Optimal Portfolios without Bounds on Holdings

Characteristics of Optimal Portfolios

Assume that a standard asset allocation problem has been solved and an optimal portfolio obtained. Each asset will be in one of three states, depending on the amount invested relative to its required upper and lower bounds. We can term these "down", "in" and "up", as follows:

down:     l(i) = x(i)
in:       l(i) < x(i) < u(i)
up:              x(i) = u(i)

Since the portfolio is optimal, it must be the case that the marginal utilities of all the "in-variables" are the same. If this were not the case, it would be possible to improve utility by reallocating money from an in-variable to another with a higher marginal utility. Thus:

for all in-variables: mu(i) = mup

where mup is a constant.

It must also be the case that the marginal utility of every down-variable is less than or equal to this amount. If this were not the case, it would be possible to increase the amount of money allocated to such a down-variable and reduce the amount allocated to an in-variable, thereby increasing utility. Thus:

for all down-variables: mu(i) <= mup

Finally, the marginal utility of every up-variable must be greater than or equal to that of every in-variable --otherwise utility could be increased by reducing the amount invested in an up-variable and increasing the amount invested in an in-variable. Thus:

for all up-variables: mu(i) >= mup

We will exploit the implications of all these characteristics when deriving the critical line method for solving general asset allocation problems. First, however, we will focus on cases in which upper and lower bounds on asset holdings are either absent or non-binding. In such instances it is possible to determine optimal portfolio composition analytically -- that is, without using iterative procedures such as those required in the gradient and critical line methods.

Optimal Portfolio Composition without Bounds on Holdings

If there are no bounds on portfolio holdings, one can use the gradient method to determine an optimal portfolio by setting all upper bounds to plus infinity and all lower bounds to minus infinity. In the resultant optimal portfolio, of course, all variables will be in (that is, between their bounds). As a result, all will have the same marginal utility. Thus we know that for the optimal portfolio in such a setting:

mu(i) = mup   :  for all i

Now, recall the formula for calculating the marginal utilities of a set of assets:

mu = e - (1/rt) * 2*C*x

For asset i:

mu(i) = e(i) - (1/rt) * [ 2*C(i,1)*x(1) + 2*C(i,2)*x(2) + ... + 2*C(i,n)*x(n) ]

We seek the composition of the optimal portfolio (x(1), x(2), ... x(n)) and the common marginal utility of all the assets in that portfolio (mup). This requires that all the marginal utilities be the same, that is:

e(i) - (1/rt) * [ 2*C(i,1)*x(1) + 2*C(i,2)*x(2) + ... + 2*C(i,n)*x(n) ] = mup

for assets 1,2,...n.

Note that each of these equations is linear and that there are n such equations. But there is one more requirement for a standard portfolio problem:

sum(x) = k

In the absence of liabilities, etc. this takes the familiar form:

sum(x) = 1

This is a linear equation in the x-variables, so we now have n+1 linear equations in n+1 unknowns. Barring degeneracy, this can be solved by simple matrix inversion and the optimal portfolio obtained directly.

It is useful to state the problem using a straightforward matrix equation. This will ease the task of providing a solution, facilitate extensions, and most importantly, make evident a number of characteristics of solutions to particular classes of problems.

To begin, we rewrite the requirement for asset i as:

2*C(i,1)*x(1) + 2*C(i,2)*x(2) + ... + 2*C(i,n)*x(n) + tmup = rt*e(i)

where:

tmup = rt*mup

We will treat tmup as an unknown, recognizing that mup can always be computed after the fact by dividing tmup by rt. In practice, the values of tmup and mup may be only of passing interest, since the primary goal is to determine the composition of the optimal portfolio.

Note that all the unknowns are now on the left-hand side. Let y be an {(n+1)*1}element vector that includes all the variables. Here:

y = [ x(1)
x(2)
...
x(n)
tmup  ]

To memorialize the fact that this vector contains the unknown x-variables, we name it y, which is one letter from x, a convention dating back to the computer in the film "2001" which was named HAL, one letter removed from IBM (although in the other direction)..

Now, let D be an {(n+1)*(n+1)}element matrix that includes information about the asset covariances and the constraint that the sum of the holdings equals a constant:

D = [ 2*C(1,1)  2*C(1,2) ...  2*C(1,n)   1
2*C(2,1)  2*C(2,2) ...  2*C(2,n)   1
.....
2*C(n,1)  2*C(n,2) ...  2*C(n,n)   1
1         1     ...     1       0 ]

Note that this matrix is formed by bordering two times the covariance with the coefficients from the left-hand side of the constraint, hence the name D (one letter after C).

For the remainder of the equation we need two more vectors. The first contains the right-hand side for the full-investment constraint in row (n+1) and zeros elsewhere. At the risk of some temporary confusion we will call this k. For the case in which the sum of the x values must equal 1:

k = [ 0
0
..
0
1 ]

Note that k contains the constant (hence "k") from the right-hand side of the constraint.

The last vector contains the asset expected returns in the first n rows and zero in the n+1'st row:

f = [ e(1)
e(2)
...
e(n)
0   ]

Since this contains the expected returns (e), we use the next letter (f) for its name.

We can now write a single matrix equation that contains all the conditions required for an optimal portfolio. It is both simple and elegant:

D*y = k + rt*f

Of course we seek the portfolio that makes this equation hold. Since vector y contains the portfolio, we need to solve for y. This is simply done by multiplying both sides of the equation by the inverse of D. The result is:

y = inv(D)*k + rt*inv(D)*f

For purposes of interpretation (and in some instances, implementation) it is useful to write this as:

y = mvp + rt*z

where:

mvp = inv(D)*k

and:

z = inv(D)*f

A Three-Asset Example

To illustrate the use of the formulas for optimal portfolio composition without upper and lower bounds we return to the simple three-asset (cash, bonds and stocks) case used earlier. Expected real returns, risks and correlations are:

e =[ 2.80
6.30
10.80 ]

sd =[ 1.00
7.40
15.40 ]

cc = [ 1.00    0.40    0.15
0.40    1.00    0.35
0.15    0.35    1.00 ]

The corresponding covariance matrix is:

C = (sd*sd').*cc
=    1.000    2.960    2.310
2.960   54.760   39.886
2.310   39.886  237.160 ]

We assume the Investor has a risk tolerance (rt) equal to 25 and require that the sum of the holdings equals 1.

This gives the following matrices:

D = [ 2*C        ones(3,1)
ones(1,3)      0    ]
D =    2.0000    5.9200    4.6200    1.0000
5.9200  109.5200   79.7720    1.0000
4.6200   79.7720  474.3200    1.0000
1.0000    1.0000    1.0000         0 ]
k = [ zeros(3,1)
1      ]
k = [ 0
0
0
1  ]
f = [ e
0 ]
f = [  2.80
6.30
10.80
0   ]

We can now find the components of the solution (mvp and z) and the solution (y):

mvp = inv(D)*k
mvp = [ 1.0392
-0.0396
0.0004
-1.8458  ]
z = inv(D)*f
z = [ -0.0389
0.0257
0.0132
2.6648  ]
y = mvp + rt*z
y = [ 0.0671
0.6021
0.3308
64.7731 ]

The first three elements of y contain the optimal portfolio holdings. In this case the best combination involves 6.71% in cash, 60.21% in bonds and 33.08% in stocks. Since every holding is positive, this would also be the optimal combination for an Investor unable to take short positions in assets. However, this need not always be the case. Consider, for example, an Investor with a risk tolerance of 50. For such an Investor:

y = mvp + 50*z
y = [ -0.9050
1.2439
0.6611
131.3920  ]

Here the optimal investment involves 124.39% in bonds, 66.11% in stocks and borrowing (a negative cash position) an amount equal to 90.50% of the initial value in order to finance the holdings in bonds and stocks.

Characteristics of Optimal Portfolios without Bounds on Holdings

As we have shown, the solution to the problem of portfolio choice without bounds on holdings can be written as an equation involving three vectors (y, mvp, z) and a constant (rt):

y = mvp + rt*z

We turn now to the properties of vectors mvp and z, which tell us a great deal about optimal portfolio holdings.

The Minimum variance Portfolio

Consider an Investor who wishes to minimize risk, no matter how much expected return is sacrificed in the process. Such a person will have a risk tolerance of zero. The optimal portfolio will, in turn, be given by

y = mvp + 0*z
= mvp

Thus mvp is the minimum variance portfolio (hence its name).

In our simple example:

mvp = [ 1.0392
-0.0396
0.0004
-1.8458  ]

It is easy to verify the fact that this is indeed a portfolio, since the sum of the x-values (elements 1 through 3) equals 1.0.

To see why mvp must be a portfolio in this sense, it is useful to consider the problem for which it is a solution. Start with the original problem formulation:

D*y = k + rt*f

In this case:

D*y = k + 0*f

or:

D*y = k

for which the solution is:

y = inv(D)*k

Recall the components of the matrices and vectors in this case:

 2*C(1,1) 2*C(1,2) 2*C(1,3) 1 x(1) 0 2*C(2,1) 2*C(2,2) 2*C(2,3) 1 * x(2) = 0 2*C(3,1) 2*C(3,2) 2*C(3,3) 1 x(3) 0 1 1 1 0 tmup 1

The last row requires that:

1*x(1) + 1*x(2) + 1*x(3) + 0*tmup = 1

Hence the solution must be a portfolio, that is the sum of the holdings must equal 1.

In this example, the minimum-variance portfolio involves borrowing an amount equal to 3.96% of the Investor's funds by issuing (shorting) bonds, then investing the proceeds plus all the Investor's original funds in a combination of cash and a minuscule amount of stocks. The portfolio proper is:

x = y(1:3,1)

and its variance (vp) is:

vp = x'*C*x
= 0.9229

giving a standard deviation of:

sdp = sqrt(vp)
= 0.9607

Note that this is less than the standard deviation of cash, which equals 1.0. Such is the power of diversification.

In this example, cash is not riskless, since returns are in real (inflation-adjusted) terms. Had nominal returns been used, cash could have been considered riskless (if the holding period and the investment period for the cash asset were the same). Consider, for example a case in which the variance of cash and its covariance with every other asset is zero. Retaining the original assumptions for bond and stock risks and correlations:

D =         0         0         0   -1.0000
0  109.5200   79.7720   -1.0000
0   79.7720  474.3200   -1.0000
1.0000    1.0000    1.0000         0

and

mvp = inv(D)*k
=     1
0
0
0

Not surprisingly, if there is a riskless asset, the minimum variance portfolio is invested exclusively in it.

The Optimal Swap

We turn next to vector z.

Recall that the optimal portfolio for an Investor with a risk tolerance of rt is:

y = mvp + rt*z

Now consider two Investors. One, with risk tolerance of zero, should hold portfolio y0, given by:

y0 = mvp + 0*z
= mvp

The other, with risk tolerance of 1.0, should hold portfolio y1, given by:

y1 = mvp + 1*z
= mvp + z

The differences between their portfolios are contained in the vector:

y1 - y0 = (mvp + z) - mvp  = z

Thus z is a vector of differences in holdings between two portfolios. The sum of the asset holdings will thus equal zero. In our earlier terminology, it is a zero-investment strategy. Hence the name (z).

In the original version of the three-asset example:

z =
-0.0389
0.0257
0.0132
-2.6648

Thus an Investor with a risk tolerance of 1.0 should hold 3.89% less in cash, 2.57% more in bonds and 1.32% more in stocks than an investor with the same amount of money but no tolerance for risk at all. Not surprisingly, the sum of the asset proportions equals zero, as it must if y is to be a portfolio.

Since the asset proportions in z sum to zero, it can be considered a recipe for a swap. One unit of the swap (for example, \$1) calls for the holder to pay (1) an amount equal to the return on 0.0389 units (\$0.0389) invested in cash and to receive an amount equal to the sum of (2) the return on 0.0257 (\$0.0257) invested in bonds and (3) the return on 0.0132 (\$0.0132) invested in stocks.

Of course, z is not just any swap. It is the optimal swap. An Investor with a positive tolerance for risk should, in effect, begin with the minimum variance portfolio, then take an appropriately large position in the optimal swap. In dollar terms, the swap position should equal the investor's initial fund times rt, since:

y = mvp + rt*z

Thus an Investor with a risk tolerance of 50 should take twice as large a position in the optimal swap z as should an Investor with the same wealth and a risk tolerance of 25.

One need not actually take a position in a swap contract to achieve the desired result. In most cases, the Investor would simply determine the optimal portfolio y and invest in it directly. However, recognition that this is equivalent to the results obtained by starting with mvp and then making a standard swap z in an appropriate magnitude proves useful in understanding differences among Investors' optimal portfolio holdings.

Two Fund Separation

The recipe for an optimal portfolio is clearly linear. In vector form:

y = mvp + rt*z

Row i will be of the form:

y(i) = mvp(i) + rt*z(i)

For the first n rows corresponding to the asset positions, so that

x(i) = mvp(i) + rt*z(i) : for i= 1,..,n

Now consider two portfolios, each optimal for a given risk tolerance. Let a represent the smaller of the two risk tolerances and b the larger. Then the portfolios are respectively:

ya = mvp + a*z
yb = mvp + b*z

Assume that ya and yb represent portfolios offered by two mutual funds (mfa and mfb, respectively). How might an investor with a risk tolerance equal to rt use such funds optimally? The answer is simple: place a proportion xa of wealth in fund a and a proportion 1-xa in fund b, using the following formula to compute xa:

xa = (b-rt)/(b-a)

To see why this works, note that:

y = xa*ya + (1-xa)*yb
= xa*(mvp+a*z)+ (1-xa)*(mvp+b*z)
= mvp + [xa*a+(1-xa)*b]*z

But, given our recipe for choosing xa:

[xa*a+(1-xa)*b] = rt

So the two-fund portfolio is in fact the optimal portfolio for the Investor in question.

No matter how many assets are used to form the two mutual funds, an Investor can achieve a completely optimal portfolio by allocating his or her funds between the two mutual funds, as long as each of the funds is optimal for a particular risk tolerance and the two funds are designed for different risk tolerances.

As a practical matter, of course, an Investor would have to take a short position in one of the funds if his or her risk tolerance fell outside the range covered by the funds. For this reason, it might be useful to utilize (1) a fund designed for an extremely small level of risk tolerance and (2) one designed for a very large level of risk tolerance.

If several mutual funds are available, an Investor could achieve an optimal portfolio by combining any two funds, as long as each is optimal for different level of risk tolerance and the proportions invested in the funds are chosen appropriately (that is, using the formula given above for xa).

This result is of sufficient importance to deserve a relatively grand name. It is sometimes termed "Tobin's separation theorem", since its introduction in Tobin 1958, but we will call the present result the two-fund separation theorem to differentiate it from more complex results that arise in different settings. Why the name? Because in this situation it is possible to separate the investment decision into two stages. In the first stage, two optimal mutual funds are formed. In the second, investors allocate their assets between the two funds. Moreover, two well-constructed funds are sufficient to span the set of desirable investment alternatives for all investors.

To be sure, these very strong results flow from very strong assumptions. Investors are assumed to agree on probabilistic forecasts (asset means, standard deviations and correlations) and to consider portfolio mean and variance to be sufficient statistics for selecting portfolios. Moreover, short positions are assumed to be feasible and costless, as are other transactions. Later we will consider the effects of dropping one or more of these assumptions. In the meantime, it is appropriate to pause to reflect on the simplicity and tranquillity of a world in which these conditions would hold.

Additional Linear Equality Constraints

It is a relatively simple matter to extend the analysis of the last few sections to cover cases with two or more linear equality constraints. Write the set of m such constraints as:

A*x = b

where A is an {m*n} matrix of "left-hand sides" and b is an {m*1) vector of "right-hand sides". For example, in addition to the standard full-investment constraint, assume (1) that it is desired to select a portfolio with an income yield equal to 5.5%, and (2) that the yields of cash, bonds and stocks are, respectively, 5%, 7% and 3%. Then:

A =
1     1     1
5     7     3
b =
1
5.5

Lagrange Multipliers

To formulate this problem so it can be solved efficiently, we need to utilize explicitly a mathematical procedure that we have employed implicitly already: the method of Lagrange multipliers.

The goal is to maximize portfolio utility, stated in expected return equivalent terms:

up = ep - vp/rt

As before, we choose instead to maximize utility stated in variance-equivalent terms since the optimal portfolio will be the same. In this metric, the objective function is:

vup = rt*ep - vp

Of course, we are not free to choose any asset holdings we might desire. Instead, we must meet one or more linear equality constraints:

A*x = b

For a solution to be feasible (that is, satisfy these constraints), we require that:

b - A*x = zeros(m,1)

Now the trick. We form a Lagrangean function by appending each linear constraint times an associated Lagrange multiplier to the original objective function. With two linear constraints:

L = rt*ep - vp + g1*[b(1)-A(1,:)*x] + g2*[b(2)-A(2,:)*x]

For portfolios that satisfy the two linear constraints, each of the terms in the square brackets will equal zero and the Lagrangean function L will equal the original objective function! Thus maximizing L will give the same answer as maximizing the original objective function, as long as only feasible portfolios are considered.

Since we wish to maximize L, the goal is to get to the top of a hill where the height is given by the value of L and the coordinates of the terrain are given by the values of the variables (the x values plus g1, g2 and any additional Lagrange multipliers. Since the terrain is smooth, it is flat at the top of this particular hill. Moreover, the top is the only place at which it is flat. Thus it is both necessary and sufficient for an optimal solution that the first derivatives of L with respect to each of the variables be set to zero. Note, however, that there are now n+m variables -- the n asset holdings (here, x1, x2 and x3) and the m Lagrange multipliers (here, g1 and g2). The derivatives with respect to the asset holdings are:

rt*e(1) - 2*C(1,1)*x(1) - 2*C(1,2)*x(2) - 2*C(1,3)*x(3) - g1*A(1,1) - g2*A(2,1)
rt*e(2) - 2*C(2,1)*x(1) - 2*C(2,2)*x(2) - 2*C(2,3)*x(3) - g1*A(1,2) - g2*A(2,2)
rt*e(3) - 2*C(3,1)*x(1) - 2*C(3,2)*x(2) - 2*C(3,3)*x(3) - g1*A(1,3) - g2*A(2,3)

Setting these to zero and rearranging gives equations:

2*C(1,1)*x(1) + 2*C(1,2)*x(2) + 2*C(1,3)*x(3) + g1*A(1,1) + g2*A(2,1) = 0 + rt*e(1)
2*C(2,1)*x(1) + 2*C(2,2)*x(2) + 2*C(2,3)*x(3) + g1*A(1,2) + g2*A(2,2) = 0 + rt*e(2)
2*C(3,1)*x(1) + 2*C(3,2)*x(2) + 2*C(3,3)*x(3) + g1*A(1,3) + g2*A(2,3) = 0 + rt*e(3)

The derivatives with respect to the Lagrange multipliers are:

b(1) - A(1,1)*x(1) - A(1,2)*x(2) - A(1,3)*x(3)
b(2) - A(2,1)*x(1) - A(2,2)*x(2) - A(2,3)*x(3)

Setting these to zero gives the original constraint equations:

b(1) - A(1,1)*x(1) - A(1,2)*x(2) - A(1,3)*x(3) = 0
b(2) - A(2,1)*x(1) - A(2,2)*x(2) - A(2,3)*x(3) = 0

Rearranging:

A(1,1)*x(1) + A(1,2)*x(2) + A(1,3)*x(3) = b(1)
A(2,1)*x(1) + A(2,2)*x(2) + A(2,3)*x(3) = b(2)

We now have five linear equations in five unknowns. They may be written succinctly (and somewhat familiarly) as:

D*y = k + rt*f

where:

D = [ 2*C             A'
A     zeros(m,m)  ]
k = [ zeros(n,1)
b   ]
f = [ e
zeros(m,1) ]

Note that our previous example represents a special case of this formula, with m=1 and each of the coefficients in A and b equal to 1.0.

Economic Interpretation of Lagrange Multipliers

Recall the form of the Lagrangean function that has been maximized when the solution is obtained:

L = rt*ep - vp + g1*[b(1)-A(1,:)*x] + g2*[b(2)-A(2,:)*x]

Consider the derivative of this function relative to, say, b(1). It will be:

d L/d b(1) = g1

Since the Lagrangean function will equal the original objective function for all feasible portfolios, we can interpret this derivative as the change in utility per unit change in the right-hand side of constraint number 1. Of course, the objective function is in variance-equivalent terms. To state the derivative in the standard expected return equivalent terms (up) we must divide by rt. Thus:

d up / d b(1) = g1 / rt

And similarly for any additional constraints.

In the case of the standard full investment constraint, the Lagrangean multiplier reflects the marginal utility in variance equivalent terms of additional money to invest (for example, allowing the sum of the asset holdings to equal 1.0001 instead of 1.0000). Dividing by the investor's risk tolerance gives the marginal utility of additional funds in expected return equivalent terms. The latter is, in effect, the marginal utility of the portfolio (mup). Correspondingly, the Lagrangean multiplier is rt times this. All of which explains why we assigned the multiplier the name tmup and the result obtained by dividing it by rt the name mup in our earlier example.

The result is quite general. Each Lagrangean multiplier indicates the marginal utility in variance-equivalent terms of a small change in the right-hand side of the corresponding constraint. In the case of portfolio yield, the multiplier would indicate the added utility per unit of change in the required yield. To state this in expected return equivalent terms, the Lagrangean would be divided by the investor's risk tolerance.

Lagrangean multipliers are often useful for evaluating the extent to which a given constraint limits achievement of an overall objective. The greater the multiplier (assuming that it is positive), the more costly the constraint. Of course, the values apply for only small changes, since the objective function is quadratic, but they are useful for evaluating the desirability of at least small changes in various constraints.

Optimal Portfolio Holdings with an Added Constraint

Finally, we are ready to solve the problem posed with a constraint on portfolio yield. Using the formulas derived above, we obtain:

D = [ 2*C             A'
A     zeros(m,m)  ]
=
2.0000    5.9200    4.6200    1.0000    5.0000
5.9200  109.5200   79.7720    1.0000    7.0000
4.6200   79.7720  474.3200    1.0000    3.0000
1.0000    1.0000    1.0000         0         0
5.0000    7.0000    3.0000         0         0
k = [ zeros(n,1)
b ]
=
0
0
0
1.0000
5.5000
f = [ e
zeros(m,1 ]
=
2.8000
6.3000
10.8000
0
0
mvp = inv(D)*k
=
0.8889
0.1805
-0.0695
39.8923
-8.4836

z = inv(D)*f
=
-0.0324
0.0162
0.0162
0.8723
0.3643

For rt = 25:

y = mvp + 25*z
=
0.0782
0.5859
0.3359
61.6987
0.6249

To see the effect on portfolio utility (up) of a small change in the yield constraint, we divide the corresponding multiplier by risk tolerance to obtain muy, the marginal utility of the yield constraint:

muy = 0.6249/25
= .0250

Requiring a higher yield would allow for a greater optimal portfolio utility, since this value is positive (had it been negative, a higher yield requirement would have lowered optimal portfolio utility). Note, however, that the value is not large -- optimal portfolio utility would increase at a rate of 0.0250 (2.5 basis points in expected return terms) per unit change (100 basis points) in yield. Of course, this is a rate of change for a small difference in required yield. To find the effect of a substantial change (e.g. from 5.50% to 6.50%) the optimization would have to be performed with both values and the difference in optimal utility calculated directly.

Before concluding the examination of cases in which there are no bounds on holdings we treat one further possible complication that has important implications for both portfolio construction and understanding the possible workings of capital markets. In particular, we consider an investor whose utility function has three or more arguments -- one quadratic and the others linear in the decision variables. To illustrate we use an example in which an investor associates a disutility with income yield due to its unfavorable tax treatment relative to capital gains. Letting yp be the yield of the portfolio, utility is now:

up = ep + uy*yp - vp/rt

where uy (utility from yield) is a constant indicating the investor's attitude towards yield. For concreteness we assume a negative value equal to -0.2 for uy. Thus a dollar received in the form of income (yield) will be 80% (1-0.2) as desirable as a dollar received in the form of a capital gain.

Having run out of obvious letters, we let q represent the vector of asset yields. In our example:

q = [ 5
7
3 ]

and

yp = x'*q

As before, we can convert the utility function to variance-equivalent terms by multiplying all terns by rt, giving:

vup = rt*ep +(uy*rt)*yp - vp

For our example with three assets and two constraints, the derivative of the Lagrangean function for asset i becomes:

rt*e(i)+(uy*rt)*y(i)-2*C(i,1)*x(1)-2*C(i,2)*x(2)-2*C(i,3)*x(3)-g1*A(1,i)-g2*A(2,i)

Setting this to zero and rearranging gives:

2*C(1,1)*x(1)+2*C(1,2)*x(2)+2*C(1,3)*x(3)+g1*A(1,1)+g2*A(2,1) = 0+rt*e(1)+(uy*rt)*y(i)

Putting these n equations together with the m equations for the derivatives taken with respect to the Lagrangean multipliers associated with the constraints gives a system of (n+m) linear equations of the form:

D*y = k + rt*f + (uy*rt)*r

where D, y and f are defined as before and

r = [ q
zeros(m,1) ]

Once again, the optimal portfolio can be determined simply by multiplying each term by the inverse of D.. Thus:

y = inv(D)*k + rt*inv(D)*f + (uy*rt)*inv(D)*r

Note that the first two terms on the right-hand side are unchanged from the earlier incarnation. Thus we may write:

y = mvp + rt*z + (uy*rt)*zy

The new vector is zy. Clearly, it is a swap, or zero-investment strategy. An investor who derives neither utility (positive uy) nor disutility (negative uy) from yield will be uninterested in this swap. An investor with a preference for yield will wish to take long positions in it, while an investor for whom yield provides disutility will wish to take short positions in it.

A-fund Separation

In the current example, optimal asset holdings are linear in two variables -- rt and (uy*rt). To keep the notation simple, let:

ur = uy*rt

Then:

y = mvp + rt*z + ur*zy

Now, assume that three mutual funds (a,b and c) have been formed. Fund a holds a portfolio that is optimal for an Investor with a risk tolerance of rta and a value of ur equal to ura. Fund b holds a portfolio that is optimal for an Investor with preferences given by rtb and urb, and fund c holds a portfolio optimal for an Investor with preferences rtc and urc. Thus:

ya = mvp + rta*z + ura*zy
yb = mvp + rtb*z + urb*zy
yc = mvp + rtc*z + urc*zy

Consider an Investor who places proportions xa, xb and xc of his or her wealth in the three mutual funds, with:

xa + xb + xc = 1

The resulting portfolio will be:

y = xa*ya + xb*yb + xc*yc

Or:

y = mvp + [xa*rta+xb*rtb+xc*rtc]*z + [xa*ura+xb*urb+xc*urc]*zy

The goal is to make the first bracketed expression equal to the Investor's risk tolerance (rt) and the second equal to his or her value of ur while keeping the sum of the proportions allocated to the funds equal to 1. This is a system of three linear equations in three unknowns (xa, xb, and xc):

xa*rta + xb*rtb + xc*rtc = rt
xa*ura + xb*urb + xc*urc = ur
xa     + xb     + xc     =  1

Barring degeneracy due to lack of differences among the mutual funds, it can be easily solved, providing the appropriate combination of mutual funds for the Investor to achieve his or her objectives.

We have thus shown that in this setting, three mutual funds can provide Investors with sufficient alternatives to achieve optimal portfolios. Each such mutual fund must be optimal for a particular combination of rt and uy, and the three funds must be designed for Investors with different degrees of both rt and ur. Thus three (different) funds suffice to span the space of optimal portfolios when there are three arguments in investor's utility functions (variance plus 2 linear terms). Examination of the procedures used to obtain this result (and the earlier two-fund theorem) show that the natural generalization of this result is in fact correct:. If there are A possible arguments in Investor's utility functions (variance plus A-1 linear terms), A different funds are sufficient to span the space of optimal portfolios. This may be called the A-fund separation theorem.

A world in which Investors care about more than expected return and risk requires more investment products and more care when selecting a combination of mutual funds for a particular Investor. But the magnitude of the Investor's task is still small, requiring only consideration of A (here, 3) mutual funds rather than n assets (e.g. potentially thousands of securities). Of course, this assumes away nasty realities such as transactions costs and bounds on holdings. It also assumes that the managers of the selected mutual funds do their jobs correctly (that is, construct optimal portfolios) and that Investors or their advisors know the preferences for which each mutual fund is optimal. To the extent that the real world falls short on one or more of these fronts, adjustments will have to be made before giving practical investment advice.