Consider the following information: Rate of Return If State Occurs State of Prob

Question

Consider the following information: Rate of Return If State Occurs State of Probability of Economy State of Economy Stock A Stock B Stock C Boom 0.17 0.352 0.452 0.332 Good 0.43 0.122 0.102 0.172 Poor 0.33 0.012 0.022 ?0.052 Bust 0.07 ?0.112 ?0.252 ?0.092 Requirement 1: Your portfolio is invested 32 percent each in A and C and 36 percent in B. What is the expected return of the portfolio? (Do not include the percent sign (%). Enter rounded answer as directed, but do not use the rounded numbers in intermediate calculations. Round your answer to 2 decimal places. Expected return of the portfolio ??? Requirement 2: (A) What is the variance of this portfolio? (Enter rounded answer as directed, but do not use the rounded numbers in intermediate calculations. Round your answer to 5 decimal places. Variance of the portfolio ???? (B) What is the standard deviation of this portfolio? (Do not include the percent sign (%). Enter rounded answer as directed, but do not use the rounded numbers in intermediate calculations. Round your answer to 2 decimal places. Standard deviation ????%

Explanation / Answer

OK, you really should have made your question easier to read. I am guessing that you really meant to say the following: Asset #1 has annual returns as follow: 2000 - 5 2001 - (9) 2002 - 4 2003 - 16 2004 - 14 Asset #2 has annual returns as follow: 2000 - 9 2001 - 10 2002 - (4) 2003 - (9) 2004 - 2 I will answer your question based on the above scenarios, so if this is incorrect, you know why. In order to determine the MVP, we must know the basic statistical data for each issue. The means we must determine the mean, variance, and standard deviation for each asset. In order to calculate variances, you must first calculate the mean. To calculate the mean, you sum the variables and divide by the number of variables. Here are the mean calculations for each asset: Asset #1: 5 - 9 + 4 + 16 + 14 = 30 Mean (Asset #1) = 30 / 5 = 6 Asset #2: 9 + 10 - 4 - 9 + 2 = 8 Mean (Asset #2) = 8 / 5 = 1.60 To calculate the variance, you determine the distance of each variable from the mean, square that difference, and then sum the squares. Divide this sum by the number of factors, less one (n-1). For each asset: Asset #1 (Mean = 6): 6 - 5 = 1, 1^2 = 1 6 - (9) = (15), (15)^2 = 225 6 - 4 = 2, 2^2 = 4 6 - 16 = (10), (10)^2 = 100 6 - 14 = (8), (8)^2 = 64 Variance (Asset #1): 1 + 225 + 4 + 100 + 64 = 394 394 / (5-1) = 394 / 4 = 98.50 Asset #2 (Mean = 1.60): 1.6 - 9 = (7.4), (7.4)^2 = 54.76 1.6 - 10 = (8.4), (8.4)^2 = 70.56 1.6 - (4) = 5.6, 5.6^2 = 31.36 1.6 - (9) = 10.6, 10.6^2 = 112.36 1.6 - 2 = (0.4), (0.4)^2 = .16 Variance (Asset #2): 54.76 + 70.56 + 31.36 + 112.36 + .16 = 269.20 269.20 / (5 - 1) = 269.20 / 4 = 67.30 To calculate the standard deviation, you simply take the square root of the variance. This give us: Std Dev of (Asset #1) = sqrt(98.5) = 9.92 Std Dev of (Asset #2) = sqrt(67.3) = 8.20 You won't use the Standard Deviation for this calculation, but it is important to understand what it represents. The Standard Deviation is the beta coefficient (Beta) for each asset. Beta determines riskiness of the issue. Since the baseline for Beta = 1 (usually the S&P Index), a number greater than one means the stock is riskier than the index, and a number less than one means that the stock is less risky. Both of these issues carry high risk. The goal of determining MVP is to optimize risk and expected return. If done properly, we will find the point of the Efficient Frontier, which gives us our our greatest risk/reward scenario (left-most graphical point). This is the point at which combined standard deviation is minimized. Taking into account the data we have so far, the Mean values tell us the expected return, while the variance and standard deviation figures give us the risk and range of the returns. A higher standard deviation (and variance) denotes a higher risk and a higher volatility. Now that we have our basic statistical information gathered, we can proceed to calculate MVP for the portfolio, and determine what proportion of each asset should be included. Asset #1: Expected Return = 6, Std Dev = 9.92 Asset #2: Expected Return = 1.6, Std Dev = 8.20 Now, we (finally) get to the ugly part of the math. Since you didn't provide a correlation coefficient for the securities, we will assume that they have zero correlation. If this is incorrect, just let me know. To calculate the MVP, we are determining the proportion of each Asset in your portfolio. Of course, this can be done with any number of assets, but we only have two in this case. Your calculation (as best as I can draw it here) is as follows (W = Weight or %): W(Asset #1) = Var(Asset #2) / [Var(Asset #1) + Var(Asset #2)] = 67.3 / (98.50 + 67.30) = 67.3 / 165.80 = 0.4059 W(Asset #2) = 1 - W(Asset #1) = 1 - 40.59% = 59.41% Your portfolio of $X, would consist of: 0.4059($X) x Asset #1 and 0.5941($X) x Asset #2 The workbooks linked below may help you understand better also. Somehow, I think this should be worth a lot more than 10 points! :-) Anyway, I hope I have helped you to understand this, and that you will be able to handle any problem you face with this on an exam or otherwise.

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