AMS Computational Finance

Transcription

1 ams-514-lec-13-m.nb 1 AMS Computational Finance Workshop Robert J. Frey, Research Professor Stony Brook University, Applied Mathematics and Statistics frey@ams.sunysb.edu April 2006 Overview This workshop deals with a complete end-to-end portfolio analysis. Beginning with raw price histories of the current components of the Dow Jones Industrial Index and ending up the computation of the mean-variance frontier of efficient portfolios. Table of Contents I. Software A. Set Up B. Downloading Data From A URL C. History Data From Yahoo!Finance D. Factor Models E. Portfolio Optimization II. Assignment A. Getting The Data B. Estimating The Portfolio Statistics C. Building A 2-Factor Model D. Portfolio Optimizations I. Software This section contains the descriptions and source code for the software used in the assignment. Where applicable, the underlying mathematical theory is outlined as well.

2 ams-514-lec-13-m.nb 2 A. Set Up The following standard packages are required. In addition, the setting of spelling warnings is adjusted to avoid a large number of spurious warnings. << JLink` << Miscellaneous`Calendar` << Statistics`MultiDescriptiveStatistics` << Statistics`LinearRegression` << Graphics`Graphics` Off@General::spell1 B. Downloading Data From A URL This function is based on the technical note on the Wolfram Research website: Elli, Brendan, "Using Java to get Stock Data from the Internet into Mathematica": This function reads data from the specified URL, stores it in a temporary file and returns the name of the file for further processing.

3 ams-514-lec-13-m.nb 3 xgeturl@surl_stringd := JavaBlock@Module@ 8uBuffer, icount, sotmp, ourl, siurl<, InstallJava@ sotmp = OpenTemporary@DOSTextFormat -> False H* Create a temp file. *L ourl = JavaNew@"java.net.URL", surl H* $Failed if not valid. *L siurl = ourl ü openstream@ H* Create a Java input stream. *L If@siUrl === $Failed, Return@$FailedD ubuffer = JavaNew@"@B", 5000 H* 5000 is an arbitrary size. *L While@ H* Read data from the stream saving it to the temp file. *L HiCount = siurl ü read@ubufferdl > 0, WriteString@soTmp, FromCharacterCode@Take@Val@uBufferD, icountddd siurl ü close@ H* Close the stream. *L Close@soTmpD H* Close the temp file, which also returns the filename. *L D This is a useful, general purpose function that will load a text representation of a URL's content into Mathematica. It's most straightforward application is in instances in which the URL returns its data as conveniently formatted text, such as comma separated value. It can also be used to download a page in HTML format. The challenges of parsing such data from within Mathematica are greater; however, Mathematica's ability to parse XML text using its XML capabilities is extremely useful. C. History Data From Yahoo!Fianance This collection of routines are used to build a simple function which will download stock data from Yahoo!Finance Utility Functions The formatting of the data is perfectly readable but is hardly optimized for use within Mathematica. These functions work to reformat these data into something more useful. ü Regularizing field names We want to be able to regularize field names by squeezing out spaces and extraneous (non-letter and non-digit) characters. xregularizename = StringJoin ü Pick@#, HLetterQ@#D»» DigitQ@#DL & êü #D & ü Characters@#D &; ü Date formatting The dates are returned as strings, e.g., {2006, 3, 27} is "27-Mar-06". The following set of functions convert these strings into Mathematica date vectors. The first converts 3-letter month names to integers.

4 ams-514-lec-13-m.nb 4 xmonthnametointeger@ss_stringd := Which@ ss ã "Jan", 1, ss ã "Feb", 2, ss ã "Mar", 3, ss ã "Apr", 4, ss ã "May", 5, ss ã "Jun", 6, ss ã "Jul", 7, ss ã "Aug", 8, ss ã "Sep", 9, ss ã "Oct", 10, ss ã "Nov", 11, ss ã "Dec", 12, True, $Failed This function converts a 2-digit year into a four digit year. It makes a guest at what point to switch from 1900 to 2000 based on the current year. xcompleteyear@iyear_integerd := Module@ 8iDivide<, idivide = Round@100 FractionalPart@First@Date@DD ê 100.D Which@ iyear 1900, iyear, iyear 100, If@iYear idivide, iyear , iyear D, True, $Failed D This function ties the prior two functions together to convert a Yahoo formatted date to one more usable by Mathematica. Here's a demonstration. xyahoodatetomathematicadate@ss_stringd := Module@ 8vsS<, vss = StringSplit@sS, "-" 8xCompleteYear@ToExpression@vsSP3TDD, xmonthnametointeger@vssp2td, ToExpression@vsSP1TD< xyahoodatetomathematicadate@"27-mar-06"d 82006, 3, 27<

5 ams-514-lec-13-m.nb 5 Building a URL The example in the Technical Note referenced above uses an MSN site. For various reasons, we have decided to use Yahoo!Finance as our data source primarily because its split- and dividend-adjusted "Adj Close" is in keeping with CRSP standards. This function builds a URL which will download daily data in comma-separated-value (CSV) format. Unfortunately, we don't have documentation for this; we were able to infer the required arguments from using the interactive historic download facility on Yahoo!Finance. xyahoofinancehistoryurl@ ssymbol_string, 8iStartYear_Integer, istartmonth_integer, istartday_integer<, 8iEndYear_Integer, iendmonth_integer, iendday_integer< D := Module@ 8xF, ss<, xf@ss_d := StringTake@"0" <> ToString@sSD, -2 StringJoin@ " H* Main call *L "s=" <> ssymbol, H* Symbol string *L "&a=" <> xf@istartmonth - 1D, H* Start month - 1 *L "&b=" <> ToString@iStartDayD, H* Start day *L "&c=" <> ToString@iStartYearD, H* Start year *L "&d=" <> xf@iendmonth - 1D, H* End month - 1 *L "&e=" <> ToString@iEndDayD, H* End day *L "&f=" <> ToString@iEndYearD, H* End year *L "&g=d", H* Frequency: daily *L "&ignore=.csv" D Downloading, reading in and processing the data. Given the data specification, as above, this function passes it to xyahoofinancehistoryurl, uses the URL to download the data, reads the data into a local variable and performs some postprocessing to put it into a convenient form for subsequent processing in Mathematica. The function returns a 3-vector consisting of the original arguments, a list of field names, and a data matrix.

6 ams-514-lec-13-m.nb 6 xgetyahoofinancehistorydata@ssymbol_string, vistart_, viend_d := Module@ 8vvsBuffer, vvudata, vsfields<, vvsbuffer = Import@ xgeturl@ xyahoofinancehistoryurl@ssymbol, vistart, viendd D, "CSV" vsfields = xregularizename êü First@vvsBuffer vvudata = Rest@vvsBuffer vvudatapall, 1T = xyahoodatetomathematicadate êü vvudatapall, 1T; vvudata = Sort@vvuData, HFirst@#1D , 100, 1<L < HFirst@#2D , 100, 1<L & 88sSymbol, vistart, viend<, vsfields, vvudata< Here are the data for IBM for the first half of June, Note, only trading days are retrieved. xgetyahoofinancehistorydata@"ibm", 82005, 6, 1<, 82005, 6, 15<D 88IBM, 82005, 6, 1<, 82005, 6, 15<<, 8Date, Open, High, Low, Close, Volume, AdjClose<, , 6, 1<, 75.57, 77.5, 75.57, 76.84, , 76.28<, , 6, 2<, 76.75, 77.39, 76.68, 77.35, , 76.79<, , 6, 3<, 77.06, 77.1, 75.74, 75.79, , 75.24<, , 6, 6<, 75.8, 75.9, 74.92, 75., , 74.45<, , 6, 7<, 75., 76.09, 75., 75.04, , 74.49<, , 6, 8<, 75.04, 75.4, 74.63, 74.8, , 74.26<, , 6, 9<, 74.58, 75.47, 74.23, 74.93, , 74.38<, , 6, 10<, 74.75, 75.05, 74.1, 74.77, , 74.23<, , 6, 13<, 74.5, 75.93, 74.45, 75.05, , 74.5<, , 6, 14<, 75.05, 75.43, 74.73, 74.89, , 74.34<, , 6, 15<, 75.7, 76.5, 75.15, 76.3, , 75.74<<< D. Factor Models Theory Given an n µ n covariance matrix Q, the factor model of order k is based on the n µ k factor exposure matrix B and diagonal error matrix D such that Q > BB T + D Typically, k << n. The factor model above is valid for factors returns which are uncorrelated and of unit variance. This function computes a factor model of order iorder from the covariance matrix mncov. Termination occurs when either the cosine of the angle between successive estimates of the main diagonal of D is less than ntol or the iteration

7 ams-514-lec-13-m.nb 7 limit imaxiter is reached. The function returns the iteration count, "error" cosine, factor exposures and main diagonal of the error covariance. The vector d is the main diagonal of the error covariance D. The main iteration starts with B i.b i T = Q DiagonalMatrix[d i-1 ] The matrix B i is computed using the singular value decomposition of B i.b i T. For a factor model of order k, the first k components of the svd are used. A new estimate of the main diagonal of D can now be computed. d i = Tr[Q B i.b i T, List] This continues until an iteration limit is reached or the cosine between successive values of d falls at least at some tolerance t. Total@d i d i-1 D ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ "#################### Total@d 2 i D "################ Total@d ########## 2 i-1 D ÅÅÅÅÅÅÅÅÅÅÅÅ t The default tolerance is The process must also be initialized and a value of d 0 = ÅÅÅÅ 1 Tr[Q, List] seems to work 2 well in practice. Code xfitfactormodel@mncov_, iorder_, ntol_: H L, imaxiter_: 10D := ModuleA 8d, f, i, n, p, r<, d = Tr@mnCov, ListD ê 2.; i = 0; n = 0; WhileAHi < imaxiterl && Hn ntoll, p = d; r = mncov - DiagonalMatrix@d f = IFirst@#D. è!!!!!!!!!!! #P2T M &@ SingularValueDecomposition@r, iorderd d = Tr@mnCov - f.transpose@fd, List Total@p dd n = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ è!!!!!!!!!!!!!!!!!!!!!! ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ Total@p 2 D è!!!!!!!!!!!!!!!!!!!!!! ; Total@d 2 D i++ E; 8i, n, f, d< E; E. Portfolio Optimization Theory Our standard form for mean-variance portfolio optimization will be allocate one unit of capital with no short sales allowed:

8 ams-514-lec-13-m.nb 8 naïve [l, Q, r] = max x 8 1 ÅÅÅÅ 2 xt Q x l r T x» x 0, T x = 1< If the covariance matrix above can be stated as a factor model then the variance portion of the objective function can be restated as follows. Q = B B T + D ï x T Q x = x T B B T x + x T D x We can next make a substitution in which a vector y represents the factor exposure of the portfolio. B T x = y ï x T B B T x + x T D x = y T y + x T D x This allows us to restate the original optimization in the following form. factor [l, B, D, r] = max i k j x y z y{ 9 ÅÅÅÅ 1 2 J x T y N J D 0 0 I N J x y N l rt x x 0, T x = 1, B T x = y= ƒ The advantage here is that the covariance matrix the Hessian of the objective function can be expressed as a diagonal matrix. This considerably simplifies many computations. The disadvantage is that we have added a large number of "accounting" constraints to get the factor exposures y right. Generally, the second form of the quadratic program requires much less memory and computation time when reasonable effort is taken to use the simplifications available, so despite the large number of constraints added, it usually a good idea to factor the covariance matrix. The runtime required in the first case increases cubicly in the size of the problem. In the second case, the runtime increases slightly worse than linear in the size of the problem. This means that large problems that can not be practically handled with a naïve covariance model are easily solved with an appropriate factorization. Other factorizations will do. For example, a Cholesky decomposition of Q = CC T could be used to define constraints C T x = y and the new formulation would be Cholesky [l, C, r] = max i k j x y z y{ 8 ÅÅÅÅ 1 2 yt y l r T x» x 0, T x = 1, C T x = y< Factor models have other benefits, however. The fact that they represent the covariance structure of the trading universe in a more parsimonious form than the full covariance matrix gives them greater numerical and statistical stability. Code Although it is not an ideal platform, the built-in Mathematica function NMinimize works quite well for small to medium sized portfolio optimization problems. Its practical use in this regard is considerably extended by employing the approach above. The first function below assumes that the covariance structure of the market can be represented by a factor model. The second assumes a naïve covariance matrix. Review NMinimize for additional insight into how they work. The inputs to the first function are: the risk tolerance l, a vector of strings giving the names of the instruments, a vector of strings giving the names of the factors, the mean vector r, the factor loading matrix B, and a vector representing the main diagonal of D. The inputs to the second function are: the risk tolerance l, a vector of strings giving the names of the instruments, the mean vector r and the covariance matrix Q. Both return their result in the same form: a 2-vector whose first element is a list containing the portfolio variance and mean and whose last element is a list containing the portfolio given in the same order as the description vector for the instruments. Although the same symbol, stdqpsolve, is used for both functions, Mathematica distinguishes between them based on the pattern of their arguments.

9 ams-514-lec-13-m.nb 9 stdqpsolve@ nrisktol_, vsinstrumentdesc_, vsfactordesc_, vninstrumentmean_, mnfactorloading_, vndiagonal_ D := Module@ 8cons, conscapital, consnonnegativity, port, obj, sol, vars, varsfactor, varsinstrument<, varsinstrument = Unique@# <> "$" & êü vsinstrumentdesc varsfactor = Unique@# <> "$" & êü vsfactordesc vars = Join@varsInstrument, varsfactor consfactorloading = H# ã 0 &L êü HHH#.varsInstrument &L êü HmnFactorLoading LL - varsfactorl; consnonnegativity = H# 0 &L êü varsinstrument; conscapital = 8Plus üü varsinstrument ã 1<; cons = And üü Join@consFactorLoading, consnonnegativity, conscapital obj = 0.5 HTotal@Join@HvarsInstrument 2 vndiagonall, varsfactor 2 DDL - nrisktol HvnInstrumentMean.varsInstrumentL; sol = NMinimize@8obj, cons<, vars, PrecisionGoal Ø 6 port = varsinstrument ê. solp2t; 88H#.#L &@port.mnfactord + Hport 2.vnDiagonalL, port.vnmean<, port< stdqpsolve@ nrisktol_, vsdesc_, vnmean_, mncov_ D := Module@ 8cons, port, obj, sol, vars<, vars = Unique@# <> "$" & êü vsdesc cons = And üü Join@8Plus üü vars ã 1<, H# 0 &L êü vars obj = 0.5 Hvars.mnCov.varsL - nrisktol HvnMean.varsL; sol = NMinimize@8obj, cons<, vars, PrecisionGoal Ø 6 port = vars ê. solp2t; 88port.mnCov.port, port.vnmean<, port< II. Assignment The assignment was to plot the efficient frontier for the current members of the Dow Jones Industrial Index based on the price history that was available in common. This was to be done using a 2-factor model first. These results were to be contrasted with using the full covariance matrix.

10 ams-514-lec-13-m.nb 10 A. Getting the Data Stock Universe The list of tickers of the Dow Jones Industrial Average. vsdowjonessymbol = 8"MMM", "AA", "MO", "AXP", "AIG", "T", "BA", "CAT", "C", "KO", "DD", "XOM", "GE", "GM", "HPQ", "HD", "HON", "INTC", "IBM", "JNJ", "JPM", "MCD", "MRK", "MSFT", "PFE", "PG", "UTX", "VZ", "WMT", "DIS"< 8MMM, AA, MO, AXP, AIG, T, BA, CAT, C, KO, DD, XOM, GE, GM, HPQ, HD, HON, INTC, IBM, JNJ, JPM, MCD, MRK, MSFT, PFE, PG, UTX, VZ, WMT, DIS< Downloading Raw Daily History Data Ultimately, we want only month ending dates and, as was established in earlier classes, it appears that we can only as far back as {1986, 7, 9}. There also isn't much point including anything past {2006, 3, 31} since we're still in April. audataabstract = xgetyahoofinancehistorydata@#, 81986, 7, 9<, 82006, 3, 31<D & êü vsdowjonessymbol; We only need the Date and AdjClose fields, so we'll restructure dataabstract to that effect. audataabstract = 8#P1T, #P2, 81, 7<T, #P3, All, 81, 7<T< & êü audataabstract; We only need the month ending dates. The Split function can be used to split a list into sublists of "same" elements. Here we employ Split with a function that defines two entries as the same if the month of their dates are the same. Applied to a single element of the top-level dataabstract list this returns sublists of month data. We then simply pull out the last element of each sublist as the last trading day of the month. Pulling Out Month Ending Dates aumonthlyabstract = 8#P1T, #P2T, Last êü Split@#P3T, H#1P1, 2T ã #2P1, 2TL &D< & êü audataabstract; We want to be very careful and check that all of the dates across all of the stocks are equal. They are. SameQ üü H#P3, All, 1T & êü aumonthlyabstractl True

11 ams-514-lec-13-m.nb 11 Finally, we build a returns data structure consisting of three elements, The first is a list of dates as row names. The second is a list of tickers as column nams. The third is a matrix of returns over the dates and tickers. Building A Return Data Structure vureturns = 9 Rest@auMonthlyAbstractP1, 3, All, 1TD, aumonthlyabstractpall, 1, 1T, Rest@#D ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ Most@#D - 1 &@ Transpose@#P3, All, 2T & êü aumonthlyabstractd D =; Dimensions êü vureturns 88236, 3<, 830<, 8236, 30<< B. Estimating the Portfolio Statistics The vector of expected returns is vnmean = Mean@vuReturnsP3T Dimensions@vnMeanD 830< The covariance matrix of returns is mncovariance = CovarianceMatrix@vuReturnsP3T Dimensions@mnCovarianceD 830, 30< C. Building A 2-Factor Model Estimating The Model First, we estimate the factor model of order 2.

12 ams-514-lec-13-m.nb 12 8iIter, nfit, mnfactor, vndiag< = xfitfactormodel@mncovariance, 2 8iIter, nfit< 86, 1.< Then, we construct the approximate covariance matrix. mnapproxcov = HmnFactor.Transpose@mnFactorDL + DiagonalMatrix@vnDiag Evaluating The Model We generate a sample of portfolio allocation vectors x = {x 1, x 1,, x i,, x n } such that x i 0 and T x ã 1. For each element of the sample we compute the portfolio standard deviation that we would expect given the factor model and original covariance, respectively. mnsample = 9 è!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! #.mnapproxcov.#, è!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! #.mncovariance.#!!!!!!!!! = & êü TableA # ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ &@Table@Random@D, 830<DD, Total@#D 81000< E; Here is a plot of the factor model versus the full covarinace. The red line represents the line y = x.

13 ams-514-lec-13-m.nb 13 mnsample, PlotRange Ø All, PlotStyle Ø Frame Ø True, FrameLabel Ø 8 "Factor Model", "Full Covariance", StyleForm@"Portfolio s Comparison", FontSize Ø 16D, "" <, ImageSize Ø 400 D, Plot@ x, 8x, Min@Flatten@mnSampleDD, Max@Flatten@mnSampleDD<, PlotStyle Ø 8RGBColor@1, 0, 0D, Thickness@0.005D< D < Portfolio s Comparison F ull Covarianc e Factor Model The next plot is a histogram of the ratio between the factor model and full covariance standard deviation.

14 ams-514-lec-13-m.nb 14 HistogramA è!!!!!!!!!!!!!!!!!!!!!!!! #P1T ê #P2T & êü mnsample, PlotLabel Ø "ApproxêFull Portfolio Variance", FontSize Ø 16 D, ImageSize Ø 400 E; 150 ApproxêFull Portfolio Variance A 30 µ 30 covariance matrix has n(n + 1)/2 = 465 unique parameters (allowing for symmetry). The 2-factor model contains (k n) + n = (2 µ 30) + 30 = 90 parameters. When dealing with larger stock universes the differences are even greater. For example, if we had been working with the S&P 500, then a 2-factor model requires 1,500 parameters while the full covariance matrix contains 125,250 parameters. Thus, the factor model would contain 1.2% of the parameters of a naïve variance model. D. Portfolio Optimizations Runtime Comparison The relative runtime improvement of the optimization based on the factor model versus that of one based on the full covariance matrix varies depending upon the geometry of the problem at hand; however, even for a small problem such as this it is quite significant.

15 ams-514-lec-13-m.nb 15 vsdowjonessymbol, 8"Factor1", "Factor2"<, vnmean, mnfactor, vndiagd D Second, , <, 80., 0., , 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., , 0., , 0., 0., 0., 0., 0., , 0., 0., 0., 0., 0., 0.<<< Timing@ stdqpsolve@0.5, vsdowjonessymbol, vnmean, mncovarianced D Second, , <, 80., µ 10-8, , 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., , 0., , 0., 0., 0., 0., 0., , 0., 0., 0., 0., 0., 0.<<< There are small differences in the allocations, but the portfolio variance and mean are quite close to one another. Efficient Frontier Using A Factor Model After a little offline experimentation, the following list of risk tolerances appear to reasonably span the risk space from l = 0 (minimum variance, i.e., only risk considered) to l > 4 (all capital allocated to the single asset with the highest return, i.e., only expected return considered). vnrisktol = Prepend@2. Range@-4,2,0.5D, 0.D 80., , , 0.125, , 0.25, , 0.5, , 1., , 2., , 4.< We can now run through a series of optimizations at various risk levels. vusolutions = stdqpsolve@#, vsdowjonessymbol, 8"Factor1", "Factor2"<, vnmean, mnfactor, vndiagd & êü vnrisktol; Pulling out the efficient frontier is straightforward. mnefficientfrontier = First êü vusolutions; The efficient frontier is plotted below.

16 ams-514-lec-13-m.nb 16 mnefficientfrontier, Frame Ø True, PlotJoined Ø True, PlotRange Ø 880, 0.014<, 80, 0.035<<, ImageSize Ø Efficient Frontier Using Naïve Covariance Given the difference in runtimes, a coarser set of risk tolerances seems reasonable. vnalttol = Prepend@2. Range@-4,2,2D, 0.D 80., , 0.25, 1, 4.< As before we run through a series of optimizations at various risk levels. vualtsolutions = stdqpsolve@#, vsdowjonessymbol, vnmean, mncovarianced & êü vnalttol; And, again, pull out the efficient frontier from the results. mnaltefficientfrontier = First êü vualtsolutions;

17 ams-514-lec-13-m.nb 17 Here we plot the results using the naïve covariance approach as points against the efficient frontier computed using a factor model. There are small differences but the results are essentially the same. DisplayTogether@8 ListPlot@ mnefficientfrontier, Frame Ø True, PlotJoined Ø True, PlotRange Ø 880, 0.014<, 80, 0.035<<, ImageSize Ø 400 D, ListPlot@ mnaltefficientfrontier, PlotRange Ø 880, 0.014<, 80, 0.035<<, PlotStyle Ø 8PointSize@0.015D, RGBColor@1, 0, 0D< D <