as.sr: Compute the Sharpe ratio.

Description

Computes the Sharpe ratio of some observed returns.

Usage

as.sr(x, c0 = 0, ope = 1, na.rm = FALSE, epoch = "yr", higher_order = FALSE)
# S3 method for default
as.sr(x, c0 = 0, ope = 1, na.rm = FALSE, epoch = "yr", higher_order = FALSE)
# S3 method for matrix
as.sr(x, c0 = 0, ope = 1, na.rm = FALSE, epoch = "yr", higher_order = FALSE)
# S3 method for data.frame
as.sr(x, c0 = 0, ope = 1, na.rm = FALSE, epoch = "yr", higher_order = FALSE)
# S3 method for lm
as.sr(x, c0 = 0, ope = 1, na.rm = FALSE, epoch = "yr", higher_order = FALSE)
# S3 method for xts
as.sr(x, c0 = 0, ope = 1, na.rm = FALSE, epoch = "yr", higher_order = FALSE)
# S3 method for timeSeries
as.sr(x, c0 = 0, ope = 1, na.rm = FALSE, epoch = "yr", higher_order = FALSE)

Arguments

vector of returns, or object of class data.frame, xts, or lm.

the 'risk-free' or 'disastrous' rate of return. this is assumed to be given in the same units as x, not in 'annualized' terms.

ope

the number of observations per 'epoch'. For convenience of interpretation, The Sharpe ratio is typically quoted in 'annualized' units for some epoch, that is, 'per square root epoch', though returns are observed at a frequency of ope per epoch. The default value is 1, meaning the code will not attempt to guess what the observation frequency is, and no annualization adjustments will be made.

na.rm

logical. Should missing values be removed?

epoch

the string representation of the 'epoch', defaulting to 'yr'.

higher_order

a Boolean. If true, we compute cumulants of the returns to leverage higher order accuracy formulae when possible.

Value

a list containing the following components:

sr: the annualized Sharpe ratio.
df: the t-stat degrees of freedom.
c0: the risk free term.
ope: the annualization factor.
rescal: the rescaling factor.
epoch: the string epoch.

cast to class sr.

Details

Suppose $x_i$ are $n$ independent returns of some asset. Let $\bar{x}$ be the sample mean, and $s$ be the sample standard deviation (using Bessel's correction). Let $c_0$ be the 'risk free rate'. Then $$z = \frac{\bar{x} - c_0}{s}$$ is the (sample) Sharpe ratio.

The units of $z$ are $\mbox{time}^{-1/2}$. Typically the Sharpe ratio is annualized by multiplying by $\sqrt{\mbox{ope}}$, where $\mbox{ope}$ is the number of observations per year (or whatever the target annualization epoch.)

Note that if ope is not given, the converter from xts attempts to infer the observations per year, without regard to the name of the epoch given.

References

Sharpe, William F. "Mutual fund performance." Journal of business (1966): 119-138. https://ideas.repec.org/a/ucp/jnlbus/v39y1965p119.html

Lo, Andrew W. "The statistics of Sharpe ratios." Financial Analysts Journal 58, no. 4 (2002): 36-52. https://www.ssrn.com/paper=377260

Examples

Run this code

# NOT RUN {
# Sharpe's 'model': just given a bunch of returns.
asr <- as.sr(rnorm(253*3),ope=253)
# or a matrix, with a name
my.returns <- matrix(rnorm(253*3),ncol=1)
colnames(my.returns) <- c("my strategy")
asr <- as.sr(my.returns)

# given an xts object:
if (require(xts)) {
 data(stock_returns)
 IBM <- stock_returns[,'IBM']
 asr <- as.sr(IBM,na.rm=TRUE)
}

# on a linear model, find the 'Sharpe' of the residual term
nfac <- 5
nyr <- 10
ope <- 253
set.seed(as.integer(charToRaw("determinstic")))
Factors <- matrix(rnorm(ope*nyr*nfac,mean=0,sd=0.0125),ncol=nfac)
Betas <- exp(0.1 * rnorm(dim(Factors)[2]))
Returns <- (Factors %*% Betas) + rnorm(dim(Factors)[1],mean=0.0005,sd=0.012)
APT_mod <- lm(Returns ~ Factors)
asr <- as.sr(APT_mod,ope=ope)
# try again, but make the Returns independent of the Factors.
Returns <- rnorm(dim(Factors)[1],mean=0.0005,sd=0.012)
APT_mod <- lm(Returns ~ Factors)
asr <- as.sr(APT_mod,ope=ope)

# compute the Sharpe of a bunch of strategies:
my.returns <- matrix(rnorm(253*3*4),ncol=4)
asr <- as.sr(my.returns)  # without sensible colnames?
colnames(my.returns) <- c("strat a","strat b","strat c","strat d")
asr <- as.sr(my.returns)
  
# }

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