Question

15

What is Jensen's alpha of a portfolio comprised of 45 percent portfolio A and 55 percent of portfolio B?

Portfolio	Average Return		Standard Deviation		Beta
A	18.9	%	21.6	%	1.92
B	13.2		12.8		1.27
The risk-free rate is 3.1 percent and the market risk premium is 6.8 percent.

		2.04 percent
		0.47 percent
		1.08 percent
		1.46 percent
		−1.25 percent

Answer 1

Solution :

The Jensen’s alpha of a Portfolio is calculated using the formula

Jensen's alpha = Portfolio Return − [Risk Free Rate of Return + ( Portfolio Beta * (Market Rate of Return − Risk Free Rate of Return ) ) ]

As per the information given in the question we have

Risk free rate of return = 3. 1%

In order to find the Jensen’s alpha we have to first deduce the following from the information given in the question :

1. Portfolio Return
2. Portfolio Beta
3. Market Rate of Return

a. Calculation of Portfolio Return :

The formula for calculation of Portfolio Return is

E(R_P) = ( R_A * W_A )+ ( R_B * W_B )

Where

E(R_P) = Portfolio Return

R_A = Average Return of Portfolio A    ;    W_A = Weight of Investment in Portfolio A

R_B = Average Return of Portfolio B    ;    W_B = Weight of Investment in Portfolio B

As per the information given in the question we have

R_A = 18.9 %   ; W_A = 45 % = 0.45    ;    R_B = 13.2 %    ;    W_B = 55 % = 0.55

Applying the values in the formula we have

= ( 18.9 % * 0.45 ) + ( 13.2 % * 0.55 )

= 8.5050 % + 7.2600 % = 15.7650 %

Thus the Portfolio Return = 15.7650 %

b. Calculation of Portfolio Beta:

The formula for calculating the Portfolio Beta is

Β_P = [ ( W_A * β_A ) + ( W_B * β_B ) ]

where

β_P = Portfolio Beta ;

W_A = Weight of Investment in Portfolio A = 45 % = 0.45 ; β_A = Beta of Portfolio A = 1.92    ;

W_B = Weight of Investment in Portfolio B = 55 % = 0.55 ; β_B = Beta of Portfolio B = 1.27    ;

Applying the above vales in the formula we have

= ( 0.45 * 1.92 )   + ( 0.55 * 1.27 )

= 0.8640 + 0.6985

= 1.5625

Thus the Portfolio Beta is = 1.5625

C. Calculation of Market rate of return :

We know that Market Risk Premium = Market rate of return - Risk free rate

As per the Information given in the Question we have

Market Risk Premium = 6.8 % ;    Risk free rate = 3. 1 %   ; Market rate of return = To find

Applying the above information in the Market Risk Premium formula we have

6.8 % = Market rate of Return - 3.1 %

Thus Market rate of return = 6.8 % + 3.1 % = 9.9 %

Thus, we now have the following information

Risk free rate of return = 3.1% ; Portfolio Return = 15.7650 %

Portfolio Beta = 1.5625 ; Market Rate of Return = 9.9 %

Applying the above values in the Jensen’s Alpha formula we have

Jensen's alpha = Portfolio Return − [Risk Free Rate of Return + ( Portfolio Beta * (Market Rate of Return − Risk Free Rate of Return )) ]

= 15.7650 % - [ 3.1 % + ( 1.5625 * ( 9.9 % - 3.1 % ) ) ]

= 15.7650 % - [ 3.1 % + ( 1.5625 * 6.8 % ) ]                 

= 15.7650 % - [ 3.1 % + 10.6250 % ]

= 15.7650 % - 13.7250 %

= 2.0400 %

= 2.04 % ( when rounded off to two decimal places )

Thus the Jensen's alpha of a portfolio comprised of 45 percent portfolio A and 55 percent of portfolio B = 2.04 %

The solution is Option 1 = 2.04 Per cent