Case Study #1
1) In order to calculate the expected return, risk premium, and standard deviation of theportfolio invested partly in the market and partly in Pioneer, we first needed to devise a table with all of the known variables:
Table 1
Pioneer Gypsum (X) Market (Y)
Expected Return 11.0% 12.5%
Standard Dev. 32% 16%
Beta 0.65 N/A
The calculation of the expected return, risk premium and the standard deviation of theportfolio are dependent upon the amount that John wants to invest. For example, if Johnwanted to allocate 2.5% of his funds in Pioneer and the rest in the market, the expected return, the risk premium and the standard deviation would look something like this:
Table 2:
% Pioneer Invested % Market Invested Expected Return Risk Premium Std Dev
2.50% 97.50% 12.46% 7.46% 0.15878
For the explanation of calculations, see below:
If we call Pioneer as Stock A, market as Stock M, and we are to mix fractions of A and M, then A + M = 1, since all investments must equal 100%. Per table 1 above, the expected return of A is 11% and M is 12.5%.
We can then calculate John’s portfolio’s expected return by utilizing the following formula:
Expected return = A*11% + y*12.5%
If we use the example in Table 2, then our formula would look something like this:
Expected return = (0.025*11% + 0.975*12.5%) which equals to 12.46%
Risk Premium can be calculated by utilizing the following formula:
Risk Premium = Expected Return – Risk Free Interest Rate (5%)
= 12.46% - 5% which equals 7.46%
To calculate Standard Deviation, John could use the following formula: δ = √(A2*σa2+M2*σm2+2*A*M*(Ρ* σa* σm))
Where we derived (Ρ* σa* σm) from β = σam/σm2, in which the equation for β can be expanded to (Ρ* σa* σm)/ σm2.
Therefore if we use the example from Table 2, we would get an equation that looks something like this: δ =