fBasics (version 3011.87)

maxdd: Drawdown Statistics

Description

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: ll{ dmaxdd the Density function, pmaxdd the Distribution function, rmaxdd the random number generator, maxddStats the expectation of drawdowns. }

Usage

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)

Arguments

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).

Value

• 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.

References

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.

Examples

## 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))