Document

## how to measure portfolio diversification

Talking about Eggs and Baskets when It comes to Diversification? Here is the science
behind Diversification. You can use the process of Diversification Calculation
to reduce investor naivety and meet or evaluate fiduciary standards.

See in the example how even though we started with 500 assets as we refine the
measurement of diversification to account for asset concentration &
commonality (aka correlations) 500 becomes misleading.

The example is the S&P 500. A market cap weighted index of the largest 500 U.S listed stocks.

But is it really diversified?

Applying a process such as Principal Component Analysis to a weighted correlation matrix or asset time series produces several dimensions which prove to measure diversification effectively.

Scientific

#### Ambient

Dimension

Description

##### 500 ##### Significant ##### NAIVIETY

Simple Count of the Number of Assets in the Portfolio.

##### Description

The quantity of assets can be a highly misleading method to measure portfolio diversification.

##### dISCUSSION

500 assets in the S&P 500 Index, naively assumed to be a diversified index.

Scientific

#### Spanning

Dimension

Description

##### 194 ##### DEscription

This is a measure of concentration. Equally weighted portfolios have the maximum possible EWE for the number of portfolio assets. Decreases in the EWE increase portfolio concentration. The concentration of the weighting scheme is without regard to commonality of the holdings. Normalize asset count for weighting scheme to produce equally weighted equivalent asset count.

##### dISCUSSION

The top 50 companies comprise 1/2 the weight of the S&P 500 Index. On average the portfolio behaves like it has 194 equally weighted assets.

Scientific

#### Intrinsic

Dimension

Description

##### 21.53 ##### DEscription

Reduce the spanning dimensionality by the Gini-Coefficient - which serves as a measure of the % of commonality - reveals the Intrinsic Dimension of the portfolio. The Quantity deduced is the equivalent number of completely independent assets. Normalize asset count for weighting scheme to produce equally weighted equivalent asset count.

##### dISCUSSION

The top 50 companies comprise 1/2 the weight of the S&P 500 Index. On average the portfolio behaves like it has 194 equally weighted assets.

## Understanding how dimensions measure diversification

This chart shows how much
diversification my be in any
portfolio.

The Asset Count is the top end of
the triangle and is also called the
ambient dimension. In the S&P 500
example, this is 500

The peak value shows how many dimensions it takes to span the portfolio with 100% of the informaiton included. More dimension = more diversification. If the top value is less than the diagonal, then there is some amount of complete redundancy in the portfolio. This is often greater in larger portfolio especially index strategies. The peak value of the curve is the spanning dimension of the portfolio and is approximately equal to the Equally Weighted Equivalent. In the S&P 500 example this drops all the way to 194 indicating large amounts of redundancy The graph would fill the diagonal exactly if all of the assets were uncorrelated and equally weighted. As systematic commonality and weighting concentrations pervade the strategy the graph will dip down, lowering diversification

The extent which the graphs file the diagonal is called the Gini Co-efficient. The Gini Co-efficient is a measure of how evenly things are distributed. For the S&P 500 example the graph only covers 11% of the triangle. The Gini Co-efficient answers the question, " How much diversification does this portfolio have for a given count of investments.

The chart integrates idiosyncratic (asset specific) diversification (AKA holding count) with the systemic commonality of the positions (the Gini Co-Efficient). Multiplying the two yields the Intrinsic Dimension.