*does*make some valid criticisms of MPT but, as we shall see, falls far short of being in the same league when it comes to providing a comprehensive toolbox for decision making regarding portfolio allocation.

**Major Criticisms of MPT**

The criticisms of MPT laid out by PMPT are basically twofold:

1. MPT equates portfolio risk with standard deviation. This makes no allowance for the investor's greater aversion to variances in portfolio returns on the downside. Excessive variance on the upside, that is, greater gains than anticipated, are not negatively considered when risk is evaluated by the investor. Harry Markowitz, the father of MPT himself, made reference to a concept such as semi-variance that he deems would have been preferable.

2. The symmetry of the standard deviation is a result of the assumption that investment returns follow a Gaussian or Normal distribution. PMPT asserts that a three-parameter log-normal distribution is more appropriate. In this distribution, the logarithms of the investment returns follow a Normal distribution. This type of distribution is characterized by the allowance for skewness of investment returns with a tail in one direction.

Today's discussion will focus on the first criticism while the second one will be addressed in Part III on Friday.

**Downside risk defined**

PMPT provides a replacement for the standard deviation as a measure of risk. It is known as

*downside risk*and is given by the following formula:

where,

d is the downside risk as a percentage,

t is the annual Required Return as a percentage,

r is the random variable for the range of investment returns,

f(r) is the best three-parameter log-normal distribution that fits the data.

d is the downside risk as a percentage,

t is the annual Required Return as a percentage,

r is the random variable for the range of investment returns,

f(r) is the best three-parameter log-normal distribution that fits the data.

The integration in this formula is performed from negative infinity up to the target Required Return. In this way it measures only the probability of returns that the investor would consider to be on the downside. To obtain a more accurate value for the downside risk, PMPT requires the use of annualized monthly asset class performance data rather than more limited annual data to provide sufficient confidence for the computation of the estimates of the three parameters of the log-normal distribution.

Note that the above formula is in integral form which means that it's what is mathematically known as a

*continuous function*. PMPT prefers this continuous function over the commonly used discrete version which simply uses the raw time series data for asset class performance evaluation over time. For comparison purposes, the discrete version is given by:

where,

d is again the downside risk as a percentage,

3.464 is the monthly to annual conversion factor, the square root of 12,

E is the Expected Value operator used to obtain the average return,

t is again the annual Required Return as a percentage,

r is again the random variable for the range of investment returns,

n is the total number of monthly returns in the data.

d is again the downside risk as a percentage,

3.464 is the monthly to annual conversion factor, the square root of 12,

E is the Expected Value operator used to obtain the average return,

t is again the annual Required Return as a percentage,

r is again the random variable for the range of investment returns,

n is the total number of monthly returns in the data.

**Why PMPT prefers a continuous function**

The preference for the continuous version of the expression for downside risk is a consequence of the aforementioned requirement of using monthly data to quantify returns. This can increase the level of perceived risk by imposing more frequent periodic performance goals. Annualizing returns with a continuous formula tends to smooth out any monthly fluctuations and present the investor with the impression of lower risk, or so the argument goes. Another reason PMPT prefers a continuous function is that it permits forward-looking predictions as well as back testing to be conducted by creating a prediction model from the data.

**Conclusion**

In summary, what PMPT does is to fit a three parameter log-normal distribution to monthly data, annualize the numbers, then apply the above formula to assess the downside risk.

The next installment of this article will describe additional aspects of PMPT.

*Posted by Prof. Pat*

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