In order to calculate the expected return of a security, it can be calculated by:
ERi is the expected return on a Security i.
In order to measure how much the outcomes differ from the mean, the average squared deviation is our Variance(Ïƒ2i) when each return is equally likely can be shown as follows:
and the square root of the variance is the Standard Deviation(Ïƒi).
We can therefore see that security UNP has a higher return (0.68%) over the three year period but at the same time also a higher risk (0.22%). Security COP has a lesser return (0.34%) however less volatility and risk (0.14%).
Calculating the variance of the portfolio after having simplified it:
Below are the results of the expected return and standard deviation of the portfolio of various :
Using the numbers from the example above we can see that by combining the two securities, the variance could be held as low as 0.14% but would result in a higher return of 0.49% taking the weights of 45% UNP and 55% COP not allowing any short sales.
In summary, in order to create an optimal combination of the portfolio I am proposing to maximize the return on the investment, measured by its mean, and minimize risk, measured by its variance which would result in a return of 0.49%, based on historic observation of three years from 12.1.2009- 07.01.2012 mentioned and minimizing the variation or risk to 0.14%% which is achieved by the mix of 45% UNP and 55% COP. It has to be noted, as we have calculated the returns over period of three years the calculation it is only indicative for the expected return of future three years period.
Using the above explained formulas the result for the portfolio in the time between 17/1/2012-07/01/2013 is as follows:
Looking further now, comparing with the weights 45% and 55% as chosen before, this seems to be the ideal combination also in the later example:
As we can see we have been able to reduce the overall risk with the lowest being 0.05% variance to 0.04% and achieving a mean return on the portfolio of 0.33%. Therefore by combining the two securities, we have achieved a lower risk profile and higher return than if we had only invested into COP in that time period.
Though when compared to the three year period before, it becomes clear that previously we have been able to obtain a higher return on the same weightings though with the result of a higher risk.
The single Index Model is in simple words a facilitation of the Markowitz Model as it needs less estimates. It assumes for example that there is only 1 macroeconomic factor (m) that causes the systematic risk affecting all stock returns. This factor can be the rate of return on a market index, such as the NYSE Composite Index in our example. It is therefore faster to consult but as it works with estimates less accurate.
The single index model distinguishes between the expected excess return of the individual stock due to firm-specific factors written as alpha coefficient (Î±), the return due to macroeconomic events that affect the market, and the unexpected microeconomic events that affect only the firm.
As we have seen in Q1a our securities moved together as their covariance was positive. Supposing we can summarize all the common effects into one macroeconomic variable the ri = return of security i. We could say that the sum of the return on i is formed by two components: E xpected (ri) â€“ which is the expected part and ei, which is the unexpected part.
And the return on stock i can...