Question

You are considering investing in two stocks, stock X and stock Y. Given your research, you expect two possible scenarios for the future: a bull market and a bear market. You also uncovered the return distribution of X and Y:

Scenarios	Probabilities	Return for Stock X	Return for Stock Y
Bull	0.3	0.8	-0.3
Bear	0.7	0.4	0.1

1. Compute the expected return of X and Y. 

2. Compute the standard deviation of X and Y. 

3. Compute the Sharpe ratio of X and Y. Assume the risk-free rate is 1%. 

4. Compute the covariance and correlation coefficient between X and Y. 

Answer 1

Part 1:

Expected Ret = Sum [ Prob * ret ]

Stock X:

Scenario	Prob	Ret	Prob * Ret
Bull	0.3000	0.8000	0.2400
Bear	0.7000	0.4000	0.2800
Expected Ret			0.5200

Stock Y:

Scenario	Prob	Ret	Prob * Ret
Bull	0.3000	(0.3000)	(0.0900)
Bear	0.7000	0.1000	0.0700
Expected Ret			(0.0200)

Standard Deviation:

Standard deviation is a measure of amount of variation or dispersion of set of values. It spcifies the risk of set of values.

SD = SQRT [ SUm [ Prob * (X-AVgX)^2 ] ]

Stock X:

State	Prob	Ret (X)	(X-AvgX)	(X-AvgX)^2	Prob * (X-Avg X)^2
Bull	0.3000	0.8000	0.2800	0.078400	0.02352
Bear	0.7000	0.4000	(0.1200)	0.014400	0.01008
Sum[ Prob * ( X-AvgX)^2 ) ]					0.03360
SD = SQRT [ [ Sum[ Prob * ( X-AvgX)^2 ) ] ] ]					0.18330

SD is 18.33%

STock Y:

State	Prob	Ret (X)	(X-AvgX)	(X-AvgX)^2	Prob * (X-Avg X)^2
Bull	0.3000	(0.3000)	(0.2800)	0.078400	0.02352
Bear	0.7000	0.1000	0.1200	0.014400	0.01008
Sum[ Prob * ( X-AvgX)^2 ) ]					0.03360
SD = SQRT [ [ Sum[ Prob * ( X-AvgX)^2 ) ] ] ]					0.18330

SD is 18.33%

Part C:

Sharpe Ratio:
The ratio is the average return earned in excess of the risk-free rate per unit of total risk or Volatality. It is alsoknown as Reward to Variability ratio.

Sharpe ratio = [ Expected Ret - Rf ] / SD

Rf - Risk free Ret

Higher Ratio will be given better ranking.

Stock X:

Particulars	Amount
Expected Ret	52.00%
Risk Free Ret	1.00%
SD	18.33%

Sharpe ratio = [ Expected Ret - Rf ] / SD
= [ 52 % - 1 % ] / 18.33 %
= [ 51 % ] / 18.33 %
= 2.7823

Stock Y:

Particulars	Amount
Expected Ret	-2.00%
Risk Free Ret	1.00%
SD	18.33%

Sharpe ratio = [ Expected Ret - Rf ] / SD
= [ -2 % - 1 % ] / 18.33 %
= [ -3 % ] / 18.33 %
= -0.1637

Part D:

Covariance:

Covariance is a statistical tool that is used to determine the relationship between the movement of two assets/ Stocks.
Covariance = Sum [ prob * (X-Avg X)(Y-Avg Y) ]

Scenario	Prob	Ret (X)	(X-AvgX)	Ret (Y)	(Y-AvgY)	(X-AVgX)(Y-AvgY)	Prob* (X-AVgX)(Y-AvgY)
Bull	0.3000	80.00%	28.00%	-30.00%	-28.00%	-0.0784	-0.02352
Bear	0.7000	40.00%	-12.00%	10.00%	12.00%	-0.0144	-0.01008
Covariance = Sum [Prob * (X-AvgX)(Y-AvgY) ]							-0.03360

Correlation:

Particulars	Values
Covariance (X , Y)	-0.0336
SD of X	18.33%
SD of Y	18.33%

Correlation (r ) = Covariance ( X , Y ) / [ SD of X * SD of Y ]
= -0.0336 / [ 0.18 * 0.18 ]
= -0.0336 / [ 0.03 ]
= -1

Correaltion (X , Y ) is -1