FFE Notes 14 The Two-Fund Theorem

Financial Economics Notes 14

The Two-Fund Theorem

This is from Luenberger, chapter 6 (with changes in notation).

Consider the following portfolio frontier, in which unrestricted short selling is possible.

E(R)

The two-fund theorem states that any point on the frontier (such as point c ) can be found as a linear combination of any two other points on the frontier (such as a and b ). Multiply the asset weights in portfolio a by α and the asset weights in portfolio b by )1(α-. Adding these weights then defines the weights of the asset in portfolio c .

1. How do we know that the weights are α and )1(α- (that is, they sum to 1)? We know that any portfolio must satisfy a restriction on the asset weights:

∑==n

i i

x

1

1.

This is therefore true for portfolios such as c , formed from a and b . Consider a two asset case with portfolio a having weights 1a x and 2a x , and portfolio b having

weights 1b x and 2b x . If portfolio c is a linear combination of the weights for portfolios a and b , the weights for portfolio c are 111b a c x x x βα+= 222b a c x x x βα+=. Adding the weights for portfolio c gives ()()1212121=+++=+b b a a c c x x x x x x βα. But this means that 1=+βα. In other words, the weights on the two funds, a and b can be written as ()()αα-1,.

In the linear combination the weights must sum to 1. They do not need to be positive and they can have any magnitude.

2. How do we know that this linear combination of frontier portfolios leads to another frontier portfolio?

The portfolio variance minimisation problem is

{}()???? ??-+???? ??-*+????

??????+=∑∑∑∑∑====≠=n i i n i i i n i n i n j i j ij j i i i x R E x E x x x L w 121111122121 minimise λλσσ.

The first-order conditions of this problem give the solution equations ()02111133122211=--++++λλσσσσR E x x x x n n for n i 1=

(1) ()∑==n

i i

i

E R E x 1*

(2)

∑==n

i i

x

1

1

(3)

giving the matrix equation

()()()()??

?

??

??

?

????????++++=???????????????????????????

??

???2123122121132123

2

1323

3231223

2

221113

122

1

λλλλλλλλσσσσσσσσσσσσσσσσn n n n n n n n n R E R E R E R E x x x x

so that solving twice (for two different initial values of *E ) will yield two optimal

weight vectors, x a and x b .

Changing the value of *E will change the solution values of 1λ and 2λ, so the two solutions for the optimal weights will be different, and we can define the solution

values for 1λ and 2λ as a a

21,λλ and b b 21,λλ respectively. Now combine these two sets of optimal weights to find a third portfolio x c , using the linear combination

()b x a x c x αα-+=1 The expected return to this portfolio is

()*1

c

i

n

i c i

E R E x =∑=

which is a legitimate expected return satisfying first-order condition (2) of the solution. This portfolio has weights that sum to 1, as we have already shown, and therefore first-order condition (3) of the solution is satisfied.

Finally, note that both x a and x b satisfy first-order condition (1) of the solution. The linear combination of these portfolios therefore yields x c also satisfying equation (1) of the solution.

(Note that the solution values of a a

21,λλ and b b 21,λλ in the combined first-order

conditions imply ()b a c 1111λααλλ-+= and ()b a

c 2221λααλλ-+=.)

Thus, x c satisfies all the first-order conditions of the minimisation problem and is therefore a solution to the problem. This means that the portfolio defined by the weights x c is a frontier portfolio .

To find the entire frontier, therefore, all you have to do is to take successive values of ()()αα-1, and apply them to the weights of any two frontier portfolios.

Clearly, this involves computation, but could be done with spreadsheet software packages, such as Excel.

It is often convenient to take the global minimum variance portfolio as one of the two frontier portfolios. Luenberger suggests using the portfolios for which 11=a λ and 02=a λ

01=b λ and 12=b λ.

This allows the first-order conditions to be solved with relative ease, although these

solutions may lead to weights that do not sum to unity. In such a case these weights are subsequently normalised so that they sum to unity (see Luenberger p164).

Reference

Luenberger, D.G (1998) Investment Science , Oxford University Press.