###### By Beau Kramer, CFA | Solovis Quantitative Analyst

We can all appreciate the common dictum not to put all our eggs in one basket. Adequate diversification reduces the effects of uncompensated risk on a portfolio’s performance. Measuring diversification allows us to understand better the efficiency of a portfolio and the impact of possible changes. However, quantifying the “number of baskets” we’re putting our eggs in is not straightforward.

At Solovis, we believe there is no single right way to view your portfolio. Varying contexts and decisions demand different lenses. To that end, we actively explore new techniques and metrics. This article explores a method for measuring the diversification of a portfolio.

### Effective Number of Bets

Developed by Attilio Meucci, the Effective Number of Bets (ENB) captures the distribution of a portfolio’s exposure to uncorrelated risk factors. A seemingly well-diversified allocation across asset classes may constitute nothing more than exposure to a single risk factor. ENB would help uncover this concentration and enable the creation of a more diversified portfolio.

We think this concept is easier to grasp in a non-investment context first. Suppose a sports fan decides to make some friendly wagers with a few friends during March Madness. She bets on the outcomes of four different games in the first round. The result of each game is independent of the others. So the ENB is four. Contrast this with participating in four simultaneous bracket contests with nearly identical brackets. A single lost game could easily wreck all four brackets. Here the ENB is closer to one.

Application in an investment context is not as intuitive because asset returns are usually not independent. However, we can find a set of independent sub-portfolios for a given portfolio using minimal linear torsion (MLT). Similar to principal component analysis (PCA), minimal linear torsion extracts a set of independent factors for a set of assets. MLT improves on PCA in that it does not bias explanatory power on the first few components and attempts to minimize distortions from the original data. In effect, MLT produces an uncorrelated set of latent factors that are as close to the original data as possible.

MLT allows us to calculate the effective number of bets by mapping a portfolio into exposures to a set of underlying factors. Then we calculate the factors’ percent contribution to portfolio variance. Finally, ENB is the Shannon entropy of these contributions to variance. For a portfolio with N assets, the effective number of bets ranges between 1 and N.

### Sample Portfolio ENB

Let’s look at an example to get a sense of how this works. Below are two sample policy portfolios. Policy Portfolio B reduces Policy Portfolio A’s equity exposure in exchange for increased exposure to Long-Term Credit Focus and Hedge Funds.

Asset Class | Policy Portfolio A Weight (%) | Policy Portfolio B Weight (%) |
---|---|---|

Cash | 3.0 | 3.0 |

Short-Term Investment Grade | 18.0 | 18.0 |

Long-Term Credit Focus | 3.0 | 10.0 |

Hedge Funds | 8.0 | 11.0 |

Public Equity | 50.0 | 40.0 |

Private Equity | 5.0 | 5.0 |

Venture Capital | 6.0 | 6.0 |

Real Estate | 4.0 | 4.0 |

Infrastructure | 3.0 | 3.0 |

We can see there is a diverse set of asset classes present. Each asset class has unique characteristics that distinguish it from others. However, there are common factors that drive significant portions of returns across asset classes. For example, Real Estate might have its idiosyncratic component, but economic growth and interest rates may explain the vast majority of its and other assets’ performance. This subtlety is where the effective number of bets shines. It quantifies whether an investment is diversifying or just doubling up on existing risk exposures.

Below we can compare the diversification of the two portfolios. Despite roughly similar volatility, we can see that Portfolio B has a higher ENB. As a portfolio, it represents a more diversified approach than Portfolio A. Given the increased allocation of assets that diversify equity risk, this difference matches our intuition.

Policy Portfolio A | Policy Portfolio B | |
---|---|---|

Annual Volatility (%) | 14.5 | 13.7 |

Effective Number of Bets | 2.3 | 3.0 |

### Should We Add This to Our Portfolio?

For a more practical application of ENB, suppose we are considering adding a new asset class to Policy Portfolio A. We believe that this asset class will help diversify our existing portfolio. But how can we put some weight behind that belief? And how should we size this addition to our portfolio? Below we can see a plot where we vary the weight of this new mystery asset class in the portfolio while decreasing pro-rata the weights of other assets. Adding to this asset would improve the diversification of the portfolio. However, we can see that the diversification benefits of this new asset diminish beyond a certain weight.

#### Effective Number of Bets vs Weight in New Asset

### Count Your Baskets

Diversification is a foundational element in portfolio management. After all, it is the rare “free lunch” that significantly reduces risk without sacrificing return. The effective number of bets provides a means of measuring diversification in the context of a portfolio. In a single number, it communicates the efficacy of decisions across the investment process. From manager selection to portfolio construction, the effective number of bets provides a new lens in the investment process. At Solovis, we actively explore concepts that can help us better understand a portfolio’s risk drivers and diversification. This research informs both the risk factors we choose to model your portfolio as well as our feature enhancement pipeline.

**About the author:**

Beau Kramer, CFA, is a Quantitative Analyst in Solovis’ Research & Development group. Prior to Solovis, Beau worked as a financial analyst at ICONIQ Capital and UBS, focusing on portfolio construction and implementation. Beau holds a BSc in Finance from Santa Clara University and a Master’s of Information and Data Science from U.C. Berkeley.