Suppose there are three securities (A, B, and C) to choose from to create your p

Question

Suppose there are three securities (A, B, and C) to choose from to create your portfolio. Next year the economy will be in an expansion, normal, or recession state with probabilities 0.46, 0.36, and 0.18, respectively. The returns (%) on the securities in these states are as follows: Security A {expansion = +8, normal = +7, recession = +2}; Security B {-1, -1, +5}; Security C {+14, +7, -8}. You are considering 4 potential portfolios of these 3 securities, with the following specific weights on each: Portfolio I (0.20, 0.40, 0.40); Portfolio II (0.38, 0.31, 0.31); Portfolio III (0.52, 0.24, 0.24); Portfolio IV (0.72, 0.14, 0.14); where the numbers inside parentheses are weight of A, weight of B and weight of C, respectively. Which portfolio has the lowest risk?

Explanation / Answer

We have to calculate the variance and the standard deviation of each portfolio to arrive at the riskiness of the portfolio.

First we will calculate the expected return from each portfolio.

Expected return = (w1r1) + (w2r2) + (w3r3)

Where w = weight in the portfolio and r = return of the security.

We will calculate the expected return of each portfolio under all scenarios.

Under the Expansion scenario with probability of 0.46 of 46%

Expected return of Portfolio I = (0.20*0.08) + (0.40*(-0.01)) + (0.40*0.14) = 0.068 = 6.8%

Expected return of Portfolio II = 0.0707 = 7.07%

Expected return of Portfolio III = 0.728 = 7.28%

Expected return of Portfolio IV = 0.0758 = 7.58%

Under the Normal scenario with probability of 0.36 or 36%

Expected return of Portfolio I = 0.038 = 3.8%

Expected return of Portfolio II = 0.0452 = 4.52%

Expected return of Portfolio III = 0.0508 = 5.08%

Expected return of Portfolio IV = 0.0588 = 5.88%

Under the Recession scenario with probability of 0.18 or 18%

Expected return of Portfolio I = -0.008 = -0.8%

Expected return of Portfolio II = -0.0017 = -0.17%

Expected return of Portfolio III = 0.0032 = 0.32%

Expected return of Portfolio IV = 0.0102 = 1.02%

Now we calculate the portfolio variance considering every scenario. The portfolio with minimum variance is the least risky.

Given below is calculation for the variance of the portfolios. From the below calculations it can be seen that Portfolio IV is the least risky with variance of 0.000557.

Portfolio I Scenario Probability Expected Return Average Deviation Variance Portfolio Variance Expansion 0.46 0.068 0.04352 0.02448 0.00059927 0.000764 Normal 0.36 0.038 -0.00552 0.0000304704 Recession 0.18 -0.008 -0.05152 0.00265431 Portfolio II Scenario Probability Expected Return Average Deviation Variance Portfolio Variance Expansion 0.46 0.0707 0.048488 0.022212 0.000493373 0.000684 Normal 0.36 0.0452 -0.00329 0.0000108109 Recession 0.18 -0.0017 -0.05019 0.002518835 Portfolio III Scenario Probability Expected Return Average Deviation Variance Portfolio Variance Expansion 0.46 0.0728 0.052352 0.020448 0.000418121 0.000628 Normal 0.36 0.0508 -0.00155 0.0000024087 Recession 0.18 0.0032 -0.04915 0.002415919 Portfolio IV Scenario Probability Expected Return Average Deviation Variance Portfolio Variance Expansion 0.46 0.0758 0.057872 0.017928 0.000321413 0.000557 Normal 0.36 0.0588 0.000928 0.0000008612 Recession 0.18 0.0102 -0.04767 0.00227262

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