Question

Assume we have equally invested in two different companies; ZICTA and AIRTEL. We anticipate that there is a 15% chance that next year’s stock returns for ZICTA will be 6%, a 60% probability that they will be 8% and a 25% probability that they will be 10%. In addition, we already know the expected value of returns is 8.2%, and the standard deviation is 1.249%. We also anticipate that the same probabilities and states are associated with a 4% return for AIRTEL, a 5% return, and a 5.5% return. The expected value of returns is then 4.975 and the standard deviation is 0.46%.

Calculate the portfolio standard deviation

Answer 1

Calculation of Portfolio standard deviation:

Given,

ZICTA	AIRTEL
Probability	Return	Probability	Return
15%	6%	15%	4%
60%	8%	60%	5%
25%	10%	25%	5.5%
Expected return	8.2%	Expected return	4.975%
Standard Deviation	1.249%	Standard Deviation	0.46%

Let stock of ZICTA be represented by X

Stock of AIRTEL be represented by Y

and Portfolio be represented by XY

Given Investment made in 2 stocks equally

Therefore Proportion of stock X (PX) = 50%

Proportion of stock Y (PY) = 50%

Covariance of XY:

Cov(X,Y) = Σ(Probability )(Return of X - Expected return of X)(Return of Y - Expected return of Y)

Cov(X,Y) = (0.15)(6 - 8.2)(4 - 4.975) + (0.60)(8 - 8.2)(5 - 4.975) +( 0.25)(10 - 8.2)(5.5 - 4.975)

= 0.15(-2.2)(-0.975) + 0.60(-0.2)(0.025) + 0.25(1.8)(0.525)

= 0.32175-0.003+0.23625

= 0.555

Correlation coefficient between X and Y:

r(XY) = Cov(X,Y)÷ (σX)(σY)

r(XY) = 0.555 ÷ (1.249)(0.46)

r(XY) = 0.555 ÷ 0.57454

r(XY) = 0.966

Standard Deviation of Portfolio:

√(PX)^2(σX)^2 + (PY)^2(σY)^2 + 2(PX)(PY)(σX)(σY)(r(XY))

= √(0.5)^2(1.249)^2 + (0.5)^2(0.46)^2 + 2(0.5)(0.5)(1.249)(0.46)(0.966)

= √( 0.25)(1.56) + (0.25)(0.2116) + 0.2775

= √ 0.39 + 0.0529 + 0.2775

= √ 0.7204

= 0.849%