Manolis Anastopoulos @ University of Leicester - Risk Management
2.3 Risk (volatility) of a portfolio and the efficient frontier
The previous analysis concluded that the higher volatility increases the risk. We have also accepted in our analysis the regime of the risk–adverse investors, which mean the desire of minimizing the volatility in portfolios. Certainly, such a decision includes an effect in the opposite direction; if we limit the volatility (risk) we are at the same time limiting the prospective rate of return.
Consequently, in order to limit the volatility, it would be to our benefit to include in our portfolio securities with negative or low correlation, i.e. different risk’s securities in a way that prospective fluctuations of individual securities would cancel each other to a satisfactory extent. For instance, an increase in a specific raw material might be good for a producing company but not for a company that uses the specific raw material. Thus, it would be to our benefit to include both companies’ shares into our portfolio.
Therefore, it can be argued that through diversification, i.e. a combination of a satisfactory number of different assets or shares, we can attain a portfolio of a high average rate of return with less fluctuation. As stated in an recent article (JWH Journal 2005), ‘ As the number of stocks in the portfolio increases, there will be a reduction in the idiosyncratic risk or unique risk associated with the individual stocks, and the overall risk of the portfolio should more closely match the systematic risk of the market.’
Such a diversified portfolio that succeeds the highest average rate of return for a given level of risk, or, in other words, the portfolio that manages risk and return is called efficient portfolio.
Finally, we conclude that there are ‘efficient’ portfolios that ‘optimally’ balance risk and return. These optimal portfolios, according to the Modern Portfolio Theory (MPT), are lying across an upward-sloping curve called the efficient frontier of portfolios (fig. 2.3).
The above chart makes clear that for any given value of volatility (risk), investors can choose an ‘optimal’ portfolio that gives them the greatest possible rate of return.
Certainly, the portfolio, in order to give the maximum return for the amount of risk investors wish to bear, should lie on the efficient frontier, rather than lower in the interior of the curve. In addition, it is also worth mentioning, that as an investor moves higher up the curve, he undertakes proportionately more risk for a lower return, and, on the other hand, as he moves lower down the curve, he prefers a combination of low risk and low return.
2.3.1 Indifference curves and risk
Given that, as earlier stated, the majority of investors are risk–averse, it is anticipated that their indifference curves (x1, x2, z1, z2) will slope upwards from left to right (fig. 2.3). The curve’s slope indicates the fact that risk–averse investors are requesting proportionally more expected return than the risk level they are to undertake. On the contrary, the curve will slope downwards, when we consider risk–taking investors.
We conclude that - as Markowitz stated, investors ask for different portfolios of assets based on how risk-averse they are. For example, as illustrated in (fig. 2.3), an investor with an indifference curve of x1 (who is less risk-averse than the other with indifference curve of Z1) choose the point a that represents a portfolio with a less volatility (risk) and less expected return, as well.
2.3.2 Portfolios with equities and cash
When investing in risk-free assets, i.e. cash, we have a ‘pure’ return Rf, say 5%, with an σk (standard deviation) which equal to zero - point A in (fig.2.3.2). Certainly, we have the option, instead of investing in cash, to invest in ‘risky’ assets, i.e. securities or bonds, with a higher expected return and a higher σk, as well – point B in (fig. 2.3.2).
Certainly, investors should choose any point between A and B or right of B close to C, which means a combination of risk-free assets and a share mix.
Undoubtedly, as we move to the right, we increase the expected return of our portfolio and the standard deviation as well. As a result, an investor might think of borrowing an (X) amount at the risk-free rate of say 4% in order to buy a share mix of the same amount (X) that theoretically will offer him a higher expected return, say 12%. In such a case he would benefit from the difference of 8% (12% - 4%).
Finally, as illustrated in (fig.2.3.2), the straight line ABC, - which is called the Capital market line, can be regarded as the ‘new’ efficient frontier. As stated in his book (King 1999), ‘The capital market line shows the highest possible expected return for each possible degree of risk, once lending or borrowing at the risk-free rate is permitted.’
The point B, where the line that starts from the Rf touches the ‘original’ efficient line, represents the market portfolio, which is the ‘target’ of all investors, or, in other words, – as stated in his book (King 1999:84), ‘All investors want an identical portfolio of risky assets which, on its own, would take them to B.’
However, such a market portfolio - which is a risky portfolio – that succeeded through diversification to eliminate all the risk factor that was possible to eliminate, should (among others) contain all companies’ shares quoted on the stock exchange, which is undoubtedly unattainable. The aforementioned unattainable concept of including all shares can be overcome by selecting a maximum of 20 shares, this number of which has been proved to reduce significantly the risk that was possible to eliminate.
As stated in their book (Lumby & Jones 2002), ‘ Several studies have shown that constructing a randomly selected portfolio of shares consisting of only between fifteen and twenty different securities results in the elimination of around 90% of the maximum amount of risk which it would be possible to eliminate through diversification.’
This theory that actually expanded the Markowitz work, since it ‘offered’ the option to investors of including risk-free assets in their portfolios, was introduced by James Tobin in 1958. The theory called separation theorem.