| maxdd {fBasics} | R Documentation |
This is a collection and description of
functions which compute drawdown statistics.
Included are density, distribution function,
and random generation for the maximum drawdown
distribution. In addition the expectation of
drawdowns for Brownian motion can be computed.
The functions are:
dmaxdd | the Density function, |
pmaxdd | the Distribution function, |
rmaxdd | the random number generator, |
maxddStats | the expectation of drawdowns. |
dmaxdd(x, sd = 1, horizon = 100, N = 1000) pmaxdd(q, sd = 1, horizon = 100, N = 1000) rmaxdd(n, mean = 0, sd = 1, horizon = 100) maxddStats(mean = 0, sd = 1, horizon = 1000)
x, q |
a numeric vector of quantiles. |
n |
an integer value, the number of observations. |
mean, sd |
two numeric values, the mean and standard deviation. |
horizon |
an integer value, the (run time) horizon of the investor. |
N |
an integer value, the precession index for summations. Before you change this value please inspect Magdon-Ismail et. al. (2003). |
dmaxdd
returns for a trendless Brownian process mean=0 and
standard deviation "sd" the density from the probability
that the maximum drawdown "D" is larger or equal to "h"
in the interval [0,T], where "T" denotes the time horizon
of the investor.
pmaxdd
returns for a trendless Brownian process mean=0 and
standard deviation "sd" the the probability
that the maximum drawdown "D" is larger or equal to "h"
in the interval [0,T], where "T" denotes the time horizon
of the investor.
rmaxdd
returns for a Brownian Motion process with mean mean and
standard deviation sd random variates of maximum drawdowns.
maxddStats
returns the expectation Value E[D] of maximum drawdowns of
Brownian Motion for a given drift mean, variance sd,
and runtime horizon of the Brownian Motion process.
Currrently, for the functions dmaxdd and pmaxdd only the
trend or driftless case is implemented.
Magdon-Ismail M., Atiya A.F., Pratap A., Abu-Mostafa Y.S. (2003); On the Maximum Drawdown of a Brownian Motion, Preprint, CalTech, Pasadena USA, p. 24.
## rmaxdd -
# Set a random seed
set.seed(1953)
# horizon of the investor, time T
horizon = 1000
# number of MC samples, N -> infinity
samples = 1000
# Range of expected Drawdons
xlim = c(0, 5)*sqrt(horizon)
# Plot Histogram of Simulated Max Drawdowns:
r = rmaxdd(n = samples, mean = 0, sd = 1, horizon = horizon)
hist(x = r, n = 40, probability = TRUE, xlim = xlim,
col = "steelblue4", border = "white", main = "Max. Drawdown Density")
points(r, rep(0, samples), pch = 20, col = "orange", cex = 0.7)
## dmaxdd -
x = seq(0, xlim[2], length = 200)
d = dmaxdd(x = x, sd = 1, horizon = horizon, N = 1000)
lines(x, d, lwd = 2)
## pmaxdd -
# Count Frequencies of Drawdowns Greater or Equal to "h":
n = 50
x = seq(0, xlim[2], length = n)
g = rep(0, times = n)
for (i in 1:n) g[i] = length (r[r > x[i]]) / samples
plot(x, g, type ="h", lwd = 3,
xlab = "q", main = "Max. Drawdown Probability")
# Compare with True Probability "G_D(h)":
x = seq(0, xlim[2], length = 5*n)
p = pmaxdd(q = x, sd = 1, horizon = horizon, N = 5000)
lines(x, p, lwd = 2, col="steelblue4")
## maxddStats -
# Compute expectation Value E[D]:
maxddStats(mean = -0.5, sd = 1, horizon = 10^(1:4))
maxddStats(mean = 0.0, sd = 1, horizon = 10^(1:4))
maxddStats(mean = 0.5, sd = 1, horizon = 10^(1:4))