# Drawdown statistics

`stats-maxdd.Rd`

A collection 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.

## 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).

## Details

`dmaxdd`

computes the density function of the maximum drawdown
distribution. `pmaxdd`

computes the distribution function.
`rmaxdd`

generates random numbers from that distribution.
`maxddStats`

computes the expectation of drawdowns.

`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 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 expected 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))
#> [1] 6.841696 52.000000 502.000000 5002.000000
maxddStats(mean = 0.0, sd = 1, horizon = 10^(1:4))
#> [1] 3.963327 12.533141 39.633273 125.331414
maxddStats(mean = 0.5, sd = 1, horizon = 10^(1:4))
#> [1] 2.529253 4.566413 6.809237 9.101853
```