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Number of Combinations of Investment Alternatives

Suppose you have 11 candidate investments that are all economically viable. That is, each has an expected monetary value (EMV) greater than zero. If you have sufficient funds and risk aversion is not an issue, then you should invest fully in each.

If you are capital constrained, then you want to find the combination of investments that maximizes EMV for the funds available. You will need to consider all combinations of investments. How many combinations are there?

Suppose we have 4 candidate investments, {A,B,C,D}. The combinations are:

0 investments	1	-none-
1 investment	4	A, B, C, D
2 investments	6	AB, AC, AD BC, BD CD
3 investments	4	ABC, ABD, ACD, BCD
4 investments	1	ABCD
	16	combinations

The combination coefficient calculates the number of ways to select from n things taken x at a time. For example, how may ways are there to pick two people for a project from a pool of five available persons?:

C(n,x) = n!/(n-x)! where n! = n x (n-1) x (n-2) x ... x 2 x 1

C(5,2) =5!/(5-2)! = (5x4x3x2x1)/(3x2x1) = 10 ways to arrange 5 things in groups of two, without regard to selection order.

The number of investment combinations to be examined increases dramatically with the number of investment choices. For example, 11 candidate investments provides 2048 combinations (including all and none cases).

To see the details, here is a BASIC routine that calculates the number of investment combinations:

    n = 11 ' no candidate investments
    comb = 0
    FOR i = 0 TO n
      comb = comb + Combination(n, i)
    NEXT i
    PRINT comb

Where Combination(n, i) calculates n!/(i-n)!

A "branch-and-bound" search algorithm would be more efficient than examining each possible combination. Imagine a decision tree structure with 11 columns of nodes, with two branches at each. There are 2¹¹ = 2048 paths in the tree. The search algorithm truncates a branch as soon as the budget limit is reached.

Of course, the possibilities are endless if we are allowed fractional investments. Is the problem now impossible? Actually, optimizing the portfolio becomes much easier! Simply rank your investment alternatives by the ratio:

Profitability Index = PI = EMV/Investment (this ratio has alias names and sometimes has different denominators)

Here is the optimization procedure:

Compute PI for each candidate investment
Take all of the highest-ranking project as your available funds allow.
If funds remain, return to Step 2, with the next-highest-ranking investment.

This procedure usually provides a near-optimal solution even when the fractional investments are not available.

Most companies (and individuals) consider themselves to be either 'capital-constrained' or 'opportunity constrained,' and this periodically changes. Many times, the actual constraint is people. The ranking 'bank-for-the-buck' criterion can use Number of People or units of some other scarce, constraining resource in the denominator. For further details you might want to obtain a copy of "Slaying the Capital Investment Constraint" from SPE.

Portfolio management is concerned with investments of different expectation and risk characteristics. The procedure described above works well when the organization or decision-maker is risk-neutral or approximately so. For widely-held companies, the risk is spread over a great many investors, most of whom are broadly diversified. Thus, risk-aversion is of minor importance, and the EMV decision policy is a good one for maximizing shareholder value. My shareholder value model, which argues for using a lower present value discount rate, is described in "Rational Is Practical: Better Evaluations through the Logic of Shareholder Value" available from SPE.

—John Schuyler, January 1999