Package {smqf}

Title:	Statistical Methods for Quantitative Finance
Version:	1.1-7
Description:	Provides data and functions used in the book "Statistical Methods for Quantitative Finance" by David Ardia (2026).
License:	GPL-3
LazyData:	true
LazyDataCompression:	xz
Encoding:	UTF-8
Depends:	R (≥ 4.1.0), xts
URL:	https://github.com/ArdiaD/smqf-package
BugReports:	https://github.com/ArdiaD/smqf-package/issues
Suggests:	zoo, testthat (≥ 3.0.0), PerformanceAnalytics, glmnet, quadprog, copula, Matrix, readxl, MASS, mvtnorm
Config/testthat/edition:	3
Imports:	graphics, stats, nloptr, pracma
Config/roxygen2/version:	8.1.0
NeedsCompilation:	no
Packaged:	2026-08-23 03:23:33 UTC; ardiad
Author:	David Ardia [aut, cre, cph, fnd], Marius Hofert [ctb, cph] (Original author of qrmdata, from which 15 market datasets were ported under GPL-2 | GPL-3), Kurt Hornik [ctb, cph] (Original author of qrmdata, from which 15 market datasets were ported under GPL-2 | GPL-3), Alexander J. McNeil [ctb, cph] (Original author of qrmdata, from which 15 market datasets were ported under GPL-2 | GPL-3), Dries Cornilly [ctb, cph] (Original author of mvskPortfolios, from which f_mvsk_portfolio() was ported under GPL-2 | GPL-3), Kris Boudt [ctb, cph] (Original author of mvskPortfolios, from which f_mvsk_portfolio() was ported under GPL-2 | GPL-3)
Maintainer:	David Ardia <david.ardia.ch@gmail.com>
Repository:	CRAN
Date/Publication:	2026-08-23 05:20:02 UTC

smqf: Statistical Methods for Quantitative Finance

Description

'smqf' is an R package providing datasets and functions used in the book "Statistical Methods for Quantitative Finance" by Ardia (2026).

Datasets

All datasets bundled with 'smqf' are illustrative: they are static snapshots provided so that the book's examples remain reproducible, and some have been modified for that purpose (e.g., 'SP500_const' is thinned to weekly frequency, and several series are truncated or re-indexed). They are not maintained data feeds and should not be relied upon for research or production use.

Index conventions: daily series (e.g., 'SP500', 'FTSE', 'VIX', 'TermStructure') use a 'Date' index; monthly series ('FungHsieh', 'GoyalWelch', 'Fred') use a 'zoo::yearmon' index (a calendar month with no day-of-month, so they merge by month and avoid first-/end-of-month ambiguity); 'FamaFrench' is a weekly series with a 'Date' index.

Acknowledgments

The market datasets 'SP500', 'SP500_const', 'DJ', 'DJ_const', 'FTSE', 'FTSE_const', 'EURSTOXX', 'EURSTX_const', 'DAX', 'CAC', 'NIKKEI', 'SMI', 'HSI', 'GOLD', and 'VIX' are redistributed from the **qrmdata** package by Marius Hofert, Kurt Hornik, and Alexander J. McNeil under the same GPL (\ge 2) license. The original data was collected by the qrmdata authors from Yahoo Finance and FRED (via Quandl) on 2016-01-03. See <https://CRAN.R-project.org/package=qrmdata> for the upstream package and the individual dataset help pages for details.

Author(s)

David Ardia

References

David Ardia (2026). *Statistical Methods for Quantitative Finance*. CRC Press.

Hofert, M., Hornik, K., & McNeil, A. J. *qrmdata: Data Sets for Quantitative Risk Management Practice*. <https://CRAN.R-project.org/package=qrmdata>.

See Also

Useful links:

• https://github.com/ArdiaD/smqf-package

• Report bugs at https://github.com/ArdiaD/smqf-package/issues

CAC 40 Index (Daily, xts)

Description

Daily adjusted close prices of the Cotation Assistée en Continu (CAC 40) stock index (ticker symbol ^FCHI), from its first date of availability on Yahoo Finance to 2015-12-31.

Usage

data("CAC")

Format

An xts object with 6549 daily observations and a single column ^FCHI containing adjusted close prices in index points. The time index spans from 1990-03-01 to 2015-12-31.

Details

Originally distributed as CAC in the qrmdata package (Hofert, Hornik, & McNeil), ported into smqf so that the book's examples remain reproducible without an extra dependency. The data is redistributed here under the same GPL (\ge 2) license as qrmdata.

Source

Yahoo Finance, downloaded on 2016-01-03 via qrmtools::get_data().

References

Hofert, M., Hornik, K., & McNeil, A. J. qrmdata: Data Sets for Quantitative Risk Management Practice, https://CRAN.R-project.org/package=qrmdata.

Examples

data("CAC")
class(CAC)        # "xts" "zoo"
dim(CAC)
head(CAC)

DAX Index (Daily, xts)

Description

Daily adjusted close prices of the Deutscher Aktienindex (DAX) stock index (ticker symbol ^GDAXI), from its first date of availability on Yahoo Finance to 2015-12-30.

Usage

data("DAX")

Format

An xts object with 6355 daily observations and a single column ^GDAXI containing adjusted close prices in index points. The time index spans from 1990-11-26 to 2015-12-30.

Details

Originally distributed as DAX in the qrmdata package (Hofert, Hornik, & McNeil), ported into smqf so that the book's examples remain reproducible without an extra dependency. The data is redistributed here under the same GPL (\ge 2) license as qrmdata.

Source

Yahoo Finance, downloaded on 2016-01-03 via qrmtools::get_data().

References

Hofert, M., Hornik, K., & McNeil, A. J. qrmdata: Data Sets for Quantitative Risk Management Practice, https://CRAN.R-project.org/package=qrmdata.

Examples

data("DAX")
class(DAX)        # "xts" "zoo"
dim(DAX)
head(DAX)

Dow Jones Industrial Average Index (Daily, xts)

Description

Daily adjusted close prices of the Dow Jones Industrial Average (ticker symbol ^DJI), from its first date of availability on Yahoo Finance to 2015-12-31.

Usage

data("DJ")

Format

An xts object with 7797 daily observations and a single column ^DJI containing adjusted close prices in index points. The time index spans from 1985-01-29 to 2015-12-31.

Details

Originally distributed as DJ in the qrmdata package (Hofert, Hornik, & McNeil), ported into smqf so that the book's examples remain reproducible without an extra dependency. The data is redistributed here under the same GPL (\ge 2) license as qrmdata.

Source

Yahoo Finance, downloaded on 2016-01-03 via qrmtools::get_data().

References

Hofert, M., Hornik, K., & McNeil, A. J. qrmdata: Data Sets for Quantitative Risk Management Practice, https://CRAN.R-project.org/package=qrmdata.

Examples

data("DJ")
class(DJ)        # "xts" "zoo"
dim(DJ)
head(DJ)

Dow Jones Industrial Average Constituents (Daily, xts)

Description

Daily adjusted close prices for the 30 constituents of the Dow Jones Industrial Average as of 2016-01-03.

Usage

data("DJ_const")

Format

An xts object with 13595 daily observations and 30 columns, one per constituent (e.g., AAPL, IBM, JPM, XOM). The time index spans from 1962-01-02 to 2015-12-31.

Details

Originally distributed as DJ_const in the qrmdata package (Hofert, Hornik, & McNeil), ported into smqf so that the book's examples remain reproducible without an extra dependency. The data is redistributed here under the same GPL (\ge 2) license as qrmdata.

Missing values

The vast majority of the 101,285 NAs are leading: a constituent is NA on every date before it first traded. A small number, however, occur after a series has started, so trimming the leading block does not guarantee a complete matrix. In this dataset there are 25 such interior NAs, confined to three dates (1985-09-27, when the NYSE was closed; 1981-08-10; and 1983-09-23) and spread over 24 of the 30 columns.

Handle both cases explicitly, for example with stats::cor(x, use = "pairwise.complete.obs"), or by restricting to rows with rowSums(is.na(x)) == 0. See EURSTX_const, where interior NAs are far more common.

Source

Yahoo Finance, downloaded on 2016-01-03 via qrmtools::get_data().

References

Hofert, M., Hornik, K., & McNeil, A. J. qrmdata: Data Sets for Quantitative Risk Management Practice, https://CRAN.R-project.org/package=qrmdata.

Examples

data("DJ_const")
class(DJ_const)        # "xts" "zoo"
dim(DJ_const)
head(colnames(DJ_const))

Euro Stoxx 50 Index (Daily, xts)

Description

Daily adjusted close prices of the Euro Stoxx 50 stock index (ticker symbol ^STOXX50E), from its first date of availability on Yahoo Finance to 2015-12-23.

Usage

data("EURSTOXX")

Format

An xts object with 7445 daily observations and a single column ^STOXX50E containing adjusted close prices in index points. The time index spans from 1986-12-31 to 2015-12-23.

Details

Originally distributed as EURSTOXX in the qrmdata package (Hofert, Hornik, & McNeil), ported into smqf so that the book's examples remain reproducible without an extra dependency. The data is redistributed here under the same GPL (\ge 2) license as qrmdata.

Source

Yahoo Finance, downloaded on 2016-01-03 via qrmtools::get_data().

References

Hofert, M., Hornik, K., & McNeil, A. J. qrmdata: Data Sets for Quantitative Risk Management Practice, https://CRAN.R-project.org/package=qrmdata.

Examples

data("EURSTOXX")
class(EURSTOXX)        # "xts" "zoo"
dim(EURSTOXX)
head(EURSTOXX)

Euro Stoxx 50 Constituents (Daily, xts)

Description

Daily adjusted close prices for the 50 constituents of the Euro Stoxx 50 stock index as of 2016-01-03.

Usage

data("EURSTX_const")

Format

An xts object with 4174 daily observations and 50 columns, one per constituent (e.g., SAP.DE, BNP.PA, SAN.MC). The time index spans from 2000-01-03 to 2015-12-31.

Details

Originally distributed as EURSTX_const in the qrmdata package (Hofert, Hornik, & McNeil), ported into smqf so that the book's examples remain reproducible without an extra dependency. The data is redistributed here under the same GPL (\ge 2) license as qrmdata.

Missing values

Of the 8,270 NAs, 1,938 occur after a series has started, and every one of the 50 columns is affected. Trimming the leading block is therefore not enough to obtain a complete matrix.

The interior NAs are mostly local trading holidays: the constituents are listed on exchanges with different calendars (Paris, Frankfurt, Madrid, Amsterdam, Milan, Brussels), so a day that is a holiday in one country still appears in the index with NA for that country's shares. They span 1,015 distinct dates, the largest being Good Friday 2011-04-22 and Easter Monday 2011-04-25 (48 series each), Labour Day 2006-05-01 (46) and Christmas 2009-12-25 (43). A few series also have long individual gaps, notably UL.PA with 679 interior NAs.

Consequently stats::cov(x) returns NAs on this dataset. Use use = "pairwise.complete.obs" and project the result onto the positive semidefinite cone with as.matrix(Matrix::nearPD(S)$mat), or restrict to rows with rowSums(is.na(x)) == 0 for a single common estimation window.

Source

Yahoo Finance, downloaded on 2016-01-03 via qrmtools::get_data().

References

Hofert, M., Hornik, K., & McNeil, A. J. qrmdata: Data Sets for Quantitative Risk Management Practice, https://CRAN.R-project.org/package=qrmdata.

Examples

data("EURSTX_const")
class(EURSTX_const)        # "xts" "zoo"
dim(EURSTX_const)
head(colnames(EURSTX_const))

FTSE 100 Index (Daily, xts)

Description

Daily adjusted close prices of the FTSE 100 stock index (ticker symbol ^FTSE), from its first date of availability on Yahoo Finance to 2015-12-31.

Usage

data("FTSE")

Format

An xts object with 8333 daily observations and a single column ^FTSE containing adjusted close prices in index points. The time index spans from 1984-01-03 to 2015-12-31.

Details

Originally distributed as FTSE in the qrmdata package (Hofert, Hornik, & McNeil), ported into smqf so that the book's examples remain reproducible without an extra dependency. The data is redistributed here under the same GPL (\ge 2) license as qrmdata.

Source

Yahoo Finance, downloaded on 2016-01-03 via qrmtools::get_data().

References

Hofert, M., Hornik, K., & McNeil, A. J. qrmdata: Data Sets for Quantitative Risk Management Practice, https://CRAN.R-project.org/package=qrmdata.

Examples

data("FTSE")
class(FTSE)        # "xts" "zoo"
dim(FTSE)
head(FTSE)

FTSE 100 Constituents (Daily, xts)

Description

Daily adjusted close prices for 98 constituents of the FTSE 100 stock index as of 2016-01-03.

Usage

data("FTSE_const")

Format

An xts object with 7198 daily observations and 98 columns, one per constituent (e.g., HSBA.L, BP.L, VOD.L). The time index spans from 1988-05-03 to 2015-12-31.

Details

Originally distributed as FTSE_const in the qrmdata package (Hofert, Hornik, & McNeil), ported into smqf so that the book's examples remain reproducible without an extra dependency. The data is redistributed here under the same GPL (\ge 2) license as qrmdata. Only 98 of the 100 constituents were available at the time of the download.

Missing values

Most of the 163,821 NAs are leading: a constituent is NA on every date before it first traded. But 4,634 of them occur after a series has started, spread over 96 of the 98 columns, so trimming the leading block does not produce a complete matrix.

Handle both cases explicitly. Two idioms cover most needs: restrict to the columns that are complete over your window, as the book does with x <- x[, colSums(is.na(x)) == 0] — over 2013–2015 this keeps 95 of the 98 names — or estimate pairwise with stats::cov(x, use = "pairwise.complete.obs") and project the result onto the positive semidefinite cone. See EURSTX_const, where interior NAs are far more common, and DJ_const, where they are rare.

Source

Yahoo Finance, downloaded on 2016-01-03 via qrmtools::get_data().

References

Hofert, M., Hornik, K., & McNeil, A. J. qrmdata: Data Sets for Quantitative Risk Management Practice, https://CRAN.R-project.org/package=qrmdata.

Examples

data("FTSE_const")
class(FTSE_const)        # "xts" "zoo"
dim(FTSE_const)
head(colnames(FTSE_const))

Fama–French Factors (Weekly, xts)

Description

Weekly time series of the three Fama–French equity risk factors and the risk-free rate, as provided by Kenneth French's data library. Values are expressed in percentage points (e.g., 1.60 means a return of 1.60%).

Usage

data("FamaFrench")

Format

An xts object with 4\,834 weekly observations (from 1926-07-02 to 2019-02-22) and 4 columns:

mkt_rf

Excess return on the market (market return minus risk-free rate), in %.

smb

Small-Minus-Big size factor return, in %.

hml

High-Minus-Low value factor return, in %.

rf

Risk-free rate (weekly), in %.

Details

The time index is a weekly Date, stamped at the end of each week. In the modern sample that is a Friday, shifted back to the Thursday when the Friday is a holiday (3,535 Fridays and 137 Thursdays in total). In the early history, when Saturday trading sessions were still held, the stamp is a Saturday: 1,158 observations, running to the late 1950s. Do not assume a Friday stamp throughout. Divide by 100 to convert to decimal returns.

Note also that rf is slightly negative in 60 Depression-era weeks (1933 and 1938), with a minimum of -0.016%. That is a property of the source data, not an error.

Aligning with other weekly series

Weekly price series downloaded from vendors are often stamped at the start of the week instead, so the two indices interleave rather than coincide — see extdata for the prices used in Chapter 1, which carry Tuesday stamps. Merging such a series with these factors and then filling the gaps, for instance with two zoo::na.locf() passes, silently lags the factors by one week and can flip the sign of an estimated market beta. Pair the two series by position over a common span, or restamp one onto the other's convention, and assert that the lengths match before regressing.

Downsampling with FamaFrench[xts::endpoints(FamaFrench, "months"), ] selects the last weekly observation of each month; it does not compound the weeks into a monthly return. To obtain monthly factor returns, aggregate the weekly ones explicitly.

Source

Kenneth R. French Data Library, https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html.

References

Fama, E. F., & French, K. R. (1993). Common Risk Factors in the Returns on Stocks and Bonds. Journal of Financial Economics, 33(1), 3–56.

Examples

data("FamaFrench")
class(FamaFrench)        # "xts" "zoo"
dim(FamaFrench)          # 4834 x 4
head(FamaFrench)

# Convert to decimal and extract the three equity factors
ff <- FamaFrench[, c("mkt_rf", "smb", "hml")] / 100
rf <- FamaFrench[, "rf"] / 100

FRED-MD Macro Factors and Dow Jones Index Changes (Monthly, 2015–2019)

Description

An xts object holding 128 FRED-MD macroeconomic predictors and the one-month-ahead change in the Dow Jones Industrial Average, aligned over the period 2015-01 to 2019-12. Used to illustrate high-dimensional regularised regression (Lasso, Ridge) in a prediction context with more predictors than observations.

Usage

data("Fred")

Format

An xts object with 60 monthly observations (index Jan 2015 to Dec 2019, class zoo::yearmon) and 129 columns:

columns 1 to 128

FRED-MD macro variables, transformed (differenced or log-differenced) to achieve stationarity following the FRED-MD transformation codes. Column names are FRED-MD series codes (e.g., "INDPRO", "CPIAUCSL", "GS10"). The series are not rescaled: each keeps the units of its own transform, and the column standard deviations span roughly eight orders of magnitude, from CONSPI (about 0.0009) to BOGMBASE (about 73,000). Any penalized fit must therefore standardize — which glmnet::glmnet() does by default, but only while standardize = TRUE.

DJI.Adjusted (final column)

One-month-ahead change in the Dow Jones Industrial Average, aligned so that row t of the predictors corresponds to row t of the response. This is a difference in index points, not a return: it ranges from about -2211 to +1785 with a standard deviation near 780. Because the index level roughly doubles over the sample, the series is heteroskedastic by construction. Divide by the lagged index level before interpreting any result as a return.

Details

The response is a difference in index points, not a return. See the DJI.Adjusted entry under Format before interpreting any coefficient.

The 128 macro predictors (columns 1 to 128) were downloaded from the McCracken–Ng FRED-MD database and transformed according to the recommended stationarity codes. The target column DJI.Adjusted was obtained from Yahoo Finance via quantmod::getSymbols("^DJI") and differenced one month ahead, giving a change in index points rather than a return: its standard deviation is about 780 and it ranges from roughly -2211 to +1785. All series are restricted to the 60-month window 2015-01 to 2019-12 and temporally aligned so that the predictor columns in row t can be used to predict DJI.Adjusted in row t.

Source

• McCracken, M. W., & Ng, S. FRED-MD database, https://www.stlouisfed.org/research/economists/mccracken/fred-databases.

• Dow Jones daily prices via quantmod::getSymbols("^DJI"), Yahoo Finance.

References

McCracken, M. W., & Ng, S. (2016). FRED-MD: A Monthly Database for Macroeconomic Research. Journal of Business & Economic Statistics, 34(4), 574–589.

See Also

The book chapter on Dimension Reduction (Chapter 4 of Statistical Methods for Quantitative Finance) introduces Lasso and Ridge regression on the lower-dimensional GoyalWelch dataset; Fred provides a complementary high-dimensional (p \gg n) test bed for the same regularised-regression workflow.

Examples

data("Fred")
class(Fred)                                    # "xts" "zoo"
y <- Fred[, "DJI.Adjusted"]                    # response (60 x 1)
X <- Fred[, colnames(Fred) != "DJI.Adjusted"]  # 128 macro predictors
dim(X)

# Lasso with cross-validation (requires glmnet, which expects matrices)
if (requireNamespace("glmnet", quietly = TRUE)) {
  set.seed(1234)
  fit <- glmnet::cv.glmnet(as.matrix(X), as.numeric(y), alpha = 1)
  coef(fit, s = "lambda.min")[coef(fit, s = "lambda.min")[,1] != 0, , drop = FALSE]
}

Fung–Hsieh Factors (Monthly, xts)

Description

Monthly time series of commonly used Fung–Hsieh style and macro factors. The object is an xts matrix indexed by month (class zoo::yearmon) with the columns listed below. Typical use cases include hedge-fund replication and factor attribution.

Usage

data("FungHsieh")

Format

An xts object with 276 monthly observations (Jan 1994 to Dec 2016), indexed by zoo::yearmon, and 8 columns.

All columns are expressed in percentage points, never in basis points or decimals. Divide by 100 before combining them with decimal return series such as PerformanceAnalytics::edhec.

EMKT

Equity market factor, % per month. Mean 0.62, sd 4.24 (14.7% annualized).

RF

Risk-free rate, % per month. Always non-negative; maximum 0.56, i.e. 6.7% annualized.

SS

Size spread (small minus large), % per month. Mean 0.05, sd 3.26, correlation with EMKT of 0.08 — the near-zero mean and low market correlation confirm a long–short spread rather than a directional return.

CST10Y

Change in the 10-year U.S. Treasury constant-maturity yield, in percentage points per month (range -1.11 to 0.65). Not a return.

BAA

Change in the Moody's Baa credit spread, in percentage points per month (range -0.99 to 1.45). Not a return.

PTFSBD

Fung–Hsieh primitive trend-following straddle: bond, % per month. Mean -1.67, sd 15.23.

PTFSCOM

Fung–Hsieh primitive trend-following straddle: commodity, % per month. Mean -0.55, sd 14.28.

PTFSFX

Fung–Hsieh primitive trend-following straddle: currency, % per month. Mean -0.85, sd 19.48.

Details

The three PTFS* series are lookback-straddle returns. They are strongly right-skewed with large positive tails (up to +89.8% in a single month) and negative means, which is the expected signature of a long-option payoff, not a data error.

Two of the eight columns — CST10Y and BAA — are changes in yields rather than returns. Do not rescale them as if they were returns, and interpret their loadings as sensitivities per percentage point of yield change.

The time index is of class zoo::yearmon (a calendar month with no day-of-month), so the series aligns with other monthly data by month, irrespective of any day-of-month convention. To merge with a Date-indexed series, coerce its index with as.yearmon().

Provenance

This object is a static snapshot compiled for the book's examples, not a maintained feed, and the limits of what is documented about it should be stated plainly. The original download URLs and the vintage of each input series were not recorded when the snapshot was taken, so data-raw/FungHsieh.R is a contract harness rather than a reconstruction script: it pins every property the book relies on and will fail loudly if a future rebuild diverges, but it cannot regenerate the object from source.

The consequence is that the column semantics described above are inferred from the behaviour of the shipped series and from the standard Fung–Hsieh construction, not verified against an archived input file. Some of them are corroborated by the data: BAA correlates -0.51 with CST10Y and -0.27 with EMKT, which is the signature of a credit spread change (spreads widen as equities fall and as safe yields drop) and rules out a raw Baa yield change, which would track the ten-year yield positively. Others are not: the exact index pair underlying SS, the day-of-month sampling convention behind CST10Y, and the precise straddle construction behind the PTFS* series cannot be settled from the object alone.

Treat the factor identifications as adequate for the illustrative regressions in the book, and re-derive the factors from https://people.duke.edu/~dah7/HFRFData.htm before publishing any attribution result that turns on the definition of a particular factor.

Source

Compiled from public factor sources commonly used in the Fung–Hsieh literature: David A. Hsieh's hedge-fund data library (https://people.duke.edu/~dah7/HFRFData.htm) for the PTFS* straddle factors, and FRED/H.15 for the yield and credit series. See data-raw/FungHsieh.R for the assembly steps.

References

Fung, W., & Hsieh, D. A. (2004). Hedge Fund Benchmarks: A Risk-Based Approach. Financial Analysts Journal, 60(5), 65–80. Fung, W., & Hsieh, D. A. (2001). The Risk in Hedge Fund Strategies: Theory and Evidence from Trend Followers. Review of Financial Studies, 14(2), 313–341.

Examples

data("FungHsieh")
class(FungHsieh)              # "xts" "zoo"
head(FungHsieh)
colnames(FungHsieh)
# Quick plot of the three PTFS factors
if (requireNamespace("zoo", quietly = TRUE)) {
  zoo::plot.zoo(FungHsieh[, c("PTFSBD", "PTFSCOM", "PTFSFX")], screens = 1, col = 1:3)
}

Gold Price (Daily, xts)

Description

Daily World Gold Council gold price in USD per troy ounce, from 1970-01-01 to 2015-12-31.

Usage

data("GOLD")

Format

An xts object with 9691 daily observations and a single column GOLD containing USD prices per troy ounce.

Details

Originally distributed as GOLD in the qrmdata package (Hofert, Hornik, & McNeil), ported into smqf so that the book's examples remain reproducible without an extra dependency. The data is redistributed here under the same GPL (\ge 2) license as qrmdata.

Source

Federal Reserve Economic Data (FRED) via Quandl, downloaded on 2016-01-03 with qrmtools::get_data().

References

Hofert, M., Hornik, K., & McNeil, A. J. qrmdata: Data Sets for Quantitative Risk Management Practice, https://CRAN.R-project.org/package=qrmdata.

Examples

data("GOLD")
class(GOLD)        # "xts" "zoo"
dim(GOLD)
head(GOLD)

Goyal–Welch Predictive Variables (Monthly, U.S.)

Description

Monthly time series of equity predictors and bond/credit variables commonly used in return predictability studies following Goyal & Welch (2008). Stored as an xts object indexed by month (class zoo::yearmon).

Usage

data("GoyalWelch")

Format

An xts object with monthly observations (Dec 1979 to Dec 2018 in this snapshot), indexed by zoo::yearmon, and 15 variables:

Index

Broad U.S. equity price index level (e.g., S&P 500).

D12

Trailing 12-month cash dividends on the index (level).

E12

Trailing 12-month earnings on the index (level).

b/m

Aggregate book-to-market ratio.

tbl

3-month Treasury bill rate, annualized, as a decimal (0.1204 = 12.04% per year). Observed monthly.

AAA

Moody’s AAA corporate bond yield, annualized decimal.

BAA

Moody’s BAA corporate bond yield, annualized decimal.

lty

Long-term government bond yield, annualized decimal.

ntis

Net equity expansion: shares issued less repurchases scaled by total equity (a supply measure).

Rfree

Risk-free rate over the month, as a decimal. This is exactly tbl / 12 — see the warning under Details.

infl

Inflation (monthly change in price level).

ltr

Long-term government bond total return over the month.

corpr

Corporate bond total return over the month.

svar

Stock market variance proxy (rolling sum of daily squared returns).

csp

Cross-sectional premium of Polk, Thompson and Vuolteenaho (2006). Available from Dec 1979 to Dec 2002 only; the remaining 192 months are NA. It is not the corporate bond return spread, which is corpr - ltr.

Details

This object mirrors the variables used in Goyal & Welch (2008) and subsequent updates. From these levels you can form the standard ratios used in the literature, for example dp = log(D12/Index), ep = log(E12/Index), dfy = BAA - AAA, dfr = corpr - ltr and tms = lty - tbl.

Beware that the classical predictor list contains further exact redundancies of the same kind as tbl/Rfree: the payout ratio de = dp - ep and the term spread tms = lty - tbl are differences of other members. These predictors are used one at a time in univariate predictive regressions; entering them jointly with their components yields a singular design matrix.

Index is a price level, not a return. The equity premium is formed as (Index[t] + D12[t]/12) / Index[t-1] - 1 - Rfree[t].

Exact redundancy between tbl and Rfree

Rfree equals tbl / 12 to the last representable digit for every one of the 469 months: max(abs(GoyalWelch[,"tbl"]/12 - GoyalWelch[,"Rfree"])) is exactly zero. The two columns therefore carry the same information, and at most one of them may enter a regression. Passing both produces a rank-deficient design: lm() detects it and returns NA for the aliased coefficient, but penalized fitters such as glmnet::glmnet() do not — they split the effect arbitrarily between the two columns and report coefficients that can differ in sign from the identified solution. In predictive regressions Rfree normally belongs on the left-hand side, subtracted from the return to form the excess return.

Missing values

csp is missing for 192 of the 469 months (Jan 2003 onward). Every other column is complete. Selecting predictors positionally, for instance GoyalWelch[, 2:14], silently excludes csp; select by name instead so the exclusion is deliberate and visible.

Source

Compiled from the public Goyal–Welch data library (predictable stock returns) and standard fixed-income sources (e.g., FRED). See data-raw/GoyalWelch.R for the contract harness that pins the structural properties documented above.

References

Goyal, A., & Welch, I. (2008). A Comprehensive Look at The Empirical Performance of Equity Premium Prediction. Review of Financial Studies, 21(4), 1455–1508.

Examples

data("GoyalWelch")
class(GoyalWelch)         # "xts" "zoo"
head(GoyalWelch)

# Rfree is exactly tbl / 12: never use both as predictors
max(abs(GoyalWelch[,"tbl"] / 12 - GoyalWelch[,"Rfree"]))

# The response: total return on the index, in excess of the risk-free rate
ix  <- GoyalWelch[,"Index"]
erp <- (ix + GoyalWelch[,"D12"] / 12) / xts::lag.xts(ix, 1) - 1 -
  GoyalWelch[,"Rfree"]
head(erp, 3)

# Construct common predictors:
dp  <- log(GoyalWelch[,"D12"]  / GoyalWelch[,"Index"])  # dividend–price
ep  <- log(GoyalWelch[,"E12"]  / GoyalWelch[,"Index"])  # earnings–price
dfy <- GoyalWelch[,"BAA"] - GoyalWelch[,"AAA"]          # default yield spread
dfr <- GoyalWelch[,"corpr"] - GoyalWelch[,"ltr"]        # default return spread
tms <- GoyalWelch[,"lty"] - GoyalWelch[,"tbl"]          # term spread

Hang Seng Index (Daily, xts)

Description

Daily adjusted close prices of the Hang Seng stock index (ticker symbol ^HSI), from its first date of availability on Yahoo Finance to 2015-12-31.

Usage

data("HSI")

Format

An xts object with 7214 daily observations and a single column ^HSI containing adjusted close prices in index points. The time index spans from 1986-12-31 to 2015-12-31.

Details

Originally distributed as HSI in the qrmdata package (Hofert, Hornik, & McNeil), ported into smqf so that the book's examples remain reproducible without an extra dependency. The data is redistributed here under the same GPL (\ge 2) license as qrmdata.

Source

Yahoo Finance, downloaded on 2016-01-03 via qrmtools::get_data().

References

Hofert, M., Hornik, K., & McNeil, A. J. qrmdata: Data Sets for Quantitative Risk Management Practice, https://CRAN.R-project.org/package=qrmdata.

Examples

data("HSI")
class(HSI)        # "xts" "zoo"
dim(HSI)
head(HSI)

NIKKEI 225 Index (Daily, xts)

Description

Daily adjusted close prices of the NIKKEI 225 stock index (ticker symbol ^N225), from its first date of availability on Yahoo Finance to 2015-12-30.

Usage

data("NIKKEI")

Format

An xts object with 7880 daily observations and a single column ^N225 containing adjusted close prices in index points. The time index spans from 1984-01-04 to 2015-12-30.

Details

Originally distributed as NIKKEI in the qrmdata package (Hofert, Hornik, & McNeil), ported into smqf so that the book's examples remain reproducible without an extra dependency. The data is redistributed here under the same GPL (\ge 2) license as qrmdata.

Source

Yahoo Finance, downloaded on 2016-01-03 via qrmtools::get_data().

References

Hofert, M., Hornik, K., & McNeil, A. J. qrmdata: Data Sets for Quantitative Risk Management Practice, https://CRAN.R-project.org/package=qrmdata.

Examples

data("NIKKEI")
class(NIKKEI)        # "xts" "zoo"
dim(NIKKEI)
head(NIKKEI)

Swiss Market Index (Daily, xts)

Description

Daily adjusted close prices of the Swiss Market Index (SMI) stock index (ticker symbol ^SSMI), from its first date of availability on Yahoo Finance to 2015-12-30.

Usage

data("SMI")

Format

An xts object with 6350 daily observations and a single column ^SSMI containing adjusted close prices in index points. The time index spans from 1990-11-09 to 2015-12-30.

Details

Originally distributed as SMI in the qrmdata package (Hofert, Hornik, & McNeil), ported into smqf so that the book's examples remain reproducible without an extra dependency. The data is redistributed here under the same GPL (\ge 2) license as qrmdata. Note: the MSGARCH package ships a different dataset also named SMI (SMI log-returns); to use that one, load it explicitly with data("SMI", package = "MSGARCH").

Source

Yahoo Finance, downloaded on 2016-01-03 via qrmtools::get_data().

References

Hofert, M., Hornik, K., & McNeil, A. J. qrmdata: Data Sets for Quantitative Risk Management Practice, https://CRAN.R-project.org/package=qrmdata.

Examples

data("SMI")
class(SMI)        # "xts" "zoo"
dim(SMI)
head(SMI)

S&P 500 Index (Daily, xts)

Description

Daily adjusted close prices of the Standard & Poor's 500 stock index (ticker symbol ^GSPC), from its first date of availability on Yahoo Finance to 2015-12-31.

Usage

data("SP500")

Format

An xts object with 16607 daily observations and a single column ^GSPC containing adjusted close prices in index points. The time index spans from 1950-01-03 to 2015-12-31.

Details

Originally distributed as SP500 in the qrmdata package (Hofert, Hornik, & McNeil), ported into smqf so that the book's examples remain reproducible without an extra dependency. The data is redistributed here under the same GPL (\ge 2) license as qrmdata.

Source

Yahoo Finance, downloaded on 2016-01-03 via qrmtools::get_data().

References

Hofert, M., Hornik, K., & McNeil, A. J. qrmdata: Data Sets for Quantitative Risk Management Practice, https://CRAN.R-project.org/package=qrmdata.

Examples

data("SP500")
class(SP500)        # "xts" "zoo"
dim(SP500)
head(SP500)

S&P 500 Constituents (Weekly, xts)

Description

Weekly adjusted close prices for the 505 constituents of the S&P 500 index (membership as of the 2016-01-03 snapshot).

Usage

data("SP500_const")

Format

An xts object with 2818 weekly observations (the last available trading day of each week) and 505 columns, one per constituent (e.g., AAPL, MSFT, XOM). Missing values are common, especially in the early part of the sample, since a given name only has data once it became available. The time index spans from 1962-01-05 to 2015-12-31.

Unlike the DJ_const, FTSE_const and EURSTX_const panels (which are daily), this dataset is stored at weekly frequency to keep the package within the CRAN size limit. Weekly data is sufficient for the high-dimensional covariance and factor-model examples in the book, where a three-year window already has more constituents than observations.

Details

Originally distributed as SP500_const in the qrmdata package (Hofert, Hornik, & McNeil), thinned to weekly frequency and ported into smqf so that the book's examples remain reproducible without an extra dependency. The data is redistributed here under the same GPL (\ge 2) license as qrmdata. The panel reflects index membership at the snapshot date and is therefore subject to survivorship bias.

Source

Yahoo Finance, downloaded on 2016-01-03 via qrmtools::get_data().

References

Hofert, M., Hornik, K., & McNeil, A. J. qrmdata: Data Sets for Quantitative Risk Management Practice, https://CRAN.R-project.org/package=qrmdata.

Examples

data("SP500_const")
class(SP500_const)        # "xts" "zoo"
dim(SP500_const)
head(colnames(SP500_const))

Daily Term Structure (1m–30y)

Description

A compact term-structure object holding daily yield curves across 11 standard maturities from mid-2006 to late-2008. The object is an xts matrix of annualized yields (rows = dates, columns = maturities); the maturity grid in years is attached as the "tau" entry of xts::xtsAttributes().

Usage

data("TermStructure")

Format

An xts object of annualized yields (percent, not decimals) with dimension 622 \times 11. The index runs daily from 2006-05-12 to 2008-10-31; columns are the 11 maturities: X1mo, X3mo, X6mo, X1yr, X2yr, X3yr, X5yr, X7yr, X10yr, X20yr, X30yr.

The maturity grid in years, c(1/12, 1/4, 1/2, 1, 2, 3, 5, 7, 10, 20, 30), is stored as a user attribute and retrieved with xts::xtsAttributes(TermStructure)$tau.

Details

Dates run from 2006-05-12 to 2008-10-31 (about 622 business days in this snapshot). Yields are curve levels for each maturity on each date and can be used directly for fitting/parsing term-structure models (e.g., Nelson–Siegel or Svensson), computing spreads (term, slope, butterfly), or building zero curves.

Examples

data("TermStructure")
class(TermStructure)         # "xts" "zoo"
dts <- index(TermStructure)  # observation dates
colnames(TermStructure)

# Example: 10Y vs 2Y term spread (in percentage points)
sprd_10y2y <- TermStructure[, "X10yr"] - TermStructure[, "X2yr"]
head(sprd_10y2y)

# Maturity grid (years), stored as a user attribute
xts::xtsAttributes(TermStructure)$tau

CBOE Volatility Index VIX (Daily, xts)

Description

Daily close values of the Chicago Board Options Exchange (CBOE) volatility index VIX (ticker symbol ^VIX), from its first date of availability on Yahoo Finance to 2015-12-31.

Usage

data("VIX")

Format

An xts object with 6553 daily observations and a single column ^VIX containing the VIX level in percent (annualized implied volatility). The time index spans from 1990-01-02 to 2015-12-31.

Details

Originally distributed as VIX in the qrmdata package (Hofert, Hornik, & McNeil), ported into smqf so that the book's examples remain reproducible without an extra dependency. The data is redistributed here under the same GPL (\ge 2) license as qrmdata. The VIX is typically used as a market-based measure of volatility, expressed in percent.

Source

Yahoo Finance, downloaded on 2016-01-03 via qrmtools::get_data().

References

Hofert, M., Hornik, K., & McNeil, A. J. qrmdata: Data Sets for Quantitative Risk Management Practice, https://CRAN.R-project.org/package=qrmdata.

Examples

data("VIX")
class(VIX)        # "xts" "zoo"
dim(VIX)
head(VIX)

Raw Data Files Shipped with the Book's Import Examples

Description

Chapter 1 of the book opens by reading a plain file from disk, so that readers practise importing data before they meet packaged datasets. Those files are installed with smqf under extdata and can be located with system.file.

Details

Copy them into a data/ folder of your own project to follow the chapter exactly:

dir.create("data", showWarnings = FALSE)
file.copy(dir(system.file("extdata", package = "smqf"), full.names = TRUE),
          "data", overwrite = FALSE)

Files

stocks.txt

Whitespace-delimited, header row, 574 weekly observations from 2008-01-01 to 2018-12-25. Columns Date, AAPL, AXP, BA, CAT, CSCO: split- and dividend-adjusted close prices in USD for five constituents of the Dow Jones Industrial Average. Read with read.table(f, header = TRUE).

stocks.csv

The same 574 x 6 table, comma-separated. Read with read.csv() or readr::read_csv().

stocks.xlsx

The same table as an Excel workbook, for the readxl::read_excel() example.

prices.txt

Whitespace-delimited, 6,109 daily observations from 1990-02-01 to 2013-07-02, no missing values. Columns Date, S&P 500, FTSE 100, CAC 40, DAX 30: index levels in local currency. Not used in Chapter 1; supplied for open-ended exercises. Note the non-syntactic column names — read with check.names = FALSE to preserve them.

Weekly timestamp convention

The dates in stocks.txt, stocks.csv and stocks.xlsx are all Tuesdays, spaced exactly seven days apart, because the series was downloaded at weekly periodicity starting on 2008-01-01 (a Tuesday). The timestamp labels the week, not the day the price was observed: the price recorded against a Tuesday is the closing price at the end of that calendar week.

This matters whenever the series is combined with weekly data that is stamped at the end of the week, as the Fama–French factors in FamaFrench are (Fridays). The return computed for Tuesday t spans the week ending on the Friday of that same calendar week, so the two series pair one-to-one in order:

returns   2017-01-03  2017-01-10  2017-01-17   (Tuesday stamps)
factors   2017-01-06  2017-01-13  2017-01-20   (Friday stamps)
          |___________|___________|__________|  same weeks, paired in order

Do not align them by carrying observations forward across a merged union index (for instance with two zoo::na.locf() passes): because the two stamps interleave, every Tuesday would inherit the preceding Friday's factors and the whole factor series would lag the returns by one week. Match by position, or restamp one series onto the other's convention, and assert that the two lengths agree before regressing.

See Also

FamaFrench for the weekly factors these prices are regressed on in Chapter 1.

Examples

f <- system.file("extdata", "stocks.txt", package = "smqf")
stocks <- read.table(f, header = TRUE)
dim(stocks)
head(stocks, 3)
unique(weekdays(as.Date(stocks$Date)))   # all Tuesdays

Clayton Copula PDF (2-Dimensional)

Description

Computes the value of the bivariate Clayton copula probability density function (PDF) at a specified point (u_1, u_2) for a given dependence parameter \theta.

Usage

f_clayton_copula_2d_pdf(u, theta, log = FALSE)

Arguments

Details

The Clayton copula is an Archimedean copula with lower-tail dependence \lambda_L = 2^{-1/\theta} and no upper-tail dependence (\lambda_U = 0). When \theta = 0, the copula reduces to the independence copula, whose PDF equals 1 for all (u_1, u_2) \in [0,1]^2.

The PDF is given by:

c(u_1, u_2; \theta) = (1 + \theta)\,(u_1 u_2)^{-(1 + \theta)}\, \left(u_1^{-\theta} + u_2^{-\theta} - 1\right)^{-2 - 1/\theta}.

Numerical computation may lose precision for large \theta (typically above about 34).

Value

A numeric value corresponding to the Clayton copula PDF evaluated at the specified point.

References

- Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman & Hall. - Patton, A. (2006). Modelling Asymmetric Exchange Rate Dependence. International Economic Review, 47(2), 527–556.

See Also

f_gumbel_copula_2d_pdf, f_normal_copula_pdf, f_student_copula_pdf

Examples

# Example: Evaluate Clayton copula PDF at (u1, u2) = (0.5, 0.5)
f_clayton_copula_2d_pdf(c(0.5, 0.5), theta = 2)

Display a Bivariate Copula Surface

Description

Evaluates and displays a bivariate copula function (typically a PDF or CDF) on a 2D grid, producing a static 3D surface via base R's persp().

Usage

f_display_copula(my_copula, grid_1, grid_2, plot = TRUE, zlab = "Pdf", ...)

Arguments

Details

This function provides a simple way to visualize the surface of a bivariate copula (e.g., Clayton, Gumbel, Gaussian). The copula function should accept a vector of two uniform values (u_1,u_2) and return its density or CDF value.

Value

Invisibly returns the matrix of evaluated copula values f_U, with rows corresponding to grid_1 and columns to grid_2.

Examples

# Example: Display the Clayton copula PDF surface
grid <- seq(0.05, 0.95, length.out = 25)
f_display_copula(
  my_copula = function(u) f_clayton_copula_2d_pdf(u, theta = 2),
  grid_1 = grid,
  grid_2 = grid
)

Compute a Long-Only Mean–Variance Efficient Frontier

Description

Solves a sequence of quadratic programs to trace the long-only, fully-invested Markowitz efficient frontier between the minimum-variance portfolio and the maximum-return portfolio.

Usage

f_efficient_frontier(mu, Sigma, n_ptf, targets = NULL)

Arguments

Details

Portfolios are obtained by minimizing variance for a grid of target returns under the constraints \sum_i w_i = 1 and w_i \ge 0.

Each QP solves

\min_{w}\; w^\top \Sigma\, w \quad \text{s.t. } \mathbf{1}^\top w = 1,\; w \ge 0

and, for every point beyond the minimum-variance portfolio, additionally

\mu^\top w = \mu^\star,

where \mu^\star spans a linear grid between the min-variance portfolio return and \max(\mu).

The maximum-return end point is obtained from the same constrained QP rather than by assigning weights by hand. This matters when several assets tie for max(mu): spreading weight equally across them is feasible but generally not variance-minimal, so the resulting portfolio can be strictly dominated and would not lie on the frontier.

Sigma is checked for symmetry and positive semidefiniteness. A matrix whose smallest eigenvalue falls clearly below zero is rejected rather than silently repaired; project it first, for example with as.matrix(Matrix::nearPD(Sigma)$mat).

If a QP fails to converge, the corresponding point is returned as NA with status == "failed" and a warning, instead of being filled with a neighbouring solution that does not meet its target return.

Requires pracma for quadprog.

Value

A list with components:

weights

Numeric matrix N \times n_ptf: portfolio weights. Rows are named after mu when it carries names; columns are labelled ptf1, ptf2, ... Columns of failed solves are NA.

volatility

Named numeric of length n_ptf: portfolio standard deviations, NA where the solve failed.

expected_returns

Named numeric of length n_ptf: portfolio expected returns, NA where the solve failed.

frontier

Data frame with columns point (integer index), expected_return, volatility, target and status for each frontier portfolio.

targets

Numeric vector of length n_ptf: the target-return grid used to trace the frontier.

status

Character vector of length n_ptf: "ok" for a solved point, "failed" otherwise.

References

Markowitz, H. (1952). Portfolio Selection. Journal of Finance, 7(1), 77–91.

See Also

f_ptf_max_U, f_portfolio_moments, quadprog

Examples

set.seed(1)
N <- 4
mu <- c(0.08, 0.10, 0.12, 0.09)
M  <- matrix(rnorm(N*N), N); Sigma <- crossprod(M) / N  # PSD covariance
ef <- f_efficient_frontier(mu, Sigma, n_ptf = 20)
# Inspect end points
ef$expected_returns[c(1, 20)]
ef$volatility[c(1, 20)]
all(ef$status == "ok")

# Ties for the maximum expected return are resolved by minimizing variance,
# not by equal weighting: assets 2 and 3 both reach 0.2, and the variance-
# minimal mix is c(0, 1/101, 100/101) with variance 100/101, well below the
# 25.25 of an equal split and below the 1 of asset 3 on its own.
ef_tie <- f_efficient_frontier(mu = c(0.1, 0.2, 0.2),
                               Sigma = diag(c(1, 100, 1)), n_ptf = 5)
ef_tie$weights[, 5]

# A common target grid makes two frontiers comparable point by point
mu2 <- mu + c(0.01, -0.01, 0.00, 0.02)
grid <- seq(0.095, 0.105, length.out = 5)
e1 <- f_efficient_frontier(mu,  Sigma, targets = grid)
e2 <- f_efficient_frontier(mu2, Sigma, targets = grid)
# Same targets, so the two weight vectors at point 3 are directly comparable
cbind(e1$weights[, 3], e2$weights[, 3])

Gumbel Copula CDF (2-Dimensional)

Description

Computes the bivariate Gumbel copula cumulative distribution function (CDF) at (u_1,u_2) for dependence parameter \theta \ge 1.

Usage

f_gumbel_copula_2d_cdf(u, theta)

Arguments

Details

The Gumbel copula exhibits upper-tail dependence \lambda_U = 2 - 2^{1/\theta} and no lower-tail dependence (\lambda_L = 0). When \theta = 1, it reduces to the independence copula with C(u_1,u_2)=u_1 u_2.

The CDF is

C(u_1,u_2;\theta) = \exp\!\left( -\left[(-\log u_1)^{\theta} + (-\log u_2)^{\theta}\right]^{1/\theta} \right), \qquad \theta \ge 1.

Value

A numeric scalar: C(u_1,u_2;\theta).

References

Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman & Hall. Nelsen, R. B. (2006). An Introduction to Copulas (2nd ed.). Springer.

See Also

f_gumbel_copula_2d_pdf, f_clayton_copula_2d_pdf, f_normal_copula_pdf, f_student_copula_pdf

Examples

f_gumbel_copula_2d_cdf(c(0.5, 0.8), theta = 2)
f_gumbel_copula_2d_cdf(c(0.7, 0.7), theta = 1)  # independence: ~0.49

Gumbel Copula PDF (2-Dimensional)

Description

Computes the bivariate Gumbel copula probability density function (PDF) at (u_1,u_2) for dependence parameter \theta \ge 1.

Usage

f_gumbel_copula_2d_pdf(u, theta)

Arguments

Details

The PDF can be written as

c(u_1,u_2;\theta) = C(u_1,u_2;\theta)\; \frac{ \{[-\log u_1]\;[-\log u_2]\}^{\theta-1} }{u_1 u_2}\; \left( (-\log u_1)^{\theta} + (-\log u_2)^{\theta} \right)^{1/\theta - 2}\; \left( \left\{ (-\log u_1)^{\theta} + (-\log u_2)^{\theta} \right\}^{1/\theta} + \theta - 1 \right).

When \theta = 1, c(u_1,u_2;1) \equiv 1 (independence).

Value

A numeric scalar: c(u_1,u_2;\theta).

Note

Numerical stability may degrade as u_i \to 0^+ (large -\log u_i).

References

Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman & Hall. Nelsen, R. B. (2006). An Introduction to Copulas (2nd ed.). Springer.

See Also

f_gumbel_copula_2d_cdf, f_clayton_copula_2d_pdf, f_normal_copula_pdf, f_student_copula_pdf

Examples

f_gumbel_copula_2d_pdf(c(0.5, 0.8), theta = 2)
f_gumbel_copula_2d_pdf(c(0.7, 0.7), theta = 1)  # independence: 1

Most Diversified Portfolio

Description

Compute the weights of the most diversified portfolio (MDP) of Choueifaty and Coignard (2008), which maximizes the diversification ratio

DR(w) = \frac{w' \sigma}{\sqrt{w' \Sigma w}}\,,

where \sigma = (\sqrt{\Sigma_{11}}, \ldots, \sqrt{\Sigma_{dd}})' is the vector of asset volatilities. Without further constraints the solution is available in closed form, w \propto \Sigma^{-1} \sigma, normalized to sum to one.

Usage

f_mdp(Sigma, tol = 1e-08)

Arguments

Details

The solution is unconstrained beyond the budget constraint: weights may be negative, and in large universes typically are, often substantially. A long-only most diversified portfolio is a different object and requires a constrained quadratic program; it is not what this function returns. Inspect sum(abs(w)) to see the gross exposure implied by the solution.

The diversification ratio is scale invariant in w, so the normalization \sum_i w_i = 1 is a choice of representative and does not change the portfolio.

Value

A named numeric vector of d portfolio weights summing to one. Names are taken from dimnames(Sigma) when available.

References

Choueifaty, Y., & Coignard, Y. (2008). Toward Maximum Diversification. Journal of Portfolio Management, 35(1), 40–51.

See Also

f_efficient_frontier, f_ptf_max_U

Examples

Sigma <- matrix(c(0.04, 0.01, 0.00,
                  0.01, 0.09, 0.02,
                  0.00, 0.02, 0.16), 3, 3,
                dimnames = list(c("A", "B", "C"), c("A", "B", "C")))
w <- f_mdp(Sigma)
w
sum(w)

# Diversification ratio of the MDP against the equally weighted portfolio
f_dr <- function(w, Sigma) {
  sum(w * sqrt(diag(Sigma))) / sqrt(drop(crossprod(w, Sigma %*% w)))
}
c(mdp = f_dr(w, Sigma), equal = f_dr(rep(1 / 3, 3), Sigma))

Higher-Moment Efficient Portfolios

Description

Tilts a reference portfolio towards more preferable moments — lower variance, higher skewness, lower kurtosis and optionally higher mean — while allowing the reference criterion to deteriorate by at most a margin kappa. This implements the algorithmic tilting of Boudt, Cornilly, Van Holle & Willems (2020).

Usage

f_mvsk_portfolio(
  m1 = NULL,
  M2 = NULL,
  M3 = NULL,
  M4 = NULL,
  w0 = NULL,
  g = NULL,
  lb = NULL,
  ub = NULL,
  lin_eq = NULL,
  lin_eqC = NULL,
  nlin_eq = NULL,
  lin_ieq = NULL,
  lin_ieqC = NULL,
  nlin_ieq = NULL,
  href = NULL,
  kappa = NULL,
  relative = FALSE,
  param = NULL,
  options = list(),
  mompref = NULL
)

Arguments

Details

The co-moment matrices must be in the matrix form returned by PerformanceAnalytics::M3.MM(x) and M4.MM(x) with the default as.mat = TRUE. If you hold the compact vector form, convert it first with PerformanceAnalytics::M3.vec2mat() and M4.vec2mat().

delta is the largest improvement attainable within the margin. It is non-decreasing in kappa — a wider margin never shrinks the feasible set — but it need not increase strictly, and at kappa = 0 it is zero only up to solver tolerance.

Value

A list with the tilted weights (one row per kappa), the achieved improvement delta, the resulting portfolio moms, the constraint values constr, and the solver status for each margin.

Provenance

This function is a port of mvskPortfolio() from the mvskPortfolios package by Dries Cornilly and Kris Boudt (https://github.com/c3jg/mvskPortfolios), redistributed here under the same GPL (\ge 2) licence so that the book's examples install from CRAN alone. The algorithm is unchanged; the internal calls to non-exported PerformanceAnalytics symbols have been replaced by the equivalent helpers of this package, and the solver status is now reported.

Author(s)

Dries Cornilly and Kris Boudt, ported by David Ardia.

References

Boudt, K., Cornilly, D., Van Holle, F., & Willems, J. (2020). Algorithmic portfolio tilting to harvest higher moment gains. Heliyon, 6(3).

Briec, W., Kerstens, K., & Jokung, O. (2007). Mean-variance-skewness portfolio performance gauging: a general shortage function and dual approach. Management Science, 53(1), 135–149.

Choueifaty, Y., & Coignard, Y. (2008). Toward maximum diversification. Journal of Portfolio Management, 35(1), 40–51.

See Also

f_ptf_max_U, f_portfolio_moments

Examples

if (requireNamespace("PerformanceAnalytics", quietly = TRUE)) {
  data(edhec, package = "PerformanceAnalytics")
  x  <- edhec[, 1:5]
  M2 <- stats::cov(x)
  M3 <- PerformanceAnalytics::M3.MM(x)
  M4 <- PerformanceAnalytics::M4.MM(x)

  res <- f_mvsk_portfolio(M2 = M2, M3 = M3, M4 = M4, w0 = "DR",
                          g = "vsk", href = "DR",
                          kappa = c(0, 0.005, 0.01))
  round(100 * res$weights, 2)
  res$delta
}

Multivariate Normal Copula PDF

Description

Computes the probability density function (PDF) of the Gaussian (normal) copula at a specified point u \in [0,1]^N, given mean vector \mu and covariance matrix \Sigma of the underlying multivariate normal distribution.

Usage

f_normal_copula_pdf(u, mu, Sigma, log = FALSE)

Arguments

Details

The Gaussian copula density is

c(u; \mu, \Sigma) = \frac{ \phi_N\!\left( \Phi^{-1}(u); \mu, \Sigma \right) }{ \prod_{i=1}^N \phi_1\!\left( \Phi^{-1}(u_i); \mu_i, \sigma_i^2 \right) },

where \phi_N and \phi_1 are multivariate and univariate normal densities respectively, and \Phi^{-1} denotes the inverse normal CDF applied componentwise. The resulting function is a valid copula density on the unit hypercube [0,1]^N.

Typically, the copula is defined for \mu = 0 and correlation matrix R, but the implementation here generalizes to arbitrary mean and covariance.

Value

A numeric scalar: the value of the Gaussian copula density c(u; \mu, \Sigma) at the point u, or its logarithm when log = TRUE. The value is returned as a plain numeric scalar (not a 1 \times 1 matrix).

References

Joe, H. (1997). *Multivariate Models and Dependence Concepts.* Chapman & Hall. Nelsen, R. B. (2006). *An Introduction to Copulas* (2nd ed.). Springer. McNeil, A. J., Frey, R., & Embrechts, P. (2015). *Quantitative Risk Management.* Princeton University Press.

See Also

f_student_copula_pdf, f_clayton_copula_2d_pdf, f_gumbel_copula_2d_pdf

Examples

# Example: 2D Gaussian copula
Sigma <- matrix(c(1, 0.7, 0.7, 1), 2, 2)
mu <- c(0, 0)
f_normal_copula_pdf(c(0.5, 0.8), mu, Sigma)

# Compare with independence (Sigma = I)
f_normal_copula_pdf(c(0.5, 0.8), mu, diag(2))

# The independence copula has density 1 everywhere, in any dimension
d <- 300
f_normal_copula_pdf(rep(0.99, d), rep(0, d), diag(d))

Portfolio Co-Moments (Variance, Co-Skewness, Co-Kurtosis)

Description

Compute the portfolio's centred 2nd, 3rd and 4th moments (variance, co-skewness, co-kurtosis) from the asset-level co-moment matrices and a weight vector.

Usage

f_portfolio_moments(w, M2 = NULL, M3 = NULL, M4 = NULL)

Arguments

Details

The three quantities are defined as

m_2(w) = w' M_2 w, \quad m_3(w) = w' M_3 (w \otimes w), \quad m_4(w) = w' M_4 (w \otimes w \otimes w),

where \otimes denotes the Kronecker product. These are the building blocks of the mean-variance-skewness-kurtosis (MVSK) framework of Jondeau, Poon and Rockinger (2007).

The function is a thin, exported wrapper around the package's internal helpers .portm2, .portm3, .portm4 (which themselves replace the unexported PerformanceAnalytics:::portm* family). It exists so the textbook can demonstrate co-moment computation without reaching into non-exported symbols of the package via :::.

Value

A named list with components m2, m3, m4 (NULL for any component whose corresponding co-moment matrix was not supplied).

References

Jondeau, E., Poon, S.-H., & Rockinger, M. (2007). Financial Modeling Under Non-Gaussian Distributions. Springer.

See Also

f_ptf_max_U, f_efficient_frontier

Examples

set.seed(1)
R  <- matrix(rnorm(200 * 4), 200, 4)
d  <- ncol(R)
M2 <- cov(R)
w  <- rep(1 / d, d)
# Variance of the equal-weighted portfolio:
f_portfolio_moments(w, M2 = M2)$m2

# Co-skewness / co-kurtosis matrices in d x d^2 / d x d^3 form (e.g. from
# PerformanceAnalytics::M3.MM() / M4.MM()) can be passed as M3, M4.

Maximize Truncated MVSK Utility with Box & Budget Constraints

Description

Solves for portfolio weights that (approximately) maximize expected utility using a fourth-order moment expansion in mean–variance–skewness–kurtosis (MVSK). The problem is solved via SLSQP with the constraints \sum_i w_i = 1 and 0 \le w_i \le w_max.

Usage

f_ptf_max_U(gamma, w_max, M1, M2, M3, M4)

Arguments

Details

The objective implemented is the negative of the truncated MVSK utility series, so minimizing it is equivalent to maximizing the utility:

\text{EU}(w) \approx \mu_p - \frac{\gamma}{2}\sigma_p^2 + \frac{\gamma(\gamma+1)}{6}\,S_p - \frac{\gamma(\gamma+1)(\gamma+2)}{24}\,K_p,

where \mu_p, \sigma_p^2, S_p, K_p are the portfolio's first four centralized moments and \gamma \ge 0 is a risk-aversion parameter.

The equality constraint is enforced via eval_g_eq and box constraints via lb/ub. Gradients are supplied analytically using internal helper functions '.derportm2', '.derportm3', and '.derportm4' defined in 'portfolio-moments.R'.

Value

A list with:

w

Numeric vector of optimal portfolio weights (length d).

EU

Scalar: value of the (approximate) expected utility at w (sign-corrected).

status

Integer: the nloptr termination status code. Positive values indicate success; a warning is issued (and the possibly invalid solution returned) when the status is negative, i.e., when the optimizer failed to converge.

Implementation note

Portfolio moment calculations (portm2/3/4 and their gradients) are performed by internal helpers in ‘R/portfolio-moments.R’, vendored from the standard kronecker-product formulas (Jondeau et al., 2007). M3 and M4 must be co-moment matrices (d \times d^2 and d \times d^3), not higher-dimensional arrays.

References

Jondeau, E., Poon, S.-H., & Rockinger, M. (2007). Financial Modeling Under Non-Gaussian Distributions. Harvey, C. R., Liechty, J. C., Liechty, M. W., & Müller, P. (2010). Portfolio selection with higher moments.

See Also

f_portfolio_moments, f_efficient_frontier, nloptr

Examples

set.seed(1)
d <- 3
M1 <- c(0.06, 0.08, 0.07)
A  <- matrix(rnorm(d*d), d); M2 <- crossprod(A)/d
M3 <- matrix(0, d, d^2)
M4 <- matrix(0, d, d^3)
res <- f_ptf_max_U(gamma = 5, w_max = 0.8, M1, M2, M3, M4)
res$w; res$EU

Multivariate Student-t Copula PDF

Description

Computes the probability density function (PDF) of the multivariate Student-t copula at a specified point u \in [0,1]^N, given location vector \mu, scatter matrix \Sigma, and degrees of freedom \nu.

Usage

f_student_copula_pdf(u, mu, Sigma, nu, log = FALSE)

Arguments

Details

The multivariate Student-t copula density is given by

c(u; \mu, \Sigma, \nu) = \frac{ t_N\!\left(t_{\nu}^{-1}(u); \mu, \Sigma, \nu \right) }{ \prod_{i=1}^N t_1\!\left(t_{\nu}^{-1}(u_i); \mu_i, \sigma_i^2, \nu \right) },

where t_N and t_1 denote the multivariate and univariate Student-t densities, respectively, and t_{\nu}^{-1} is the quantile function of the univariate Student-t with \nu degrees of freedom.

For \nu \to \infty, this copula converges to the Gaussian copula.

Value

A numeric scalar: the value of the Student-t copula density c(u; \mu, \Sigma, \nu) at u, or its logarithm when log = TRUE. The value is returned as a plain numeric scalar (not a 1 \times 1 matrix).

References

Demarta, S., & McNeil, A. J. (2005). The t Copula and Related Copulas. *International Statistical Review*, 73(1), 111–129. Joe, H. (1997). *Multivariate Models and Dependence Concepts*. Chapman & Hall. Nelsen, R. B. (2006). *An Introduction to Copulas* (2nd ed.). Springer.

See Also

f_normal_copula_pdf, f_clayton_copula_2d_pdf, f_gumbel_copula_2d_pdf

Examples

# Example: 2D t-copula density
mu <- c(0, 0)
Sigma <- matrix(c(1, 0.7, 0.7, 1), 2, 2)
f_student_copula_pdf(c(0.6, 0.8), mu, Sigma, nu = 5)

# Compare to Gaussian copula (nu large)
f_student_copula_pdf(c(0.6, 0.8), mu, Sigma, nu = 100)

# High dimension: use the log scale, the density itself overflows
d <- 300
f_student_copula_pdf(rep(0.99, d), rep(0, d), diag(d), nu = 5, log = TRUE)

Empirical Tail Dependence and Exceedance Correlation

Description

Estimates the empirical tail‐dependence coefficient and the exceedance correlation between two series at a given quantile threshold \alpha for either the lower or upper tail.

Usage

f_tail_dependence(x, y, alpha, side = c("lower", "upper"))

Arguments

Details

The function classifies observations in the \alpha‐quantile tail of each margin and computes:

• the standard empirical tail‐dependence coefficient: the proportion of joint tail events conditional on x being in its tail (lambda), and

• the sample correlation between the values of x and y for which both lie in their respective tails (excorr).

If the threshold selects no observation of x at all — which happens when x is constant or heavily tied — both components are NA and a warning is issued: an empty tail carries no information about tail dependence, and reporting zero there would describe a perfect match as independence.

If joint tail events are absent while the marginal tail is not empty, lambda = 0 and excorr = NA. If fewer than two joint-tail observations are available, excorr is also NA (the sample correlation is undefined). When the joint-tail subset is degenerate (e.g., x and y are identical so that the subset has zero variance), cor returns NA with a warning.

Value

A list with components:

lambda

Empirical tail‐dependence coefficient. For side = "lower":

\hat{\lambda}_L = \frac{\#\{X < q_\alpha^X \text{ and } Y < q_\alpha^Y\}} {\#\{X < q_\alpha^X\}}.

For side = "upper":

\hat{\lambda}_U = \frac{\#\{X > q_{1-\alpha}^X \text{ and } Y > q_{1-\alpha}^Y\}} {\#\{X > q_{1-\alpha}^X\}}.

excorr

Exceedance correlation between x and y in the joint tail (computed only if there are joint exceedances).

References

Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman & Hall. Embrechts, P., McNeil, A. J., & Straumann, D. (2002). Correlation and dependence in risk management. Risk Management: Value at Risk and Beyond. Cambridge University Press.

See Also

f_efficient_frontier

Examples

set.seed(1)
x <- rnorm(1000)
y <- 0.7 * x + sqrt(1 - 0.7^2) * rnorm(1000)

f_tail_dependence(x, y, alpha = 0.05)
f_tail_dependence(x, y, alpha = 0.05, side = "upper")