Tesing Porfolio Efficiency wih Non-Traded Asses: Taking ino Accoun Labor Income, Housing and Liabiliies Roy Kouwenberg Mahidol Universiy and Erasmus Universiy Roerdam Thierry Pos Erasmus Universiy Roerdam This version: 4-May-007 Absrac This sudy exends he classical Gibbons, Ross and Shanken (989) es for mean-variance efficiency of a given porfolio o include linear equaliy resricions on he weighs of a subse of resriced asses. The resriced asses can be hough of as illiquid or non-raded. This includes he relevan applicaions of esing porfolio efficiency while aking ino accoun non-raded labor income, housing and pension liabiliies. Assuming a condiional normal disribuion for he asse reurns, we show ha he es saisic follows an F-disribuion in small samples. JEL Classificaion: G, G Keywords: Porfolio efficiency, Mulivariae es, Asse pricing, Non-raded asses Pos is corresponding auhor: Erasmus Universiy Roerdam, Finance Deparmen, P.O. Box 738, 3000 DR, Roerdam, The Neherlands, email: gpos@few.eur.nl, el: +3-040848, fax: +3-0408965. Financial suppor by Tinbergen Insiue, Erasmus Research Insiue of Managemen and Erasmus Cener of Financial Research is graefully acknowledged. Pim van Vlie is credied for making he daa maerial available. Any remaining errors are he auhors responsibiliy.
Tesing Porfolio Efficiency wih Non-Traded Asses: Taking ino Accoun Labor Income, Housing and Liabiliies Absrac This sudy exends he classical Gibbons, Ross and Shanken (989) es for mean-variance efficiency of a given porfolio o include linear equaliy resricions on he weighs of a subse of resriced asses. The resriced asses can be hough of as illiquid or non-raded. This includes he relevan applicaions of a porfolio efficiency es while aking ino non-raded labor income, housing and pension liabiliies. Assuming a condiional normal disribuion for he asse reurns, we show ha he es saisic follows an F-disribuion in small samples. JEL Classificaion: G, G Keywords: Porfolio efficiency, Mulivariae es, Asse pricing, Non-raded asses
. Inroducion Tess for mean-variance efficiency of a given porfolio are useful ools for porfolio managemen applicaions and empirical asse pricing research. Early efficiency ess such as he classical mean-variance efficiency ess of Jobson and Korkie (980, 98), Gibbons (98), Kandel (984), MacKinlay (987), Shanken (985, 986), and Gibbons, Ross and Shanken (GRS; 989) focus on he case where he porfolio weighs are unresriced. In his paper we consider a seing where he rading of a subse of asses is resriced by linear consrains. The resriced subse of asses can be hough of as illiquid, or when he porfolio weighs are fixed a given values, as non-raded. Applicaions include ess of porfolio efficiency for invesors wih a subsanial invesmen in housing, labor income, or non-raded liabiliies. An example of a relevan applicaion is esing household porfolio efficiency while aking ino accoun an illiquid invesmen in housing ha canno be adjused in he shor-erm. Flavin and Yamashia (00), Cocco (004) and Hu (005) show ha a subsanial invesmen in housing ypical for mos individuals can crowd socks ou of he invesor s porfolio. Pelizzon and Weber (003) es he efficiency of more han 5000 Ialian household porfolios under he assumpion ha he individual s invesmen in housing is fixed. The resuls show ha he consrain on he housing invesmen plays an imporan role in deermining wheher he porfolios are efficien. Our es is no only applicable in he growing field of household finance (for an overview, see Campbell, 006), bu also useful for insiuional invesors wih non-raded liabiliies, such as he liabiliies arising from defined benefis pension schemes (see, e.g., Berkelaar and Kouwenberg, 003). Our analysis sars from he opimaliy condiions for mean-variance efficiency of a given porfolio under consrains. We formulae he null hypohesis of efficiency and propose a es saisic for measuring deviaions from he null. Under he assumpion of a normal
disribuion for he excess asse reurns, we prove ha he es saisic follows an F-disribuion. The unresriced classical GRS es is a special case wihin our framework. Apar from generalizing he GRS es, he conribuion of he paper o he lieraure is ha he es saisic is easily compued and suied for small samples, whereas available ess for efficiency under resricions ypically rely on approximaions, large sample heory or compuer simulaion of he poserior disribuion. This paper aims o enrich he se of mehods for esing mean-variance efficiency under consrains available in he lieraure. Wang (998) exends he Bayesian approach for examining porfolio efficiency of Kandel e al. (995) o include general resricions on he porfolio weighs. Similar o our paper, Wang (998) assumes ha asse reurns are normally disribued. The poserior disribuion of he efficiency measure is compued numerically wih simulaions. An advanage of he numerical approach is ha he es can handle many differen ypes of consrains. Furher, Wang (998) uses direc measures of he degree of porfolio efficiency, such as he maximum improvemen in mean reurn given he variance of he evaluaed porfolio. On he oher hand, simulaions can be ime-consuming and some researchers migh prefer he classical approach of hypohesis esing over he Bayesian approach (which uses poserior odds raios, insead of p-values). We will no ener he debae abou he relaive meris of classical and Bayesian saisics here. Raher, our purpose is o exend he classical approach o esing mean-variance efficiency wih a es ha applies under resricions on he porfolio weighs. Basak, Jagannahan and Sun (00) develop a direc es for porfolio efficiency subjec o shor sale consrains. Similar o Wang (998), Basak e al. (00) measure he maximum improvemen in variance ha can be achieved by forming a porfolio of he primiive asses wih he same mean as he benchmark porfolio. Basak e al. (00) es wheher he poenial improvemen is significanly greaer han zero using a classical 3
saisical approach, complemening he Bayesian approach followed by Wang (998). Basak e al. (00) prove ha he sampling disribuion of he esimaed efficiency measure converges o a normal disribuion as he number of observaions goes o infiniy. In order o derive his asympoic resul he paper applies a linear approximaion mehod. Basak e al. (00, p. 3) repor ha he esimaed efficiency measure is a non-linear funcion of he daa in applicaions wih shor sale consrains and he linear approximaion mehod migh herefore inroduce large errors. Gouriéroux and Jouneau (999) develop a mean-variance efficiency es for an invesmen seing where he porfolio weighs of a subse of asses are fixed a given weighs. Under he assumpion of a mulivariae normal asse reurn disribuion, he es saisic proposed by Gouriéroux and Jouneau (999) follows a chi-square disribuion asympoically. The es includes he unresriced mean-variance es of Jobson and Korkie (980, 98) as a special case. Our paper complemens he work of Gouriéroux and Jouneau (999) by generalizing he classical es of Gibbons, Ross and Shanken (989) o an invesmen seing wih a subse of illiquid or non-raded asses. An advanage of our approach is ha we find he exac small sample disribuion of he es saisic (F-disribuion). Furher, our invesmen seing is slighly more general, as i includes linear resricions on he enire subse of resriced asses. A relevan example of such a consrain is a binding limi on foreign invesmen. Following GRS and ohers, we assume ha he asse reurns follow a join normal disribuion. As shown in Affleck-Graves and McDonald (989), monhly US sock reurns are reasonably normal and he GRS es is robus o he exising non-normaliies. Neverheless, for oher asse reurn series (for example, derivaives or high-frequency daa), deviaions from normaliy can be more severe. In hese cases, we can use, for example, he asympoic mean-variance efficiency es of MacKinlay and Richardson (99) or Zhou s 4
(993) generalizaion of he GRS es under an ellipical disribuion. However, if reurns do no follow an ellipical disribuion, he economic meaning of he mean-variance crierion is no well-defined o begin wih (see, e.g., Chamberlain, 983) and in ha case we would advise he use of more general sochasic dominance efficiency ess (see, e.g., Pos, 003; Kuosmanen, 004). Like he original GRS es, our es does no use condiioning informaion. There exiss mouning evidence in favor of ime-varying risk and ime-varying risk aversion (see, e.g., Ferson e al., 987; Jagannahan and Wang, 996; Leau and Ludvigson, 00). Condiional efficiency generally does no imply uncondiional efficiency (see, e.g., Hansen and Richard, 987), and condiional ess are needed in case of ime variaion. We refer o he recen paper of Ferson and Siegel (006) for ess ha use condiional informaion efficienly and generalizaions of earlier work. On he oher hand, given he lack of heoreical guidance for selecing he appropriae specificaion, condiional ess also enail risk of specificaion error (see, for example, Ghysels, 998). In his paper we focus on uncondiional efficiency and we leave he developmen of a condiional version of he es for fuure research. As a parial remedy, researchers and praciioners applying our uncondiional es can use ad hoc approaches o conrol for ime variaion, including he formaion of es porfolios ha are periodically rebalanced, and moving or rolling window analysis. Finally, we would also like o menion a number of oher papers ha are indirecly relaed o our work. The formulaion of our es saisic for mean-variance efficiency is inspired by he work of Shapiro and Homem-de-Mello (998). Higle and Sen (99) and Shapiro and Homem-de-Mello (998) derive general asympoic ess for he opimaliy of a candidae soluion o a sochasic opimizaion problem. De Roon, Nyman and Werker (00) develop asympoic ess for mean-variance spanning under shor sale consrains and ransacion coss, using a similar es saisic. We refer o Korkie and Turle (00) for an 5
exensive mean-variance analysis of self-financing porfolios, including he derivaion of spanning and efficiency ess under self-financing resricions. Wihin our framework selffinancing consrains can be imposed as well, bu only on a sub-se of he risky asses. The remainder of his sudy is srucured as follows. Secion formulaes he null hypohesis of mean-variance efficiency in an invesmen seing wih a subse of resriced asses. Secion 3 derives our generalizaion of he GRS es saisic and is small sample disribuion. Secion 4 analyzes he size and power of he es. Secion 5 applies our es o wo relevan pracical cases: assessing porfolio efficiency in he presence of non-raded labor income and non-raded liabiliies. Secion 6 ess wheher a value-weighed US sock-bond porfolio is mean-variance efficien, while aking ino accoun a subsanial posiion in nonraded human capial. Finally, Secion 7 presens our conclusions and suggesions for furher research. Throughou he ex, we will use R N for an N-dimensional Euclidean space and N R + for he posiive orhan. To disinguish beween vecors and scalars, we use a bold fon for vecors and a regular fon for scalars. Furher, all vecors are column vecors and we use r for he ranspose of r. Finally, 0 N and N denoe a (xn) zero vecor and a (xn) uniy vecor.. Null Hypohesis of he Tes The invesmen universe includes N risky asses and a riskless asse. Invesors can consruc N porfolios λ R. We assume ha he firs N R risky asses can be raded freely by he invesor, bu ha rading of he las R asses is resriced, e.g. due o lack of liquidiy. We spli he porfolio weigh vecor λ = λ λ ] up ino he N R weighs of he unconsrained [ asses N R R λ R and he R resriced weighs λ R. The porfolio weighs λ of he resriced asses are subjec o a se of K equaliy consrains: A λ = b, wih K R A R, K b R and K R. The resriced asses could for example include he invesor s human 6
capial (labor income), he invesor s house or he liabiliies of a pension fund. In hese hree cases he porfolio weigh is ypically fixed a a paricular value: he consrain marix hen reduces o an ideniy marix, i.e. A = I K, while b specifies he values of he fixed porfolio weighs, i.e. λ = b. Assuming ha he marke is incomplee and no perfec hedge is available o undo he fixed porfolio weighs, we will refer o hese asses as non-raded. The se of feasible porfolios is defined as λ R N N R R Λ : λ R, λ R, Aλ = b. λ The special case A = and b =, represens a es wihou resricions on he porfolio weighs, i.e. he radiional GRS es, while A = I K represens he special case wih nonraded asses wih given porfolio weighs λ = b. Le R N r denoe he excess reurns of he risky asses. The reurns follow a join disribuion wih mean µ E[r] and covariance marix Ω E[( r µ )( r µ ) ]. We make a disincion beween he expeced excess reurns of he raded asses N -R µ R and he resriced asses R µ R, wih = µ µ µ. We pariion he covariance marix Ω Ω similarly: Ω =, wih Ω Ω ( N R) ( N R) Ω R he covariance marix of he raded asses, R R Ω R he covariance marix of he resriced asses and Ω R R ( N R) collecing he covariance erms beween he raded and resriced asses. Invesors choose invesmen porfolios o maximize a mean-variance objecive funcion g r) = E[ r] ζvar[ r], where ζ 0 is a risk aversion parameer. The porfolio ( choice problem is λ Λ { λ } { λ λ } { µ ( ) = max E[ r ] Var[ r ] = max λ ζλ Ωλ} λ Λ ζ λ Λ max g () 7
A given porfolio τ Λ is efficien if and only if i is an opimal soluion of () and saisfies he firs-order Karush-Kuhn-Tucker (KKT) condiions of he consrained opimizaion problem. Before we show he KKT condiions, we firs define he alphas of he asses as he firs-order derivaives of he objecive funcion () wih respec o he porfolio weighs, evaluaed a he given porfolioτ Λ : d α g( λ) dλ λ = τ = µ ζ Ωτ () Using he expression for he alphas, he KKT opimaliy condiions for he efficiency of he given porfolioτ are: α µ ζ ( Ω α = = α µ ζ ( Ω τ + Ω τ ) 0 N R = τ + τ ) A Ω ρ K (3) K wih ρ K R a vecor of Lagrange mulipliers for he K equaliy consrains on λ. The KKT condiions are necessary and sufficien for he quadraic maximizaion problem subjec o linear consrains (), as he covariance marix is posiive definie. α = 0 N In he unresriced case he KKT condiions reduce o he familiar Euler equaion, i.e. all alphas should equal zero. Noe ha in he case wih resricions, even if some resriced asses have non-zero alphas, he evaluaed porfolio can sill be mean-variance efficien. More specifically, he following polyhedral cone gives he se of admissible alphas: 8
N 0 N R K C (A) { z R : z =, ρ K R } (4) A ρ K This sudy develops a es for he null hypohesis ha he evaluaed porfolio is efficien, H : α C(A), agains he alernaive hypohesis of inefficiency, H : α C(A). In he 0 unresriced case we find C( ) = 0 and he null reduces o H : α = 0 N. N 0 A remaining problem is he specificaion of he risk aversion parameer ζ of he invesor holding porfolio τ. The GRS es implicily chooses a value for his parameer by seing he alpha of he evaluaed porfolio equal o zero, ha is, α τ = 0, which gives ζ GRS ( µ τ )( τ Ωτ ). 3 The alphas can hen be expressed as α GRS µ ζ GRSΩτ = µ ( µ τ ) β, wih β ( Ω τ )( τ Ωτ ). This approach is generally no consisen wih he null hypohesis in he case wih resricions on he porfolio weighs, as he alpha of he evaluaed porfolio does no necessarily has o equal zero. However, noe ha in he resriced case he alphas of he N R unresriced asses sill need o be zero: α = 0 N R. Hence, we can infer he invesor s risk aversion parameer ζ from his porfolio of unresriced asses, by solving he equaion α τ 0. This approach is consisen wih he null hypohesis under resricions = and gives he following risk aversion parameer: ζ = ( µ τ )( τ Ω τ + τ Ω τ. ) Afer subsiuing he expression for ζ in he KKT condiions (3), we obain: α µ ( µ τ )( βτ ) α = = α µ µ τ β ( )( τ ) β 0 N R = β A ρ K (5) wih he vecor of beas defined as usual 9
β β + ( Ω τ Ω τ )( τ Ωτ ) Ω τ ( τ Ωτ ), = Ωτ ( τ Ωτ ) = (6) β + ( Ω τ Ω τ )( τ Ωτ ) = We will refer o he alphas defined by (5) as generalized alphas, because under porfolio weigh resricions hey may differ from he classical alphas α GRS. The relaion beween he generalized alphas and he GRS alphas is as follows: α = α GRS + ( µ τ ψ ) β, wih ψ ( µ τ β τ he Treynor raio of he porfolio of unresriced asses. )( ) 3. Empirical Tesing An empirical es of mean-variance efficiency is based on a imeseries of risky asse excess reurns r observed a ime =, L, T, where N r R is a (Nx) vecor of reurns. By analogy o GRS, we define he daa generaing process (DGP) as r = α + β (r τ ) + ε, =, L, T (7) GRS We assume ha he regression errors ε are serially independen and idenically disribued random draws from a mulivariae normal disribuion wih mean 0 N and covariance marix N N Σ ε R, condiional on he reurns ) ( r τ of he invesor s porfolio a ime. Leas squares esimaion of he DGP (7) gives esimaes of he classical beas β and alphas α GRS, bu no an esimae of he generalized alphas α under resricions. To esimae he generalized alphas we use he relaion α = α GRS + ( µ τ ψ ) β, and replace α GRS in (7) by α ( µ τ ψ ) β. Afer some rearranging of he erms, we find: 0
r = α + β ) + u, =, L, T (8) ( βτ ) ( r, τ wih he error erm u defined as u = ε + β r τ µ τ ) + β ( β τ ) ( µ τ r τ ), =, K,. (9) (, T The error erm u follows a mulivariae normal disribuion wih E[ u ] = 0, condiional on he reurns ( r τ ) of he invesor s enire porfolio τ including he R resriced asses and he τ reurns ( r κ ) on he porfolio κ = of N R unresriced asses. We define he 0 R covariance marix of he regression errors as Σ = E u u ]. [ Given he esimaed beas βˆ, we propose he following unbiased esimaor for he generalized alphas based on (8): ˆ ˆ α ˆ µ ˆ( β β τ ) ( ˆ µ τ ). (0) Since he errors are joinly normally disribued, he esimaed generalized alphas also follow a join normal disribuion, condiional on he reurns of he porfolios τ and κ : ˆ ~ (, ( ˆ α N α T +θ ) Σ) () wih ˆ θ S ˆ ˆ, where Ŝ is he Sharpe raio of he unresriced asse porfolio κ, and = ρ τκ ρˆ τκ he esimaed correlaion beween he reurns of he porfolio τ including he R resriced
asses and he reurns of he unresriced asse porfolio κ. The full derivaion of () is in he Appendix. 4 As a es saisic, we will use he smalles disance beween he esimaed generalized alphas and he cone of admissible alphas (4): ξ ( A) ˆ min ( + θ ) z C ( A) ( ˆ α z) Σˆ ( ˆ α z) () = where ˆ T Σ ( T ) ( uˆ uˆ ) is an unbiased esimaor of Σ, based on he empirical regression errors ˆ u ˆ r ˆ α ˆ( β β κ ) ( r κ ). The es saisic is a resriced version of he classical Hoelling s T saisic used in mulivariae saisical analysis. GRS derive he small sample disribuion of he unresriced es saisic ˆ ) ˆ α ˆ ξ ( ) = ( + θ ˆ GRSΣ α GRS. The esimaes αˆ GRS and (T ) Σˆ are independen and follow he normal disribuion in () and a Wishar disribuion wih parameer marix Σ and (T ) degrees of freedom, respecively. I follows ha a simple ransformaion of he es saisic follows an F-disribuion: T ( T N ) N( T ) ξ ( ) ~ f N,( T N ),λ (3) wih non-cenraliy parameer ˆ λ Σ α (4) T ( + θ ) α GRS ε GRS
For he case wih R resriced asses, we will now derive he exac small sample disribuion of he es saisic. We define he augmened consrain marix K N A R as A = [ O K, N R A], where K ( N R) O K, N R R denoes a zero marix. The null hypohesis is H α 0N R α = = = A ρ. Le α A ρ K 0 : K N ( N K ) Μ R denoe a marix whose columns form a basis se for he null space of A. Noe ha he range of he marix Μ, denoed by R (Μ), is equal o he null space of he marix A, denoed by N (A): R (Μ) = N (A). According o he fundamenal heorem of linear algebra, N (Μ ) = R ( A ). The null hypohesis, H :α R( A ), is herefore equivalen o H 0 : α N( Μ ), i.e. H 0 : Μ α = 0N K. We can now formulae he es saisic as follows (see he Appendix for he full derivaion): 0 ξ (A) = ˆ ( ) ˆ α ( ˆ + θ Μ Μ ΣΜ) Μ ˆ α (5) The vecor Μ αˆ follows a (N K)-dimensional mulivariae normal disribuion wih mean Μ α and covariance marix + ˆ T ( θ ) Μ ΣΜ. Hence, he disribuion of ξ (A) is known: T( T N + K ) ( N K)( T ) ξ ( A) ~ f( N K ),( T N + K ), λ A (6) wih non-cenraliy parameer λ + ˆ ) α Μ ( Μ ΣΜ ) Μ A T( θ α (7) 3
Under he null hypohesis, λ = 0 and he es saisic follows a cenral F-disribuion wih A (N K) and (T N + K ) degrees of freedom. The mos relevan applicaions of our efficiency es under resricions involve nonraded asses wih a fixed porfolio weigh, such as he invesor s labor income, housing or he liabiliies of a pension fund. For hese applicaions he porfolio weigh resricions are λ = b, K=R, and he consrain marix reduces o an ideniy marix: A = I R. Given he simple I srucure of he consrain marix, i is sraighforward o show ha Μ = O Μ α ˆ = αˆ and Μ ΣΜ ˆ = Σ ˆ, and herefore he es saisic ξ I ) reduces o ( R N R R, N R. Noe ha ( I R ) = ( + ˆ θ ) ˆ α Σˆ ˆ α (8) ξ Hence, for he special case of non-raded asses, he expression for he es saisic can be simplified considerably. A firs sigh, i migh appear ha he alphas ˆα and regression errors for he N R unresriced asses deermine he value of he es saisic compleely, while he R non-raded asses play no obvious role. Noe, however, ha he esimaed alphas ˆα of he unresriced asses depend explicily on he covariance beween he excess reurns of he unresriced asses and he non-raded asses. The same holds for θˆ and ˆΣ. 4. Size and Power of he Tes We will now invesigae he size and power of our efficiency es under resricions. The small sample disribuion of he es saisic in his paper and in GRS is derived under he assumpion of a condiional mulivariae normal reurn disribuion, given he reurns of he porfolio ha we would like o assess. An uncondiional mulivariae normal disribuion for 4
he asse reurns, reaing he reurns of he given porfolio as a funcion of he random individual asse reurns, i.e. as a random variable, seems more appropriae. Forunaely, Jobson and Korkie (985) show ha he GRS es-saisic follows an F-disribuion as well under he assumpion of an uncondiional mulivariae normal reurn disribuion. Furher, numerical resuls in Jobson and Korkie (98) and Campbell e al. (997) demonsrae ha he GRS es performs much beer in a mulivariae normal seing in erms of size and power han alernaive asympoic ess of porfolio efficiency, such as he Wald es saisic of Jobson and Korkie (98, JK). Given ha our efficiency es is an exension of GRS, a priori we would expec our es o perform well in small samples, regardless of wheher he underlying reurn disribuion is condiionally or uncondiionally normal. On he oher hand, he asympoic es for efficiency under resricions proposed by Gouriéroux and Jouneau (999) is an exension of he asympoic Wald es of JK, and for his reason we do no expec i o perform well in small samples. We will now conduc simulaion experimens o verify hese premises. As a saring poin for he simulaion we use sock and bond reurn daa from he US, consising of he Ibboson long-erm governmen bond index, he Ibboson long-erm corporae bond index and six Fama and French porfolios resuling from a wo by hree double-soring of all US socks based on size and value (source: homepage of Kenneh French). We refer o Table for descripive saisics of annual oal reurn daa from he period 956-005. Afer esimaing he sample mean and covariance marix of he reurns, we calculae he weighs of he unconsrained ex pos angency porfolio (w u ) wih maximum Sharpe raio. As an example of weigh consrains, we fix he porfolio weigh of long-erm governmen bonds a 40% and he weigh of long-erm corporae bonds a 0%. We calculae he weighs of he ex pos angency porfolio subjec o hese consrains (w c ). Nex, we draw random samples of lengh T from a uncondiional mulivariae normal disribuion wih mean 5
and covariance marix fixed a he sample values. We calculae he reurns of he unconsrained porfolio w u and es is efficiency wih he GRS F saisic and he JK Wald saisic. We calculae he reurns of he consrained porfolio w c and es is efficiency wih he F es derived in his paper and he es of Gouriéroux and Jouneau (999). The simulaion is repeaed a oal of S imes o replicae he empirical disribuion of he es saisics. One imporan difference in he implemenaion of he F ess and he ess is ha he ess ideally should include all primary asses ha are par of he given porfolio, as oherwise he es saisic erroneously could ake on negaive values. On he oher hand, he F ess should never use all primary asses in he given porfolio as es asses, as in ha case he residual covariance marix of he regressions in (7) and (8) is singular and he es saisic canno be compued. For his reason we use all N = 8 primary asses (wo bond porfolios and six FF porfolios) o implemen he ess, while we calculae he F ess wih N = 6 primary asses, excluding he mid-cap value porfolio and he mid-cap size porfolio from he se of FF porfolios. Overall, he uncondiional normal simulaion seing wih known opimal porfolio weighs favors he Wald ess, as boh he JK and GJ es were derived under hese assumpions. 5 We also assess he power of he various ess in he simulaion runs. For his purpose we es he efficiency of an equally weighed porfolio of he unresriced asses, which is clearly inefficien based on he ex pos Sharpe raio. Table 3 shows he resuls of he simulaions. For each es he columns of he able show he mean and he variance of he simulaed es saisic, he size of he es a he %, 5% and 0% level (rejecion rae of he ex pos efficien porfolio) and he power of he es a he %, 5% and 0% level (rejecion rae of he equally weighed porfolio). Direcly below each row of simulaion resuls we show he mean and variance of he heoreical es saisic disribuion for comparison. Wih a small 6
sample size of T = 50 observaions, he size of he GRS F es is nearly idenical o he pre-se significance level, while he JK Wald es has a much larger Type I error (3.8% a he 5% significance level and 5.% a he % level). For he ess under porfolio weigh consrains we find similar resuls: wih T = 50 observaions he F es derived in his paper has a size ha is very close o he desired significance level, while he Wald es of Gouriéroux and Jouneau (999) rejecs he null hypohesis oo ofen (e.g., a 0.7% rejecion rae a he 5% significance level). A small sample sizes, i.e. T = 50 and T = 00, he F ess perform much beer han in erms of size han he Wald ess, while in larger samples (T = 00, T = 400 and T = 800) he performance of he Wald es gradually improves. The power of he Wald ess is generally slighly higher han he power of he F ess in small samples, bu his is no a big advanage, given he corresponding large Type I error: he Wald ess rejec he null hypohesis more ofen, regardless of wheher he null is rue or no. In samples of T = 00 and larger, he power of he F and Wald ess is similar. Please noe ha he esimaed mean and variance in Table 3 indicae ha he simulaed disribuion of our es saisic follows he heoreical F disribuion closely in small samples. This is no he case for he Gouriéroux and Jouneau (999) saisic, which has a much higher mean and variance in small samples han he heoreical (asympoic) disribuion. Overall, hese simulaion resuls indicae ha our F es for mean-variance efficiency wih non-raded asses has similar favorable properies as he GRS es in small samples, performing beer han he asympoic Wald es of Gouriéroux and Jouneau (999). Furher, if run he Wald ess wih a reduced se of N = 6 primary asses, insead of he complee se of 8 asses, hen he es saisic is no well-defined (can become negaive) and he simulaed disribuion becomes compleely differen from heoreical disribuion, wih poor simulaed es size resuls (resuls no repored o save space, bu available upon reques). 7
5. Tesing he Efficiency of Porfolios wih Non-Traded Asses In his secion we show how our es for porfolio efficiency wih resriced asses can be applied in he presence of non-raded liabiliies, as well as in he case of non-raded labor income. We do no discuss he relevan case of a non-raded posiion in housing o save some space, bu he approach follows he same seps as in he wo examples in his secion. 5. Asse-Liabiliy Managemen An ineresing applicaion of our mean-variance es under resricions is o es he efficiency of porfolios ha are evaluaed relaive o an exogenous, non-raded, sochasic benchmark. For example, he risk and reurn of he invesmen porfolio of a defined benefi pension plan are usually measured relaive o he growh of he plan liabiliies L, defined as he ne presen value of all fuure pension paymens. The plan surplus, S, is defined as he difference beween he value of he asses, A, and he liabiliies: S = A L. Given a fixed level of plan conribuions, in he shor-erm he fund managers of he plan ypically make a rade-off beween maximizing he expeced value of he plan surplus E[S + ] and avoiding unpredicable flucuaions in he surplus ha migh lead o plan deficis (S + < 0). This rade-off can be formalized wih he following mean-variance surplus managemen problem: max E[ S ] ζ Var[ S + ] (9) + Le s assume for ease of exposiion ha a risk free asse wih reurn R 0 exiss. Le r L denoe he random reurn on he liabiliies from ime o +, in excess of he risk free rae. The (Ix) vecor r denoe he excess reurns of a se of unresriced risky asses available o he pension fund porfolio manager. Given he (Ix) vecor of invesmen porfolio weighs λ, he surplus a ime + is equal o S ) L + = ( + r λ + R0 ) A ( + rl + R0 8
= + R ) S + A ( r λ ( L ( 0 / A ) rl now be reduced o he following equivalen formulaion: ). The mean-variance surplus managemen problem can ~ max E[ r λ ( L / A ) rl ] ζ Var[ r λ ( L / A ) rl ] (0) ~ wih ζ = A ζ. The surplus managemen problem as defined above in (0) has been proposed and sudied by Sharpe and Tin (990). Suppose ha he plan manager would like o evaluae he mean-variance surplus efficiency of he given (Ix) risky asse porfolio τ, assuming no consrains on he risky asse weighs. The firs order condiions for mean-variance surplus efficiency of τ are: ~ α µ ζ ) () Ω ( τ ( L / A σ L ) = 0 I where he (Ix) row vecor µ denoes expeced excess reurns of he risky asses and Ω he corresponding (IxI) covariance marix, while he (Ix) vecor σ L measures he reurn covariance beween he risky asses and he liabiliies. So far, he risk aversion parameer ζ ~ has no been specified ye. To give he plan s fund manager he benefi of he doub, we se he value of ζ ~ such ha he evaluaed porfolio ~ τ has zero alpha. i.e. ζ = τ /[ τ Ω τ ( L / ) σ ]. The firs-order efficiency condiion µ A Lτ now is ( Ω α µ µ τ ( τ Ω τ ( L / A ) σ L ) = 0 I τ ( L / A ) σ τ ) L () 9
Please noe he equivalence beween he firs order condiions for he N R unresriced asses in (5) and he firs order condiions of he surplus managemen problem () above. Our mean-variance es under resricions can be applied o derive an unbiased esimaor for he alphas α and a mulivariae es saisic. Wihin our framework we simply rea he I risky asses as N R unresriced asses wih weighs λ and excess reurns r, and he plan liabiliies as a single resriced asse, i.e. wih R =, wih excess reurn r = r L and porfolio weigh λ. The plan s shor posiion in he liabiliies can be modeled wih he single equaliy consrain A λ = b wih A = and b = L / A ). Noe ha N = I + and K =. ( To implemen he empirical es, we firs esimae he radiional marke model (7) relaive o he reurns ( τ ) r on he pension fund s augmened porfolio = [ τ ( L / )] A τ : r = α + r +, =, L, T (3),, GRS β( τ ( L / A ) rl, ) ε, Nex, he generalized alphas are esimaed wih equaion (0): ˆ ˆ ˆ α = ˆ µ β( β τ ) ( ˆ µ τ ) (4) We calculae he residuals û, corresponding o (4) and esimae he covariance marix ˆΣ. Nex, we compue he value of es saisic as = ( ˆ ) ˆ α ˆ ˆ + θ Σ α, wih ˆ ˆ θ S ρ, where Ŝ is he Sharpe raio of he risky asse porfolio τ and ξ = ˆ τ ˆρ τ he esimaed correlaion beween he reurns of he augmened porfolio τ including he shor posiion in he liabiliies and he reurns of he risky asse porfolio τ. The es saisic for he mean variance surplus 0
efficiency of porfolio τ follows an F-disribuion wih ( N K) = I and ( T N + K ) = ( T I ) degrees of freedom. 5. Mean-Variance Tes wih Non-Traded Labor Income A second relevan applicaion of our mean-variance es under resricions is o es he efficiency of porfolios of individuals wih non-radable labor income. We consider a nonreired individual invesor. A ime he individual s overall wealh W consiss of a liquid invesmen porfolio A invesed in bonds, socks, ec.. and he expeced ne presen value of fuure labor income, denoed by Y. 6 The ne presen value of labor income a ime + is defined as: Y + = ( + r Y + R 0 )Y, wih r Y a normally disribued random variable. The (Ix) vecor r denoes he excess reurns on he risky asses available o he individual, following a mulivariae normal disribuion. Given he (Ix) vecor of invesmen porfolio weighs λ, he individual s wealh a ime + is W A Y R A r R ) Y + = + + + = ( + r λ + 0 ) + ( + Y + 0 = R W A W r λ ( Y / W ) r ) W. ( + 0 ) + (( / ) + Y ime +, The individual invesor s aim is o inves in an efficien porfolio in erms of wealh a max E[ W + ] ζ Var[ W+ ] (5) which is equivalen o he maximizing he following objecive, ~ max E [( A / W ) r λ + ( Y / W ) ry ] ζ Var[( A / W ) r λ + ( Y / W ) ry ] (6) ~ wih ζ = W ζ.
Following he same seps as before, we can derive he firs order condiions for he mean-variance efficiency of a given (Ix) risky asse porfolio τ : (( A / W ) Ωτ + ( Y / W ) σ Y ) α µ µ τ = 0 I (( A / W ) τ Ω τ + ( A / W )( Y / W ) σ τ ) Y (7) wih he (Ix) vecor σ Y measuring he covariance beween he excess asse reurn and he change in he presen value of labor income. Our mehodology can be applied o derive an unbiased esimaor for he alphas α and a mulivariae es saisic for he mean-variance efficiency of he porfolio τ, given he invesor s non-radable labor income. Wihin our framework we rea he I risky asses as N R unresriced asses wih porfolio weighs λ and we ake he ne value presen value of labor income as a single resriced asse λ, subjec o he consrain λ = Y / W. To esimae he classical beas ˆβ we use he marke model (7) relaive o he reurns on he individual s overall porfolio including he value of labor income and we use equaion (0) o esimae he generalized alphas ˆα. The es saisic, = wih I and ( T I ) degrees of freedom. ˆ ξ ( + θ ) ˆ α ˆ Σα ˆ, follows an F-disribuion 6. Empirical Applicaion In his secion we will illusrae our mean-variance efficiency es under resricions wih an empirical applicaion. We will examine US sock marke daa o es if a proxy for he marke porfolio is mean-variance efficien for an individual invesor wih labour income. For various reasons, marke porfolio efficiency is an ineresing hypohesis. Firs, he Sharpe-Linner- Mossin CAPM predics ha he marke porfolio is efficien. Second, marke porfolio
efficiency seems consisen wih he populariy of passive muual funds and exchange raded funds ha rack broad value-weighed indexes. As a proxy for he marke porfolio we consruc a porfolio ha invess 50% in US bonds and 50% in he CRSP all-share index, which is he value-weighed average of all common socks lised on he NYSE, AMEX and NASDAQ markes and covered by CRSP. The 50% porfolio weigh of bonds consiss of an invesmen of 5% in long-erm US governmen bonds and 5% in long-erm corporae bonds, boh represened by oal reurn indices of Ibboson and Associaes. The 50% percen porfolio weigh ha we assign o bonds is no based on prior informaion abou he oal marke value of US long-erm bonds relaive o he oal marke value of US equiy, bu serves as an example and crude approximaion. We use wo ses of es asses. The firs se consiss of value-weighed indusry porfolios from he daa library on he homepage of Kenneh French. The second se of es asses consiss of he Ibboson long-erm governmen bond index, he Ibboson long-erm corporae bond index and four Fama and French porfolios: small socks wih low price o book (SL), small socks wih high price o book (SH), big socks wih low price o book (BL) and big socks wih high price o book (BH). The four Fama and French porfolio were seleced from six porfolios ha resul from a wo by hree double-soring of all US socks based on size and value, available from he daa library on he homepage of Kenneh French. We use annual reurn daa from he pos-war era 956-005, a oal of 50 observaions. We use annual daa for hree reasons. Firs, as argued by Benarzi and Thaler (995), we expec ha many invesors have an invesmen horizon of one year. Second, we know he exac disribuion of he es saisic under normaliy and we would like o exploi his advanage of he es in a small sample seing. Third, annual reurns follow a normal disribuion more closely han asse reurns of higher frequency (e.g. monhly, weekly or daily reurns), which is imporan given ha we assumed normaliy o derive he disribuion of he 3
es saisic. Table and Table display descripive saisics of he excess reurn series, as well as he esimaed correlaions beween he excess reurns. 6. Mean-Variance Tes Resuls wih Non-Traded Labor Income We firs es he efficiency of our proxy for he marke porfolio, consising of 50% bonds and 50% socks, relaive o he se of indusry porfolios wih he unconsrained GRS es, wihou aking he individual s labor income ino accoun. Table 4 shows ha he p-value of he unconsrained GRS es is equal o 0.97, indicaing ha he efficiency of he given porfolio canno be rejeced. We now addiionally ake ino accoun he esimaed value of he individual s labor income, assuming ha i canno be hedged perfecly and ha is weigh in oal wealh is fixed a Y /W. For he growh rae of he individual s labor income we use he yearly change in he series Average hourly earnings of producion workers in he manufacuring secor from he US Bureau of Labor Saisics (hp://www.bls.gov/). Table and show descripive saisics for his series. We choose his paricular series mainly as an illusraion, expecing i o capure sysemaic flucuaions in labor income in he secor ha are relevan for porfolio choice. We would like menion for he sake of compleeness ha individual labor income has a volaile idiosyncraic componen due o he career pah of he individual ha is no fully capured in an average hourly earnings series. We refer readers ineresed in a careful panel esimaion of he individual labor income process o Cocco, Gomes and Maenhou (005). Wha value should we give o Y /W, he presen value of labor income divided by oal wealh? This raio will vary srongly from one individual o anoher, bu some quick back-ofhe-envelope calculaions show ha he presen value of labor income will dominae oher sources of wealh for mos wage-earners. For example, consider a relaively wealhy individual, 0 years from reiremen, wih a liquid invesmen porfolio of $800,000 (assume 4
no homeownership for he sake of simpliciy) and an annual income of $00,000 growing a 3% per year on average. Seing he discoun rae for fuure labor income a 5%, we find ha he presen value of labor income is $90,00, and he raio Y /W = 53%. Considering he same individual a 5 or 5 years from reiremen, he raios are Y /W = 37% and Y /W = 6%, respecively. For a young individual, 40 years from reiremen, wih an annual income of $30,000 and an iniial asse porfolio of $50,000, he raio Y /W is 89%. A 5 and 5 years from reiremen, he raio is 80% and 86%, respecively. Given hese and similar esimaes, we expec he labor-o-oal-wealh raio o be relaively high for he ypical (median) wage-earner and we use 50%, 70% and 90% as base cases for Y /W. Table 4 shows he es resuls for he efficiency of he sock-bond marke porfolio proxy, relaive o he indusry porfolios, given a resriced invesmen in labor income growh rae based on he BLS manufacuring average earnings wih Y /W equal o 50%, 70% and 90%, respecively. In all hree cases we find ha efficiency of he given porfolio canno be rejeced (p-values 0.9, 0.93 and 0.6), as in he unresriced GRS case wihou labor income. Ineresingly, hough, he esimaed alphas of some of he indusry porfolios change considerable once labor income is aken ino accoun. For example, as he presen value of labor income from working in he manufacuring indusry becomes a larger componen of he individual s oal wealh, he esimaed alpha of he Manufacuring indusry porfolio urns from posiive (0.3% per year) o srongly negaive (-3.4% per year). This effec arises due o he relaively high correlaion of 0.9 beween he excess reurns of he Manufacuring indusry porfolio and he change of average hourly earnings in he indusry (measured in excess of he risk-free rae), repored in Table. We find a similar posiive correlaion, and hence decreasing alpha a higher levels of Y /W, for he indusry porfolios Business Equipmen, Durable Consumer Goods and Energy. For he indusry porfolios Non- 5
Durable Consumer Goods, Healh Care and Telecommunicaions, on he oher hand, he correlaion is negaive and he alpha increases a higher levels of Y /W. Table 4 shows ha he average deviaion of he esimaed alphas from he null hypohesis value of zero defined as ˆ α i is relaively high a Y /W = 0.90, bu on he oher hand he value of es saisic is relaively low and he null hypohesis canno be rejeced (p-value 0.6). Basically, as he value of labor income sars o dominae he individual s oal wealh, he sandard deviaion of he regression errors associaed wih he esimaed alphas become larger, as wage growh is no very srongly correlaed wih he indusry porfolio reurns. This laer increasing error effec dominaes he increase in he deviaion of he alphas from zero and overall he value of he es saisic decreases, leading o lower es significance. In Table 5 we repea he efficiency ess, using as es asses he Ibboson long-erm governmen bond index, he Ibboson long-erm corporae bond index and four Fama and French value/size porfolios (Fama and French, 99). No surprisingly, due o he presence of srong size and value effecs in his se of reurns, he unconsrained GRS es wihou considering labor income srongly rejecs he efficiency of our sock-bond marke porfolio proxy (p-value of 0.00). The small value porfolio sicks ou wih an esimaed alpha of 7.9% per annum, followed a some disance by he porfolio of large-cap value socks wih an alpha of 3.6% per annum. Afer aking ino accoun he individual s labor income, he esimaed alpha of he small value porfolio shrinks o 3.0% per year a a labor-income-o-oal-wealh raio of Y /W = 0.90. A he same ime, he esimaed alpha of he small growh company porfolio (SL) drops sharply from -0.4% in he GRS-case o -8.3% in he presence of labor income (a Y /W = 0.90). Furher, bonds become more aracive in he presence of labor income. These effecs are driven by he posiive correlaion beween he reurns of small socks boh value and growh and he growh of average hourly earnings in he 6
manufacuring indusry (measured in excess of he risk-free rae) and he negaive correlaion beween bonds reurns wih labor income growh. Alhough he esimaed alphas change in he presence of labor income, he overall deviaion of he esimaed alphas from zero remains of similar magniude and efficiency of he marke porfolio proxy is srongly rejeced. In he case Y /W = 0.90 he regression errors of he alpha esimaes are relaively high, leading o a somewha higher p-value of 0.09 for he efficiency es, bu efficiency is sill clearly rejeced a he 5% level. 7. Conclusions This paper exends he classical Gibbons, Ross and Shanken (989) es for mean-variance efficiency of a given porfolio o include linear equaliy resricions on he weighs of a subse of resriced asses. Our es can be applied o es porfolio efficiency while aking ino accoun invesmens in non-raded labor income, housing and pension liabiliies. We derive he exac small sample disribuion of he es saisic under boh he null hypohesis and he alernaive hypohesis, under he assumpion of a condiional mulivariae normal disribuion for he excess asse reurns. The unresriced GRS es is a special case wihin our framework. Simulaion experimens demonsrae ha our es performs well: he ype I error of he es is very close o he desired significance level, while he asympoic Wald es of Gouriéroux and Jouneau (999) rejecs he null oo ofen in small samples (wih 50 or 00 observaions). As an illusraion, we apply our es o assess he mean-variance efficiency of a welldiversified US sock-bond porfolio for an individual invesor wih non-raded labor income. We use wo ses of primiive es asses. The firs se consiss of indusry porfolios and he second se consis of four Fama and French size and value porfolios and wo Ibboson longerm bond porfolios. For he growh rae of he individual s labor income we use series Average hourly earnings of producion workers in he US manufacuring secor. Exploiing 7
he suiabiliy of our es for small samples, we use 50 years of annual reurn daa for he efficiency ess. In line wih exising evidence, we find ha mean-variance efficiency of he broad sock-bond porfolio canno be rejeced relaive o he indusry porfolios, while efficiency is srongly rejeced when size and value sored porfolios are used as es asses. Taking ino accoun he non-raded fuure labor income of he invesor does no change he conclusions regarding porfolio efficiency, bu i does considerably affec he magniude, and even he sign, of he esimaed alphas. For example, he esimaed alpha of he Manufacuring indusry porfolio changes from 0.3% per year o -3.4% per year, once we ake labor income linked o average wage growh in he manufacuring secor ino accoun. Following GRS, our es assumes a serially-iid normal asse reurn disribuion, wihou incorporaing condiioning informaion abou he sae-of-he-world. Furher research could focus on deriving a version of he es in a seing wih condiioning informaion, following, for example, MacKinlay and Richardson (99), Zhou (993), Jagannahan and Wang (996) and Ferson and Siegel (006). Finally, we would like o sress ha he meanvariance model can fail o disinguish beween efficien and inefficien porfolios if he reurn disribuion is no ellipical (see, for example, Chamberlain, 983). To avoid possible specificaion error, we advise he empirical researcher o use mean-variance efficiency ess in combinaion wih more general sochasic dominance efficiency ess. 8
Appendix In his appendix we prove ha he esimaor of he generalized alphas defined in (0) follows a join normal disribuion condiional on he reurns of he porfolios τ and κ. Le N T R R denoe a marix conaining he sample reurns: R r Lr ). Using µ T RT, ( T ˆ ~ ~ ~ ~ R R µˆ, R R = RR T ~ ~ and δ ( R τ )( τ RR τ ), we can wrie he OLS esimaor for he beas in (7) as βˆ R δ. Furher, i follows ha ˆ β κ = ˆ σ τκ ˆ σ τ, wih R ~ ~ ˆ σ τ ( τ R τ ) and ˆ σ τκ ( κ R ~ R ~ τ ). We can now wrie he esimaor for he generalized alphas in (0) as: α = ˆ µ ˆ( β ˆ β κ ) ( δ ( ˆ σ ˆ σ )( ˆ µ κ )) ˆ τ τκ ( ˆ µ κ ) = R T T (A) Reformulaing he DGP in (8) in marix noaion as R = ( α + T β ( β κ ) κ R + U), wih N T U R denoing he marix of regression errors U u Lu ) generalized alphas are a linear funcion of he errors U: ( T, we can now show ha he ( δ ( ˆ σ ˆ σ )( ˆ µ κ )) ˆ = α + T T τ τκ α U (B) Proof of ˆ α = α + ( δ ( ˆ σ ˆ σ )( ˆ µ κ )) U T T : τ τκ Using R = ( α + T β ( β κ ) κ R + U), δ = 0 and κ R δ = κ ˆ β = ˆ σ α = R( T T ( ˆ µ κ )( ˆ β κ ) ˆ δ ) T ˆ τκ στ = ( α + β ( β κ ) κ T R + U)( T T, we find ( ˆ µ κ )( ˆ β κ ) δ ) = α α( ˆ µ κ )( ˆ β κ ) + U( T T δ + β ( β κ ) T ( ˆ µ κ )( ˆ β κ ) ( ( ˆ µ κ ) ˆ σ ˆ δ ) δ ) = α + U T T τ σ τκ. τκ ( T κ R T ( ˆ µ κ )( ˆ β κ ) κ Rδ ) 9
Condiional on he reurns of he porfolios τ and κ, i follows from (B) ha he esimaor αˆ follows a join normal disribuion wih E [ ˆ] α = α. Below we provide he proof of [ ˆ ] ( ˆ Var α α = T +θ )Σ. Proof of [ ˆ ] ( ˆ Var α α = T +θ ) Σ : ~ ~ Using δ = 0 and δ = ( τ R R τ ) T δ = T ˆ σ τ Var[ ˆ α α] = ( T ( ˆ µ κ ) ˆ σ ˆ σ δ )( T T τ τκ T, we find ( ˆ µ κ ) ˆ σ ˆ σ τ τκ δ )Σ 4 = ( T T ( ˆ µ κ ) ˆ σ ˆ δ ( ˆ µ κ ) ˆ ˆ τ σ τκ + σ τ σ τκ δ T δ )Σ τκ = ( T + T ( ˆ µ κ ) ( ˆ σ ˆ τσ τκ ) )Σ τκ ( ˆ T + θ )Σ, wih θ ˆ µ κ ( ˆ σ ˆ σ ) = ˆ = τ τκ S ˆ ˆ = ρ τκ Proof of ˆ ( ) ( ) ˆ α ( ˆ ξ A = + θ Μ Μ ΣΜ) Μ ˆ α : The es saisic is he soluion o an unresriced minimizaion problem, ha is, ˆ ( ) min ( ) ( ˆ ) ˆ ξ A = + θ α ρ A Σ ( ˆ α A ρ). The soluion o his problem is K ρ R * ρ ˆ ( ) ˆ A Σ A AΣ ˆ α and ˆ α ( ˆ ˆ ( ) ( ) ˆ ( ˆ ) ˆ ξ A = + θ Σ Σ A AΣ A AΣ ) ˆ α. Using Khari s (966) lemma, we find ( ˆ ˆ ( ˆ ) ˆ Σ Σ A AΣ A AΣ ) = Μ ( Μ ΣΜ ˆ ) Μ and hus ˆ ( ) ( ) ˆ α ( ˆ ξ A = + θ Μ Μ ΣΜ) Μ ˆ α. 30
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