投资组合经理

为了解决这个问题，可以使用“星与条”方法。把每 100 万美元看作一颗星，把资产类别之间的分隔看作一根隔板。

基金总额为 7500 万美元，且以 100 万美元为单位投资，因此共有 75 颗星。5 类资产需要 4 根隔板来分隔。

于是问题变成：在 75 颗星和 4 根隔板组成的 79 个位置中，选择 4 个位置放隔板。分配方式数为

$$\binom{79}{4}=\binom{79}{75}=\frac{79!}{75!4!}.$$

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Original Explanation

To solve this question, we can use the "Stars and Bars" approach. This approach is an elegant way to solve many counting problems that follow the pattern, "how many ways can you place $x$ unique values into $k$ bins".

For this question, let each $1M investment be a star and let each bar be a delimiter between the amount of money we allocate to each asset class. Since the fund is worth $75M and the manager can invest in 5 asset classes, we have 75 stars and 4 bars (remember that this is a delimiter between asset classes, and therefore is 1 less than the number of asset classes that we have; only 4 dividers are necessary to partition a space into 5 groups).

In total there are 79 symbols (bars + stars) and the problem is simplified to finding the number of ways we can choose 4 of the 79 symbols to be bars. Therefore our answer is ${79 \choose 4} = {79 \choose 75} = \frac{79!}{75!4!}$.

这是上一问的一个变体。不能再任意放置隔板，因为若两个隔板距离太近，某一资产类别会少于 500 万美元。

处理方法是先给 5 类资产各预留 500 万美元，共预留 2500 万美元。剩余 5000 万美元仍以 100 万美元为单位自由分配。

于是新的星与条问题为：50 颗星、4 根隔板。无论剩余部分如何分配，最后再给每类资产加回 500 万美元，就能满足最低 500 万美元约束。

因此分配方式数为

$$\binom{54}{4}=\binom{54}{50}=\frac{54!}{50!4!}.$$

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Original Explanation

The problem of distributing the $75M such that each asset class has a minimum of $5M is a slight variation of the previous stars and bars problem. We can't simply choose the bars to be at any of the 79 possible locations now, because placing two bars within 5 of each other would imply an asset class with less than $5M total allocation.

The way to solve this problem, however, is to automatically assume a $5M allocation to each of the 5 asset classes. This leaves $50M left to be allocated accordingly, forming our new stars and bars problem: 50 stars, 4 bars. And whatever the resultant combination implies for each asset class (let's say 'Equity Options' - $13M, 'Commodity Futures' - $18M, 'Fixed Income' - $14M, 'Event Contracts' - $15M, 'Cryptocurrencies' - $0M), we simply add $5M to each value to get our true allocation with a minimum of $5M.

Using stars and bars to solve this gives us ${54 \choose 4} = {54 \choose 50} = \frac{54!}{50!4!}$.