Copyrght by Zh Da and Rav Jagannathan Teachng Note on For Model th a Ve --- A tutoral Ths verson: May 5, 2005 Prepared by Zh Da * Ths tutoral demonstrates ho to ncorporate economc ves n optmal asset allocaton n the frameork of for model. The tutoral s structured n a queston-and-anser format. Please try to solve each queston before lookng at ts anser. All orgnal data and detaled calculaton are contaned n an excel spreadsheet --- forve.xls. In addton, MarkotzII s requred for the optmal portfolo calculaton. Basc problem: You are nterested n nvestng n sx portfolos, formed by sortng all stocks accordng to ther market captalzaton and book-to-market rato. Book-to-market Market Cap Lo Medum Hgh Small Portfolo : Small groth Portfolo 2: Small neutral Portfolo 3: Small value Bg Portfolo 4: Bg groth Portfolo 5: Bg neutral Portfolo : Bg value You beleve that the returns are affected by macro economc varables and the dynamcs can be captured by a four-for model, here the excess return of portfolo (or asset) can be rtten as: R = a + b F + b F + b F + b F + e. () t 2 2 3 3 4 4 t The frst for s the excess return on the stock market ndex portfolo. The second for s the change n the slope of the term structure. The thrd for s the change n the yeld spread beteen Baa and Aaa bonds. The fourth for s Ol nflaton (percentage change n Ol prce). The quarterly tme seres of the four fors and excess returns of the sx portfolos 2 from 994 to 2003 are also provded n forve.xls. Today s Dec 3, 2003 and you are currently holdng a passve portfolo of the sx assets. Hoever one energy expert recently told you that the ol prce next year s gong to be a lot hgher than you mght expect. Specfcally, hs forecast of ol nflaton for year 2004 s 25% hgher than the consensus forecast. You are very confdent n hs forecast and therefore ant to ncorporate ths ve n your asset allocaton decson. * Ths tutoral s prepared under the supervson of Prof Rav Jagannathan for the teachng of FINC40 nvestment. Detaled descrpton on the sx portfolos and ther returns can be found n Prof Ken French s ebste: http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_lbrary.html. 2 The excess return s defned as portfolo return mnus rsk free rate.
Queston : What are the for loadngs for each portfolo? Equaton () can be estmated usng regresson n Excel. To do that, go to Tools --- Data Analyss --- Regresson. In ths case, Y corresponds to excess returns of one of the sx portfolos and X corresponds to the four fors. The regresson coeffcents are the for loadngs. Repeat the regresson sx tmes to obtan the for loadngs for all sx portfolos as reported n Table. In general, the market for s sgnfcant for all portfolos. In addton, the term spread for s sgnfcant for the small groth portfolo and the ol for s sgnfcant for the bg value portfolo. In another ords, the bg value portfolo ll be affected the most by the ol prce shock. Queston 2: What s the varance of each of the sx portfolo returns? What s the varance due to for exposure,.e., systematc varance? What s the resdual or specfc varance? What s the varance of the value-eghted market portfolo? What s the specfc and systematc varance of the market portfolo? Usng the for loadngs, e can decompose the total varance of any asset nto systematc varance and resdual varance. Let b denote a vector contanng for loadngs for asset,.e., b = [ b b2 b3 b4 ]. Let Σ denote the covarance matrx of the four fors. Then, e have: Var = SysVar + Re svar, SysVar = b Σb '. We can also do the same varance decomposton for the market portfolo. Let denote a vector contanng the market portfolo s current eghts n the sx assets and Ω denote the covarance matrx of the sx assets. Usng the market values of the sx portfolos at the end of year 2003, e can compute the eghts to be: Mkt port eght Portfolo : Small groth Portfolo 2: Small neutral Portfolo 3: Small value Portfolo 4: Bg groth Portfolo 5: Bg neutral Portfolo : Bg value 2.95% 3.89% 2.23% 58.07% 24.0% 8.7% Then e can compute the total varance of the market as: Var = Ω'. To compute the systematc varance of the market, e need to fnd out the for loadngs of the market, hch are smply eghted averages of for loadngs of ndvdual asset: b, j = SysVar = b j = b, j Σb, and '. 2
To compute the covarance matrx Σ and Ω, go to Tools --- Data Analyss --- Covarance. To carry out matrx operaton, use excel functon MMULT() for matrx multplcaton and TRANSPOSE() for matrx transpose. 3 Notce, the market portfolo have a postve resdual. In ths example, all fors are macro varables (rather than rsk fors) hch are convenent for convertng ves on economy to changes n expected returns. Queston 3: Assume the expected returns of the sx portfolos for year 2004 are: Expected return Portfolo : Small groth Portfolo 2: Small neutral Portfolo 3: Small value Portfolo 4: Bg groth Portfolo 5: Bg neutral Portfolo : Bg value 0.47 0.0 0.0 0.23 0.097 0.093 In addton, e assume every nvestor (ncludng yourself) has a rsk averson of 2.407 and the rsk free rate s 3%. What ll be the optmal portfolo eghts? We solve ths problem usng the MarkotzII spreadsheet. Compute standard devatons and correlatons of the sx portfolos usng the hstorcal returns. Input expected returns, standard devatons, correlatons, rsk averson and rsk free rate nto the MarkotzII and use solver to fnd the optmal portfolo eghts that maxmze the slope of the CAL. Number of securtes: Fll n Names No Name Fron Expected Standard Correlatons 2 3 4 5 0 Return Devaton port port 3 port 5 port 0.03 0.47 0.30 port 2 port 4 port 2 port 2 0.033 0.07 0.208 port 0.89 0.80 0.8 0.7 0.8 3 port 3 0.02 0.02 0.20 port 2.00 0.97 0.73 0.87 0.82 4 port 4 0.58 0.2 0.97 port 3.00 0.5 0.85 0.8 5 port 5 0.242 0.097 0.74 port 4.00 0.7 0.74 port 0.088 0.093 0.9 port 5.00 0.94 port.00 Corr OK? YES Results: Portfolo's Expected Return 0.075 Portfolo's Standard Devaton 0.795 Rsk Free Rate Slope of CAL 0.0300 Rsk Averson Coeffcent: A= 2.407 0.439 Weght on optmal rsky portfolo: x*=.00 All nvestor ll hold only the optmal rsky portfolo (x*=) and the eghts of the optmal rsky portfolo concde th those of the current market portfolo. In another ords, under the expected returns n ths questons, all nvestor chooses to hold the same market portfolo. We call these expected returns --- the consensus expected returns. Queston 4: You expect the ol nflaton to be 25% hgher than the consensus forecast. Ho ll your ve on the ol for affect your expectaton (measured as devaton from the consensus) on other three fors? 3 Remember to press CRTL+SHIFT+ENTER at the same tme for matrx operaton n Excel. 3
Snce all fors are correlated. If you expect ol nflaton to devate from consensus, you ould also expect other fors to devate. To determne the mp of for surprse on ol nflaton on other fors, e compute the ol beta as: ol β = cov( F, F ) / var( F ). The ol beta of for s just the slope coeffcent hen e regress for on ol nflaton for. Intutvely, t captures the senstvty of other for to changes n ol nflaton. Therefore e have: expected change n F = The for surprses are computed as: ol β ol ol * expected change n ol nflaton. Mkt excess return change n term spread change n credt spread Ol nflaton ol beta -0.47-0.0048-0.0032.0000 for surprse -3.7% -0.2% -0.08% 25.00% Queston 5: What are the ne expected returns of the sx portfolos and the market portfolo under your ve? The ne expected returns are just consensus expected returns plus changes due to for surprses. E 4 b, j F j. = ( R ve) = Consensus ER + The market expected return s agan s eghted-average of expected returns of the sx portfolos. Queston : Suppose you can only nvest n the market portfolo and the rsk free asset. Gven the ve on the ol nflaton, hat ll be the fron nvested n the market, and ho much ll be nvested n the rsk free asset? Compute the expected return on your portfolo and ts standard devaton. What ould be the expected return and the standard devaton f you are forced to hold the market portfolo nstead? What s the certanty equvalent return on the to portfolos -- the combnaton of the market and the rsk free you chose to hold gven the ve, and f you are forced to hold the market? Input the ne expected returns under the ve nto MarkotzII and read off the eght on the optmal rsky portfolo (x*) to be 0.45. Ths s the fron you should nvest n the market portfolo gven your ve. 4
Number of securtes: Fll n Names No Name Fron Expected Standard Correlatons 2 3 4 5 0 Return Devaton port port 3 port 5 port 0.03 0.098 0.30 port 2 port 4 port 2 port 2 0.033 0.0 0.208 port 0.89 0.80 0.8 0.7 0.8 3 port 3 0.02 0.058 0.20 port 2.00 0.97 0.73 0.87 0.82 4 port 4 0.58 0.07 0.97 port 3.00 0.5 0.85 0.8 5 port 5 0.242 0.05 0.74 port 4.00 0.7 0.74 port 0.088 0.042 0.9 port 5.00 0.94 port.00 Corr OK? YES Results: Expected return 0.045935 Portfolo's Expected Return 0.052 Std dev 0.0834 Portfolo's Standard Devaton 0.795 Certanty Equvalent return 0.03797 Rsk Free Rate Slope of CAL 0.0300 Rsk Averson Coeffcent: A= 2.407 0.958 Weght on optmal rsky portfolo: x*= 0.45 Table 2 summarzes the man results for varous cases. Frst thng to note s that the Sharpe rato comes don sgnfcantly under your ve on ol nflaton snce the ol shock has a substantal negatve mp on expected returns. Consequently, the rsky asset becomes less favorably aganst the rsk free asset. That s hy you choose to nvest more than half of your ealth n rsk free asset hen the rsk free asset s alloed. The certanty equvalent return s 3.80%. Hoever f you are forced to hold the market portfolo only (x=), the certanty equvalent return drops to 2.4%. Ths means that smply by the rsk free asset, there s an mprovement of.% (3.80%-2.4%) n terms of certanty equvalent return. For a total nvestment of $500,000, ths s the same as an ncrease n value at the end of the year of $500,000*.% = $5,782, hch s not a small amount. Queston 7: Suppose you can no n addton choose a dfferent combnaton of the sx portfolos to construct the optmal rsky asset: hat ould be the eghts assgned to each portfolo? What are the expected return and the standard devaton of ths portfolo gven the ve? What fron ll you hold n the rsk free and ths rsky asset? What are the expected return and the standard devaton of your portfolo? What s the certanty equvalent? Ho much do you gan by beng able to use a dfferent rsky asset than beng forced to choose a combnaton of the market and the rsk free? In ths case, nvestor ll form a ne optmal rsky portfolo by optmally nvestng n the sx assets to maxmze the slope of CAL under the ne expected returns. 5
Number of securtes: Fll n Names No Name Fron Expected Standard Correlatons 2 3 4 5 0 Return Devaton port port 3 port 5 port 3.48 0.098 0.30 port 2 port 4 port 2 port 2 -.234 0.0 0.208 port 0.89 0.80 0.8 0.7 0.8 3 port 3-2.00 0.058 0.20 port 2.00 0.97 0.73 0.87 0.82 4 port 4 0.395 0.07 0.97 port 3.00 0.5 0.85 0.8 5 port 5 2.4 0.05 0.74 port 4.00 0.7 0.74 port -2.209 0.042 0.9 port 5.00 0.94 port.00 Corr OK? YES Results: Expected return 0.07294 Portfolo's Expected Return 0.300 Std dev 0.3302 Portfolo's Standard Devaton.02 Certanty Equvalent return 0.05482 Rsk Free Rate Slope of CAL 0.0300 Rsk Averson Coeffcent: A= 2.407 0.32 Weght on optmal rsky portfolo: x*= 0.3 Both Sharpe rato and certanty equvalent return ncrease as a result of re-optmzaton after changes n expected returns (Table 2). Compare to the case hen you can only hold the market and rsk free asset (n queston ), the certanty equvalent return ncreases by.35% (5.4%-3.80%). Hoever, you observe extreme postons n some assets hch may not be desrable. 4 Queston 8: Sho that t s possble to thnk of the ne optmal rsky portfolo as a portfolo of the current market portfolo and an ACTIVE portfolo that s managed usng the ve. What s the composton of the ACTIVE portfolo; ts expected return; standard devaton and Sharpe Rato? Snce the market portfolo eght n asset and the ACTIVE portfolo eght n asset should add up to be the optmal rsky portfolo s eght n asset, or, +, = opt,, e can solve for the ACTIVE portfolo eght as:, = opt,,. In addton, e kno: hch mples: =, = = =, =, In another ords, the ACTIVE portfolo s a zero-nvestment strategy here the long and short poston net out and the net nvestment s zero. Once e have the eghts of the ACTIVE portfolo, e can compute the expected return and standard devaton and Sharpe rato n the usual ay: opt, = 0. 4 In f, no the optmal eghts solver calculates are more senstve to the precson level settng n solver. You mght ant to choose a hgh precson level. To do that, go to solver --- optons and choose a very small number n the precson feld.
E ( R ve) = =, E( R ve), std. dev = Ω ' and SR = E( R ve) Rf. std. dev Queston 9: What s the alpha th respect to the CAPM of the ACTIVE portfolo from the perspectve of the consensus ve? What s the alpha (gven the expected return on the market gven your ve)? What s the beta? The CAPM alpha s defned as: α = ER Rf β α ve = E ( ER Rf ), [ Rf ]. ( R ve) Rf β E ( R ve) All the expected returns can be computed as usual. To determne the CAPM beta, make use of ts defnton: β cov( R, R = var( R ) ) = Ω Ω ' '. Under the consensus ve, CAPM holds th respect to the current market portfolo. Therefore all asset should have zero alpha ncludng the ACTIVE portfolo. Hoever, the current marker portfolo s no longer the market portfolo under the ol ve (as seen n Q7, you ould rather hold a dfferent optmal rsky portfolo), therefore CAPM does not hold th the current market portfolo and the ACTIVE portfolo has a huge postve alpha of more than 20%. Queston 0: Suppose you can not short sell any asset. What ould be the ne optmal rsky portfolo? Agan sho that ths can be decomposed nto the current market portfolo plus an ve portfolo. What s the expected return, standard devaton, Sharpe Rato of ths ve portfolo? What s the alpha? What s the beta? What s the composton of the nvestor's portfolo? What s the certanty equvalent return? By ho much has the certanty equvalent of the nvestor's portfolo come don by the no short sale constrant gven the ve? Ths queston s smlar to Q7 th addtonal constrants that all eghts must be greater or equal to 0. Usng solver, e obtan the eghts of the constraned optmal rsky portfolo hch nvests only n to assets. Alpha, beta and Sharpe rato of the ACTIVE portfolo n ths case can be computed n a smlar fashon as n Q8 and Q9. 7
Number of securtes: Fll n Names No Name Fron Expected Standard Correlatons 2 3 4 5 0 Return Devaton port port 3 port 5 port 0.4 0.098 0.30 0.034 0.2388 0.09703 0.093092 port 2 port 4 port 2 port 2 0.000 0.0 0.208 port 0.89 0.80 0.8 0.7 0.8 3 port 3 0.000 0.058 0.20 port 2.00 0.97 0.73 0.87 0.82 4 port 4 0.38 0.07 0.97 port 3.00 0.5 0.85 0.8 5 port 5 0.000 0.05 0.74 port 4.00 0.7 0.74 port 0.000 0.042 0.9 port 5.00 0.94 port.00 Corr OK? YES Results: Expected return 0.05094 Portfolo's Expected Return 0.087 Std dev 0.09327 Portfolo's Standard Devaton 0.254 Certanty Equvalent return 0.04047 Rsk Free Rate Slope of CAL 0.0300 Rsk Averson Coeffcent: A= 2.407 0.22 Weght on optmal rsky portfolo: x*= 0.3 The no-short-sale constrant reduces the certanty equvalent return from 5.4% n Q7 to 4.05%. Hoever, t s stll slghtly hgher than that n Q here nvestor s restrcted to only the current market portfolo and the rsk free asset. 8
Table : For Loadngs and T-values for the sx portfolos For Loadngs port term spread credt spread ol.439 5.487 0.279 0.043 2 0.872 3.49-0.08-0.09 3 0.78 3.925.250-0.03 4.022-0.02 2.7-0.00 5 0.749-0.22-2.34-0.02 0.72-0.795 -.202-0.3 T-values port term spread credt spread ol 3.22 3.004 0.028 0.34 2 8.49.993-0.002-0.298 3.472.94 0.3-0.480 4 30.92 -.09 0.873-0.49 5 8.4-0.52-0.270 -.53 7.39-0.542-0.50-2.070 9
Table 2: Optmal eghts, expected returns, standard devatons, certanty equvalent returns and Sharpe ratos under varous cases hold market under consensus ve (Q3) hold market + rf under the ol ve (Q) hold market only under the ol ve (Q) hold optmal rsky port + rf under the ol ve (Q7) hold optmal rsky port (th no short sale constrant + rf under the ol ve (Q0) eght on rsky port 00.00% 45.37% 00.00% 3.0% 3.37% eght on rskfree 0.00% 54.3% 0.00% 8.99% 3.3% Expected return 0.75% 4.59%.52% 7.29% 5.09% std-dev 7.94% 8.4% 7.94% 3.34% 9.33% certanty equ ret.88% 3.80% 2.4% 5.4% 4.05% Sharpe rato 0.43 0.20 0.20 0.32 0.22 0