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.
Simple
description
Asset
Count
Scientific
Ambient
Dimension
Description
Example
500
NAIVIETY
High
FIDUCIARY RISK EXPOSURE
Significant
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.
Simple
description
Equally
Weighted
Equivalent
(EWE)
Scientific
Spanning
Dimension
Description
Example
194
NAIVIETY
Medium
FIDUCIARY RISK EXPOSURE
Material
MaterialDEscription
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.
Task
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.
Simple
description
Independence
Equivalent
Scientific
Intrinsic
Dimension
Description
Example
21.53
NAIVIETY
None
FIDUCIARY RISK EXPOSURE
Low
LowDEscription
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.
Task
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.