Combinations/Permutations Formula Help - Total Potential Portfolio Allocations

Question

If one was to invest in whole-number percentages (1%, 2%, ...), a 2-asset portfolio has 99 different combinations to weight the portfolio. (1%,99% to 99%,1%).

What is the formula to find the total number of combinations if one was to have a 10 asset portfolio?

3 example portfolios: Assets: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 Weightings: 91%, 1%, 1%, 1%, 1%, 1%, 1%, 1%, 1%, 1% Weightings: 1%, 91%, 1%, 1%, 1%, 1%, 1%, 1%, 1%, 1% Weightings: 18%, 15%, 1%, 4%, 50%, 1%, 2%, 2%, 1%, 6%

10 assets must be used (>=1%), order matters, whole number percentages only.

Please help! Thanks!

Answer 1

This is known as the composition problem. For n indistinguishable balls and k labeled boxes, you have $n-1 \choose k-1$ possibilities of distributing the balls among the boxes when no box is empty. Since there are 100 1's that you can distribute in 10 asset portfolios, the answer would be $99 \choose 9$ which is 1731030945644 ways.

Answer 2

You can do this with the stars and bars method. First note that each asset needs to have at least $1$% allocated, so set aside $10$% for that allocation. We want to find out then, how to split the remaining $90$% among $10$ different assets.

Let each $1$% of the $90$% be represented by a star, meaning that we have 90 stars. To split the 10 assets, we need to use $10 - 1 = 9$ bars. Consider all possible arrangements of these stars and bars. All the stars to the left of the first bar will be invested in asset $1$, all the starts to the left of the second bar and to the right of the first bar will be invested in asset $2$, between the second and third into asset $3$, and so on... up to the stars to the right of the ninth bar being invested in asset $10$.

To solve this problem then, we just need to consider the number of possible arrangements of the $9$ bars and $90$ stars. See if you can finish the problem from here.

For further reference on stars and bars, you can check out this link: https://brilliant.org/wiki/integer-equations-star-and-bars/#stars-and-bars