# The Post Modern Portfolio Theory in financial structuring

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Nowadays, many investors are using some theory in making investment decision. The introducing of Post-modern Portfolio Theory offers a structure that helps to recognize the desirable for upside and downside volatility. It extent the traditional Modern Portfolio Theory, also referred to as Mean-Variance Optimization.

Over years, some important limitations of Modern Portfolio Theory have been founded. The causes of the unsatisfied aspect of Modern Portfolio Theory are the assumption of "variance of portfolios returns is the correct measure of investment risk" and "the investment returns of all securities and portfolio can be adequately represented by the normal distribution". It can be say that the results of Modern Portfolio Theory provide are limited by the measure of return and risk, which means that it do not always stand for the realities of investment markets.

With the advance in portfolio and financial theory as well as increased in electronic computing, the limitation of Modern Portfolio Theory can be solved. New risk-return paradigm being established which is known as the Post-Modern Portfolio Theory.

According to Harry Markowitz, Post-modern Portfolio Theory is a refinement to Modern Portfolio Theory in order to maximize usefulness of the theory for advisors in the effort to improve investment results for their clients. Post-modern Portfolio Theory assist investor in considering how a risky asset should be priced and how rational investors should use diversification to maximize their portfolios. Similar to Modern Portfolio Theory, Post Modern Portfolio Theory also advises how the rational investors can use diversification tools to optimize their portfolio returns and how the risky assets should be priced.

Besides, Post-modern Portfolio Theory recognizes that investment risk should be tied to each investor's particular goals and the outcomes above this goal do not represent economic of financial risk. Post-modern Portfolio Theory's downside measure creates an obvious distinction between downside and upside volatility. In Post-modern Portfolio Theory only volatility below the investor's target returns incurs risk, all returns above this target cause 'uncertainty', which is nothing more than riskless opportunity for unexpectedly high returns. (Brian M. Rom & Kathleen W. Ferguson, 1994)

## Background Research

According to Vern Sumnicht (2008), Fifty years ago, the authors of Modern Portfolio Theory understood that the limitations of their work to its ability define and quantify risk. The reason is that the founders of Modern Portfolio Theory did not use, for example, a downside deviation measure of risk was the modern computer technology wasn't available to them, and the calculations that perform the mathematic functions for such measurements were too complicated.

By Pete Swisher (2005), in 1959, Harry Markowitz, the "father of modern portfolio theory," published Portfolio Selection, in which he proposed that investors expect to be compensated for taking additional risk, and that an infinite number of "efficient" portfolios exist along a curve defined by three variables: standard deviation, correlation coefficient, and return. The efficient-frontier curve consists of portfolios with the maximum return for a given level of risk or the minimum risk for a given level of return. From Harry Markowitz (1959) Harry Markowitz formalized what investors already knew when they looked to have placement profitability correspond to the level of risk. But he was the first person to mathematically establish that the total risk of a portfolio is inferior to the sum of the individual risk for each element of a portfolio. By taking periodical performances as random variables, it was possible to calculate performance expectations, standard deviation and correlations.

The work on modern portfolio theory won Markowitz his shore of a Nobel Prize. Merton Miller along with Harry Markowitz and William Share, were awarded the 1990 Nobel Prize in Economics for research on theories of "Financial Economics." Vern Sumnicht (2008)

In 1993, Vijay K. Chopra and William T. Ziemba show that it is essentially assessment errors on performance expectations that have an impact on the setting up of portfolios. The assessment errors on variances and co-variances clearly have less impact. The efforts of the portfolio advisors put into good assessment of the performance. In the same year of 1993, Brian M. Rom and Kathleen W. Ferguson published an article on the postmodern theory of portfolio in the "The Journal of Investing". The introduction of the Sortino ratio, of risk loss, of the MAR (minimum asset return) and other parameters give way to new avenues for refining of the average-variance model.

## Literature Review

Both theoretical and empirical discussion on the suitability and appropriate application of has result in a vast amount research over the last decades.

According to Brian M. Rom &Kathleen W. Ferguson (1994), "It has been a generation since Harry Markowitz laid the foundations and built much of the structure of what we now know as Modern Portfolio Theory (MPT)." Modern Portfolio Theory has a big contribution on given a formal risk and return framework for investor in making investment decision. Markowitz (1952) helps investors in making investment decision by provide mathematical formulation of the concept of diversification in investing, the target is to select a collection of investment asset that has collectively lower risk than any individual asset.

Many researchers found that there are some signify limitation to the traditional Modern Portfolio Theory calculation. "Under certain conditions, the mean-variance approach can be shown to lead to unsatisfactory predictions of behavior. Markowitz suggests that a model based on the semi-variance would be preferable; in light of the formidable computational problems, however, he bases his analysis on the variance and standard deviation" (Noteworthy among these Sharpe, 1964). "But the recent market turmoil is causing some question whether Markowitz's playbook needs to be rewritten. Upon further review, the simple and elegant math of Modern Portfolio Theory requires some assumptions that bear little resemblance to market reality". (Nolan Bean, 2009). Modern Portfolio Theory is inefficient in measure the risk and return which do not reflect the reality of investment market.

The theory that overcomes these problems is known as Post-Modern Portfolio Theory. Two of the most important enhancements offered by the Post-Modern Portfolio Theory formulation are downside risk and asymmetrical return distributions (Brian M. Rom &Kathleen W. Ferguson, 1994). Post-Modern Portfolio Theory provides analysts with flexibility and accuracy in constructing efficient portfolios which was unattainable under Markowitz mean-variance methodology. Brian uses some examples of policy decisions using these two methods illustrate how Mean Variance Optimization can produce illogical and counter-intuitive results and the ability of Post-Modern Portfolio Theory to overcome these problems.

According to Vern Sumnicht, 2008, "Advances in computer technology, research in portfolio management and behavioral science allow us to carefully reconsider the founding framework of Modern Portfolio Theory. These advances, and the development of a profession dedicated to investment management, are beginning to challenge the status quo." It is needed to create an asset allocation models which is robust enough to consider additional capital as well as economic factors and put on them to asset allocation and re-balancing decisions. (Vern Sumnicht, 2008)

Post-Modern Portfolio Theory has given a great contribution to the applications and technologies that can meliorate investment results and transfer the Modern Portfolio Theory principle to a new level of usefulness (Vern Sumnicht, 2008). All of these improvements make available for the investment advisors to apply in order to assist investor to reach their investment objective.

## Total variability of return

By using Modern Portfolio theory, the risk is defined as the total variability of returns around the mean return. It handle all uncertainty the same. "Risk is not symmetrical-it is severely skewed, with the greatest concern going to the downside." (Brian M. Rom &Kathleen W. Ferguson, 1994)This importance of skew lies in the fact that the more non-normal a return series is, the more its true risk will be distorted by Modern Portfolio Theory measure.

At the time, the new introducing of Post-Modern Portfolio Theory is able to capture more accurate information contained in the returns under consideration. "PMPT recognizes that investment risk should be tied to each investor's specific goals and that any outcomes above this goal do not represent economic or financial risk." (Brian M. Rom & Kathleen W. Ferguson, 1994) "In Post-Modern Portfolio Theory, only volatility below the investor's target returns incurs risk; all returns above this target cause "uncertainty" when there is nothing more than riskless opportunity for unexpectedly high returns." (Brian M. Rom &Kathleen W. Ferguson, 1994)

In the theory of post modern portfolio, the investor's target rate of return is called the Minimum Acceptable Return. "Minimum Acceptable Return represents the rate of return that must be earned to avoid failing to achieved some important financial objective" (Brian M. Rom & Kathleen W. Ferguson, 1994). Because Minimum Acceptable Return is investor specific, it means that there are an infinite number of efficient frontiers, one for each minimum acceptable return. This means that Post-Modern Portfolio Theory is more accurately reflects the reality that there have different aims and appetite for risk in different investor.

## Downside Risk and Mean-Variance Optimization

There has a simple question that been asked today: what is the best return an investor can get for a given level of risk? Conversely, what is the least risk an investor can take for a given level of return?

The first study of the concept of portfolio optimization related to Downside Risk instead of the traditional Mean- variance optimization is introduced to the field of real estate research by Sivitanides (1998) and. Sing an Ong (2000). Sivitanides had analyzes the return to a Downside Risk profile of portfolios based on the four property types, which is office, retail, Research and development, and warehouse direct real estate investments in investments. While Sing and Ong (200) had examines the mixed asset portfolio allocations which containing stocks, bonds, and direct real estate. They had determined that how investor risk can aversion can incorporate with the downside risk asset optimization model. These researches focus on the comparison between portfolios realized by the Mean-variance or Downside Risk framework.

The author, Ping Cheng (2001) has examined Mean-Variance Optimization as well as Downside Risk Optimization to explain which can superior portfolio. These Mean-Variance Optimization and Downside Risk Optimization have the best tradeoff within their own risk-return spaces, so he offers a measure based on the common dimension of two spaces which is return to make a cogent comparison between them. This measurement determine among these two approaches, which can create portfolio that provide higher returns. To compare these two distinct approaches, the author using bootstrapping procedure. The result shows that Downside Risk Optimization produces the portfolio combination which is more realistic and accurate to the practice of institutional investors in terms of real estate allocation. these results from Downside Risk Optimization methods are in demand to those investors who welcome every bit of downside risk deduction. (Cheng.P, 2001)

The concept of downside risk used in Post-Modern Portfolio Theory has been proposed as an alternative approach use in to the traditional Mean Variance Optimization used in Modern Portfolio Theory. According to Peter Swisher (2005), the traditional Mean variance optimization attempts to answer these questions using standard deviation as the definition of risk. Whereas, downside risk optimization has differences the definition of risk. "Standard deviation has some limitations and is not the best surrogate of risk." (Antto Alenius, n.d). "However standard deviation assumes the returns of the fund to be normally distributed, which can be misleading when interpreting result."(Eling and Schuhmacher, 2007: 2632). Instead of using standard deviation, Downside Risk Optimization uses downside risk as estimation. Is the Downside Risk Optimization models in is superior to Mean-Variance Optimization in Modern Portfolio Theory?

Pete Swisher (2005) "Standard deviation can lead to nonsensical result whent used as a risk proxy, whereas variance downside risk capture more closely; even if volatility were a perfect representation of risk, it's still would not work perfectly because financial asset return do not follow a normal distribution; when we put Downside Risk Optimization and Mean-Variance Optimization head to head and compare portfolio, Downside Risk Optimization wins. Specifically Downside Risk Optimization outputs make intuitive sense well Mean-Variance Optimization outputs often do not, and Mean-Variance Optimization outputs frequently reach risk conclusions opposite those of Downside Risk Optimization." Downside risk is efficient than standard deviation that used in mean variance optimization because it supply different views of risk (Riddles Neil, 2001).

According to Anton Abdulbasah Kamil & khalipah Ibrahim research, (2005), "The popularity of downside risk in Post-Modern Portfolio Theory among investors is growing and mean-return-downside risk portfolio selection models seem to oppress the familiar mean-variance approach." The model has success because it separate return fluctuation into downside risk and upside potential. Because of in mean-variance model, upside potential is same as downside risk, so this led Markowitz to propose the downside risk measure which is semi-variance to replace variance as the risk measure. Anton had compared the returns of the optimal portfolio to the performance of the model with the other models. In the comparison result, it shows that the performance of the model with Downside Risk Optimization model is efficient than Mean-Variance Optimization model.

According to Ang, Chen & Xing (2005), they analyzed downside risks premium in the cross section of stock returns. The result shows that cross section of stock returns reflected a premium for downside risks. "Stocks that co-vary strongly with the market, conditional on market declines have high average return." (Ang, Chen & Xing, 2005)

## Engineering Returns and Risks

Ray Dalio, founder of bridgewater Associates (2005) "The traditional application of Modern Portfolio Theory first combines asset classes based on their expected returns, risks and correlation, and ones the asset allocation mixed is determined identify the best managers in each asset class. By contrast, Post-Modern Portfolio Theory differs in three key ways: first, returns from alpha and beta are separated; second, the sized are altered to more desirable levels; finally, far more diversified portfolios of each are derived."

In a consequence, Post-Modern Portfolio Theory portfolio not only focus on risks and returns but also more suitable to the investors objective. Ray Dalio discus three basic building blocks: the risk free returns, returns in alpha, and returns in beta and describes how they can fit together. Ray Dalio believe that produce more diversified beta and alpha portfolios calibrated to one's targeted returns, investors can dramatically improve to achieve their investment objective. (Ray Dalio, 2005)

From the literature review, it can be conclude that Post-Modern Portfolio Theory provide the investors with more reality and accurate in the form of efficient portfolio that is unavailable under traditional Modern portfolio theory which introduce by Markowitz.

## Supporting Theory

## What is the Post-Modern Portfolio Theory?

As a strategy seeking to maximize absolute return and minimize risk, the Post-Modern Portfolio Theory opens up brand new perspectives to investors on the top of those already offered by the traditional Modern Portfolio Theory. The Post-Modern Portfolio Theory promotes a greater flexibility in the management of the asset classes without resorting to alternative investments and it can cover most of the portfolio's assets and significantly contribute to its total return. (Olivier Hoang, 2004)

## What are the technical features of PMPT?

This is why the Post-Modern Portfolio Theory favours indexed vehicles invested in large caps equities and government bonds. This strategy also sticks to long-only positions. In order to get rid of any bias in the forecasting and implementation of a market scenario, the decision-making process has to be quantitative. It is generally recognized that the most obvious outcome is not necessarily the most likely. The Post-Modern Portfolio Theory teaches us that the volatility and the tracking error are not full-blown risk measures. In the end, the Post-Modern Portfolio Theory leads to revisiting the international diversification notion thanks to the principle of the encapsulation of the currency risk, which makes that geographical allocations are independently managed. (Olivier Hoang, 2004)

## Downside risk Optimization

Post-Modern Portfolio Theory presents a new method of asset allocation that optimizes portfolio based on returns versus downside risk called Downside Risk Optimization instead of Mean-Variance optimization.

By using downside risk, formula that combining these three elements have been established.

1. Downside frequency - The frequency, expressed as a percentage, of returns below Minimal

Acceptable Return.

2. Average downside deviation - The average size of the deviation below the Minimal

Acceptable Return.

3. Downside magnitude - The worst-case scenario, represented by the return below Minimal

Acceptable Return at the 99th percentile,

These three statistics have combined into a single downside risk measure. Each of these measures is defined with reference to an investor- specific minimal acceptable return. (Pete Swisher, 2005). The result is expressed as a percentage, much like standard deviation, and the values themselves might even be similar.

## LPMn=

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)

n= degree of the moment

It can be says that when n=2, LMPâ‚‚ is called semi-variance. For the square root of semi-variance is known as semi-deviation. Downside risk is an estimation of a security's potential to endure a decline in price when the market conditions get worse. It can think as an estimation of the potential losses that may occurs on investment or stocks. (Ping Cheng, 2001)

## There have several ways to view downside risk.

View the 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 an exponential rate.

There are two formulas for Downside risk

Continuous form

t= annual target return

r= random variable representing the return for the distribution of annual returns f(r).

f(r) = normal or three parameter lognormal distribution.

Discrete form

3.464*

3.464 = the square roof of 12, the factor used to annualize

the monthly downside risk.

E = mathematical Expectation operator

t = monthly target return

r = random variable representing operator monthly return

n = total number of monthly returns observed

The continuous form is more preferable because it permits all the calculation to be made using annual returns. It helps investor to specify their investment target. While for discrete formulas, it requires monthly returns, which in turn requires investors to converting the annual target into a monthly target. (Frank A. Sortino & Stephen E. Satchell, 2001)

Downside Risk Optimization model in Post- Modern Portfolio Theory can be say that is more efficient than Variance Optimization in Modern Portfolio Theory. It is because Downside Risk Optimization produces the portfolio combination which is more realistic and accurate to the practice of institutional investors in terms of real estate allocation. Compare to Variance Optimization portfolio, Downside Risk Optimization method is more consistent with investor's risk conceptions that encourage investor who are more anxiety with downside risks. Not only that, it also seems to improve portfolio performance with higher median returns. (Cheng.P, 2001)

## Volatility skewness

World is change over year, many researchers have state that not all the distribution are normal. In Modern Portfolio Theory, this theory allows only the normal distribution. In using normal distribution to model the pattern of investment returns, it creates the investment that result with more upside than downside returns appear more risky than they actually are, and vice versa for returns with more a preponderance of downside returns. It can be concluding that using traditional Modern Portfolio Theory for measuring the portfolio are often distorts the investment reality.

Fortunately, with the recent advent of hedging and derivative strategies, asymmetrical are designed and used in Post-Modern Portfolio Theory. Volatility is another concept that introduced by Post-Modern Portfolio Theory enthusiasts. Post-Modern Portfolio Theory is able to capture significantly more true information. It examines 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. (Brian M. Rom, 1994)

According to Brian M. Rom (1994) research, the table 1 shows the several assets classes over different periods and skewness ratio. The ratio which greater than 1.0 signify a positive skewness which imply that distribution with more returns occurring above the median return. In contrast, the ratio which is less than 1.0 signifies a negative skewness. From table 1, it shows that skewness ration is different from 1.0 over the time periods.

Table 1

Skewness of Major Asset Classes and Inflation

Asset

Periods Ending 12/31/92

10 Yrs

20 Yrs

Large-Cap Stocks

1.80

1.23

Small-Cap Stocks

1.07

1.22

Foreign Stocks

0.92

1.10

Bonds

0.83

0.94

Cash

0.64

1.25

Inflation

0.82

1.35

Skewness equals(High 10t Percentile Return - Median Return)/(Median Return - Low 10th

Fortunately, Post-Modern Portfolio Theory formulation reduces this puzzle and also utilizes a broader class of asymmetrical distribution. Three-parameter lognormal distribution permits the positive as well as negative skewness. It accommodates all asset classes which comprise options, derivatives and hedge funds. This can be concluding that three-parameter lognormal distribution can better representation of the shape of investment returns.

## Basic Mathematical Formulas for the Three Parameter Lognormal. (Hal Forsey, 2006)

Sample mean, sample standard deviation and extreme value is the three basic parameters to estimate.

There are several auxiliary parameters:

## Dif = |Mean -|

## Formula for the lognormal curve f(x)

If the extreme value is a minimum and x is greater than the extreme value :

If the extreme value is a maximum and x is less than the extreme value then

Formula for the lognormal cumulative distribution function F(x)

## f(x) = 1 âˆ’

## f(x) = 1 âˆ’

## Sortino Ratio

Sortino ratio was introduced by Sortino and Price (1994), and it is used to measure risk adjusted returns for the target and downside risk. It is a modified version of Sharpe ratio. It assists investment manager or investor to estimate portfolio risk. For the Sharpe ratio was developed by Nobel Laureate economist William Sharpe, this ratio measures risk adjusted performance. (Richard Loth, 2010) It measures the excess return or called Risk Premium per unit for an investment stock pr strategy. It quantifies the return (alpha) over the volatility (beta) that assumed in the portfolio. The Sharpe ratio is to interpreted as the risk premium per unit of total risk. Because it can be computed and interpreted easily, so the Sharpe ratio is often employed in practice as well as in theoretical research (Modigliani and Modigliani, 1997).

## Sortino Ratio

## Sharpe ratio

## S =

R = annual rate of return for the investment

T = required rate of return

DR = downside risk, square roof of the target

semi-variance.

## S =

R = asset return

Rf= return on a benchmarck asset, such as risk

free rate

Standard deviation of the asset.

(Brian M. Rom & Kathleen W. Ferguson, 1994)

However, the modified version- Sortina ratio only comprises downside risk as a deviation from the norm of minimum acceptable return. Compare to Sharpe ratio which penalizes both upside and downside volatility equally, Sortina ratio penalizes only those return falling below a user-specified target. Thus, measures of risk adjusted return that treat risk more realistic than the Sharpe ratio.

## Conclusion

According to Nolan Bean (June 4, 2009), in conclusion to have real confidence in a portfolio's diversification, those investors have to understand the primary market risk factor of each asset in their portfolio and diversify those risks. Institutional investors need to find themselves with an asset allocation policy that containing some of the categories, such as global equity, global fixed income, real assets and diversifying strategies. This can help them to better captures risk and is a more intuitive way to construct a portfolio. By Cynthia Harrington ( 2002 ), Many advisors use . Post-Modern Portfolio Theory to some extent recently. It's well known that investors are emotional and must weigh the risk and returns of reaching specific goals. However, fewer advisors use some of the post-modern tools like Sortino ratios and downside risk measures.

As mentioned above by Vern Sumnicht (2008) it should carefully reconsider the founding framework of Modern Portfolio Theory. For example, now we understand that the equating risk implies the clients are indifferent to an investment's upside volatility or downside volatility. Certain asset classes are showing signs of increasing correlation convergence. Clients can't reduce their risk through diversification without investing in asset classes with low correlation. Using concepts from the fields of post-modern portfolio theory when appropriate, this need to guide the clients through the complexities of the financial arena teaching the clients to use the financial system to their advantage.

Vern Sumnicht (2008) Post-Modern Portfolio Theory and research in Behavioral Finance have pointed the way to applications that can improve investment results and catapult the MPT principles to a new level of usefulness. These improvements are available today for professional investment advisors to apply in order to improve the lives of those who rely on them to reach their financial objectives.

According to Brain M. Rom & Kathleen W. Ferguson, (1994), Post-Modern Portfolio Theory is originally used to improve portfolio optimization and asset allocation. Furthermore, it is increasingly being applied to measuring the investment performance of portfolio, investment managers and mutual funds. And it is particular emphasis on performance measurement. It points the way to an improved science of investing that incorporates not only Downside Risk Optimization but also behavioral finance and any other innovation that leads to better outcomes.