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# Portfolio Combinatorics

July 21, 2017

A portfolio manager (PM) seeks to adjust her portfolio's contract percent shares in order to obtain the minimum 1 in 200 year return period loss for the same return.

The portfolio comprises of 100 contracts and for the purpose of this example, each contract percent share can be selected to be any value between 0% and 100%.  The PM decides to discretise the available percent shares into 1% steps for each contract, however there are no other constraints that the PM wishes to consider at this point.

The PM has developed a routine to simulate 100 million different portfolio constructions (randomly combine different contract percent shares) and record only those which have the lowest 1 in 200 RPL for their given return. What confidence does the PM have that the recorded portfolios have the lowest possible level of risk?  To help inform us, we can compare the number of simulations to the number of possible combinations.

• The number of possible portfolio combinations = (no.of percent share steps for each contract)^(no of contracts) = 100^100 = 10^200

• This is concise notation for an extremely large number (1 with 200 zeros after it).  The oft quoted no of atoms in the universe is approximately 10^80

• 100 million simulations = 10^8 simulations

• This is 10^-190 % of all possible combinations.  A vanishingly small percentage!

The simulations will of course illuminate many better candidate portfolios. However, they are by no means optimal (i.e. have the absolute minimum risk). Even for a relatively modest portfolio consisting of only 100 contracts, the number of possible portfolio combinations is simply too large for simulation or 'brute force' approaches to reach such a conclusion.