1- # ' Market timing models
2- # '
3- # ' Allows to estimate Treynor-Mazuy or Merton-Henriksson market timing model.
4- # ' The Treynor-Mazuy model is essentially a quadratic extension of the basic
5- # ' CAPM. It is estimated using a multiple regression. The second term in the
6- # ' regression is the value of excess return squared. If the gamma coefficient
7- # ' in the regression is positive, then the estimated equation describes a
8- # ' convex upward-sloping regression "line". The quadratic regression is:
9- # ' \deqn{R_{p}-R_{f}=\alpha+\beta (R_{b} - R_{f})+\gamma (R_{b}-R_{f})^2+
10- # ' \varepsilon_{p}}{Rp - Rf = alpha + beta(Rb -Rf) + gamma(Rb - Rf)^2 +
11- # ' epsilonp}
12- # ' \eqn{\gamma}{gamma} is a measure of the curvature of the regression line.
13- # ' If \eqn{\gamma}{gamma} is positive, this would indicate that the manager's
14- # ' investment strategy demonstrates market timing ability.
15- # '
16- # ' The basic idea of the Merton-Henriksson test is to perform a multiple
17- # ' regression in which the dependent variable (portfolio excess return and a
18- # ' second variable that mimics the payoff to an option). This second variable
19- # ' is zero when the market excess return is at or below zero and is 1 when it
20- # ' is above zero:
21- # ' \deqn{R_{p}-R_{f}=\alpha+\beta (R_{b}-R_{f})+\gamma D+\varepsilon_{p}}{Rp -
22- # ' Rf = alpha + beta * (Rb - Rf) + gamma * D + epsilonp}
23- # ' where all variables are familiar from the CAPM model, except for the
24- # ' up-market return \eqn{D=max(0,R_{b}-R_{f})}{D = max(0, Rb - Rf)} and market
25- # ' timing abilities \eqn{\gamma}{gamma}
26- # ' @param Ra an xts, vector, matrix, data frame, timeSeries or zoo object of
27- # ' the asset returns
28- # ' @param Rb an xts, vector, matrix, data frame, timeSeries or zoo object of
29- # ' the benchmark asset return
30- # ' @param Rf risk free rate, in same period as your returns
31- # ' @param method used to select between Treynor-Mazuy and Henriksson-Merton
32- # ' models. May be any of: \itemize{ \item TM - Treynor-Mazuy model,
33- # ' \item HM - Henriksson-Merton model} By default Treynor-Mazuy is selected
34- # ' @param \dots any other passthrough parameters
35- # ' @author Andrii Babii, Peter Carl
36- # ' @seealso \code{\link{CAPM.beta}}
37- # ' @references J. Christopherson, D. Carino, W. Ferson. \emph{Portfolio
38- # ' Performance Measurement and Benchmarking}. 2009. McGraw-Hill, p. 127-133.
39- # ' \cr J. L. Treynor and K. Mazuy, "Can Mutual Funds Outguess the Market?"
40- # ' \emph{Harvard Business Review}, vol44, 1966, pp. 131-136
41- # ' \cr Roy D. Henriksson and Robert C. Merton, "On Market Timing and Investment
42- # ' Performance. II. Statistical Procedures for Evaluating Forecast Skills,"
43- # ' \emph{Journal of Business}, vol.54, October 1981, pp.513-533 \cr
44- # ' @examples
45- # '
46- # ' data(managers)
47- # ' MarketTiming(managers[,1,drop=FALSE ], managers[,8,drop=FALSE ], Rf=.035/12, method = "HM")
48- # ' MarketTiming(managers[80:120,1:6], managers[80:120,7,drop=FALSE ], managers[80:120,10,drop=FALSE ])
49- # ' MarketTiming(managers[80:120,1:6], managers[80:120,8:7], managers[80:120,10,drop=FALSE ], method = "TM")
50- # '
51- # ' @export
52- MarketTiming <- function (Ra , Rb , Rf = 0 , method = c(" TM" " HM"
53- { # @author Andrii Babii, Peter Carl
54-
55- # FUNCTION
56-
57- Ra = checkData(Ra )
58- Rb = checkData(Rb )
59- if (! is.null(dim(Rf )))
60- Rf = checkData(Rf )
61- Ra.ncols = NCOL(Ra )
62- Rb.ncols = NCOL(Rb )
63- pairs = expand.grid(1 : Ra.ncols , 1 )
64- method = method [1 ]
65- xRa = Return.excess(Ra , Rf )
66- xRb = Return.excess(Rb , Rf )
67-
68- mt <- function (xRa , xRb )
69- {
70- switch (method ,
71- " HM" = { S = xRb > 0 },
72- " TM" = { S = xRb }
73- )
74- R = merge(xRa , xRb , xRb * S )
75- R.df = as.data.frame(R )
76- model = lm(R.df [, 1 ] ~ 1 + . , data = R.df [, - 1 ])
77- return (coef(model ))
78- }
79-
80- result = apply(pairs , 1 , FUN = function (n , xRa , xRb )
81- mt(xRa [, n [1 ]], xRb [, 1 ]), xRa = xRa , xRb = xRb )
82- result = t(result )
83-
84- if (ncol(Rb ) > 1 ){
85- for (i in 2 : ncol(xRb )){
86- res = apply(pairs , 1 , FUN = function (n , xRa , xRb )
87- mt(xRa [, n [1 ]], xRb [, i ]), xRa = xRa , xRb = xRb )
88- res = t(res )
89- result = rbind(result , res )
90- }
91- }
92-
93- rownames(result ) = paste(rep(colnames(Ra ), ncol(Rb )), " to" Rb ), each = ncol(Ra )))
94- colnames(result ) = c(" Alpha" " Beta" " Gamma"
95- return (result )
1+ # ' Market timing models
2+ # '
3+ # ' Allows to estimate Treynor-Mazuy or Merton-Henriksson market timing model.
4+ # ' The Treynor-Mazuy model is essentially a quadratic extension of the basic
5+ # ' CAPM. It is estimated using a multiple regression. The second term in the
6+ # ' regression is the value of excess return squared. If the gamma coefficient
7+ # ' in the regression is positive, then the estimated equation describes a
8+ # ' convex upward-sloping regression "line". The quadratic regression is:
9+ # ' \deqn{R_{p}-R_{f}=\alpha+\beta (R_{b} - R_{f})+\gamma (R_{b}-R_{f})^2+
10+ # ' \varepsilon_{p}}{Rp - Rf = alpha + beta(Rb -Rf) + gamma(Rb - Rf)^2 +
11+ # ' epsilonp}
12+ # ' \eqn{\gamma}{gamma} is a measure of the curvature of the regression line.
13+ # ' If \eqn{\gamma}{gamma} is positive, this would indicate that the manager's
14+ # ' investment strategy demonstrates market timing ability.
15+ # '
16+ # ' The basic idea of the Merton-Henriksson test is to perform a multiple
17+ # ' regression in which the dependent variable (portfolio excess return and a
18+ # ' second variable that mimics the payoff to an option). This second variable
19+ # ' is zero when the market excess return is at or below zero and is 1 when it
20+ # ' is above zero:
21+ # ' \deqn{R_{p}-R_{f}=\alpha+\beta (R_{b}-R_{f})+\gamma D+\varepsilon_{p}}{Rp -
22+ # ' Rf = alpha + beta * (Rb - Rf) + gamma * D + epsilonp}
23+ # ' where all variables are familiar from the CAPM model, except for the
24+ # ' up-market return \eqn{D=max(0,R_{b}-R_{f})}{D = max(0, Rb - Rf)} and market
25+ # ' timing abilities \eqn{\gamma}{gamma}
26+ # ' @param Ra an xts, vector, matrix, data frame, timeSeries or zoo object of
27+ # ' the asset returns
28+ # ' @param Rb an xts, vector, matrix, data frame, timeSeries or zoo object of
29+ # ' the benchmark asset return
30+ # ' @param Rf risk free rate, in same period as your returns
31+ # ' @param method used to select between Treynor-Mazuy and Henriksson-Merton
32+ # ' models. May be any of: \itemize{ \item TM - Treynor-Mazuy model,
33+ # ' \item HM - Henriksson-Merton model} By default Treynor-Mazuy is selected
34+ # ' @param \dots any other passthrough parameters
35+ # ' @author Andrii Babii, Peter Carl
36+ # ' @seealso \code{\link{CAPM.beta}}
37+ # ' @references J. Christopherson, D. Carino, W. Ferson. \emph{Portfolio
38+ # ' Performance Measurement and Benchmarking}. 2009. McGraw-Hill, p. 127-133.
39+ # ' \cr J. L. Treynor and K. Mazuy, "Can Mutual Funds Outguess the Market?"
40+ # ' \emph{Harvard Business Review}, vol44, 1966, pp. 131-136
41+ # ' \cr Roy D. Henriksson and Robert C. Merton, "On Market Timing and Investment
42+ # ' Performance. II. Statistical Procedures for Evaluating Forecast Skills,"
43+ # ' \emph{Journal of Business}, vol.54, October 1981, pp.513-533 \cr
44+ # ' @examples
45+ # '
46+ # ' data(managers)
47+ # ' MarketTiming(managers[,1], managers[,8], Rf=.035/12, method = "HM")
48+ # ' MarketTiming(managers[80:120,1:6], managers[80:120,7], managers[80:120,10])
49+ # ' MarketTiming(managers[80:120,1:6], managers[80:120,8:7], managers[80:120,10], method = "TM")
50+ # '
51+ # ' @export
52+ MarketTiming <- function (Ra , Rb , Rf = 0 , method = c(" TM" " HM"
53+ { # @author Andrii Babii, Peter Carl
54+
55+ # FUNCTION
56+
57+ Ra = checkData(Ra )
58+ Rb = checkData(Rb )
59+ if (! is.null(dim(Rf )))
60+ Rf = checkData(Rf )
61+ Ra.ncols = NCOL(Ra )
62+ Rb.ncols = NCOL(Rb )
63+ pairs = expand.grid(1 : Ra.ncols , 1 )
64+ method = method [1 ]
65+ xRa = Return.excess(Ra , Rf )
66+ xRb = Return.excess(Rb , Rf )
67+
68+ mt <- function (xRa , xRb )
69+ {
70+ switch (method ,
71+ " HM" = { S = xRb > 0 },
72+ " TM" = { S = xRb }
73+ )
74+ R = merge(xRa , xRb , xRb * S )
75+ R.df = as.data.frame(R )
76+ model = lm(R.df [, 1 ] ~ 1 + . , data = R.df [, - 1 ])
77+ return (coef(model ))
78+ }
79+
80+ result = apply(pairs , 1 , FUN = function (n , xRa , xRb )
81+ mt(xRa [, n [1 ]], xRb [, 1 ]), xRa = xRa , xRb = xRb )
82+ result = t(result )
83+
84+ if (ncol(Rb ) > 1 ){
85+ for (i in 2 : ncol(xRb )){
86+ res = apply(pairs , 1 , FUN = function (n , xRa , xRb )
87+ mt(xRa [, n [1 ]], xRb [, i ]), xRa = xRa , xRb = xRb )
88+ res = t(res )
89+ result = rbind(result , res )
90+ }
91+ }
92+
93+ rownames(result ) = paste(rep(colnames(Ra ), ncol(Rb )), " to" Rb ), each = ncol(Ra )))
94+ colnames(result ) = c(" Alpha" " Beta" " Gamma"
95+ return (result )
9696}
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