Portfolio Management with the R Programming Language: 2015

Sunday, July 26, 2015

[12] Portfolio Optimization: Black-Litterman Model

This section is the summary of Reference (1). Also, some modified sample codes are added so that readers can easily follow and understand it.

1. Overview

The Black-Litterman model's goal is to create a systematic method of specifying and then incorporating analyst / portfolio manager views into the estimation of market parameters (e.g., returns of assets or securities).
R = {r₁, r₂, ... , r_n}

a set of returns (random variables) of n assets or securities
R ~ N(μ, Σ): multivariate normal distribution with mean μ and standard deviation Σ

μ itself is a random variable which is normally distributed.
μ ~ N(π, τΣ): its dispersion is proportional to that of the market.
π: underlying parameters which can be determined by the analyst using some established procedure (obtained from the intercepts of the CAPM)

Next, the analyst forms subjective views on the actual mean of the returns for the holding period. This is the part of the model that allows the analyst or portfolio manager to include his or her views. BL proposed that views should be made on linear combinations (i.e., portfolios) of the asset return variable means μ. Each view would take the form of a "mean plus error." Thus, for example, a typical view would look like as follows:

p_i1μ₁ + p_i2μ₂ + ... + p_inμ_n = q_i+ ε_i
ε_i ~ N(0, σ_i²)

The variance σ_i² of each view could be taken as controlling the confidence in each view.

Collecting these views into a matrix we will call the "pick" matrix (a vector of long and short for each asset / security), we obtain the "general" view specification:

Pμ ~ N(μ, Ω)
Ω is the diagonal matrix diag(σ₁², σ₂², ..., σ_n²).
It can be shown, based on Bayes' Law, that the posterior distribution of the market mean conditional on these views is:

μ|_q;Ω ~ N(μ_BL, Σ_BL^μ)
μ_BL = ((τΣ)^-1+ P^TΩ^-1P)^-1((τΣ)^-1π + P^TΩ^-1q)
Σ_BL^μ= ((τΣ)^-1+ P^TΩ^-1P)^-1

We can then obtain the posterior distribution of the market by taking R|_q;Ω = μ|_q;Ω + Z, and Z ~ N(0, Σ) is independent of μ. One then obtains E[R] = μ_BL and Σ_BL = Σ + Σ_BL^μ (See Reference (2), page 5.).

2. Importing BLCOP

BLCOP
http://cran.r-project.org/package=BLCOP

Windows binaries:	r-release: BLCOP_0.3.1.zip
OS X Snow Leopard binaries:	r-oldrel: BLCOP_0.3.1.tgz
OS X Mavericks binaries:	r-release: not available

RUnit
http://cran.r-project.org/package=RUnit

Windows binaries:	r-release: RUnit_0.4.28.zip
OS X Mavericks binaries:	r-release: RUnit_0.4.28.tgz

timeSeries, fBasics, fMultivar, and fPortfolio
In the previous session, we have already installed these packages.

**sample code(s)**
# RUnit: R Unit Test Framework *install.packages("~/Google Drive/Finance/R/RUnit_0.4.28.tgz", repos = NULL, type="source")* library(RUnit) # BLCOP: Black-Litterman and Copula Opinion Pooling Frameworks *install.packages("~/Google Drive/Finance/R/BLCOP_0.3.1.tgz", repos = NULL, type="source")* library(BLCOP)

3. BLCOP: Black-Litterman Model Optinization - a simple example

The implementation of the Black-Litterman model in BLCOP is based on objects that represent:

views on the market, and the
posterior distribution of the market after blending the views above.

Suppose that an analyst wishes to form views on 6 stocks, 2 of which are technology stocks and the other 4 of which are from the financial sector.

Initially, you believe that the average of 2 tech stocks (e.g., DELL, IBM) will outperform one of the 4 financial stocks (e.g., MS), say, (1/2)(IBM + DELL) - MS = (1/2)(IBM) - MS + (1/2)(DELL), ~ N(0.06, 0.01), from return perspective. We have 3 stocks in total here.

We will create a BLViews class object w/ the BLViews constructor function. Its arguments are the "pick" matrix P, the vector "q" (0.06 here), and the names of the assets in one's "universe."

**sample code(s)**
c(1/2, -1, 1/2, rep(0,3))
[1] 0.5 -1.0 0.5 0.0 0.0 0.0
pickMatrix <- matrix(c(1/2, -1, 1/2, rep(0, 3)), nrow = 1, ncol = 6) views <- BLViews(P=pickMatrix, q=0.06, confidences=100, assetNames=colnames(monthlyReturns)) views
1 : 0.5IBM+-1MS+0.5*DELL=0.06 + eps. Confidence: 100

Next, we need to determine the "prior" distribution of these assets. For instance, the mean is set to zero, and then calculate the variance-covariance matrix of these through some standard estimation procedure (e.g., exponentially weighted moving average with a half-life of 6-month). Here we use cov.mve from the MASS package.

**sample code(s)**
rep(0,6)
[1] 0 0 0 0 0 0
priorMeans <- rep(0,6) monthlyReturns
IBM MS DELL C JPM BAC 1998-02-02 0.057620253 0.195786228 0.4066773934 0.1224778047 0.157384084 0.143954576 1998-03-02 -0.005457679 0.043833262 -0.5156562768 0.0785547367 0.087215863 0.064817518 1998-04-01 0.115529027 0.082338411 0.1918819188 0.0198333333 0.027283511 0.041952290 1998-05-01 0.014067489 -0.010270065 0.0205572755 0.0009805524 -0.018908776 -0.006578947 1998-06-01 -0.022893617 0.170509864 0.1261982769 -0.0101224490 -0.444607915 0.015761589 1998-07-01 0.154080655 -0.047170844 0.1700247818 0.1091868712 0.001589404 0.039900900 1998-08-03 -0.150037736 -0.333103607 -0.0791048900 -0.3400743494 -0.305739222 -0.278996865 1998-09-01 0.141005150 -0.257147778 -0.3425000000 -0.1550247859 -0.178476190 -0.069565217 1998-10-01 0.155642023 0.507071644 -0.0038022814 0.2533333333 0.317180617 0.074766355 1998-11-02 0.111986532 0.071076923 -0.0716030534 0.0691489362 0.116704805 0.133739130 1998-12-01 0.116574820 0.019821890 0.2035849367 -0.0111442786 0.119167718 -0.077619267 1999-01-04 -0.006128647 0.220000000 0.3663068725 0.1281948078 0.083661972 0.112090471 1999-02-01 -0.073669850 0.044793350 -0.1988000000 0.0479843025 0.034832337 -0.023328847 1999-03-01 0.044182622 0.104309392 -0.4897653520 0.0873191489 0.021979402 0.081304548 1999-04-01 0.180197461 -0.007504503 0.0075831703 0.1720413275 0.013887182 0.013310677 1999-05-03 -0.445480185 -0.027119669 -0.1638747269 -0.1151328970 -0.122787879 -0.096003354 1999-06-01 0.114224138 0.063419689 0.0743321719 -0.2830188679 0.195246649 0.133250889 1999-07-01 -0.027543520 -0.120541805 0.1048648649 -0.0618947368 -0.109132948 -0.094666485 1999-08-02 -0.008990373 -0.049196676 0.1939823875 -0.0026929982 0.086036854 -0.088443574 1999-09-01 -0.028580604 0.039389349 -0.1434132350 -0.0099009901 -0.099414506 -0.079504132 1999-10-01 -0.188016529 0.236797847 -0.0401817747 0.2329545455 0.157622396 0.157119770 1999-11-01 0.048956743 0.093463875 0.0715175679 -0.0068202765 -0.114613181 -0.090471757 1999-12-01 0.046671842 0.183468745 0.1860465116 0.0335931700 0.005695793 -0.143661491 2000-01-03 0.040604431 -0.535901926 -0.2462745098 0.0235230742 0.038615008 -0.034867503 2000-02-01 -0.084632517 0.063245283 0.0616545265 -0.0921052632 -0.013260627 -0.050371594 2000-03-01 0.152019465 0.176462237 0.3217348689 0.1571014493 0.095076614 0.140000000 2000-04-03 -0.058038354 -0.073850609 -0.0706340378 -0.0146960588 -0.172840922 -0.065598780 2000-05-01 -0.037578475 -0.068403909 -0.1396369439 0.0540677966 0.035635053 0.131428571 2000-06-01 0.020967291 0.164335664 0.1432877348 -0.0311947258 -0.383317713 -0.224386724 2000-07-03 0.024552756 0.096096096 -0.1089028595 0.1701244813 0.081415545 0.101860465 2000-08-01 0.176124722 0.178958904 -0.0070550751 -0.1717730496 0.121863080 0.130856902 2000-09-01 -0.146947432 -0.150027886 -0.2938345175 -0.0741565337 -0.173407301 -0.022396417 2000-10-02 -0.125377375 -0.121719160 -0.0425186628 -0.0264520903 -0.014938298 -0.082474227 2000-11-01 -0.050761421 -0.210808119 -0.3474576271 -0.0535816074 -0.189450549 -0.168955472 2000-12-01 -0.090909091 0.250394446 -0.0940259740 0.0250953624 0.232104121 0.148723085 2001-01-02 0.317647059 0.070031546 0.4977064220 0.0961613788 0.210167254 0.173060157 2001-02-01 -0.108035714 -0.231957547 -0.1627105666 -0.1213149902 -0.151482088 -0.057971014 2001-03-01 -0.037237237 -0.178565945 0.1746684957 -0.0854005693 -0.037719674 0.079881657 2001-04-02 0.197130381 0.173644860 0.0214091086 0.0927078702 0.068596882 0.022831050 2001-05-01 -0.029008164 0.035355948 -0.0716463415 0.0427263479 0.024385160 0.058035714 2001-06-01 0.015205725 -0.011998154 0.0734811166 0.0310243902 -0.095218718 0.013164557 2001-07-02 -0.073039648 -0.068659505 0.0298279159 -0.0497728993 -0.026309872 0.059803432 2001-08-01 -0.049995248 -0.108157807 -0.2060898626 -0.0878311093 -0.090069284 -0.033322854 2001-09-04 -0.082341171 -0.131208997 -0.1333021515 -0.1157205240 -0.133248731 -0.050406504 2001-10-01 0.178259922 0.055447681 0.2941176471 0.1239506173 0.035431918 0.010102740 2001-11-01 0.069584529 0.134505315 0.1647206005 0.0522847100 0.066742081 0.040515342 2001-12-03 0.046457306 0.007927928 -0.0268528464 0.0538622129 -0.036320255 0.025578364 2002-01-02 -0.108052249 -0.016803718 0.0114054452 -0.0610142631 -0.063273728 0.001270850 2002-02-01 -0.090555195 -0.106909091 -0.1018552201 -0.0453586498 -0.140969163 0.014596224 2002-03-01 0.059926620 0.166734528 0.0575131632 0.0943646409 0.218803419 0.063643471 2002-04-01 -0.194615385 -0.167335544 0.0088088855 -0.1256058158 -0.015427770 0.065568950 2002-05-01 -0.039517670 -0.047359598 0.0193621868 -0.0027713626 0.024216524 0.045943709 2002-06-03 -0.105034183 -0.052353718 -0.0264432030 -0.1025937934 -0.056467316 -0.071890252 2002-07-01 -0.022222222 -0.063370474 -0.0462892119 -0.1344516129 -0.264150943 -0.054860716 2002-08-01 0.070738636 0.058736059 0.0677898115 -0.0235539654 0.057692308 0.053834586 2002-09-03 -0.226452640 -0.206928839 -0.1168294515 -0.0946564885 -0.280681818 -0.089611872 2002-10-01 0.353798662 0.148760331 0.2169289664 0.2462057336 0.092680358 0.094043887 2002-11-01 0.101089435 0.162384378 -0.0003495281 0.0522327470 0.213012048 0.004011461 2002-12-02 -0.108375518 -0.117595049 -0.0650349650 -0.0949074074 -0.046483909 -0.007277397 2003-01-02 0.009032258 -0.050601202 -0.1077038145 -0.0238704177 -0.027500000 0.006899526 2003-02-03 -0.003196931 -0.027704485 0.1299245599 -0.0294032023 -0.028277635 -0.011563169 2003-03-03 0.006157793 0.040705563 0.0129821958 0.0332933413 0.045414462 -0.034662045 2003-04-01 0.082493944 0.166883963 0.0611497620 0.1393323657 0.237874315 0.107869539 2003-05-01 0.036984688 0.022346369 0.0824706694 0.0450955414 0.119591141 0.002025658 2003-06-02 -0.062925943 -0.065573770 0.0149824673 0.0433934666 0.040170420 0.065094340 2003-07-01 -0.015151515 0.109707602 0.0577889447 0.0467289720 0.025453482 0.044793117 2003-08-01 0.009353846 0.028456998 -0.0314726841 -0.0323660714 -0.023680456 -0.040208308 2003-09-02 0.077063773 0.034228325 0.0245248314 0.0498269896 0.003214494 -0.015268139 2003-10-01 0.013019359 0.087395957 0.0771992819 0.0415293342 0.045732595 -0.029600205 2003-11-03 0.011846223 0.007472207 -0.0397222222 -0.0078059072 -0.013927577 -0.003961442 2003-12-01 0.023635962 0.046852388 -0.0170668209 0.0321071656 0.037570621 0.066286623 attr(,"na.action") 1998-02-02 1 attr(,"class") [1] "omit" attr(,"control") method "r"
priorVarcov <- cov.mve(monthlyReturns)$cov priorVarcov
IBM MS DELL C JPM BAC IBM 0.010743588 0.006899941 0.006875766 0.006333900 0.005729608 0.002179980 MS 0.006899941 0.013216783 0.010115206 0.007579162 0.008996890 0.003032132 DELL 0.006875766 0.010115206 0.022510315 0.006662863 0.008328882 0.003206410 C 0.006333900 0.007579162 0.006662863 0.007640900 0.006616338 0.003439520 JPM 0.005729608 0.008996890 0.008328882 0.006616338 0.011606268 0.004166217 BAC 0.002179980 0.003032132 0.003206410 0.003439520 0.004166217 0.004733423

"Posterior" distribution of assets

**sample code(s)**
# The posteriorEst takes as parameters the view object, the prior covariance and mean, and "tau (τ)." The procedure to define τ is controversial, but let's say it's 1/2 here. marketPosterior <- posteriorEst(views = views, sigma = priorVarcov, mu = priorMeans, tau = 1/2) marketPosterior
Prior means: IBM MS DELL C JPM BAC 0 0 0 0 0 0 Posterior means: IBM MS DELL C JPM BAC 0.0040991717 -0.0101081324 0.0098261425 -0.0023198538 -0.0042234721 -0.0007275142 Posterior covariance: IBM MS DELL C JPM BAC IBM 0.016050145 0.010510778 0.010157271 0.009537769 0.008661627 0.003281547 MS 0.010510778 0.019428497 0.015558420 0.011277704 0.013329592 0.004519647 DELL 0.010157271 0.015558420 0.033390619 0.010082793 0.012654443 0.004837369 C 0.009537769 0.011277704 0.010082793 0.011440456 0.009886468 0.005152728 JPM 0.008661627 0.013329592 0.012654443 0.009886468 0.017340150 0.006237397 BAC 0.003281547 0.004519647 0.004837369 0.005152728 0.006237397 0.007098080

Now suppose that we wish to add another view, this time on the average of the four financial stocks. This can be done conveniently with addBLViews as in the following example:

**sample code(s)**
finViews <- matrix(ncol = 4, nrow = 1, dimnames = list(NULL, c("C", "JPM", "BAC", "MS" ))) finViews
C JPM BAC MS [1,] NA NA NA NA
rep(1/4,4)
[1] 0.25 0.25 0.25 0.25
finViews[,1:4] <- rep(1/4,4) finViews[,1:4]
C JPM BAC MS 0.25 0.25 0.25 0.25
# addBLViews(finViews, Mean, Confidence, views) # q: 0.15 # Confidence: 90 views <- addBLViews(finViews, 0.15, 90, views) views
1 : 0.5IBM+-1MS+0.5DELL=0.06 + eps. Confidence: 100 2 : 0.25MS+0.25C+0.25JPM+0.25*BAC=0.15 + eps. Confidence: 90

We will now reproduce the posterior, but this time using the CAPM to compute the "prior" means. Rather than manually computing these, it is convenient to use the BLPosterior wrapper function. It will compute these "alphas," as well as the variance-covariance matrix of a returns series, and will the call posteriorEst automatically.

**sample code(s)**
marketPosterior <- BLPosterior(as.matrix(monthlyReturns), views, tau = 1/2, marketIndex = as.matrix(sp500Returns), riskFree = as.matrix(US13wTB)) marketPosterior
Prior means: IBM MS DELL C JPM BAC 0.020883598 0.059548398 0.017010062 0.014492325 0.027365230 0.002829908 Posterior means: IBM MS DELL C JPM BAC 0.06344562 0.07195806 0.07777653 0.04030821 0.06884519 0.02592776 Posterior covariance: IBM MS DELL C JPM BAC IBM 0.021334221 0.010575532 0.012465444 0.008518356 0.010605748 0.005281807 MS 0.010575532 0.031231768 0.017034827 0.012704758 0.014532900 0.008023646 DELL 0.012465444 0.017034827 0.047250599 0.007386821 0.009352949 0.005086150 C 0.008518356 0.012704758 0.007386821 0.016267422 0.010968240 0.006365457 JPM 0.010605748 0.014532900 0.009352949 0.010968240 0.028181136 0.011716834 BAC 0.005281807 0.008023646 0.005086150 0.006365457 0.011716834 0.011199343

Both BLPosterior (2nd) and posteriorEst (1st) have a kappa parameter which may be used to replace the matrix Ω of confidences in the posterior calculation. If it is greater than 0, then Ω is set to κP^TΣP rather than diag(σ₁², σ₂², ..., σ_n²). This choice of Ω is suggested by several authors, and it leads to the confidences being determined by volatilities of the asset returns.

A user may also be interested in comparing allocations that are optimal under the prior and posterior distributions. The fPortfolio package of the Rmetrics project ([RmCTWu09]), for example, has a rich set of functionality available for portfolio optimization. The helper function optimalPortfolios.fPort was created to wrap these functions for exploratory purposes.

**sample code(s)**
library(fPortfolio) optPorts <- optimalPortfolios.fPort(marketPosterior, optimizer = "tangencyPortfolio") optPorts
$priorOptimPortfolio Title: MV Tangency Portfolio Estimator: getPriorEstim Solver: solveRquadprog Optimize: minRisk Constraints: LongOnly Portfolio Weights: IBM MS DELL C JPM BAC 0.0765 0.9235 0.0000 0.0000 0.0000 0.0000 Covariance Risk Budgets: IBM MS DELL C JPM BAC Target Returns and Risks: mean mu Cov Sigma CVaR VaR 0.0000 0.0566 0.1460 0.0000 0.0000 Description: Fri Jul 17 16:41:29 2015 by user: xxxxxxx $posteriorOptimPortfolio Title: MV Tangency Portfolio Estimator: getPosteriorEstim Solver: solveRquadprog Optimize: minRisk Constraints: LongOnly Portfolio Weights: IBM MS DELL C JPM BAC 0.3633 0.1966 0.1622 0.0000 0.2779 0.0000 Covariance Risk Budgets: IBM MS DELL C JPM BAC Target Returns and Risks: mean mu Cov Sigma CVaR VaR 0.0000 0.0689 0.1268 0.0000 0.0000 Description: Fri Jul 17 16:41:29 2015 by user: xxxxxxx attr(,"class") [1] "BLOptimPortfolios"
mfcol = c(2, 1) par(mfcol = c(2, 1)) mfcol
[1] 2 1
weightsPie(optPorts$priorOptimPortfolio) weightsPie(optPorts$posteriorOptimPortfolio)

Additional parameters may be passed into function to control the optimization process. Users are referred to the fPortfolio package documentation for details.

**sample code(s)**
optPorts2 <- optimalPortfolios.fPort(marketPosterior, constraints = "minW[1:6]=0.1", optimizer = "minriskPortfolio") optPorts2
rOptimPortfolio Title: MV Minimum Risk Portfolio Estimator: getPriorEstim Solver: solveRquadprog Optimize: minRisk Constraints: minW Portfolio Weights: IBM MS DELL C JPM BAC 0.1137 0.1000 0.1000 0.1098 0.1000 0.4764 Covariance Risk Budgets: IBM MS DELL C JPM BAC Target Returns and Risks: mean mu Cov Sigma CVaR VaR 0.0000 0.0157 0.0864 0.0000 0.0000 Description: Fri Jul 17 17:02:23 2015 by user: xxxxxxx $posteriorOptimPortfolio Title: MV Minimum Risk Portfolio Estimator: getPosteriorEstim Solver: solveRquadprog Optimize: minRisk Constraints: minW Portfolio Weights: IBM MS DELL C JPM BAC 0.1000 0.1000 0.1000 0.1326 0.1000 0.4674 Covariance Risk Budgets: IBM MS DELL C JPM BAC Target Returns and Risks: mean mu Cov Sigma CVaR VaR 0.0000 0.0457 0.1008 0.0000 0.0000 Description: Fri Jul 17 17:02:23 2015 by user: xxxxxxx attr(,"class") [1] "BLOptimPortfolios"

Finally, density plots of marginal prior and posterior distributions can be generated with densityPlots. As we will see in the next section, this gives more interesting results when used with copula opinion pooling.

**sample code(s)**
# Please close the former graph window before running the following code. densityPlots(marketPosterior, assetsSel = "JPM")

4. Overview of Copula Opinion Pooling (COP)

Copula opinion pooling is an alternative way to blend analyst views on market distributions that was developed by Attilio Meucci towards the end of 2005. It is similar to the Black-Litterman model in that it also uses a "pick" matrix to formulate views. However, it has several advantages including the following:

Views are made on realization of the market, not on market parameters as in the original formulation of BL.
The joint distribution of the market can be any multivariate distribution.
Views are NOT restricted to the normal distribution.
The parameters in the model have clearer meanings.
The model can easily be generalized to incorporate the views of multiple analysts.

Nevertheless, all of this comes at a price. We can no longer use closed-form expressions for calculating the posterior distribution of the market and hence rely on simulation instead. Before proceeding to the implementation, however, let's look at the theory. Please refer to (3) for a more detailed discussion.

As before, suppose that we have a set of n assets whose returns are represented by a set of random variables R = {r₁, r₂, ..., r_n}. As in Black-Litterman, we suppose that R has some prior joint distribution whose c.d.f we will denote by Φ_A. Denote the marginals of this distributon by Φ_i. An analyst forms his views on linear combinations of future realizations of the values of R by assigning subjective probability distributions to these linear combinations. That is we form views of the form p_i,1a₁+ p_i,2a₂+...+ p_i,na_n~θi, where θi is some distribution. Denote the pick matrix formed by all of these views by P once again. Now, since we have assigned some prior distribution Φ_Ato these assets, it follows that actually the product V = PA inherits a distribution as well, say
v_i= p_i,1a₁+ p_i,2a₂+...+ p_i,na_n~θ'i
In general, θ'i≠θ'i unless one's views are identical to the market prior. Thus we must somehow resolve this contradiction. A straightforward way of doing this is to take the weighted sum of the two marginal c.d.fs, so i.e.,

, and

is a parameter representing our confidence in our subjective views. This is the actual marginal distribution that will be used to determine the market posterior.

The market posterior is actually determined by setting the marginals of distributions of V to

, while using a copula to keep the dependence structure of V intact. Let V = (v₁, v₂, ..., v_k), where k is the number of views that the analyst has formed. Then

. Let C be the copula of V so that C is the joint distribution of

if we now take the θ'i to be cumulative density functions. Next set

as the random variable with the joint distriburtion

. The posterior market distribution is obtained by rotating

back into market coordinates using the orthogonal complement of P. See Reference (3), p. 5 for details.

5. BLCOP: COP

Let us now work through a brief example to see how these ideas are implemented in the BLCOP package.
First, one again works with object that hold the view specification, which in the COP case are of class COPViews. These can again be created with a constructor function of the same name. However, a significant difference is the use of mvdistribution and distribution class objects to specify the prior distribution and view distributions respectively. We will show the use of these in the following example, which is based on the example used in Reference (3), p.9. Suppose that we wish to invest in 4 market indices (S&P500, FTSE, CAC and DAX). Meucci suggests a multivariate Student-t distribution with = 5 degrees of freedom and dispersion matrix given by:

He then sets μ = δΣ w_eqwhere w_eqis the relative capitalization of the 4 indices and δ = 2.5. For simplicity we will simply take w_eq= (1/4, 1/4, 1/4, 1/4).

**sample code(s)**
dispersion <- c(.376,.253,.360,.333,.360,.600,.397,.396,.578,.775) / 1000 sigma <- BLCOP:::.symmetricMatrix(dispersion, dim = 4) caps <- rep(1/4, 4) *mu <- 2.5 sigma %% caps* dim(mu) <- NULL # The following function could return an error message. marketDistribution <- mvdistribution("mt", mean = mu, S = sigma, df = 5 ) class(marketDistribution)
[1] "mvdistribution" attr(,"package") [1] "BLCOP"

The class mvdistribution works with R multivariate probabilty distribution "su xes". mt is the R
"name"/"su x" of the multivariate Student-t as found in the package mnormt. That is, the sampling function is given by rmt, the density by dmt, and so on. The other parameters are those required by the these functions to fully parametrize the multivariate Student-t. The distribution class works with univariate distributions in a similar way and is used to create the view distributions. We continue with the above example by creating a single view on the DAX.

**sample code(s)**
pick <- matrix(0, ncol = 4, nrow = 1, dimnames = list(NULL, c("SP", "FTSE", "CAC", "DAX"))) pick[1,"DAX"] <- 1 viewDist <- list(distribution("unif", min = -0.02, max = 0)) views <- COPViews(pick, viewDist = viewDist, confidences = 0.2, assetNames = c("SP", "FTSE", "CAC", "DAX"))

As can be seen, the view distributions are given as a list of distribution class objects, and the
con dences set the tau's described previously. Here we have assigned a U(􀀀0:02; 0) distribution to our view with con dence 0:2. Additional views can be added with addCOPViews.

**sample code(s)**
newPick <- matrix(0, 1, 2) dimnames(newPick) <- list(NULL, c("SP", "FTSE")) newPick[1,] <- c(1, -1) # add a relative view views <- addCOPViews(newPick, list(distribution("norm", mean = 0.05, sd = 0.02)), 0.5, views)

The posterior is calculated with COPPosterior, and the updated marginal distributions can be visualized with densityPlots once again. The calculation is performed by simulation, based on the ideas described in Reference (4). The simulations of the posterior distribution are stored in the posteriorSims of the class COPResult that is returned by COPPosterior.

**sample code(s)**
# The following functions could return an error message. marketPosterior <- COPPosterior(marketDistribution, views, numSimulations = 50000) densityPlots(marketPosterior, assetsSel = 4)

6. Future Developments

While mostly stable, the code is currently in need of some minor cleanup work and refactoring (e.g. pick matrices are referred to as P in some places and pick in others) as well as improvements in the documentation and examples. Attilio Meucci has also very recently proposed an even more general view-blending method which he calls Entropy Pooling and its inclusion would be another obvious extension of this package's functionality in the longer term.

Reference
(1) http://cran.r-project.org/web/packages/BLCOP/vignettes/BLCOP.pdf

(2) Meucci, Attilio. The Black-Litterman Approach: Original Model and Extensions. April 2008.
http://ssrn.com/abstract=1117574

(3) Meucci, Attilio. Beyond Black-Litterman: Views on Non-Normal Markets. November 2005.
http://ssrn.com/abstract=848407

(4) Meucci, Attilio. Beyond Black-Litterman in Practice: A Five-Step Recipe
to Input Views on non-Normal Markets. May 2006, Available at SSRN:
http://papers.ssrn.com/sol3/papers.cfm?abstract id=872577

Others
http://ssrn.com/abstract=1314585
http://datalab.morningstar.com/knowledgebase/aspx/files/Step_by_Step_Guide_to_the_Black_Litterman_Model.pdf
http://brage.bibsys.no/xmlui/bitstream/id/85285/Syvertsen,

Tuesday, July 14, 2015

[11c] Portfolio Optimization and Efficient Frontier: Import CSV files

Create csv files:

**er.csv**
SPY,EFA,LQD,EEM 0.14473728,0.06458429,0.05929884,0.07109433

**covmat.csv**
,SPY,EFA,LQD,HYG SPY,0.02141010,0.02351144,-0.00049749,0.00915216 EFA,0.02351144,0.03227444,-0.00011169,0.01105802 LQD,-0.00049749,-0.00011169,0.00323208,0.00146939 HYG,0.00915216,0.01105802,0.00146939,0.00689213

Original sample codes in [11a] or [11b] are on the left hand side. For this section, please try the sample codes of read.csv on the right hand side, and then run sample codes in [11a] and/or [11b].

**sample code(s)**
r.free <- 0.00973449 er <- c( "SPY" = 0.14473728, "EFA" = 0.06458429, "LQD" = 0.05929884, "EEM" = 0.07109433 ) covmat <- matrix(c(0.02141010, 0.02351144, -0.00049749, 0.00915216, 0.02351144, 0.03227444, -0.00011169, 0.01105802, -0.00049749, -0.00011169, 0.00323208, 0.00146939, 0.00915216, 0.01105802, 0.00146939, 0.00689213), nrow = 4, ncol = 4, byrow = TRUE) rownames(covmat) <- c("SPY", "EFA", "LQD", "HYG") colnames(covmat) <- c("SPY", "EFA", "LQD", "HYG")	r.free <- 0.00973449

**sample code(s)**
er SPY EFA LQD EEM 0.14473728 0.06458429 0.05929884 0.07109433 class(er) [1] "numeric" names(er) [1] "SPY" "EFA" "LQD" "EEM" rownames(er) NULL colnames(er) NULL dim(er) NULL	*erdf <- read.csv("H:/Quantitative Research/R/Optimization/er.csv", sep=",")* erdf SPY EFA LQD EEM 1 0.1447373 0.06458429 0.05929884 0.07109433 class(erdf) [1] "data.frame" names(erdf) [1] "SPY" "EFA" "LQD" "EEM" rownames(erdf) [1] "1" colnames(erdf) [1] "SPY" "EFA" "LQD" "EEM" dim(erdf) [1] 1 4 ermt <- as.matrix(erdf) ermt SPY EFA LQD EEM [1,] 0.1447373 0.06458429 0.05929884 0.07109433 class(ermt) [1] "matrix" names(ermt) NULL rownames(ermt) NULL colnames(ermt) [1] "SPY" "EFA" "LQD" "EEM" dim(ermt) [1] 1 4 names(ermt) <- colnames(ermt) names(ermt) [1] "SPY" "EFA" "LQD" "EEM" er <- ermt er SPY EFA LQD EEM [1,] 0.1447373 0.06458429 0.05929884 0.07109433 attr(,"names") [1] "SPY" "EFA" "LQD" "EEM" class(er) [1] "matrix" names(er) [1] "SPY" "EFA" "LQD" "EEM" rownames(er) NULL colnames(er) [1] "SPY" "EFA" "LQD" "EEM" dim(er) [1] 1 4

**sample code(s)**
covmat SPY EFA LQD HYG SPY 0.02141010 0.02351144 -0.00049749 0.00915216 EFA 0.02351144 0.03227444 -0.00011169 0.01105802 LQD -0.00049749 -0.00011169 0.00323208 0.00146939 HYG 0.00915216 0.01105802 0.00146939 0.00689213 names(covmat) NULL rownames(covmat) [1] "SPY" "EFA" "LQD" "HYG" colnames(covmat) [1] "SPY" "EFA" "LQD" "HYG" class(covmat) [1] "matrix" dim(covmat) [1] 4 4	*covmatdf <- read.csv("H:/Quantitative Research/R/Optimization/covmat.csv", sep=",", row.names=1)* covmatdf SPY EFA LQD HYG SPY 0.02141010 0.02351144 -0.00049749 0.00915216 EFA 0.02351144 0.03227444 -0.00011169 0.01105802 LQD -0.00049749 -0.00011169 0.00323208 0.00146939 HYG 0.00915216 0.01105802 0.00146939 0.00689213 names(covmatdf) [1] "SPY" "EFA" "LQD" "HYG" class(covmatdf) [1] "data.frame" dim(covmatdf) [1] 4 4 covmatmt <- as.matrix(covmatdf) covmatmt SPY EFA LQD HYG SPY 0.02141010 0.02351144 -0.00049749 0.00915216 EFA 0.02351144 0.03227444 -0.00011169 0.01105802 LQD -0.00049749 -0.00011169 0.00323208 0.00146939 HYG 0.00915216 0.01105802 0.00146939 0.00689213 names(covmatmt) NULL rownames(covmatmt) [1] "SPY" "EFA" "LQD" "HYG" colnames(covmatmt) [1] "SPY" "EFA" "LQD" "HYG" class(covmatmt) [1] "data.frame" dim(covmatmt) [1] 4 4 covmat <- covmatmt covmat SPY EFA LQD HYG SPY 0.02141010 0.02351144 -0.00049749 0.00915216 EFA 0.02351144 0.03227444 -0.00011169 0.01105802 LQD -0.00049749 -0.00011169 0.00323208 0.00146939 HYG 0.00915216 0.01105802 0.00146939 0.00689213 class(covmat) [1] "matrix" names(covmat) [1] "SPY" "EFA" "LQD" "EEM" rownames(covmat) [1] "SPY" "EFA" "LQD" "HYG" colnames(covmat) [1] "SPY" "EFA" "LQD" "EEM" dim(covmat) [1] 1 4

Try [11a] and/or [11b] after running the sample codes of read.csv on the right hand side.

Monday, July 13, 2015

[11b] Long-Short Portfolio Optimization (equal-weighted, tangency, minimum volatility, and target return portfolio) and Efficient Frontier

Create the following r source code files by Eric Zivot on a certain directory.

portfolio.r

**portfolio.r**
# portfolio.r # # Functions for portfolio analysis # to be used in Introduction to Computational Finance & Financial Econometrics # last updated: November 7, 2000 by Eric Zivot # Oct 15, 2003 by Tim Hesterberg # November 18, 2003 by Eric Zivot # November 9, 2004 by Eric Zivot # November 9, 2008 by Eric Zivot # August 11, 2011 by Eric Zivot # # Functions: # 1. efficient.portfolio compute minimum variance portfolio # subject to target return # 2. globalMin.portfolio compute global minimum variance portfolio # 3. tangency.portfolio compute tangency portfolio # 4. efficient.frontier compute Markowitz bullet # 5. getPortfolio create portfolio object # getPortfolio <- function(er, cov.mat, weights) { # contruct portfolio object # # inputs: # er N x 1 vector of expected returns # cov.mat N x N covariance matrix of returns # weights N x 1 vector of portfolio weights # # output is portfolio object with the following elements # call original function call # er portfolio expected return # sd portfolio standard deviation # weights N x 1 vector of portfolio weights # call <- match.call() # # check for valid inputs # asset.names <- names(er) weights <- as.vector(weights) names(weights) = names(er) er <- as.vector(er) # assign names if none exist if(length(er) != length(weights)) stop("dimensions of er and weights do not match") cov.mat <- as.matrix(cov.mat) if(length(er) != nrow(cov.mat)) stop("dimensions of er and cov.mat do not match") if(any(diag(chol(cov.mat)) <= 0)) stop("Covariance matrix not positive definite") # # create portfolio # er.port <- crossprod(er,weights) *sd.port <- sqrt(weights %% cov.mat %% weights)* ans <- list("call" = call, "er" = as.vector(er.port), "sd" = as.vector(sd.port), "weights" = weights) class(ans) <- "portfolio" ans } efficient.portfolio <- function(er, cov.mat, target.return) { # compute minimum variance portfolio subject to target return # # inputs: # er N x 1 vector of expected returns # cov.mat N x N covariance matrix of returns # target.return scalar, target expected return # # output is portfolio object with the following elements # call original function call # er portfolio expected return # sd portfolio standard deviation # weights N x 1 vector of portfolio weights # call <- match.call() # # check for valid inputs # asset.names <- names(er) er <- as.vector(er) # assign names if none exist cov.mat <- as.matrix(cov.mat) if(length(er) != nrow(cov.mat)) stop("invalid inputs") if(any(diag(chol(cov.mat)) <= 0)) stop("Covariance matrix not positive definite") # remark: could use generalized inverse if cov.mat is positive semidefinite # # compute efficient portfolio # ones <- rep(1, length(er)) *top <- cbind(2cov.mat, er, ones) bot <- cbind(rbind(er, ones), matrix(0,2,2)) A <- rbind(top, bot) b.target <- as.matrix(c(rep(0, length(er)), target.return, 1)) x <- solve(A, b.target) w <- x[1:length(er)] names(w) <- asset.names # # compute portfolio expected returns and variance # er.port <- crossprod(er,w) sd.port <- sqrt(w %% cov.mat %% w) ans <- list("call" = call, "er" = as.vector(er.port), "sd" = as.vector(sd.port), "weights" = w) class(ans) <- "portfolio" ans } globalMin.portfolio <- function(er, cov.mat) { # Compute global minimum variance portfolio # # inputs: # er N x 1 vector of expected returns # cov.mat N x N return covariance matrix # # output is portfolio object with the following elements # call original function call # er portfolio expected return # sd portfolio standard deviation # weights N x 1 vector of portfolio weights call <- match.call() # # check for valid inputs # asset.names <- names(er) er <- as.vector(er) # assign names if none exist cov.mat <- as.matrix(cov.mat) if(length(er) != nrow(cov.mat)) stop("invalid inputs") if(any(diag(chol(cov.mat)) <= 0)) stop("Covariance matrix not positive definite") # remark: could use generalized inverse if cov.mat is positive semi-definite # # compute global minimum portfolio # cov.mat.inv <- solve(cov.mat) one.vec <- rep(1,length(er)) # w.gmin <- cov.mat.inv %% one.vec/as.vector(one.vec %% cov.mat.inv %% one.vec)* w.gmin <- rowSums(cov.mat.inv) / sum(cov.mat.inv) w.gmin <- as.vector(w.gmin) names(w.gmin) <- asset.names er.gmin <- crossprod(w.gmin,er) *sd.gmin <- sqrt(t(w.gmin) %% cov.mat %% w.gmin)* gmin.port <- list("call" = call, "er" = as.vector(er.gmin), "sd" = as.vector(sd.gmin), "weights" = w.gmin) class(gmin.port) <- "portfolio" gmin.port } tangency.portfolio <- function(er,cov.mat,risk.free) { # compute tangency portfolio # # inputs: # er N x 1 vector of expected returns # cov.mat N x N return covariance matrix # risk.free scalar, risk-free rate # # output is portfolio object with the following elements # call captures function call # er portfolio expected return # sd portfolio standard deviation # weights N x 1 vector of portfolio weights call <- match.call() # # check for valid inputs # asset.names <- names(er) if(risk.free < 0) stop("Risk-free rate must be positive") er <- as.vector(er) cov.mat <- as.matrix(cov.mat) if(length(er) != nrow(cov.mat)) stop("invalid inputs") if(any(diag(chol(cov.mat)) <= 0)) stop("Covariance matrix not positive definite") # remark: could use generalized inverse if cov.mat is positive semi-definite # # compute global minimum variance portfolio # gmin.port <- globalMin.portfolio(er,cov.mat) if(gmin.port$er < risk.free) stop("Risk-free rate greater than avg return on global minimum variance portfolio") # # compute tangency portfolio # cov.mat.inv <- solve(cov.mat) *w.t <- cov.mat.inv %% (er - risk.free) # tangency portfolio w.t <- as.vector(w.t/sum(w.t)) # normalize weights names(w.t) <- asset.names er.t <- crossprod(w.t,er) sd.t <- sqrt(t(w.t) %% cov.mat %% w.t) tan.port <- list("call" = call, "er" = as.vector(er.t), "sd" = as.vector(sd.t), "weights" = w.t) class(tan.port) <- "portfolio" tan.port } efficient.frontier <- function(er, cov.mat, nport=20, alpha.min=-0.5, alpha.max=1.5) { # Compute efficient frontier with no short-sales constraints # # inputs: # er N x 1 vector of expected returns # cov.mat N x N return covariance matrix # nport scalar, number of efficient portfolios to compute # # output is a Markowitz object with the following elements # call captures function call # er nport x 1 vector of expected returns on efficient porfolios # sd nport x 1 vector of std deviations on efficient portfolios # weights nport x N matrix of weights on efficient portfolios call <- match.call() # # check for valid inputs # asset.names <- names(er) er <- as.vector(er) cov.mat <- as.matrix(cov.mat) if(length(er) != nrow(cov.mat)) stop("invalid inputs") if(any(diag(chol(cov.mat)) <= 0)) stop("Covariance matrix not positive definite") # # create portfolio names # port.names <- rep("port",nport) ns <- seq(1,nport) port.names <- paste(port.names,ns) # # compute global minimum variance portfolio # cov.mat.inv <- solve(cov.mat) one.vec <- rep(1,length(er)) port.gmin <- globalMin.portfolio(er,cov.mat) w.gmin <- port.gmin$weights # # compute efficient frontier as convex combinations of two efficient portfolios # 1st efficient port: global min var portfolio # 2nd efficient port: min var port with ER = max of ER for all assets # er.max <- max(er) port.max <- efficient.portfolio(er,cov.mat,er.max) w.max <- port.max$weights a <- seq(from=alpha.min,to=alpha.max,length=nport) # convex combinations we.mat <- a %o% w.gmin + (1-a) %o% w.max # rows are efficient portfolios er.e <- we.mat %% er # expected returns of efficient portfolios* er.e <- as.vector(er.e) names(er.e) <- port.names *cov.e <- we.mat %% cov.mat %% t(we.mat) # cov mat of efficient portfolios* sd.e <- sqrt(diag(cov.e)) # std devs of efficient portfolios sd.e <- as.vector(sd.e) names(sd.e) <- port.names dimnames(we.mat) <- list(port.names,asset.names) # # summarize results # ans <- list("call" = call, "er" = er.e, "sd" = sd.e, "weights" = we.mat) class(ans) <- "Markowitz" ans } # # print method for portfolio object print.portfolio <- function(x, ...) { cat("Call:\n") print(x$call, ...) cat("\nPortfolio expected return: ", format(x$er, ...), "\n") cat("Portfolio standard deviation: ", format(x$sd, ...), "\n") cat("Portfolio weights:\n") print(round(x$weights,4), ...) invisible(x) } # # summary method for portfolio object summary.portfolio <- function(object, risk.free=NULL, ...) # risk.free risk-free rate. If not null then # compute and print Sharpe ratio # { cat("Call:\n") print(object$call) cat("\nPortfolio expected return: ", format(object$er, ...), "\n") cat( "Portfolio standard deviation: ", format(object$sd, ...), "\n") if(!is.null(risk.free)) { SharpeRatio <- (object$er - risk.free)/object$sd cat("Portfolio Sharpe Ratio: ", format(SharpeRatio), "\n") } cat("Portfolio weights:\n") print(round(object$weights,4), ...) invisible(object) } # hard-coded 4 digits; prefer to let user control, via ... or other argument # # plot method for portfolio object plot.portfolio <- function(object, ...) { asset.names <- names(object$weights) barplot(object$weights, names=asset.names, xlab="Assets", ylab="Weight", main="Portfolio Weights", ...) invisible() } # # print method for Markowitz object print.Markowitz <- function(x, ...) { cat("Call:\n") print(x$call) xx <- rbind(x$er,x$sd) dimnames(xx)[[1]] <- c("ER","SD") cat("\nFrontier portfolios' expected returns and standard deviations\n") print(round(xx,4), ...) invisible(x) } # hard-coded 4, should let user control # # summary method for Markowitz object summary.Markowitz <- function(object, risk.free=NULL) { call <- object$call asset.names <- colnames(object$weights) port.names <- rownames(object$weights) if(!is.null(risk.free)) { # compute efficient portfolios with a risk-free asset nport <- length(object$er) sd.max <- object$sd[1] sd.e <- seq(from=0,to=sd.max,length=nport) names(sd.e) <- port.names # # get original er and cov.mat data from call er <- eval(object$call$er) cov.mat <- eval(object$call$cov.mat) # # compute tangency portfolio tan.port <- tangency.portfolio(er,cov.mat,risk.free) x.t <- sd.e/tan.port$sd # weights in tangency port rf <- 1 - x.t # weights in t-bills *er.e <- risk.free + x.t(tan.port$er - risk.free) names(er.e) <- port.names we.mat <- x.t %o% tan.port$weights # rows are efficient portfolios dimnames(we.mat) <- list(port.names, asset.names) we.mat <- cbind(rf,we.mat) } else { er.e <- object$er sd.e <- object$sd we.mat <- object$weights } ans <- list("call" = call, "er"=er.e, "sd"=sd.e, "weights"=we.mat) class(ans) <- "summary.Markowitz" ans } print.summary.Markowitz <- function(x, ...) { xx <- rbind(x$er,x$sd) port.names <- names(x$er) asset.names <- colnames(x$weights) dimnames(xx)[[1]] <- c("ER","SD") cat("Frontier portfolios' expected returns and standard deviations\n") print(round(xx,4), ...) cat("\nPortfolio weights:\n") print(round(x$weights,4), ...) invisible(x) } # hard-coded 4, should let user control # # plot efficient frontier # # things to add: plot original assets with names # tangency portfolio # global min portfolio # risk free asset and line connecting rf to tangency portfolio # plot.Markowitz <- function(object, plot.assets=FALSE, ...) # plot.assets logical. If true then plot asset sd and er { if (!plot.assets) { y.lim=c(0,max(object$er)) x.lim=c(0,max(object$sd)) plot(object$sd,object$er,type="b",xlim=x.lim, ylim=y.lim, xlab="Portfolio SD", ylab="Portfolio ER", main="Efficient Frontier", ...) } else { call = object$call mu.vals = eval(call$er) sd.vals = sqrt( diag( eval(call$cov.mat) ) ) y.lim = range(c(0,mu.vals,object$er)) x.lim = range(c(0,sd.vals,object$sd)) plot(object$sd,object$er,type="b", xlim=x.lim, ylim=y.lim, xlab="Portfolio SD", ylab="Portfolio ER", main="Efficient Frontier", ...) text(sd.vals, mu.vals, labels=names(mu.vals)) } invisible()** }

Source the r source codes above.

**sample code(s)**
source("H:/Quantitative Research/R/Optimization/portfolio.r")

Sample ticker code:

SPY SPDR S&P 500 ETF TRUST
EFA ISHARES MSCI EAFE ETF
LQD ISHARES IBOXX INVESTMENT GRA
HYG ISHARES IBOXX HIGH YIELD COR

Weekly basis stats (in decimal numbers):

(1) Risk-free return

Risk-free

0.00018631

(2) Average Return and Standard Deviation of Returns

Weekly	SPY	EFA	LQD	HYG
Average	0.00260290	0.00120427	0.00110844	0.00132166
SD	0.02029120	0.02491309	0.00788386	0.01151264

(3) Covariance matrix

Cov	SPY	EFA	LQD	HYG
SPY	0.00041173	0.00045214	-0.00000957	0.00017600
EFA	0.00045214	0.00062066	-0.00000215	0.00021265
LQD	-0.00000957	-0.00000215	0.00006216	0.00002826
HYG	-0.00000617	-0.00000570	0.00000461	-0.00000108

Annual basis stats (in decimal numbers):

(1') Geometrical-Average Risk-free return

Annual

Risk-free

0.00973449

(2') Geometrical-Average Return and Standard Deviation of Returns (SD figures of (2) are multiplied by 52^0.5.)

Annual	SPY	EFA	LQD	HYG
Average	0.14473728	0.06458429	0.05929884	0.07109433
SD	0.14632190	0.17965088	0.05685135	0.08301885

(3') Covariance matrix --- all the numbers of the covariance of (3) are multiplied by 52 (weeks).

Cov	SPY	EFA	LQD	HYG
SPY	0.02141010	0.02351144	-0.00049749	0.00915216
EFA	0.02351144	0.03227444	-0.00011169	0.01105802
LQD	-0.00049749	-0.00011169	0.00323208	0.00146939
HYG	0.00915216	0.01105802	0.00146939	0.00689213

Stats on a annual basis:

**sample code(s)**
# annual r.free <- 0.00973449 r.free [1] 0.00973449 er <- c( "SPY" = 0.14473728, "EFA" = 0.06458429, "LQD" = 0.05929884, "EEM" = 0.07109433 ) er SPY EFA LQD EEM 0.14473728 0.06458429 0.05929884 0.07109433 covmat <- matrix(c(0.02141010, 0.02351144, -0.00049749, 0.00915216, 0.02351144, 0.03227444, -0.00011169, 0.01105802, -0.00049749, -0.00011169, 0.00323208, 0.00146939, 0.00915216, 0.01105802, 0.00146939, 0.00689213), nrow = 4, ncol = 4, byrow = TRUE) rownames(covmat) <- c("SPY", "EFA", "LQD", "HYG") colnames(covmat) <- c("SPY", "EFA", "LQD", "HYG") covmat SPY EFA LQD HYG SPY 0.02141010 0.02351144 -0.00049749 0.00915216 EFA 0.02351144 0.03227444 -0.00011169 0.01105802 LQD -0.00049749 -0.00011169 0.00323208 0.00146939 HYG 0.00915216 0.01105802 0.00146939 0.00689213

Equal-weighted Portfolio

**sample code(s)**
# [1 1 1 1] rep(1,4) # equally-weighted ew = rep(1,4)/4 ew [1] 0.25 0.25 0.25 0.25 equalWeight.portfolio = getPortfolio(er=er,cov.mat=covmat,weights=ew) equalWeight.portfolio Call: getPortfolio(er = er, cov.mat = covmat, weights = ew) Portfolio expected return: 0.08492869 Portfolio standard deviation: 0.09777922 Portfolio weights: SPY EFA LQD EEM 0.25 0.25 0.25 0.25 plot(equalWeight.portfolio) # If you want to change the color, then try: # plot(equalWeight.portfolio, col="blue")

Tangency Portfolio (long-short)

**sample code(s)**
tan.port <- tangency.portfolio(er, covmat, r.free) # You can also run without a "shorts" option. # tan.port <- tangency.portfolio(er, covmat, r.free) tan.port Call: tangency.portfolio(er = er, cov.mat = covmat, risk.free = r.free, shorts = TRUE) Portfolio expected return: 0.1371406 Portfolio standard deviation: 0.06950249 Portfolio weights: SPY EFA LQD EEM 0.9766 -0.5690 0.8118 -0.2194 print(tan.port) Call: tangency.portfolio(er = er, cov.mat = covmat, risk.free = r.free, shorts = TRUE) Portfolio expected return: 0.1371406 Portfolio standard deviation: 0.06950249 Portfolio weights: SPY EFA LQD EEM 0.9766 -0.5690 0.8118 -0.2194 plot(tan.port)

Minimum Volatility Portfolio (long-short)

**sample code(s)**
gmin.port <- globalMin.portfolio(er, covmat) gmin.port Call: globalMin.portfolio(er = er, cov.mat = covmat, shorts = TRUE) Portfolio expected return: 0.07863945 Portfolio standard deviation: 0.05111289 Portfolio weights: SPY EFA LQD EEM 0.2210 -0.1071 0.7990 0.0872 attributes(gmin.port) $names [1] "call" "er" "sd" "weights" $class [1] "portfolio" print(gmin.port) Call: globalMin.portfolio(er = er, cov.mat = covmat, shorts = TRUE) Portfolio expected return: 0.07863945 Portfolio standard deviation: 0.05111289 Portfolio weights: SPY EFA LQD EEM 0.2210 -0.1071 0.7990 0.0872 plot(gmin.port)

Target Return Portfolio (long-short)

**sample code(s)**
target.return = er["SPY"] target.return SPY 0.1447373 e.port.SPY = efficient.portfolio(er, covmat, target.return) e.port.SPY Call: efficient.portfolio(er = er, cov.mat = covmat, target.return = target.return, shorts = TRUE) Portfolio expected return: 0.1447373 Portfolio standard deviation: 0.0737838 Portfolio weights: SPY EFA LQD EEM 1.0747 -0.6290 0.8135 -0.2592 plot(e.port.SPY)

Efficient Frontier (long-short)

**sample code(s)**
ef = efficient.frontier(er, covmat, alpha.min = -2, alpha.max = 1.5, nport = 20) attributes(ef) $names [1] "call" "er" "sd" "weights" $class [1] "Markowitz" summary(ef) Frontier portfolios' expected returns and standard deviations port 1 port 2 port 3 port 4 port 5 port 6 port 7 port 8 port 9 port 10 ER 0.2769 0.2648 0.2526 0.2404 0.2282 0.2161 0.2039 0.1917 0.1795 0.1673 SD 0.1676 0.1583 0.1491 0.1399 0.1308 0.1219 0.1130 0.1044 0.0960 0.0878 port 11 port 12 port 13 port 14 port 15 port 16 port 17 port 18 port 19 ER 0.1552 0.1430 0.1308 0.1186 0.1065 0.0943 0.0821 0.0699 0.0578 SD 0.0801 0.0728 0.0662 0.0604 0.0558 0.0526 0.0512 0.0516 0.0538 port 20 ER 0.0456 SD 0.0576 Portfolio weights: SPY EFA LQD EEM port 1 2.7822 -1.6727 0.8425 -0.9520 port 2 2.6249 -1.5766 0.8399 -0.8882 port 3 2.4676 -1.4804 0.8372 -0.8244 port 4 2.3104 -1.3843 0.8345 -0.7606 port 5 2.1531 -1.2882 0.8318 -0.6968 port 6 1.9958 -1.1920 0.8292 -0.6330 port 7 1.8386 -1.0959 0.8265 -0.5692 port 8 1.6813 -0.9998 0.8238 -0.5053 port 9 1.5240 -0.9036 0.8211 -0.4415 port 10 1.3668 -0.8075 0.8185 -0.3777 port 11 1.2095 -0.7114 0.8158 -0.3139 port 12 1.0522 -0.6152 0.8131 -0.2501 port 13 0.8950 -0.5191 0.8104 -0.1863 port 14 0.7377 -0.4230 0.8078 -0.1225 port 15 0.5804 -0.3268 0.8051 -0.0587 port 16 0.4232 -0.2307 0.8024 0.0051 port 17 0.2659 -0.1346 0.7998 0.0689 port 18 0.1086 -0.0385 0.7971 0.1327 port 19 -0.0486 0.0577 0.7944 0.1966 port 20 -0.2059 0.1538 0.7917 0.2604 plot(ef, plot.assets=T, col="blue", pch=16) points(gmin.port$sd, gmin.port$er, col="green", pch=16, cex=2) points(tan.port$sd, tan.port$er, col="red", pch=16, cex=2) text(gmin.port$sd, gmin.port$er, labels="GLOBAL MIN", pos=2) text(tan.port$sd, tan.port$er, labels="TANGENCY", pos=2) sr.tan = (tan.port$er - r.free)/tan.port$sd abline(a=r.free, b=sr.tan, col="green", lwd=2)