Steven E. Pav Steven@cerebellumcapital
Portfolio inference with this one weird trickSteven E. Pavsteven@cerebellumcapital.comCerebellum CapitalMay 16, 2014Steven Pav (Cerebellum Capital)Portfolio Inference .May 16, 20141 / 36
One Weird TrickA weird .Consider p-vector of asset returns, x. Let:µ : E [x] ,Σ : Var (x) . Prepend a ‘1’ to the vector: x̃ : 1, x .The second moment of x̃ contains the first two moments of x: hi 1µ Θ : E x̃x̃ .µ Σ µµ Steven Pav (Cerebellum Capital)Portfolio Inference .May 16, 20142 / 36
One Weird Trick. trick 1then: Θ1 µ Σ 1 µ µ Σ 1 Σ 1 µΣ 1 1 ζ 2 ν , ν Σ 1 ν is the Markowitz portfolio,ζ is the Sharpe ratio of ν (cf. Hotelling’s T 2 ),Σ 1 is the ‘precision matrix’.The portfolio is ‘optimal’, solving e.g., Roy’s problem: [17]ν argmaxν: ν Σν R 2ν µ r0 ,ν Σνi.e., “maximize Sharpe with a bound on risk.”Steven Pav (Cerebellum Capital)Portfolio Inference .May 16, 20143 / 36
One Weird TrickBut is it useful?Steven Pav (Cerebellum Capital)Portfolio Inference .May 16, 20144 / 36
Portfolio InferenceSample estimator Since Θ E x̃x̃ the simple estimator is unbiased: 1 X1µ̂ x̃i x̃i .Θ̂ : µ̂ Σ̂ µ̂µ̂ n1 i nThe inverse contains the sample estimates: 21 ζ̂ ν̂ .Θ̂ 1 ν̂ Σ̂ 1Steven Pav (Cerebellum Capital)Portfolio Inference .May 16, 20145 / 36
Portfolio InferenceAsymptotics IBy the Central Limit Theorem: n vech Θ̂ vech (Θ)N (0, Ω) , where Ω : Var vech x̃x̃ .We can estimate Ω from the sample, call it Ω̂:It’s just sample covariance of vech x̃i x̃i , for 1 i n.Use the delta method: n vech Θ̂ 1 vech Θ 1 N 0, UΩU .Here U is some ‘ugly’ derivative, depending on Θ.Steven Pav (Cerebellum Capital)Portfolio Inference .May 16, 20146 / 36
Portfolio InferenceAsymptotics IIIgnoring details about symmetry, etc., the derivative is: [7, 12] dX 1 X X 1 .dX(This generalizes the scalar derivative!)Steven Pav (Cerebellum Capital)Portfolio Inference .May 16, 20147 / 36
Portfolio InferenceI can make a hat or a brooch or a pterodactyl. Θ̂ 1 1 ζ̂ 2 ν̂ ν̂ Σ̂ 1 . What is the use for Var vech Θ̂ 1 ?Perform inference on elements of ν via Wald statistic.(Compare elements of ν to their standard errors.)Perform inference on the maximal Sharpe ratio, ζ .Equivalently, Hotelling’s T 2 test. (tests hypothesis: µ is all zeros)Portfolio shrinkage.Estimate the covariance of ν̂ and Σ̂ 1 . (Attribute portfolio error toreturns or covariance.) [5]Steven Pav (Cerebellum Capital)Portfolio Inference .May 16, 20148 / 36
Portfolio InferenceImplementation: trust but verifyrequire(MarkowitzR)set.seed(2014)X - matrix(rnorm(1000 * 5), ncol 5) # toy dataism - MarkowitzR::mp vcov(X)walds - function(ism) ism W/sqrt(diag(ism What))print(t(walds(ism))) # Wald stats##X1X2X3X4X5## Intercept 0.89 -0.22 -1.6 -2.4 -0.49# c.f. Britten-Jones, http://jstor.org/stable/2697722y - rep(1, dim(X)[1])print(t(summary(lm(y X - 1)) coefficients[, 3]))##X1X2X3X4X5## [1,] 0.89 -0.22 -1.6 -2.5 -0.48Steven Pav (Cerebellum Capital)Portfolio Inference .May 16, 20149 / 36
Portfolio InferenceGame over?Steven Pav (Cerebellum Capital)Portfolio Inference .May 16, 201410 / 36
Portfolio InferenceSelling this weird trickWhy weird trick, not Britten-Jones, or Okhrin et al.? [4, 2, 14]Fewer assumptions: fourth moments exist vs. normality of returns.Straightforward to use HAC estimator for Ω.Models covariance between return and volatility. (At a cost?)Solves a larger problem, e.g., can use for inference on ζ 2 .Steven Pav (Cerebellum Capital)Portfolio Inference .May 16, 201411 / 36
Portfolio InferenceSelling this weird trickWhy weird trick, not Britten-Jones, or Okhrin et al.? [4, 2, 14]Fewer assumptions: fourth moments exist vs. normality of returns.Straightforward to use HAC estimator for Ω.Models covariance between return and volatility. (At a cost?)Solves a larger problem, e.g., can use for inference on ζ 2 .Real question: what’s wrong with vanilla Markowitz?This trick can be adapted to deal with:Hedged portfolios.Heteroskedasticity.Conditional expected returns.Perhaps more .Steven Pav (Cerebellum Capital)Portfolio Inference .May 16, 201411 / 36
ExtensionsHedged portfolios IHedging: the goalReturns which are statistically independent from some random variables.Hedging: a more realistic goalA portfolio with zero covariance to some random variables.Hedging: an achievable goalA portfolio with zero sample covariance to some other portfolios oftradeable assets.(e.g., you may have to hold some Mkt to hedge out the Mkt.)Steven Pav (Cerebellum Capital)Portfolio Inference .May 16, 201412 / 36
ExtensionsHedged portfolios IImaxν:GΣν 0,ν Σν R 2ν µ r0 ,ν Σνwhere G is a pg p matrix of rank pg .Rows of G define portfolios against which we have 0 covariance.Typically G consists of some rows of identity matrix.i.e., “Maximize Sharpe ratio with risk bound and zero covariance to someother portfolios.”Solved by cν G, , with c to satisfy risk bound, and 1 Gµ .ν G, : Σ 1 µ G GΣG Steven Pav (Cerebellum Capital)Portfolio Inference .May 16, 201413 / 36
ExtensionsHedged portfolios III Use the weird trick! Let G̃ : 1 00 G , then, 1Θ 1 G̃ G̃ΘG̃ G̃ µ Σ 1 µ µ G 1GΣG Gµ ν G, Σ 1 ν G, G 1GΣG G . ν G, is the optimal hedged portfolio.UL corner is squared Sharpe ratio of ν G, .Also used for portfolio spanning. [16, 8, 10, 11]LR corner is loss of precision?Steven Pav (Cerebellum Capital)Portfolio Inference .May 16, 201414 / 36
ExtensionsHedged portfolios IVDelta method gives the asymptotic distribution: n vech G̃ Θ̂ 1 vech G̃ Θ 1 N 0, UΩU ,with more ugly derivatives.Steven Pav (Cerebellum Capital)Portfolio Inference .May 16, 201415 / 36
ExtensionsHedged portfolios VDownload the Fama-French 3 factor Momentum monthly data(1927-02-01 to 2014-01-01) from Quandl. [13]Add risk-free rate back to market, compute (unhedged) Markowitzportfolio, and Wald statistics.w.stats - rbind(do.both(ff4.xts[, 1:4]), wtrick.ws(ff4.xts[,1:4], vcov.func sandwich::vcovHAC))rownames(w.stats)[3] - c("weird trick w/ HAC")xtable(w.stats)Britten Jones t-statweird trick Wald statweird trick w/ HACSteven Pav (Cerebellum Capital)Mkt6.285.375.10SMB0.720.770.77Portfolio Inference .HML4.994.473.92UMD8.206.035.53May 16, 201416 / 36
ExtensionsHedged portfolios VINow hedge out Mkt:walds - function(ism) ism W/sqrt(diag(ism What))Gmat - matrix(diag(1, 4)[1, ], ncol 4)asymv - MarkowitzR::mp vcov(ff4.xts[, 1:4], fit.intercept TRUE,Gmat 7HML4.47UMD6.03And compute the spanning Wald statistic:ef.stat - function(ism) ism mu[1]/sqrt(ism Ohat[1, 1])print(ef.stat(asymv))## [1] 3.8Steven Pav (Cerebellum Capital)Portfolio Inference .May 16, 201417 / 36
ExtensionsHedged portfolios VIINow hedge out Mkt and RF:# hedge out RFR tooGmat - matrix(diag(1, 5)[c(1, 5), ], ncol 5)asymv - MarkowitzR::mp vcov(ff4.xts[, 1:5], fit.intercept TRUE,Gmat 5HML2.27UMD3.43RF-1.30And the spanning statistic:print(ef.stat(asymv))## [1] 2.1Steven Pav (Cerebellum Capital)Portfolio Inference .May 16, 201418 / 36
ExtensionsHeteroskedasticityPrior to investment decision, observe si proportional to volatility.Two competing models:(constant):(floating):E [xi 1 si ] si µVar (xi 1 si ) si 2 Σ,E [xi 1 si ] Var (xi 1 si ) si 2 Σ.µpFor (constant), ζ is µ Σ 1 µ, independent of si . (Volatility timevs. wall-clock time)pFor (floating), it is si 1 µ Σ 1 µ, higher when volatility is low.(Volatility drinks your milkshake.)Steven Pav (Cerebellum Capital)Portfolio Inference .May 16, 201419 / 36
ExtensionsHeteroskedasticityPrior to investment decision, observe si proportional to volatility.Two competing models:(constant):(floating):E [xi 1 si ] si µVar (xi 1 si ) si 2 Σ,E [xi 1 si ] Var (xi 1 si ) si 2 Σ.µpFor (constant), ζ is µ Σ 1 µ, independent of si . (Volatility timevs. wall-clock time)pFor (floating), it is si 1 µ Σ 1 µ, higher when volatility is low.(Volatility drinks your milkshake.)Why do I have to choose?(mixed):E [xi 1 si ] si µ0 µ1Steven Pav (Cerebellum Capital)Portfolio Inference .Var (xi 1 si ) si 2 Σ.May 16, 201419 / 36
ExtensionsConditional expectation. ISuppose f -vector f i observed prior to investment decision, and(conditional):E [xi 1 f i ] Bf iVar (xi 1 f i ) Σ,B is some p f matrix. [6, 9, 3]Conditional on observing f i , solveargmaxν: Var(ν xi 1 f i ) R 2 E ν xi 1 f i r0p,Var (ν xi 1 f i )for r0 0, R 0.“Maximize Sharpe, with bound on risk, conditional on f i .”Steven Pav (Cerebellum Capital)Portfolio Inference .May 16, 201420 / 36
ExtensionsConditional expectation. IIOptimal portfolio is cν withν : Σ 1 B f i .Σ 1 B generalizes the Markowitz portfolio:the coefficient of the Sharpe-optimal portfolio linear in features f i .The ‘Markowitz coefficient.’Steven Pav (Cerebellum Capital)Portfolio Inference .May 16, 201421 / 36
ExtensionsConditional expectation. III Same weird trick works! Let x̃i 1 : f i , xi 1 .The uncentered second moment is h i ΓΓf B f Θf : E x̃x̃ ,BΓf Σ BΓf B whereh iΓf : E ff .The inverse of Θf is"Θf 1 Γf 1 B Σ 1 B B Σ 1 Σ 1 BΣ 1#.Σ 1 B appears in off diagonals.B Σ 1 B related to HLT.Steven Pav (Cerebellum Capital)Portfolio Inference .May 16, 201422 / 36
ExtensionsConditional expectation. IVAgain, define sample estimator,Θ̂f : 1 X x̃i x̃i .n1 i nUse Central Limit theorem and delta method to get: 1n vech Θ̂ 1 vechΘffSteven Pav (Cerebellum Capital)Portfolio Inference . N 0, UΩU May 16, 201423 / 36
ExamplesExamples. ITake the Fama-French 3 factor Momentum monthly returns(1927-02-01 to 2014-01-01) from Quandl. [13]Add risk-free rate back to market.Use Shiller’s P/E ratio as predictive state variable.# Z-score the P/E datazsc - function(x, .) (x - mean(x, .))/sd(x, .)features.z - zsc(features, na.rm TRUE)asym - MarkowitzR::mp vcov(ff4.xts[, 1:4], features.z,fit.intercept TRUE, vcov.func sandwich::vcovHAC)xtable(signif(t(walds(asym)), digits 2))InterceptCyclically Adjusted PE RatioSteven Pav (Cerebellum Capital)Mkt3.10-1.80Portfolio Inference .SMB3.30-1.00HML2.40-0.09UMD3.503.70May 16, 201424 / 36
ExamplesExamples. IINow the same, but hedge out Mkt and RF:# hedge out Mkt and RFGmat - matrix(diag(1, 5)[c(1, 5), ], ncol 5)asym - MarkowitzR::mp vcov(ff4.xts[, 1:5], features.z,fit.intercept TRUE, Gmat Gmat, vcov.func sandwich::vcovHAC)xtable(signif(t(walds(asym)), digits 2))InterceptCyclically Adjusted PE RatioSteven Pav (Cerebellum Capital)Mkt0.552.20SMB2.10-1.20Portfolio Inference .HML2.100.12UMD2.107.50RF-1.50-1.40May 16, 201425 / 36
ExamplesWhat’s next?Constrained estimation of Θ. (Linear constraints; rank constraints?)Generalize to higher dimensions?Fancier hedging model?Conditional covariance models?Jak’s Tap?Thank You.Steven Pav (Cerebellum Capital)Portfolio Inference .May 16, 201426 / 36
AppendixBibliography I[1] Clifford S. Asness, Andrea Frazzini, Ronen Israel, and Tobias J. Moskowitz. Fact, fiction and momentum investing.Privately Published, May 2014. URL http://ssrn.com/abstract 2435323.[2] Taras Bodnar and Yarema Okhrin. On the product of inverse Wishart and normal distributions with applications todiscriminant analysis and portfolio theory. Scandinavian Journal of Statistics, 38(2):311–331, 2011. ISSN 1467-9469. doi:10.1111/j.1467-9469.2011.00729.x. URL .[3] Michael W Brandt. Portfolio choice problems. Handbook of financial econometrics, 1:269–336, 2009. URLhttp://shr.receptidocs.ru/docs/5/4748/conv 1/file1.pdf#page 298.[4] Mark Britten-Jones. The sampling error in estimates of mean-variance efficient portfolio weights. The Journal of Finance,54(2):655–671, 1999. URL http://www.jstor.org/stable/2697722.[5] Vijay Kumar Chopra and William T. Ziemba. The effect of errors in means, variances, and covariances on optimal portfoliochoice. The Journal of Portfolio Management, 19(2):6–11, 1993. URLhttp://faculty.fuqua.duke.edu/ charvey/Teaching/BA453 2006/Chopra The effect of 1993.pdf.[6] Gregory Connor. Sensible return forecasting for portfolio management. Financial Analysts Journal, 53(5):pp. 44–51, 1997.ISSN 0015198X. URL https://faculty.fuqua.duke.edu/ charvey/Teaching/BA453 2006/Connor Sensible Return Forecasting 1997.pdf.[7] Paul L. Fackler. Notes on matrix calculus. Privately Published, 2005. URLhttp://www4.ncsu.edu/ pfackler/MatCalc.pdf.[8] Narayan C. Giri. On the likelihood ratio test of a normal multivariate testing problem. The Annals of MathematicalStatistics, 35(1):181–189, 1964. doi: 10.1214/aoms/1177703740. .[9] Ulf Herold and Raimond Maurer. Tactical asset allocation and estimation risk. Financial Markets and PortfolioManagement, 18(1):39–57, 2004. ISSN 1555-4961. doi: 10.1007/s11408-004-0104-2. ] Gur Huberman and Shmuel Kandel. Mean-variance spanning. The Journal of Finance, 42(4):pp. 873–888, 1987. ISSN00221082. URL http://www.jstor.org/stable/2328296.Steven Pav (Cerebellum Capital)Portfolio Inference .May 16, 201427 / 36
AppendixBibliography II[11] Raymond Kan and GuoFu Zhou. Tests of mean-variance spanning. Annals of Economics and Finance, 13(1), 2012. 5.pdf.[12] Jan R. Magnus and H. Neudecker. Matrix Differential Calculus with Applications in Statistics and Econometrics. WileySeries in Probability and Statistics: Texts and References Section. Wiley, 3rd edition, 2007. ISBN 9780471986331. .[13] Raymond McTaggart and Gergely Daroczi. Quandl: Quandl Data Connection, 2014. URLhttp://CRAN.R-project.org/package Quandl. R package version 2.3.2.[14] Yarema Okhrin and Wolfgang Schmid. Distributional properties of portfolio weights. Journal of Econometrics, 134(1):235–256, 2006. URL 0304407605001442.[15] Steven E. Pav. MarkowitzR: Statistical significance of the Markowitz portfolio, 2014. bbychef/MarkowitzR. R package version 0.1403.[16] C. Radhakrishna Rao. Advanced Statistical Methods in Biometric Research. John Wiley and Sons, 1952. URLhttp://books.google.com/books?id HvFLAAAAMAAJ.[17] A. D. Roy. Safety first and the holding of assets. Econometrica, 20(3):pp. 431–449, 1952. ISSN 00129682. URLhttp://www.jstor.org/stable/1907413.[18] Mervyn J. Silvapulle and Pranab Kumar Sen. Constrained statistical inference : inequality, order, and shape restrictions.Wiley-Interscience, Hoboken, N.J., 2005. ISBN 0471208272. URL http://books.google.com/books?isbn 0471208272.Steven Pav (Cerebellum Capital)Portfolio Inference .May 16, 201428 / 36
AppendixCommon Questions IDoesn’t this require fourth order moments?I always use relative (or ‘percent’) returns. These are bounded. Allmoments exist. Identical distribution is a much more questionableassumption. Isn’t the complexity Ω p 4 ?Portfolio optimization for large p (bigger than 20?) is not typicallyrecommended.Won’t estimating a large number of parameters hurt performance? The covariance Var vech x̃x̃ has Ω p 4 elements, but the portfolio isconstructed only from Ω p 2 elements, as with vanilla Markowitz.Steven Pav (Cerebellum Capital)Portfolio Inference .May 16, 201429 / 36
AppendixCommon Questions III want to hedge out exposure to a non-asset.I want that as well. It does not appear to be a simple modification of theweird trick, but it may be one discovery away.I want to maximize Sharpe ratio with a time-dependent risk-free rate.I suspect that the ‘right’ way to do this is to include the RFR as an asset,then hedge out exposure to it. This effectively allows each asset to have anon-unit ‘beta’ to the risk-free, which seems like a higher bar than justhedging a constant unit of the risk-free.What was the quote about the pterodactyl?It was from the movie, Airplane.Steven Pav (Cerebellum Capital)Portfolio Inference .May 16, 201430 / 36
AppendixCommon Questions IIII want to hedge out an asset, but I do not want the mean of that asset tobe estimated.I believe this can be done with constrained estimation of Θ̂. Briefly, if thereare linear constraints one believes Θ satisfies, you can solve a least-squaresproblem to get a sample estimate which satisfies the constraints and is nottoo ‘far’ from the unconstrained estimator. I have not done the analysis,but believe it is another simple application of the delta method.The conditional expectation model is many-to-many. How do I sparseify it?Similar to the above, but I believe one would want to specify linearconstraints on the Cholesky factor of Θ. This might be more complicated.Or maybe not.Steven Pav (Cerebellum Capital)Portfolio Inference .May 16, 201431 / 36
AppendixCommon Questions IVI don’t want to deal with the headaches of symmetry! 10The Cholesky factor of Θ is. This is a lower triangularµ Σ1/2matrix and completely determines Θ. I suspect much of the analysis canbe re-couched in terms of this square root, but I do not know the matrixderivative of the Cholesky factorization.What about a mashup with Kalman Filters?Sure! This should probably be expressed as an update on the Choleskyfactor, Θ1/2 .Which portfolio managers are using the weird trick?All of them except you!Steven Pav (Cerebellum Capital)Portfolio Inference .May 16, 201432 / 36
AppendixCommon Questions VI am not comforted by the fact that ζ̂ 2ζ 2 , since the portfolio ν̂ mayachieve a much lower Sharpe ratio than optimal.Because ν is the optimal population Sharpe ratio of any portfolio, it is anupper bound on the Sharpe ratio of ν̂ . To estimate the ‘gap’ requires, Ibelieve, the second-order multivariate delta method. I have not done theanalysis.Can you shoehorn a short-sale constraint into the model?I doubt it is feasible. It is known, for example, that Hotelling’s statisticunder a positivity constraint is not a similar statistic, indicating Sharperatio is an imperfect yardstick for sign-constrained portfolio problems. [18]Steven Pav (Cerebellum Capital)Portfolio Inference .May 16, 201433 / 36
AppendixCommon Questions VIWhy maximize Sharpe ratio? Everyone else maximizes ‘utility’.No investor has ever told us their ‘risk aversion parameter,’ but they askabout our Sharpe ratio all the time. Also, read Roy for the connectionbetween Sharpe ratio and probability of a loss. [17]How do you deal with trade costs?It is not clear. One hack would be to assume trade costs quadratic in thetarget portfolio. I believe this merely leads to an inflation of the Σ̂, butthere are likely complications.Isn’t independence of x̃i suspicious?If the state variables wi depend on the previous period returns, xi ,independence will be violated. However, the CLT may apply if thesequence is weakly dependent, or ‘strongly mixing’.Steven Pav (Cerebellum Capital)Portfolio Inference .May 16, 201434 / 36
AppendixCommon Questions VIIHow do you detect outliers?This probably requires one to impose a likelihood on x̃i .Does the math simplify if you assume normal returns?In this case nΘ̂ takes a conditional Wishart distribution.But does it do big data?Computation of Θ̂ is very simple, since it is just an uncentered moment.How should a Bayesian approach estimation of Θ?I don’t know. Ask one. I suspect they would assume normal returns, thenassume some kind of conditional Wishart prior.Steven Pav (Cerebellum Capital)Portfolio Inference .May 16, 201435 / 36
AppendixCommon Questions VIIIDoes the hedged portfolio involve a projection?It does! The hedged portfolio is the optimal portfolio minus a projectionunder the metric induced by Σ.It seems that when I hedge out a single asset, only the holdings in thatasset change in the portfolio.If you look at the projection operation, the change can only occur in thecolumn space of G̃ , which in this case means only the holdings in thesingle asset will change. (This is all modulo adjustments to overall grossleverage to meet the risk budget.)Can you back out the traditional significance tests from the asymptoticdistribution of Θ̂?Possibly, but probably a bit uglier than I can stomach.Steven Pav (Cerebellum Capital)Portfolio Inference .May 16, 201436 / 36
g p matrix of rank p g. Rows of G de ne portfolios against which we have 0 covariance. Typically G consists of some rows of identity matrix. i.e., \Maximize Sharpe ratio with risk bound and zero covariance to some other portfolios." Solved by c G;;with c to satisfy risk bound, and G;: 1 G GG 1 G : Steven Pav (Cerebellum Capital) Portfolio .
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