Applications
In 1987, the Pension Research Institute at San Francisco State University developed the practical mathematical algorithms of PMPT that are in use today. These methods provide a framework that recognizes investors' preferences for upside over downside volatility. At the same time, a more robust model for the pattern of investment returns, the three-parameter lognormal distribution, [3] was introduced.
Downside risk
Downside risk (DR) is measured by target semi-deviation (the square root of target semivariance) and is termed downside deviation. It 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 quadratically. This is consistent with observations made on the behavior of individual decision-making under
where
d = downside deviation (commonly known in the financial community as 'downside risk'). Note: By extension, d² = downside variance.
t = the annual target return, originally termed the minimum acceptable return, or MAR.
r = the random variable representing the return for the distribution of annual returns f(r),
f(r) = 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.
1. The continuous form permits all subsequent calculations to be made using annual returns which is 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.
2. A second reason for strongly preferring the continuous form to the discrete form has been proposed by Sortino & Forsey (1996):
"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 (even though no negative returns may actually have been observed), or the range within which two-thirds of all returns lies (even though the specific returns identifying this range have not necessarily occurred). 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 PMPT an analogous process is followed:
- Observe the monthly returns,
- Fit a distribution that permits asymmetry to the observations,
- Bootrap these returns to a large number of pseudo-annualized returns
- Use this large number of returns to empirically generate the various PMPT statistics.
Sortino Ratio
The Sortino ratio, developed in 1993 by Rom's company, Investment Technologies, LLC, was the first new element in the PMPT rubric. It is defined as:
where
r = the annualized rate of return,
t = the target return,
d = downside risk.
The following table shows that this ratio is demonstrably superior to the traditional Sharpe ratio as a means for ranking investment results. The table shows risk-adjusted ratios for several major indexes using both Sortino and Sharpe ratios. The data cover the five years 1992-1996 and are based on monthly total returns. The Sortino ratio is calculated against a 9.0% target.
| Index | Sortino ratio | Sharpe ratio |
|---|---|---|
| 90-day T-bill | -1.00 | 0.00 |
| Lehman Aggregate | -0.29 | 0.63 |
| MSCI EAFE | -0.05 | 0.30 |
| Russell 2000 | 0.55 | 0.93 |
| S&P 500 | 0.84 | 1.25 |
As an example of the different conclusions that can be drawn using these two ratios, notice how the Lehman Aggregate and MSCI EAFE compare - the Lehman ranks higher using the Sharpe ratio whereas EAFE ranks higher using the Sortino ratio. In many cases, manager or index rankings will be different, depending on the risk-adjusted measure used. These patterns will change again for different values of t. For example, when t is close to the risk-free rate, the Sortino Ratio for T-Bill's will be higher than that for the S&P 500, while the Sharpe ratio remains unchanged.
In March 2008, researchers at the Queensland Investment Corporation and Queensland University of Technology showed that for skewed return distributions, the Sortino ratio is superior to the Sharpe ratio as a measure of portfolio risk. [4]
Volatility skewness
Volatility skewness is the second portfolio-analysis statistic introduced by Rom and Ferguson under the PMPT rubric. It measures the ratio of a distribution's percentage of total variance from returns above the mean, to the percentage of the distribution's total variance from returns below the mean. Thus, if a distribution is symmetrical ( as in the normal case, as is assumed under MPT), it has a volatility skewness of 1.00. Values greater than 1.00 indicate positive skewness; values less than 1.00 indicate negative skewness. While closely correlated with the traditional statistical measure of skewness (viz., the third moment of a distribution), the authors of PMPT argue that their volatility skewness measure has the advantage of being intuitively more understandable to non-statisticians who are the primary practical users of these tools.
The importance of skewness lies in the fact that the more non-normal (i.e., skewed) a return series is, the more its true risk will be distorted by traditional MPT measures such as the Sharpe ratio. Thus, with the recent advent of hedging and derivative strategies, which are asymmetrical by design, MPT measures are essentially useless, while PMPT is able to capture significantly more of the true information contained in the returns under consideration. Many of the common market indices and the returns of stock and bond mutual funds cannot themselves always be assumed to be accurately represented by the normal distribution.
| Index | Upside volatility (%) | Downside volatility (%) | Volatility skewness |
|---|---|---|---|
| Lehman Aggregate | 32.35 | 67.65 | 0.48 |
| Russell 2000 | 37.19 | 62.81 | 0.59 |
| S&P 500 | 38.63 | 61.37 | 0.63 |
| 90-day T-Bill | 48.26 | 51.74 | 0.93 |
| MSCI EAFE | 54.67 | 45.33 | 1.21 |
Data: Monthly returns, January, 1991 through December, 1996.
