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, ... , rn}
- 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
- 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:
- pi1μ1 + pi2μ2 + ... + pinμn = qi + εi
- εi ~ N(0, σi2)
- The variance σi2 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, ..., σn2).
- 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(μBL, ΣBLμ)
- μBL = ((τΣ)-1 + PTΩ-1 P)-1((τΣ)-1 π + PTΩ-1 q)
- ΣBLμ = ((τΣ)-1 + PTΩ-1 P)-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] = μBL and ΣBL = Σ + ΣBLμ (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.
| # 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.
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."
| 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 : 0.5*IBM+-1*MS+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.
| 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
| # 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:
| 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 : 0.5*IBM+-1*MS+0.5*DELL=0.06 + eps. Confidence: 100 2 : 0.25*MS+0.25*C+0.25*JPM+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.
| 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 κPTΣP rather than diag(σ12, σ22, ..., σn2). 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.
library(fPortfolio) 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.
| 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.
| # 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, ..., rn}. As in Black-Litterman, we suppose that R has some prior joint distribution whose c.d.f we will denote by ΦA. Denote the marginals of this distributon by Φi. 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 pi,1a1 + pi,2a2 +...+ pi,nan~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 ΦA to these assets, it follows that actually the product V = PA inherits a distribution as well, say
vi = pi,1a1 + pi,2a2 +...+ pi,nan~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.,
The market posterior is actually determined by setting the marginals of distributions of V to
as the random variable with the joint distriburtion
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:
| 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.
| 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.
| 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.
| # 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,
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