Question

Tracking Portfolio- Example

Consider three securities whose expected returns and factor sensitivities are given by:

rA = 0.45 + 1.5F1 − 4F2 + εA

rB = 0.05 + 3F1 + 2F2 + εB

rC = 0.08 + 1.2F1 + 0F2 + εC

Suppose we wish to construct a tracking portfolio with β1 = 1.8 and β2 = 1.

Determine the proportion to be invested in each security.

Simultaneous Equations

xA + xB + xC = 1

1.5xA + 3xB + 1.2xC = 1.8

−4xA + 2xB + 0 = 1

solution: xA = −0.08; xB = 0.35; xC = 0.73

Could you please explain what is happening in this example. I don't understand what the point of this is and what they are solving for and how they are getting the first simultaneous equation.

Answer 1

Assume you are investing in three securities.

If Total amount invested is say, $100 million and $20 million is invested in Security A, $30 million in B and $50million in C

XA=Weight of Security A in the portfolio=20/100=0.2

XB=Weight of Security B in the portfolio=30/100=0.3

XC=Weight of Security A in the portfolio=50/100=0.5

XA+XB+XC=0.2+0.3+0.5=1

In this problem we are trying to find out allocation of total investments to the three securities , such that the Portfolio tracks certain return with Beta1=1.8 and Beta2=1

Beta indicates the variation of security return with the return of the market.

As shown in the return equation(Coefficients of F1)

Beta1 of Security A=1.5

Beta1 of Security B=3

Beta1 of Security C=1.2

When we make a portfolio with three securities with weight XA, XB and XC,

The total of XA, XB and XC will be equal to one as shown above.

XA+XB+XC=1……..Equation (1)

Now, Portfolio Beta =Sum of (Weight * Beta )of each security in the portfolio.

Required Portfolio Beta 1=1.8

Hence,

1.5XA+3XB+1.2XC=1.8…….Equation (2)

Required Beta2=1

As shown in the return equation(Coefficients of F2)

Beta1 of Security A=-4

Beta1 of Security B=2

Beta1 of Security C=0

-4XA+2XB+0=1………..Equation (3)

The whole purpose of the exercise is to find the allocation of investments (Represented by weights XA,XB and XC) to the three securities, so that , the desired portfolio Beta1 and Beta2 will be achieved .

There are three unknowns(XA,XB and XC) and three equations .

35% of investment will be invested in security B

73% of investment will be invested in security C

8% amount will be generated by short selling Security C

In order to achieve required tracking Beta1 and Beta2