You conduct the following scenario analysis:EconomyProbabilityPortfolio XPortfolio YVery good0.125%40%Good0.210%15%Average0.45%0%Bad0.2-5%-20%Very bad0.1-15%-50% What are the expected return and standard deviation of portfolio X and Y?What are the Sharpe ratios of the portfolios based on scenario analysis, risk free rate is 1%?

Name: You conduct the following scenario analysis:EconomyProbabilityPortfoli
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Description: You conduct the following scenario analysis:EconomyProbabilityPortfolio XPortfolio YVery good0.125%40%Good0.210%15%Average0.45%0%Bad0.2-5%-20%Very bad0.1-15%-50% What are the expected return and standard deviation of portfolio X and Y?What are the S

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Author:Amos Gilat

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You conduct the following scenario analysis:

Economy	Probability	Portfolio X	Portfolio Y
Very good	0.1	25%	40%
Good	0.2	10%	15%
Average	0.4	5%	0%
Bad	0.2	-5%	-20%
Very bad	0.1	-15%	-50%

What are the expected return and standard deviation of portfolio X and Y?
What are the Sharpe ratios of the portfolios based on scenario analysis, risk free rate is 1%?

Expert Solution

Step 1

Scenario analysis is an technique of analyzing decisions or future possibilities using alternative possible outcomes.

First of all , Expected return and Standard deviation for portfolio X is ,

$\begin{array}{rcl} E x p e c t e d r e t u r n f o r X (i n %) & = & R_{x} = E (X) = \sum_{x} x P (X = x) \\ = & 25 \times 0.1 + 10 \times 0.2 + 5 \times 0.4 + - 5 \times 0.2 + - 15 \times 0.1 \\ = & 4 \\ a n d \\ S \tan d a r d d e v i a t i o n & = & \sqrt{E (X^{2}) - E {(X)}^{2}} \\ C o n i d e r, E (X^{2}) & = & \sum_{x} X^{2} P (X = x) \\ = & 25^{2} \times 0.1 + 10^{2} \times 0.2 + 5^{2} \times 0.4 + {(- 5)}^{2} \times 0.2 + {(- 15)}^{2} \times 0.1 \\ = & 120 \\ H e n c e, S t a n d a r d d e v i a t i o n & = & σ_{x} = \sqrt{E (X^{2}) - E {(X)}^{2}} \\ = & \sqrt{120 - 4^{2}} \\ = & 10.19804 \end{array}$

To find, Expected return and Standard deviation for portfolio Y ,

$\begin{array}{rcl} E x p e c t e d r e t u r n f o r Y (i n %) & = & R_{y} = E (Y) = \sum_{x} y P (Y = y) \\ = & 40 \times 0.1 + 15 \times 0.2 + 0 \times 0.4 + - 20 \times 0.2 + - 50 \times 0.1 \\ = & - 2 \\ a n d \\ S \tan d a r d d e v i a t i o n & = & \sqrt{E (Y^{2}) - E {(Y)}^{2}} \\ C o n i d e r, E (Y^{2}) & = & \sum_{x} Y^{2} P (Y = y) \\ = & 40^{2} \times 0.1 + 15^{2} \times 0.2 + 0^{2} \times 0.4 + {(- 20)}^{2} \times 0.2 + {(- 50)}^{2} \times 0.1 \\ = & 160 + 45 + 0 + 80 + 250 \\ = & 535 \\ H e n c e, S t a n d a r d d e v i a t i o n & = & σ_{y} = \sqrt{E (X^{2}) - E {(X)}^{2}} \\ = & \sqrt{535 - {(- 2)}^{2}} \\ = & 23.04344 \end{array}$

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