A fund has $$50$ million to distribute among five asset classes (equity options, commodity futures, fixed income, FX and crypto), in whole multiples of $$1$ million; a class may receive nothing. For instance $(25, 15, 5, 5, 0)$ million is one allocation. How many allocations are there?

Allocating 50 million across five asset classes

Question

A fund has $\text{\char36}50$ million to distribute among five asset classes (equity options, commodity futures, fixed income, FX and crypto), in whole multiples of $\text{\char36}1$ million; a class may receive nothing. For instance $(25, 15, 5, 5, 0)$ million is one allocation. How many allocations are there?

Short answer

$316{,}251$ . Count non-negative integer solutions of $x_1+\dots+x_5=50$ . By stars and bars (fifty units and four dividers), that is $\binom{54}{4}=316251$ .

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Lessons behind this question

Counting and Combinatorics
Forward and Futures Prices: Cost-of-Carry, Known Income, and Continuous Yields
Mean-Variance Optimisation

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