Correlation Theory for Asset Selection in a Fund
Fund managers perform a role of portfolio managers and decide the investment strategy for an asset management company. The primary task of a fund manager is to increase the return of investors but at the same time to reduce the risk. This may be achieved by ‘diversification’ into various assets (Elton and Gruber, 1997). For greater diversification, fund managers usually take correlation between assets into consideration when selecting assets to be included in a portfolio.
Correlation is defined as a statistical technique that measures the extent to which two or more variables can oscillate with each other (Yule, 1897). Correlation is a measure of linear association between two variables. It is denoted by the Greek letter ? (rho). The variables for the purpose of portfolio management are any assets that are prospective constituents of a particular portfolio. The fluctuation of such variables (or assets) is explained through correlation coefficient which is also a measurement of correlation. A correlation coefficient is primarily a measure of association between two assets. The value of correlation coefficient ranges from -1 to +1. The sign of correlation coefficient depends on the type of correlation; a positive correlation between the two variables is denoted by a positive sign, while the negative correlation is denoted by a negative sign.
There are two types of correlation coefficient used; first is Pearson’s correlation coefficient which is denoted by ‘r’ another one is Spearman rank correlation coefficient which is denoted by rho ‘?’. However, the most widely used correlation coefficient is Pearson’s coefficient ‘r’.
r= (?_(i=1)^n?(x_i-x ? )(y_(i-) y ? ) )/(?(?_(i=1)^n?(x_i-x ? )^2 ) ?_(i=)^n?(y_(i-) y ? )^2 )
Where X bar and Y bar are the sample mean of X1, X2, . . ., Xn and Y1, Y2, . . ., Yn, respectively.
As explained earlier, the correlation coefficient has a value ranging from -1 to +1. In the case of portfolio management, if the value is 0, this signifies no correlation between the two assets. There is another term called the Coefficient of Determination which explains how much one asset accounts for variation in another variable (Yule, 1897). It is denoted as r² and it is the square of a correlation coefficient. For instance, if the correlation coefficient between two assets X and Y is r= 0.20, the coefficient of determination r² =0.040. Therefore, the asset X explains 4% of the variation in the asset Y or in other words it can be said as that 4% of the variation in the asset Y can be explained by its relationship to the asset X and 96% of the variance cannot be explained in Y from the variation in X. The former one is called an explained variance and the latter one is called an unexplained variance.
The correlation between two assets can be represented by a scatter diagram. The values of variable ‘X’ can be seen on the x-axis and values of variable ‘Y’ can be seen on the y-axis.
A positive correlation is a correlation in which the two variables are directly proportional to each other. It means when the value of X increases, the value of Y will also tend to increase and vice a versa. Figure (a) illustrates both variables moving in the same direction. Figure (b) shows negative correlation. In this type of correlation, the variables are inversely proportional to each other and they move in the opposite direction. In other words, increase in the values of X will cause a reduction in the values of Y and vice a versa. The third type of correlation is zero or no correlation between the variables X and Y. Therefore, the value of correlation coefficient ‘r’ is close to 0.
(a) Positive correlation (b) Negative correlation (c) No correlation
r = 0.85 r = –0.9 r = 0.15
Fund managers use the correlation theory to understand the strength of the relationship between the assets to be included in the portfolio. In case of fund managers managing the portfolio of different assets, they need to look at the correlation between each pair of such assets. If the correlation between the two assets is high enough (close to +1), they may need to rebalance their portfolio as this may not provide a sufficient diversification. On the other hand, if the correlation between the two assets is close to -1, this will result into little or no growth in the portfolio. Therefore, it is advisable to have a correlation close to ‘0’ in order to reduce risk and achieve an optimal return (See example in Appendix 1).
Correlation may be used to make predictions about the behavior of assets in the portfolio. The portfolio managers could perform a simulator, in order to know what the change in one asset will be if certain percentage of change in the other asset takes place. This may help them to predict the return and risk of the portfolio under certain assumptions. For example, a correlation of 1 signifies that if one asset is expected to grow by 5%, the other will also grow by 5%.
It is a simple technique to assess the relationship between two assets, whether they co vary; if their relationship is positive or negative and also the degree to which they correlate (Hight, 2010). All these are very helpful information to a fund manager managing a large portfolio.
Correlation cannot be used for finding the correlation between more than two assets and therefore correlation matrix may be necessary to prepare. The correlation matrix exhibits the individual correlations between each pair of assets. This will again not define the relationship between more than two variables at a time.
Hight (2010) argues that portfolio pairs with smaller correlation coefficient values would carry less amount of risk than pairs having larger values. However, the correlation between assets and risk factors is not always obvious and therefore such assets should be selected carefully.
It is assumed that the correlation between assets are fixed and predictable which does not happen in the real world. Therefore, fund managers may need to continuously evaluate the correlation between the assets in the portfolio. Furthermore, Campbell et al. (2002) investigated the data for international equity markets and found that there is an increased correlation between them during the bear market (falling market). This may mean that the use of correlation is of little help to fund managers in bear market.
The other limitation of correlation is known as ‘sensitivity to outliers’. In other words, the outliers may significantly impact the correlation between the two assets. Therefore, the fund managers may get a false impression of correlation if there are more outliers in these two variables.
The correlation values are utilised extensively in the Modern Portfolio Theory (MPT) introduced by the Nobel Prize Winner Markowitz (1952). The MPT is a theory of investment which is based on the idea that investors who are risk-averse can form a portfolio that will optimise and maximise their profits and returns on a given level of market risk (Markowitz, 1951). Markowitz (1991) emphasize that risk comes by default with a higher reward, and therefore different amount of returns can be formed with the given level of risks. It is one of the most vital and influential theories that deal with finance and investment.
The next section argues how fund managers use this theory for designing the portfolio.
Modern portfolio theory
MPT emphasizes on the diversification of portfolio, by the fund managers, in order to reduce correlation between the returns from the selected assets in the portfolio. The primary goal of a fund manager is to optimise expected return with a given level of risk. A fund manager generally tends to select assets that have a very low correlation coefficient, so that the assets in the portfolio are less likely to lose value at the same time. In other words, by using MPT, fund managers look for correlation between expected returns and volatility of different assets.
Fund managers will select another portfolio only if they see a higher expected return or more gain from it (Mangram, 2013). However, the ultimate goal is to optimise the expected return given a certain level of risk. The importance of correlation cannot be underestimated in order to strategically select the assets that are less likely to fall at the same time (Ilmanen, 2003).
MPT attempts to analyse the interrelationship between different investments with the help of correlation to quantify the diversification effect on a portfolio performance (Markowitz, 1999). In that regard, the correlation coefficient divides the covariance by the standard deviations/risk of two assets. The values of correlation coefficient along with the weights of assets directly influence the portfolio standard deviation. For example, the higher the amount of uncorrelated assets in a portfolio, the higher the risk reduction. Therefore, correlation is an important measure which affects the diversification as it effectively measures the covariance of the returns of asset pairs. Although covariance is meaningful for its influence on portfolio risk, correlation coefficients of the portfolio is more useful for portfolio manager since it standardises covariance (Mangram, 2013).
Diversification, as indicated earlier, refers to the relationship between the portfolio risk and correlations. It is a technique to reduce the risk of portfolio and increase chances of higher expected return. This technique allows a fund manager (investor) to invest in different assets that would each react differently to the same event(s) (Blake et al,1999). For example, the news of the financial crisis in Europe causes the stock market to behave accordingly and at the same time, there are positive results on the price of a certain goods, such as: gold (Mangram, 2013). Thus, it is important for a fund manager to diversify in the assets from different asset classes such as stocks, commodities, currencies, real estate, and bonds and also diversify internationally. This technique is very important and is an effective risk reduction strategy since the risk return is attained without affecting returns (Hight, 2010). The example exhibiting the importance of correlation in portfolio risk and return is shown in the appendix (2. Example)
Criticisms of MPT
Having discussed the advantages of MPT, it is important to note a key criticism of MPT. Firstly, MPT assumes that the correlation between assets are fixed and predictable which is not true since such correlations between assets do not remain constant in the practical world. Therefore, quite paradoxically, MPT may not be useful during uncertain times when fund managers require more protection from the volatility in the market.
In essence, MPT observes the correlation between expected returns and volatility of various investments. This expected risk-reward relationship was titled “the efficient frontier” by Markowitz (1991). The efficient frontier represents the optimal correlation between risk and return in MPT.
Efficient frontier explains the relationship between the expected portfolio and risks related with that portfolio. It is represented in the form of a graph which illustrates the comparison of risk against the expected return of the portfolio (Elton and Gruber, 1997).
The portfolios which are depicted using this curve demonstrates the highest possible expected return on investment for the given amount of risk. The fund managers use the points along the curve of efficient frontier to choose the portfolio as they represent portfolios which depict the best possible arrangements of expected return along with the given level of risk from the investment in that portfolio (Mangram, 2013). Any other point below the efficient frontier is technically sub-optimal and hence is avoided by the fund managers.
As discussed, the correlation theory is quite useful for the fund managers in order to build an optimal portfolio that provides a certain amount of return and which carry the lowest risk. The theory is also used to evaluate the expected return in advance based on the correlation between the assets which is represented by correlation matrix. However, the theory is not useful to diversify a systematic or market risk as observed by higher correlation between the assets during the challenging time.