The Sortino ratio was created in 1983 by Brian M. Rom at the software development
company Investment Technologies. The ratio is named for Frank A. Sortino, an
early popularizer of downside risk optimization. It measures the
risk-adjusted return of an investment asset, portfolio, or strategy. It is a
modification of the Sharpe ratio but penalizes only those returns falling
below a user-specified target or required rate of return, while the
Sharpe ratio penalizes both upside and downside volatility equally. Though both
ratios measure an investment's risk-adjusted return, they do so in
significantly different ways that will frequently lead to differing conclusions
as to the true nature of the investment's return-generating
efficiency. The Sortino ratio is used as a way to
compare the risk-adjusted performance of programs with differing risk and return
profiles. In general, risk-adjusted returns seek to normalize the risk
across programs and then see which has the higher return unit per risk.
Definition The ratio is calculated as
where is the asset or portfolio average realized return, is the target or
required rate of return for the investment strategy under consideration,
and is the target semi-deviation, termed downside deviation. is expressed
in percentages and therefore allows for rankings in the same way as standard
deviation. An intuitive way to view downside risk
is the annualized standard deviation of returns below the target. Another is the
square root of the probability-weighted squared below-target returns. The
squaring of the below-target returns has the effect of penalizing failures at a
quadratic rate. This is consistent with observations made on the behavior of
individual decision making under uncertainty.
Here = downside deviation or "downside risk",
= the annual target return, originally termed the minimum acceptable return
MAR, = the random variable representing the
return for the distribution of annual returns , and
= the distribution for the annual returns, e.g., the three-parameter
lognormal distribution. For the reasons provided below, this
continuous formula is preferred over a simpler discrete version that determines
the standard deviation of below-target periodic returns taken from the return
series. The continuous form permits all
subsequent calculations to be made using annual returns, the natural way for
investors to specify their investment goals. The discrete form requires
monthly returns for there to be sufficient data points to make a
meaningful calculation, which in turn requires converting the annual target
into a monthly target. This significantly affects the amount of risk
that is identified. For example, a goal of earning 1% in every month of one year
results in a greater risk than the seemingly equivalent goal of earning 12%
in one year. A second reason for strongly preferring
the continuous form to the discrete form has been proposed by Sortino & Forsey:
"Before we make an investment, we don't know what the outcome will be... After
the investment is made, and we want to measure its performance, all we know is
what the outcome was, not what it could have been. To cope with this
uncertainty, we assume that a reasonable estimate of the range of possible
returns, as well as the probabilities associated with estimation of those
returns...In statistical terms, the shape of [this] uncertainty is called a
probability distribution. In other words, looking at just the discrete
monthly or annual values does not tell the whole story."
Using the observed points to create a distribution is a staple of conventional
performance measurement. For example, monthly returns are used to calculate a
fund's mean and standard deviation. Using these values and the properties of
the normal distribution, we can make statements such as the likelihood of
losing money or the range within which two-thirds of all returns lies. Our
ability to make these statements comes from the process of assuming the
continuous form of the normal distribution and certain of its
well-known properties. In post-modern portfolio theory an
analogous process is followed. Observe the monthly returns.
Fit a distribution that permits asymmetry to the observations.
Annualize the monthly returns, making sure the shape characteristics of the
distribution are retained. Apply integral calculus to the resultant
distribution to calculate the appropriate statistics.
As a caveat, some practitioners have fallen into the habit of using discrete
periodic returns to compute downside risk. This method is conceptually and
operationally incorrect and negates the foundational statistic of post-modern
portfolio theory which was built a quarter-century ago by Brian M. Rom and
Frank A. Sortino. Usage
The Sortino ratio is used to score a portfolio's risk-adjusted returns
relative to an investment target using downside risk. This is analogous to the
Sharpe ratio, which scores risk-adjusted returns relative to the risk-free rate
using standard deviation. When return distributions are near symmetrical and
the target return is close to the distribution median, these two measure
will produce similar results. As skewness increases and targets vary from
the median, results can be expected to show dramatic differences.
See also Modern portfolio theory
Modigliani Risk-Adjusted Performance Omega ratio
Post-modern portfolio theory Sharpe ratio
Upside potential ratio V2 ratio
References