stocks are cocetrated i certai idustry groups ad cocetrated i the group of stocks with higher volatility. - The secod pitfall is icremetal decisio makig. A portfolio based o a sequece of idividual asset decisios, each of them wise o the surface, ca soo become much more risky tha the portfolio maager iteded. 4. ortfolio based performace aalysis is the most sophisticated approach to distiguishig skill ad luck alog may differet dimesios. II. Skill ad Luck 1. Efficiet markets hypothesis suggests that active maagers have o skill. - Semi-strog form suggests active maagemet skill is really isider tradig. - Week form rules out techical aalysis as skilled active maagemet, but would allow for skillful active maagemet based o fudametal ad ecoomic aalysis.. Recet studies have show that the average maager matches the bechmark et of fees, that top maagers do have statistically sigificat skill, ad that positive performace may persist. III. Defiig Returs 1. Compoud total retur:. Geometric average retur: t= T R(1, T) = R( t) t= 1 t= T T (1 + g ) = R ( t) t= T 1 3. Average log retur: z = ( ) l[ R( t )] T t= 1 t= T 1 4. Arithmetic average retur: 1 + a = ( ) R( t ) T t= 1 5. Geometric average retur is compouded aually, while the average log retur is compouded cotiuously. It is always true that This does ot ecessarily say that oe measure is better to use tha the other. It t= 1 z g a. 5

26 Notes: Active ortfolio Maagemet y Zhipeg Ya does idicate that cosistecy is importat to make sure we are ot comparig apples ad orages. IV. Cross-sectioal Comparisos - Usually cotai survivorship bias, which is icreasigly severe the loger the horizo. - It does t adjust for risk. The top performer may have take large risks ad bee lucky. V. Returs-based performace aalysis: basic 1. Returs regressio: - asic returs-based performace aalysis accordig to Jese (1986) ivolves regressig the time series of portfolio excess returs agaist bechmark excess returs (separates returs ito systematic ad residual compoets, ad the aalyzes the statistical sigificace of the residual compoet). - r () t = α + β r () t + ε () t - The regressio divides the portfolio s excess retur ito the bechmark compoet ad the residual compoet. θ () t = α + ε () t α - The t-statistic is approximately: t = T ω where α ad ω are ot aualized, ad T is the umber of observatios (periods). The t measures where alpha differs sigificatly from zero.. The t-statistic measures the statistical sigificace of the retur ad skill. The iformatio ratio measures the ratio of aual retur to risk, ad relates to ivestmet value added. The iformatio ratio measures realized value added, whether statistically sigificat or ot. 3. The basic alterative to the Jese approach is to compare Sharpe ratio for the portfolio ad the bechmark. A portfolio with: r r > σ σ - We ca aalyze the statistical sigificace of this relatioship: Assumig that the stadard errors i our estimates of the meas returs r ad r domiate the errors i our estimates of σ ad σ, the stadard error of each Sharpe ratio is approximately : 1/ N - Hece, a statistically sigificat (95% cofidece level) demostratio of skill occurs whe: r r > σ σ N - Dybvig ad Ross (1985) have show that superior performace accordig to Sharpe implies positive Jese alphas, but that positive Jese alphas do ot imply positive performace accordig to Sharpe. 6

27 Notes: Active ortfolio Maagemet y Zhipeg Ya VI. Returs-based performace aalysis: advaced: 1. ayesia correlatio: allows us to use our prior kowledge about the distributio of alphas ad betas across maagers. See Vasicek (1973).. Heteroskedasticity 3. Autocorrelatio. 4. echmark timig: oe fiacially based refiemet to the regressio model is a bechmark timig compoet. The expaded model is: r () t = α + β r () t + γ Max{0, r ()} t + ε () t - The model icludes a dow-market beta, β, ad a up-market beta, β + γ. If γ is sigificatly positive, the we say there is evidece of timig skill; bechmark exposure is sigificatly differet i up ad dow cases. 5. Value added: use the cocept of value added ad ideas from the theory of valuatio (Chapter 8). 6. Style aalysis: attempts to extract as much iformatio as possible out of the time series of portfolio returs without requirig the portfolio holdigs. Like the factor model approach, style aalysis assumes that portfolio returs have the form: J r () t = h r () t + u () t j j j= 1 - The r () t are returs to J styles, the h measure the portfolio s holdigs of j those styles, ad u () t is the selectio retur, the portio of the retur which style caot explai. - Style aalysis attributes returs to several style classes ad givig maagers credit oly for the remaiig selectio returs. - Here the styles typically allocate portfolio returs alog the dimesios of value versus growth, large versus small cap, domestic versus iteratioal, ad equities versus bods. - We estimate holdigs via a quadratic program: h j Miimize Var u () t s.t. j J hj = 1, ad h j >=0 for all j. - Style aalysis requires oly the time series of portfolio returs, ad the returs to a set of style idices. The result is a top-dow attributio of the portfolio returs ito style ad selectio. VII. ortfolio-based performace aalysis. 1. ortfolio-based performace aalysis is a bottom-up approach, attributig returs ito may compoets based o the ex-ate portfolio holdigs ad the givig maagers credit for returs alog may of these compoets.. I cotrast to returs-based performace aalysis, performace-based aalysis schemes ca attribute returs to several compoets of possible maager skill. 3. The aalysis proceeds i two steps: performace attributio ad performace aalysis. VIII. erformace attributio j= 1 7

28 Notes: Active ortfolio Maagemet y Zhipeg Ya 1. erformace attributio looks at portfolio returs over a sigle period ad attributes them to factors. The uderlyig priciple is the multiple-factor model: J r () t = x () t b () t + u () t j j j= 1 - Examiig returs ex-post, we kow the portfolio s exposure, x () t, at the begiig of the period, as well as the portfolio s realized retur, r () t, ad the estimated factor returs over the period. - The retur attributed to factor j is: r () t = x () t b () t. The portfolio s specific j j j retur is u () t.. We are free to choose factors as described i Chapter 3, ad i fact we typically ru performace attributio usig the same risk model factors. However, we are ot i priciple limited to the same factors i our risk model. We wat to choose some factors for risk cotrol ad others as sources of retur. The risk cotrol factors are typically idustry or market factors. 3. I buildig risk models we always use ex-ate factors based o iformatio kow at the begiig of the period. For retur attributio we could also cosider ex-post factors based o iformatio kow oly at the ed of the period. 4. eyod the maager s returs attributed to factors will remai the specific retur to the portfolio. A maager s ability to pick idividual stocks, after cotrollig for the factors, will appear i this term. 5. We ca apply performace attributio to total, active returs, ad eve active residual retur. For active returs, the aalysis is exactly the same, but we work with active portfolio holdigs ad returs; J r () t = x () t b () t + u () t A Aj j A j= To break dow active returs ito systematic ad residual xar, j = xa, j βa x, j, where we simply subtract the active beta times the bechmark s exposure from the active exposure, ad residual holdigs similarly as: har, = ha, β A h,, substitutig these ito ad remember that u = h u A A,, we fid: J r () t = β r () t + x () t b () t + u () t A A AR, j j AR j= 1 j IX. erformace aalysis 1. erformace aalysis begis with the attributed returs each period ad aalyzes the statistical sigificace ad value added of the attributed retur series. This aalysis relies o t-statistic ad iformatio ratio to determie statistical sigificace ad value added. 8

29 Notes: Active ortfolio Maagemet y Zhipeg Ya. Cosider the attributio defied i 14.0, with active returs separated ito systematic ( β r () t ) ad residual, ad active residual returs further attributed A J to commo factors( xar, j () t bj () t ) ad specific returs( u () AR t ). Chapter 15 j= 1 echmark Timig I. Defiig bechmark timig 1. echmark timig is a active maagemet decisio to vary the maaged portfolio s beta with respect to the bechmark. If we believe that the bechmark will do better tha usual, the beta is icreased.. I its purest sese we should thik of bechmark timig as choosig the correct mixture of the bechmark portfolio ad cash. This type of bechmark timig is aki to buyig or sellig futures cotracts o the bechmark. 3. echmark timig is ot asset allocatio. Asset allocatio focuses o aggregate asset classes rather tha specific idividual stocks, bods, etc. 4. The process of selectig a target asset allocatio is called strategic asset allocatio. The variatio i asset allocatio aroud that target is called tactical asset allocatio. We oly address tactical asset allocatio here i that the priciples for active maagemet i oe equity market also apply to tactical asset allocatio. 5. Sice IRT = IC T R, a idepedet bechmark timig forecast every quarter oly leads to a breadth of 4. To geerate a bechmark timig iformatio ratio of 0.5 requires a iformatio coefficiet of 0.5! The fudametal law captures exactly why most istitutioal maagers focus o stock selectio. 6. Stock selectio strategies ca diversify bets cross-sectioally across may stocks. echmark timig strategies ca oly diversify serially, through frequet bets per year. Sigificat bechmark timig value added ca oly arise with multiple bets per year. II. Futures versus stocks - echmark timig is choosig a active beta. We ca implemet bechmark timig with futures. Whe the bechmark has o closely associated futures cotract, the potetial for addig value through bechmark timig is very small. III. Value added 1. VA[ β A Δf ] = β A Δf λt β A σ (15.3) - Where beta is the portfolio s active beta with respect to the bechmark. This is the decisio variable. - The optimal level of active beta is determied by FOC of the previous equatio. - * Δf β A = λt σ (15.4). If we look at forecast deviatio Δ f directly, we ca greatly simplify matters: - Δf = σ IC S, (15.6) 9

30 Notes: Active ortfolio Maagemet y Zhipeg Ya where - IC = iformatio coefficiet, the correlatio betwee our forecasts ad subsequet exceptioal bechmark returs that is a measure of forecastig skill. - S = score, a ormalized sigal with mea zero ad stadard deviatio equal to oe over time. - With a correlatio of IC =0.1, you would expect to be correct 55% of the time. IV. Forecastig Frequecy 1. The volatility of the bechmark over ay period t will be: - σ ( t) = σ / T - eriod by period, the forecastig rule of thumb still applies: σ IC S( t) Δf ( t) = σ ( t) IC S( t) = T - sice we ultimately keep score o a aual basis, we must aalyze the aual value added geerated by these higher frequecy forecast. It is the sum of value added each period. VA = T β ( t) Δf ( t) λ β A ( t) σ ( t) A T t= 1 t= 1 T Chapter 16 Summary I. What we have covered - The active maagemet framework begis with a bechmark portfolio, ad defies exceptioal returs relative to that bechmark. Active maagers seek exceptioal returs, at the cost of assumig risk relative to matchig the bechmark retur. - We measure value added as the risk adjusted exceptioal retur. - The key characteristic measurig a maager s ability to add value is the iformatio ratio, the amout of additioal exceptioal retur he ca geerate for each additioal uit of risk. The iformatio ratio is both a figure of merit ad a budget costrait. A maager s ability to add value is costraied by his iformatio ratio. - Give this framework, portfolio theory coects exceptioal retur forecasts retur forecasts which differ from cosesus expected returs with portfolios that differ from the bechmark. If a maager s forecasts agree with the cosesus, he will hold the bechmark. To the extet that his iformatio ratio is positive, the maager will hold a portfolio that differs from the cosesus. - The fudametal law high iformatio ratio require both skill ad breadth. II. Themes - First, active maagemet is a process. Active maagemet begis with raw ifo, refies it ito forecast, ad the optimally ad efficietly costructs portfolios balacig those forecasts of retur agaist risk. 30

31 Notes: Active ortfolio Maagemet y Zhipeg Ya - Secod, active maagemet is forecastig, ad a key to active maager performace is superior ifo. Most of this book describes the machiery for processig this superior ifo ito portfolios. - Thirdly, active maagers should forecast as ofte as possible. Give the realities of active maagemet, the best hope for a large iformatio ratio is to develop a small edge ad bet very ofte. I this search for breadth, we also advocate icludig multiple sources of ifo: the more the better. III. What s left? What this book ultimately ca t help with, is the search for superior ifo. 31