For asset A A = (0.25x19) + (0.65x15) + (0.1x11) = 15.6
For asset B B = (0.25x20) + (0.65x19) + (0.1x6) = 17.95
Here, Asset B has a higher expected rate of return however with these figures we can now find out the variance and the standard deviation. The variance of the return is a measure of risk. This is the statistic that measures the dispersion of the probability distribution of return which will tell us the risk so therefore, the bigger the dispersion the bigger the risk. “An asset which has shown a wide dispersion of actual returns around the mean is riskier than one where returns have been tightly clustered around the mean value.” (Howells & Bain 2008)
The variance is measured by the formula so for asset A;
a = 0.25(19-15.6)+0.65(15-15.6)+0.1(11-15.6)= 5.24
And for asset B;
b = 0.25(20-17.95)2 + 0.65(19-17.95)2 +0.1(6-17.95)2 = 16.0475
The variance in an asset’s rate of return measures the asset’s risk as the more volatile the return, the more difficult it is to predict. However the variance is ex-post as opposed to ex-ante which is the basis of investor’s decision. The standard deviation for each asset tells us the amount of risk that comes with the expected rate of return expressed as a percentage.
a = = 2.29%
b = = 4.01%
This means that the variance of asset B being over 3 times more than that of asset A for not that much more expected rate of return 15.6 to 17.95, investors, due to the law of diminishing marginal utility of wealth would normally favour asset A out of the 2 as it less risky expressed by a lower standard deviation of 2.29% in comparison to 4.01%. If investors replace A with B they will have more risk but also a higher return, the risk will increase by 1.72% and the expected return by 2.35%. The expected return gained per unit of risk is then 1.37. If Investors replace B with A, they will have less risk and have a lower rate of return; risk will decrease by 1.72% and expected return by 2.35%. Expected return lost per unit of risk is then 1.37.
(ii) Construct a portfolio of the two assets in such a way that the portfolio consists of 50% of each asset.
As noted earlier investors can include a percentage of each asset (all must equal 100%) into the portfolio but for this example we will be using 50% of each. To find the expected portfolio return of such we use the formula,
As = 15.6 and = 17.95, Wj is the percentage amount invested as a proportion of the portfolio.
= (0.5x15.6) + (0.5x17.95) = 16.775%
This figure is the weighted average for the expected rate of return on the assets that make up the portfolio. With this figure we can find the co-variance which measures the degree of co-movement of both assets in the portfolio. The formula is expressed as,
= 0.25(19-15.6)(20-17.95) + 0.65(15-15.6)(19-17.95) + 0.1(11-15.6)(6-17.95) = 6.83
Now with this figure of 6.83 we can find the standard deviation as a measure of risk for this portfolio. However to find this we use the formula below.
To measure the co-movement of both assets in the portfolio is called we use the correlation coefficient which tells us the degree to which the returns on each of the assets move together. This is a very important measure as it can let us know how effective diversification would be. The correlation coefficient is always between -1 and 1 where the closer the figure is to -1, the less correlated the assets are which means the more effective diversification will be. Diversification will be most effective however when the correlation coefficient is -1. To calculate our correlation coefficient we use the formula,
= = 0.74
(iii)Compare your results and discuss them in detail, drawing attention to the difference between specific, market and total risk.
The results from the equally weighted constructed portfolio in comparison to both assets alone are as follows.
The correlation coefficient as we calculated earlier is also 0.74, an important figure that we’ll revisit later.
So now we can choose to invest 100% in asset A or 100% in asset B or 50% in each as per the weightings of the previously constructed portfolio. First and foremost we can see the benefits of diversification with the portfolio risk being 2.96% which is lower than the weighted average of the asset risk of 3.15% derived from, (0.5x2.29) + (0.5x4.01). In replacing asset B with asset A we reduce risk at a rate of 0.73 (4.01-2.29) / (17.95-15.6) for every unit reduction in return but, if we replace asset B with a 50/50 portfolio we ‘buy’ the reduction of risk (diversify) at the rate of 0.9 (0.897435) derived from (4.01-2.96) / (17.95-16.78) for every one point reduction in return.
Investing 100% in asset B will get an annual rate of return of 17.95% with a standard deviation on the returns of 4.01. This is where they could get their highest return. If they invested 100% in asset A, they will get an annual rate of return of 15.6% and with a smaller standard deviation on the returns of 2.29 so risk is reduced but so is the expected rate of return. If we diversify with equal weighting as per the previously constructed portfolio then our expected rate of return is 16.78% pa but with a standard deviation of 2.96.
If investors diversify 50% into A the risk decreases by 1.05 and the expected return by 1.17 and so the expected return lost per unit of risk is 1.11 (1.17/1.05). If investors diversify 50% into asset B the risk will increase by 0.67 and the expected return by 1.18 (16.78-15.6) so the expected return gain per unit of risk is 1.76. All of this shows diversification gives a better risk to return trade-off.
However, diversification is not at its most effective in this portfolio with a correlation coefficient of 0.74 even though this shows a small reduction in risk. For it to be a zero-risk portfolio the correlation coefficient must be -1. However, even in this scenario the total risk is not eliminated, only the specific risk and the returns on the assets will still be subjected to market risk. Total risk (standard deviation) is the combination of specific risk + market risk. Only the specific risk is reduced by diversification because it is a risk unique to each asset in the portfolio. For example, if asset A is gold and there’s a discovery of a new, rarer, more precious metal, this will affect the risk of this particular asset. Specific risk is therefore the deviation in the returns of an asset caused by events which affect that particular asset.
Market risk cannot be reduced with diversification as all assets are sensitive to market risk, some more than others. Market risk is the risk arising from factors that affect the economy as a whole like the economic cycle, a change in the price of borrowing or a change in the price of currency.
The diagram shows that the more assets added to the portfolio for diversification the more the total risk is reduced because of the reduction in specific risk as the curve slopes downward. This is because as you add more assets to the portfolio there is more chance of the correlation coefficient reducing or becoming negatively correlated. This adding of assets to the portfolio works up until all that is left is the market risk where the slope straightens out and is then pointless to diversify anymore. With this diagram we can tell that our example of 2 assets of equal weighting is not enough to eliminate specific risk. We can see that it normally takes around 20 assets to eliminate specific risk in a portfolio.
Investors will look to diversify until specific risk is eliminated through the right number of and weighting of different assets in a portfolio. They’ll have a zero risk portfolio and the reward they get for the remaining risk will be related to the market risk. This isn’t easy to measure but with the CAPM (Capital Asset Pricing Model) we can measure an individual asset or security’s market risk relative to its expected return with the SML (Security Market Line) equation. In the CAPM the beta is the measure of market risk.
Diversification reduces risk to an extent and can remove specific risk depending on the correlation coefficient of the return of assets in the portfolio. The more negatively correlated the better. However it can’t remove TOTAL risk as the assets will be subject to market risks. A good strategy may be to diversify internationally as suggested by Bruno H. Solnik, to reduce market risk to some degree through offsetting.
BIBLIOGRAPHY
Howells & Bain (2009), The Economics of Money, Banking and Finance, A European Text, Prentice Hall 4th Edition
Bodie, Kane, Marcus (2008), Essentials of Investments, McGraw-Hill, 7th Edition
Rugman (1996), The theory of multinational enterprises: The selected scientific papers of Alan M. Rugman, Edward Elgar Publishing, 1st Edition
Richard G Lipsey and K Alec Chrystal (2007), The Demand for Insurance,
International Portfolio Diversification (2007)