# HHI for Diversification

**12/23/15**

Here is yet another tool investors can use to help answer the question of 'how many stocks do I need?', the Herfindahl-Hirschmann Index (HHI). The HHI is used in economics to study market concentration, and equivalent measures are used in other sciences (such as Simpson's biodiversity index).

Say you have 3 stocks and their totals:

Stock A, total $ for A

Stock B, total $ for B

Stock C, total $ for C

Calculate the grand sum of the totals, call it S. Then calculate the "market share" for A, B, and C as A/S, B/S, and C/S

The HHI is then (A/S)^{2}+(B/S)^{2}+(C/S)^{2}

If you only had stock A, then A/S = 100, and HHI = 10,000, so 10,000 is the maximum HHI, indicating a monopoly (or in the investing sense, total __non__-diversification).

Very small HHI (minimum is 10,000/number of stocks) indicate 'perfect competition', in the investing case I'd say it indicates likeness to an 'index'-y situation. An HHI of 10,000/number of stocks only occurs when there are equal dollars in each stock. Note that this is something that many people consider "diversification" or "balanced".

One can set up bounds to monitor your portfolio if your HHI is between certain numbers, telling you to consider to distribute cash around, or buy more stock, or sell stock, record keeping, or do nothing. Or one can use it to answer questions like if I do this or that what will happen to my HHI. Last, one can calculate the HHI at a total portfolio level and/or within asset classes. These are just some of the possibilities.

Consider the following table:

HHI | type of market/portfolio |
---|---|

> 1,800 | monopoly, too concentrated, not diversified |

[1,000, 1,800] | moderate concentration |

< 1,000 | perfect competition, diversified, index |

Recall that the smallest HHI can possibly be is 10,000/number of stocks, which occurs when there are equal dollars in each stock, so we get the following table (using the ranges from above):

num of stocks in portfolio | 10,000/num | type of portfolio |
---|---|---|

1 | 10,000 | not diversified |

2 | 5,000 | not diversified |

3 | 3,333.333 | not diversified |

4 | 2,500 | not diversified |

5 | 2,000 | not diversified |

6 | 1,666.667 | moderate concentration |

7 | 1,428.571 | moderate concentration |

8 | 1,250 | moderate concentration |

9 | 1,1111.111 | moderate concentration |

10 | 1,000 | moderate concentration |

11 | 909.090 | diversified |

12 | 833.333 | diversified |

So we conclude that, if the totals of the stocks in dollar terms are all equal, then:

- 1-5 stocks is too concentrated
- 6-10 stocks is moderate concentration
- and greater than 10 stocks is diversified

You can play with the bounds on the HHI from above to calibrate the outcome (so say > 25 stocks starts the "diversified" category) For example, the table

HHI | type of market/portfolio |
---|---|

> 1,300 | monopoly, too concentrated, not diversified |

[400, 1,300] | moderate concentration |

< 400 | perfect competition, diversified, index |

- 1-7 stocks is too concentrated
- 8-25 stocks is moderate concentration
- and greater than 25 stocks is diversified

You can also do potentially useful stuff like seeing how mergers/acquisitions/splits/divestitures of companies affect your portfolio diversification. If stock A has a% of your portfolio, and stock B has b% of your portfolio, then together they have a^{2} + b^{2} of the HHI. However, if company A and B merge, then that combined company's HHI contribution of your portfolio is now increased to (a+b)^{2} = a^{2} + b^{2} + 2ab.

Just thought I'd share this application of the HHI. Please let me know if you find it useful.

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