550.447 Quanttatve ortfolo Theory & erformance Analyss Wee of March 4 & 11 (snow), 013 ast Algorthms, the Effcent ronter & the Sngle-Index Model Where we are Chapters 1-3 of AL: erformance, Rs and MT Chapters 4, 5, and 11 of E&G The Opportunty Set, Effcent s and Selecton Chapters 5+ & 9 n Hull Closer Loo & revew of mean-varance calculatons for Rs, VaR, and Extreme Value Theory (EVT) 1.1 1. Assgnment or Wee of March 4 th Read: A&L, Chapter 3 (Basc Elements of MT) Read: E&G, Chapters 9 & 7 (Effcent ronter and Correlaton Structure for Sngle-Index Model ) See Supplemental on Webste roblems: H5:, 5, 9, 11; 19 (Due Mar 4 th ) roblems: H6: 6 (Due Mar 4 th ) roblems: H9: 3, 6 (Due Mar 4 th ) roblems: EG7: 1,, 4 (Due Mar 13 th ) roblems: EG9: 1, (Due Mar 13 th ) Assgnment or Wees of March 4 th / March 11 th Read: A&L, Chapter 4 (Captal Asset rcng Model and ts Applcaton to erformance Measurement) Read: E&G, Chapters 9 & 7 (Effcent ronter and Correlaton Structure for Sngle-Index Model ) See Supplemental on Webste roblems: EG7: 1,, 4 (Due Mar 13 th ) roblems: EG9: 1, (Due Mar 13 th ) 1.3 1.4 1
Assgnment Sprng Brea: March 18 3 Md Term: EW! Aprl 3 rd Last Day of Class: Wednesday, May 1 st nal: Monday, May 13 th ; 9:00am oon In the classroom: Whtehead 03 Marowtz and Sharpe We ve looed at the formulatons for the effcent fronter, the complexty of the covarance structure, and the algorthmc soluton Sharpe provdes an emprcal smplfcaton 1-factor model allows reducton of unnown parameters n an appealng fashon Let s vew the algorthmc consequence of Sharpe s approach 1.5 1.17 Suppose returns, R can be modeled as R R e m Where R m, s the return on the maret ndex, and Where R m, e are r.v. wth std dev σ m, σ e respectvely e s zero mean, and e s uncorrelated w/r m : cov( e, Rm) Ee Rm R m 0 Estmates of,, and e from tme seres, regresson analyss whch provdes an uncorrelated result (last) nally, we assume, e, e are uncorrelated: E ee 0 1.18 Then we have Mean Return R Rm Varance of Return m e Covarance of returns m If the sngle ndex model holds, then for a The expected return s R XRX X R 1 1 1 Varance X XX m 1 1 1 X m XX m X e 1 1 1 1 1.19
So for the we can wrte Gvng a defnton for alpha and beta X X 1 1 As for rs X X X X Or R X X R R m m 1 1 m m e 1 1 1 1 XX m X e XXm X e 1 1 1 1 1 1 m e 1 If we assume a large, equally weghted then m e 1 Whch s 1/ tmes the average resdual rs n the And as the becomes large, the mportance of the average resdual rs dmnshes to nsgnfcant 1/ Thus m m m X 1 So the measure of the contrbuton of a securty to the rs of a large s ts beta often used a measure of a securtes rs We wll return to ths shortly but frst X 1.0 1.1 1 1 Key s the calculaton of the Optmal ortfolo rst we ll go through the steps and then ratonalze/ustfy The steps are: 1. Identfy the unverse of canddate securtes. Select those securtes desrable to be ncluded n the optmal 3. Calculate the weghtngs for those selected 1. The man step s the selecton process Ths s done usng a measure of desrablty related to the excess return to beta rato Ths s a measure of addtonal return ganed from a canddate securty per unt of non-dversfable rs The addtonal return s relatve to the rs-free rate More formally, we use the rato R R R = the expected return on stoc R = the return on the rsless asset = the expected change n the rate of return on stoc wth a 1% change n the maret return 1.3 3
The selecton process R R When the measure s better than some threshold, the securty s ncluded or convenence, ran the unverse by excess return over beta from hghest to lowest The optmum portfolo conssts of all stocs for whch the excess return s greater than some cut-off, Ths far, the procedure s very smple were we to now the cut-off Lets loo at an example 1.4 The selecton process Consder the 10 stocs and R = 5% If = 5, then only the 1 st 5 are ncluded n optmal 1.5 The selecton process So how do we fnd The cut-off s determned successvely by assumng a monotoncally ncreasng-szed set ncludng successvely lesser securtes To do ths we desgnate C as a canddate for C s calculated wth securtes assumed n the optmal We now that we have found the optmum C, called, when all securtes used n the calculaton of C have excess return to beta above C and all securtes not used n the calculaton The selecton process ndng (result now; verfcaton later) The stocs are raned by excess return to rs from hgh to low or a of stocs, C s gven by R R m 1 e C 1 m 1 e R R But we can show that ths s ust C where: have excess returns to beta below C = exp change n ROR on stoc w/ 1% change n opt 1.6 R = exp return on opt 1.7 4
The selecton process ndng R R C But we can show that ths s ust where: = exp change n ROR on stoc w/ 1% change n opt R = exp return on opt Of course we don t now these untl the opt s found, but we can stll mae use of the expresson as R R We add securtes to the canddate as long as C Or from the above, as long as R R R R R R R R 1.8 The selecton process ndng R R R R The rhs s ust the expected excess return on a partcular stoc, based solely on the exp performance of the opt The lhs s the analysts estmated excess return on the stoc Thus, f analyss of a partcular stoc leads the manager to beleve t wll perform better than would be expected, based on the opt, t should be n the R R ow we loo at how => m 1 e can be used n our C 1 framewor to fnd m 1 1.9 e R R m 1 e C 1 m 1 e ndng return to the Table (assume m = 10%) The selecton process The 10 stocs and R = 5% Whch follows from 1.30 1.31 5
The selecton process ndng The ey pont s Columns 5 & 6 are cumulatve sums of columns 3 & 4, resp Column 7 s ust m column 5 C 1 m column 6 As soon as Col 7 reaches a maxmum, or Col 7 s greater than Col because of our orderng scheme we have found 1.3 The Optmal ortfolo Weghtngs rst we recall the case where we allow short sales w/ rsless lendng and borrowng (smlar n other cases) The crtera s to maxmze the obectve functon R R Subect to X 1 1 Thus wth R 1R XR XR 1 1 We can wrte XR R 1 X X X 1 1 1 1.33 The Optmal ortfolo Weghtngs And fndng the maxmum nvolves solvng the equatons d 0 where 1,, dx Whch can be shown (Chapter 6) as equvalent to solvng R R 1 1 for = 1,, Where the optmum proporton to nvest n each stoc s X 1 1.34 The Optmal ortfolo Weghtngs Lets loo at: R R 1 1 1 But for our sngle ndex Sharpe s model we have m e m 1 1 R R Or combnng terms R R e m 1 And solvng R R m R R * C e e 1 e Where we defne m 1 1.35 6
The Optmal ortfolo Weghtngs As an asde, to get the earler dauntng expresson for we must express R R m and m e e 1 1 In terms that do not nvoe 1 To do so, we multply the left hand equaton by beta and sum R R m 1 1 e 1 e 1 We can solve for R R 1 m 1.36 1 1 e 1 e The Optmal ortfolo Weghtngs Whch, when substtuted nto the expresson for gves R R m 1 e m 1 1 m as before 1 e 1.37 The Optmal ortfolo Weghtngs On slde 1.7 we sad we could show that ths optmal cutoff could be expressed as R R C where: = exp change n ROR on stoc w/ 1% change n opt R = exp return on opt We have defned m 1 Also, s proportonal to the optmal fracton of the the nvestor should hold n each stoc, X The proportonalty constant multplyng X s the rato of the excess return of the optmal to the varance of ts return 1.38 The Optmal ortfolo Weghtngs The proportonalty constant s the rato of the excess return of the optmal to the varance of ts return R R Thus R R m X 1 rom the sngle ndex model, we recognze the beta, so R R X R R X R R m m m 1 1 1.39 7
The Optmal ortfolo Weghtngs Multplyng by 1 ( = ) so m 1 m R R R R 1cov(, ) 1 R R R R Where s the regresson coeffcent of the return on securty to the return on the Whch gves the other form for whch we used n defnng the selecton process on 1.7 R R 1.40 The Optmal ortfolo Weghtngs We end the asde to formalze the approach and now determne the optmal weghtngs The ey equatons are X R R C * and e 1 The second equaton determnes the relatve nvestment n each securty, and The frst scales the weghts so they sum to 1: fully nvested ote the sgnfcance that the resdual varance, e, n each securty plays n determnng how much to nvest n each 1.41 Optmal Weghtngs or the example Our weghtngs are R R Col 4 Col e X Where the come from Col 4 of the frst table, and 0.091 The : The X : X1 3.46% 5.45 not 4.5 => 0.3875 0.0956 X 4.65% 0.3875 0.0775 X 3 19.98% 0.3875 0.110 X 4 8.36% 0.3875 0.01375 X 5 3.54% 0.3875 1 1.4 Important to note that the prevous result s exactly the same as would have been found had the problem been solved usng the establshed quadratc programmng result Taes a fracton of the tme Smple Calculatons Easy to debug The characterstcs of a stoc that mae t attractve and relatvely desrable are determned n advance or off-lne Desrablty of any stoc s solely a functon of ts excess return to beta rato 1.43 8
Wth Short Sales Roughly the same technque as before Ran order va excess expected return over beta rato nd and bfurcate set of stocs relatve to ow, = 10, so go to last row (see next slde) for C 10 = 4.5 Rato hgher than are longs, those lower are shorts R R Weghtngs use C * agan, pos or neg e Then % are found n two ways from ether X or X 1 1 Dependng on defnton of short sales (renvest proceeds nto longs (1 st ) or renvest nto rs-free rate, aa Lntner, ( nd 1.45 )) Wth Short Sales Returnng to the example 1.46 Wth Short Sales Returnng to the example Wth Short Sales Returnng to the example 10 1 1.379 1.47 1.48 9
Constant Correlaton assumpton and an approach to selectng the optmum Utlzng the result that the effcent fronter can be determned by solvng the smultaneous equatons R R 1 But now we assume So R R 1 1 Solvng for 1 R R 1 1.49 Constant Correlaton Solvng for 1 R R where 1 1 As before, to express n nown terms, multply by each and the and add up the equatons R R C * 1 1 1 1 Developng and solvng for yelds R R 1 R R 1 1 1 1 1 1 1 Or R R 1 1.50 1 Constant Correlaton Approach Securtes can be raned accordng to ther excess return to standard devaton quotent R R The cutoff, as used before, s found usng R R C 1 1 Lets loo at an example Constant Correlaton Approach 1.51 1.5 10
(let ρ = 0.5) Constant Correlaton Approach To determne weghtngs n each of the 3 qualfers The optmum to nvest n each chosen stoc s X where X And 1 1 R R C * 1 1.53 1.54 Constant Correlaton Approach The Example where C * = 1/4 A fnal example A fnal example 1.55 1.56 11
A fnal example Usng the earler rule; ranng s 1, 3,, 5, 4 and = 11.8 so 1 & 3 are n The last thng for the sngle ndex model s parameterzaton for each securty, The model R Rm e The alpha and beta are determned by regresson on the returns seres Rt Rt Rmt Rmt Beta m t1 m Rmt Rmt t1 Alpha R R t mt 1.57 1.58 The last thng for the sngle ndex model s parameterzaton for each securty, The model R Rm e Alpha and beta from regresson on the returns seres Model Statstcs Estmate Error 1 e Rt Rmt t1 Coeffcent of Determnaton the square of the correlaton m m m m m m a measure of how much of the varaton n a sngle stoc s due to a varaton n the maret Standard Error n beta for securty : 1.59 Hstorcal betas better predctors on than on stocs 1.60 e m How explanatory s ths parameterzaton? How much assocaton n beta one perod to the next? Beta s on large contan a great deal of nformaton about future betas not so good for ndvdual securtes 1
How explanatory? Can the predctve ablty be mproved? Blume s Results Betas tend to one 0.343 0.677 1 Adust predcton n next perod Adds undesrable modfcaton of average for a populaton of stocs ths bas on ndvdual betas should be removed orecast mproved by subtractng bas after adustng (`48 `54) 1 (`55 `61) 1.0 & under model 1.0 => 1.033 for `6-`68 or ndvdual beta: subtract 1.033, add 1.0 for `6-`68 1.61 How explanatory? Can the predctve ablty be mproved? Vasce s Results adust predcton: tae a weghted average of stoc s hstorcal beta, 1, and the prevous perod average beta, 1, over a sample of stocs ML ust taes average (equally weghted) Vascae: 1 1 1 1 1 1 1 1 1 1 = varance n dstrbuton of hstorcal perod betas over a set of stocs = varance n dstrbuton of beta for securty measured n the hstorcal perod, Ths s a Bayesan Technque 1 e1 m1 1.6 How explanatory? Can the predctve ablty be mproved? Both Blume & Vasce lead to better betas than not adustng Is beta the rght crtera? In analyss the ey nputs are expected returns, varances and correlatons, so: Better to consder ablty to gve better correlatons! Ths s what matters to optmzaton m Study of Correlaton matrx forecast Use the hstorcal correlaton matrx tself Matrx from hstorcal beta forecast Matrx from Blume-adusted betas usng pror perods Matrx from Vasce-adusted Bayesan technque 1.63 How explanatory? Can the predctve ablty be mproved? Hstorcal correlaton matrx was worst by far All sngle ndex models dd better surprsng when t mght be though that these loose nformaton n achevng smplfcaton! Bayesan Technque s best f results are forced to have statonary average correlaton to the perod where the model was ftted More can be sad about other factors that nfluence beta and how fundamentals can provde mprovements to the forecast - later 1.64 13