El teorema de Comfort-Ross

Following the analysis of the individual assets, investors have to consider whether or not they would benefit from combining these assets in portfolios.

Ultimately the investors would like to add investments to a portfolio that would lead to the highest level of diversification, therefore if an investment increases the degree of diversification of a diversified portfolio, investors should consider these assets for investment purposes.

The aim of this section is to prove that a well-diversified portfolio, consisting of equities, bonds and cash, may be diversified even more through the inclusion of the Kruger Rand asset.

In order to prove the above-mentioned hypothesis, the study will aim to construct a well-diversified portfolio consisting of a mixture between cash, equities and fixed-income investments. After considering the unique characteristics of this particular portfolio, the study will investigate the effects that the inclusion of the Kruger Rand in the portfolio will have on it in terms of risk and return.

6.2.4.1 Co-movements between the assets

As was indicated in the discussion on portfolio theory, Markowitz and other theorists underlined the fact that diversification of a portfolio can be attained by adding investments to the portfolio that have low positive or negative

Figure 6.10

Scatter plot matrix of asset returns

Source: Inet

Figure 6.10 represents a scatter plot matrix in terms of which the returns of two assets were plotted against one another. This process was completed for all possible asset combinations. An upward sloping trendline indicates a positive relationship between the returns of the individual assets, whilst a downward sloping trendline indicates a negative relationship between the asset returns. This exercise assists investors in detecting the existence of positive or negative relationships between assets, prior to conducting various measurement calculations.

Various measures of co-movement may be used to determine the relationships between the assets used to construct a portfolio, one of these

measures is covariance, which Reilly and Brown (2000:262) defines as a measure of the degree to which two variables move together relative to their individual mean values. The formula used for the calculation of covariance is cited below:

Applying the above-mentioned formula to the data gathered for each of the different assets results in the following covariance matrix:

Table 6.3 Covariance matrix

Debt Equity Cash Kruger Rands

Debt 1

Equity 0.0000762 1

Cash 0.00000042 0.000064 1

Kruger Rands -0.0000658 0.000107 -0.0000541 1 Source: Inet

Even though these figures indicate whether or not the relationships between assets are positive or negative, they are difficult to interpret. In order to address this problem, the resultant figures need to be standardized. The correlation coefficient of assets provides investors with such a standardized measure. Correlation coefficients range between –1 and +1. Where –1 indicates a perfect negative relationship between two variables and +1 indicates a perfect positive relationship between two variables. The correlation

The following formula may be used to calculate the correlation coefficient for each of the assets.

y x

xy xy

Cov σ

ρ = σ [6.6]

Table 6.4

Correlation coefficient matrix

Debt Equity Cash Kruger

Rands

Debt 1

Equity 0.230 1

Cash 0.001 -0.075 1

Kruger Rands -0.249 0.1843 -0.008 1

Source: Inet

Table 6.4 indicates that all of the assets have either a low positive correlation with one or more of the other assets, or a negative correlation with the other assets, a fact that indicates the possibility of diversification between the different asset classes.

6.2.4.2 Initial diversified portfolio

This section aims to identify the portfolio that offers the investor the highest level of return for a given level of risk, assuming that the investor invests in a mix between cash, equities and fixed-income investments.

Figure 6.11 Efficient frontier

Source: Inet

For the purpose of this study it is assumed that investors aim to identify the minimum variance portfolio, or the portfolio with the lowest level of risk. The portfolio that satisfies this requirement may be found when the following asset mix is attained:

Figure 6.12

Asset mix of a diversified portfolio

ALBI Cash

11.56%

ALSI 14.35%

Given the above mentioned asset mix it is clear that the minimum variance portfolio would have provided the holder of the portfolio with a annualised return of 5.26%, and a standard deviation of 7.41%.

6.2.4.3 The alternative portfolio

In order to avoid confusion the diversified portfolio which includes the Kruger Rand asset will be labeled the alternative portfolio.

Following the construction of the diversified portfolio the question remains as to whether or not the inclusion of an alternative asset into that portfolio will hold any additional benefits for the investor. In order to answer this question one would need to construct an efficient frontier of the alternative portfolio and identify the minimum variance portfolio. After this has been done a direct comparison between the two portfolios is possible.

Figure 6.13

Efficient frontier of the alternative portfolio

Source: Inet

Figure 6.13 illustrates the efficient frontier of the alternative portfolio. On the basis of visual inspection it would seem as though this portfolio holds a lower level of risk than the diversified portfolio, this is evident in the lower levels of standard deviation indicated by the figure.

As was the case for the diversified portfolio, the assumption is made that the investor will target the minimum variance portfolio. In doing so the following figure (figure 6.14) illustrates the asset mix of the alternative portfolio.

Figure 6.14

Asset mix of the alternative portfolio

ALSI 8.88%

Cash 10.50%

Kruger Rand 18.84%

ALBI 61.78%

Source: Inet

Figure 6.14 indicates that the alternative portfolio should be constructed in such a manner that 18.84% of its total value is made up of the Kruger Rand asset.

Given the above mentioned asset mix it is clear that the minimum variance alternative portfolio would have provided the holder of the portfolio with an annualised return of 5.59%, and a standard deviation of 6.86%.

This indicates that the alternative portfolio will enhance both return and risk.

Table 6.5 tabulates the risk and return figures for both the portfolios.

Table 6.5

Portfolio risk and return figures

Portfolio Return Risk

Diversified Portfolio 5.26% 7.61%

Alternative Portfolio 5.59% 6.86%

Source: Inet

From this we are able to conclude that investors who held diversified portfolios, could have enhanced both risk and return on their portfolios if they included Kruger Rands to their portfolios. Furthermore it may be said that portfolios that weren’t effectively diversified would also have benefited from the inclusion of the Kruger Rand. Lastly as all asset allocation decisions are based on historical data it follows that rational investors should expect that the inclusion of Kruger Rands in their portfolios should lead to increased diversification.