Step-by-step explanation:
a. Expected Return of each Asset:
The expected return of an asset is calculated by multiplying each return with its corresponding probability and then summing up the results.
For Asset R:
Expected Return (R) = (0.15 * 0.040) + (0.25 * 0.040) + (0.35 * 0.040) + (0.25 * 0.040)
Expected Return (R) = 0.006 + 0.010 + 0.014 + 0.010
Expected Return (R) = 0.040 or 4.0%
For Asset S:
Expected Return (S) = (0.15 * 0.280) + (0.25 * 0.140) + (0.35 * 0.070) + (0.25 * (-0.035))
Expected Return (S) = 0.042 + 0.035 + 0.025 - 0.009
Expected Return (S) = 0.093 or 9.3%
For Asset T:
Expected Return (T) = (0.15 * 0.450) + (0.25 * 0.275) + (0.35 * 0.025) + (0.25 * (-0.175))
Expected Return (T) = 0.068 + 0.069 + 0.009 - 0.044
Expected Return (T) = 0.102 or 10.2%
b. Variance and Standard Deviation of each Asset:
The variance of an asset is calculated by summing up the squared difference between each return and the expected return, multiplied by its corresponding probability.
For Asset R:
Variance (R) = (0.15 * (0.040 - 0.040)^2) + (0.25 * (0.040 - 0.040)^2) + (0.35 * (0.040 - 0.040)^2) + (0.25 * (0.040 - 0.040)^2)
Variance (R) = 0
Standard Deviation (R) = √Variance (R) = √0 = 0
For Asset S:
Variance (S) = (0.15 * (0.280 - 0.093)^2) + (0.25 * (0.140 - 0.093)^2) + (0.35 * (0.070 - 0.093)^2) + (0.25 * (-0.035 - 0.093)^2)
Variance (S) = 0.0110
Standard Deviation (S) = √Variance (S) = √0.0110 ≈ 0.105 or 10.5%
For Asset T:
Variance (T) = (0.15 * (0.450 - 0.102)^2) + (0.25 * (0.275 - 0.102)^2) + (0.35 * (0.025 - 0.102)^2) + (0.25 * (-0.175 - 0.102)^2)
Variance (T) = 0.0360
Standard Deviation (T) = √Variance (T) = √0.0360 ≈ 0.190 or 19.0%
c. Expected Return of a Portfolio with Equal Investment in all Three Assets:
Since the investment is equal in all three assets, the expected return of the portfolio is the average of the expected returns of the three assets.
Expected Return (Portfolio) = (0.040 + 0.093 + 0.102) / 3
Expected Return (Portfolio) = 0.078 or 7.8%
d. Portfolio's Variance and Standard Deviation with the same Asset Weights as in part (c):
When all assets have equal weights in the portfolio, the portfolio's variance can be calculated as follows:
Portfolio Variance = (Weight^2 * Variance of Asset R) + (Weight^2 * Variance of Asset S) + (Weight^2 * Variance of Asset T)
Where Weight = 1/3 (since each asset has equal weight).
Portfolio Variance = (1/3)^2 * 0 + (1/3)^2 * 0.0110 + (1/3)^2 * 0.0360
Portfolio Variance = 0.00121
Portfolio Standard Deviation = √Portfolio Variance = √0.00121 ≈ 0.035 or 3.5%
So, the portfolio's variance with equal investment in all three assets is 0.00121, and the standard deviation is approximately 3.5%.