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.

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

Explanation:

Solution

Given that:

Now,

The Jensen’s alpha of a Portfolio is computed by applying  the formula  below:

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

For the information given in the question we have the following,

The Risk free rate of return = 3. 1%

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

1. Portfolio Return

2. Portfolio Beta

3.Market Rate of Return

Thus,

(A)Calculation of Portfolio Return :

The formula for calculation of Portfolio Return is  given as:

E(RP) = ( RA * WA )+ ( RB * WB )

Where

E(RP) = Portfolio Return

RA = Average Return of Portfolio A ; WA = Weight of Investment in Portfolio A

RB = Average Return of Portfolio B ;  WB = Weight of Investment in Portfolio B

For the information given in the question we have the following:

RA = 18.9 %, WA = 45 % = 0.45, RB = 13.2 %,  WB = 55 % = 0.55

By applying the values in the formula we have

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

= 8.5050 % + 7.2600 % = 15.7650 %

(B). Calculation of Portfolio Beta:

Now,

The formula for calculating the Portfolio Beta is

ΒP = [ ( WA * βA ) + ( WB * βB ) ]

Where,

βP = Portfolio Beta

WA = Weight of Investment in Portfolio A = 45 % = 0.45 ; βA = Beta of Portfolio A = 1.92

WB = Weight of Investment in Portfolio B = 55 % = 0.55 ; βB = Beta of Portfolio B = 1.27

By Applying the above vales in the formula we have

= ( 0.45 * 1.92 )   + ( 0.55 * 1.27 )

= 0.8640 + 0.6985

= 1.5625

(C). Calculation of Market rate of return :

Now,

The Market Risk Premium = Market rate of return - Risk free rate

From the Information given in the Question we have

The Market Risk Premium = 6.8 %

Risk free rate = 3. 1 %

Market rate of return = To find

Then

By 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 %

So,

From the following  information, we gave

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

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

Now

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

The 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 )

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

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