An analyst makes the following forecasts about the economy for next year and the rates of return of a portfolio:EconomyProbabilityRate of ReturnRecession 25%-30%Normal35% 5%Recovery40%20% Calculate the standard deviation of the expected rates of return (as a percentage) for the portfolio for the one-year period.
Question
An analyst makes the following forecasts about the economy for next year and the rates of return of a portfolio:EconomyProbabilityRate of ReturnRecession 25%-30%Normal35% 5%Recovery40%20% Calculate the standard deviation of the expected rates of return (as a percentage) for the portfolio for the one-year period.
Solution 1
To calculate the standard deviation of the expected rates of return for the portfolio, we first need to calculate the expected return and then use this to find the variance, the square root of which is the standard deviation.
Step 1: Calculate the Expected Return The expected return is the sum of the product of the probability of each state of the economy and the corresponding rate of return.
Expected Return = (0.25*-30%) + (0.355%) + (0.4020%) = -7.5% + 1.75% + 8% = 2.25%
Step 2: Calculate the Variance The variance is the sum of the squared difference between each rate of return and the expected return, multiplied by the probability of each state of the economy.
Variance = 0.25*(-30% - 2.25%)^2 + 0.35*(5% - 2.25%)^2 + 0.40*(20% - 2.25%)^2
Variance = 0.25*(0.3225) + 0.35*(0.007225) + 0.40*(0.315625) = 0.080625 + 0.00252875 + 0.12625 = 0.20940375
Step 3: Calculate the Standard Deviation The standard deviation is the square root of the variance.
Standard Deviation = sqrt(0.20940375) = 45.76%
So, the standard deviation of the expected rates of return for the portfolio for the one-year period is 45.76%.
Solution 2
To calculate the standard deviation of the expected rates of return for the portfolio, we first need to calculate the expected return and then use this to find the variance, the square root of which is the standard deviation.
Step 1: Calculate the Expected Return The expected return is the sum of the product of the probability of each state of the economy and the corresponding rate of return.
Expected Return = (0.25*-30%) + (0.355%) + (0.4020%) = -7.5% + 1.75% + 8% = 2.25%
Step 2: Calculate the Variance The variance is the sum of the squared difference between each rate of return and the expected return, multiplied by the probability of each state of the economy.
Variance = 0.25*(-30% - 2.25%)^2 + 0.35*(5% - 2.25%)^2 + 0.40*(20% - 2.25%)^2
Variance = 0.25*(0.3225) + 0.35*(0.007225) + 0.40*(0.315625) = 0.080625 + 0.00252875 + 0.12625 = 0.20940375
Step 3: Calculate the Standard Deviation The standard deviation is the square root of the variance.
Standard Deviation = sqrt(0.20940375) = 45.76%
So, the standard deviation of the expected rates of return for the portfolio for the one-year period is 45.76%.
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