Thursday, June 5, 2008

PMPT Part II: Mathematical Tools--Sortino Ratio & Volatility Skewness

Today we continue with our investigation of Post Modern Portfolio Theory (PMPT) and look at a couple of its mathematical tools--the Sortino Ratio and the concept of volatility skewness.

The Sortino Ratio
The Sortino ratio measures the risk-adjusted return an of 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 risk equally. It is thus a measure of risk-adjusted returns that some people find to be more relevant than the Sharpe. The ratio is calculated as follows:

S = (r – t) / d

r is the annual rate of return for the investment,
t is the Required Rate of return,
d is the downside risk as computed above.

The Sortino Ratio is touted as providing a more realistic risk assessment tool than the Sharpe Ratio as it it based on the semi-variance notion of downside risk and the asserted enhancement of using a Log-normal model rather than a Normal model of investment performance data. The same criticisms PMPT makes against MPT are made against use of the Sharpe Ratio in favor of the Sortino Ratio. That is, rankings of investment funds within the same asset class made using the Sharpe Ratio can potentially be distorted because it ignores the preference of investors to avoid volatility on the downside while tolerating it of the upside. The Sortino Ratio is designed to specifically address this investor concern.

Volatility Skewness
Volatility skewness
is another concept promoted by PMPT enthusiasts. It is the ratio of the total variance of a log-normal distribution that occurs above its mean to that which occurs below its mean. A volatility skewness greater than 1.0 indicates positive skewness, i.e. a tailing off of the distribution toward positive returns to the right. A volatility skewness of less than 1.0 indicates a tailing off of the distribution toward negative returns to the left. There is already a formal statistical definition of skewness for a probability distribution known as the third standardized moment so it is questionable why such a new parameter is necessary. Perhaps volatility skewness is simply a more easily and intuitively understood manner in which to provide arguments in favor of PMPT for those situations in which the assumption of an unskewed Normal distribution and therefore MPT and the Sharpe Ratio can be credibly subjected to criticism.

PMPT short-comings
That is essentially the crux of PMPT. The volatility skewness and Sortino ratio are useful only as tools for fund evaluation and ranking, but there's nothing in PMPT that addresses the asset class portfolio optimization problem or computes cross-correlation among asset classes.

PMPT is contradictory to the MPT tenet that asset classes be represented by Index funds as suggested by the Efficient Market Hypothesis (EMH). The EMH was first proposed by French mathematician Louis Bachelier and developed by Dr. Eugene Fama. It contends that markets are informationally efficient in their pricing of assets and that any known information about them is already incorporated in their price. It therefore follows that it is impossible to consistently or reliably outperform market averages except through random chance. As such MPT recommends the holding of widely diversified market indices as individual asset classes represented in an overall portfolio. This concept is an anathema to PMPT.

Instead, asset allocation optimization among asset classes in PMPT is left to opaque and proprietary products from professional investment advisory houses who typically recommend expensive commissions and load-laden managed funds. That of course does nothing to rightly address the concerns of astute investors about what exactly it is that is being optimized by these products.

As such, there is little to combat the charge that PMPT is merely a contrivance invented for the singular purpose of rehabilitating the proposition that commission or fee-based professional active asset management is necessary and essential and that individual investors cannot competently nor advantageously see to their own investment management affairs as MPT would permit them to do.**

The next installment will discuss the PMPT assertion that investment returns are Log-normally distributed and present some actual distribution data to let us see for ourselves.

*See Part V of the MPT series (May 7th blog)

**Note from Dr. Kris: I do not share Prof. Pat's view that markets are efficient because there is overwhelming evidence that they aren't. However, that is not to discount MPT as a valuable and useful tool for investment management.

Posted by Dr. Pat

No comments: