Question 1a
In order to calculate the expected return of a security, it can be calculated by:
Ri=ERi=j=1MPijRij
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(σ2i) when each return is equally likely can be shown as follows:
σ2i=1nn=1n(μ-Ri)2
and the square root of the variance is the Standard Deviation(σi).
σ=1nn=1n(μ-Ri)2
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%).
The covariance determines how dependent one security is from another.
σUNP,COP=i=1nRUNPi*RCOPin-(μUNP*μCOP)
Whereby RUNPi is the return of security UNP in period i and μUNP return of security UNP. Same respectively for the second security. In our example this is 0.11%. This positive correlation proposes that the returns on assets tend to rise and fall together.
The correlation coefficient statistically measures the interrelationship between two securities. It is therefore positive when both securities move in the same direction, and negative when they move in opposite directions. This analysis is useful to determine relationships between sectors, stocks and markets. Here the correlation coefficient between eth two securities is 0.624.
As we are estimating possible outcomes and the associated probabilities based on historic data to determine the security’s return, we have to multiply the variance formula by M/(M-1) to get an estimate of the variance that is unbiased but has the disadvantage of producing poorer estimates of the true variance
Considering securities in a portfolio, here UNP and COP, full distribution of funds is assumed .Holding the securities U and C in Proportion Xithen the expected return on the portfolio can be shown as follows:
Rp=ERp=i=1NXiRi
The variance...