Modern Portfolio Theory

Introduction

Modern Portfolio Theory is the subject for which Harry Markowitz was awarded the Nobel Prize in 1990. MPT will help you to optimize your portfolio with regard to the risk of the individual assets, the correlation between the assets and the expected returns of your assets.

Expected Returns and Risk

The returns an investor will get in the end can deviate considerably from the the returns that were expected. But investment decisions must be made in advance, based on expectations of an uncertain future. It is not sufficient to base these decisions on expected returns alone because expected returns alone provide an incomplete description of the future. A further factor which is of equal importance for a well-analyzed decision is the risk of the investment, that is, a measure of how certain we can be that the expected return will be realized.

Afterwards when we know the outcome, a higher return is always preferred before a lower return, irrespective whether this has been achieved with a high or low level of risk. But hindsight is not an option when making investment decisions. Well-analyzed investment decisions are always based on both expected returns and risk, because this will describe a more complete picture of the future.

More about risk

The risk(volatility) of an equity is measured by the standard deviation of the equity's rate of return. You can think of the standard deviation as measuring how far away from the expected return the realized return is likely to be. The greater the standard deviation, the more variable the rate of return.

The 68-95 rule states that most returns lie within 2 standard deviations of the expected return. About 68% of the returns lie within one standard deviation of the expected return (between the expected return minus one standard deviation and the expected return plus one standard deviation). About 95% of the returns lie within two standard deviation of the expected return. For example, if the standard deviation is 0.5%/day and the expected price for the next day is $85, then there is a probability of 68% that the next day's price will lie in the range $84.50-$85.50.

Correlation

When estimating the risk of a portfolio it is not sufficient to estimate the risk for all individual assets in the portfolio. The risk of a portfolio also depends on the correlation between all assets. Correlation is a measure of the degree to which two assets (or investments) move together.

The correlation between two assets lies between -100% and 100%. The higher the correlation is between two assets, the more similar are the price movements of the assets. A high correlation, for example 60%, means that the assets tend to move in the same direction. If there is no relationship between the movements of two assets, then the correlation is zero and the relationship is governed by randomness. If correlation is negative the assets tend to move in opposite direction. The correlation will be closer to -100% if the negative relationship is strong and closer to zero if the relationship is weak.

Portfolio Risk and Correlation

If correlation is less than 100% the risk of the portfolio will be less than the average of the risk of the assets. Risk, measured with portfolio standard deviation, falls with correlation. The lower correlations are between assets of a portfolio, the lower the risk is of the portfolio. Diversification will thus be profitable (meaning we will get a higher return for a certain level of risk) when combining assets which are not entirely the same.

How Do We Combine Assets Into A Well-Balanced Portfolio?

A well-balanced portfolio consisting of many stocks not strongly correlated will always produce a lower risk for a certain expected return rate, but how do we combine such a portfolio?

For every level of expected return, there is one optimal asset combination which offers the lowest possible risk, and for every level of risk, there is one optimal combination which offers the highest expected return. These optimal combinations are called efficient portfolios. An efficient portfolio is one which has the smallest attainable portfolio risk for a given level of expected return (or the largest expected return for a given level of risk).

There are two demands on an efficient portfolio:

There is no other combination of assets which offers the same expected return for a smaller portfolio risk.
There is no other combination of assets which offers a higher expected return for a given portfolio risk.

These optimal combinations can be plotted on a graph, and the resulting line is called the efficient frontier.

The efficient frontier begins on the left side of the chart with the portfolio which has the smallest attainable risk of all efficient portfolios. There is no other combination of assets which can achieve a smaller portfolio risk. This portfolio is selected by default with a red solid circle in Optimal Trader. Click on another part of the curve to move the red solid circle and select another efficient portfolio. The portfolio far out to the right on the efficient frontier only tries to maximize return without any risk consideration. This portfolio is solely constituted by the stock with the highest averaged return and is of course not a realistic portfolio combination.

Which efficient portfolio you decide to select depends on your acceptable level of risk. But irrespective of your risk tolerance it is always superior to select an optimal combination of expected return and risk, that is, a portfolio on the efficient frontier.

Expected Returns Estimation

When calculating efficient portfolios on the efficient frontier, three measures are needed:

Expected returns for all portfolio assets
Risk for all portfolio assets
Correlation between all portfolio assets

Risk and correlation are fairly constant measures and do usually not change much over time. Expected returns, on the other hand, is a much more difficult factor to handle in real life.

Expected returns in Optimal Trader are by default estimated by calculating the geometric mean of the historical daily percentual returns. There are limitations with traditional expected return estimations which are discussed in the next paragraph. To eliminate this weakness, expected returns can also be estimated in Optimal Trader by a statistical classification or be proportional to Portfolio Scan results.

Traditionally and theoretically, expected returns are simply estimated by averaging the historical daily returns. But if a stock historically has yielded an average annualized return of 10% and another stock historically has yielded 5% it is bold to make the assumption that this relation will hold in the future. Although this is not a reliable estimation it is often used because there are no obvious good alternatives.

For that reason some investors only use one efficient portfolio: the one with the lowest attainable risk. This portfolio is situated furthest to the left of the efficient frontier and is special because it does not consider expected returns at all. It is simply constituted by the portfolio which has the lowest attainable risk of all possible portfolios. This portfolio is selected by default in Optimal Trader when calculating the efficient frontier and is thus marked by a solid red circle by default.

To use this strategy you should include a limited number of equities, maybe 5-15 stocks which you are certain that you want to invest in. You can use Optimal Trader's Portfolio Scan to find stocks that meet you investment criteria.

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