Summary

With elDiversificationBenefit, you can measure how much your portfolio benefits from diversification. The Add-in function can be translated as such:

$$ DB = 1 - \frac{risk \quad measure(portfolio \quad level)}{risk \quad measure(positions \quad level)} $$

The numerator is the risk measure of the portfolio. The denominator is the sum of the risk measures of each position, weighted by the position’s weights.

$$ risk \quad measure(positions \quad level) = \sum_{i} w_{i} riskMeasure_{i}. $$

The larger is the diversification benefit, the more the portfolio benefits from diversification. The risk measure can be either volatility, value-at-risk, or expected shortfall.

Intuitively, if there is no diversification in the portfolio, all the positions behave in the same way and the correlation between positions are 1. Then, the risk measure of the portfolio is equal to the risk measure of the positions, the fraction is equal to 1 and there is no diversification benefit. For the diversification benefit to increase, the positions in the portfolio should have a small correlation so that the portfolio risk is smaller than the total risk of the positions.

A simple example with risk measured by the volatility and constant correlation

Consider a simple example with a portfolio made of $n$ positions, all with the same weights $w_i = 1/n$ and the same volatility $\sigma$. All the positions share the same correlation $\rho$, namely the correlation between 2 positions $i$ and $j$ is $\rho_{i,j} = \rho$ (and the correlation is one when $i = j$). The risk measure is the volatility.

With this setting, the analytical computations are straightforward.

At the position level, the risk of each position is $\sigma$, and the total risk evaluated at the position level is $\sum_{i} w_{i} \sigma = \sigma$.

At the portfolio level, the risk is given by

\[\sigma(portfolio) = \sigma \sqrt{\rho + (1-\rho)/n}.\]

The ratio of the risks at portfolio and position levels is not depending on $\sigma$, but only on the correlation and the number of positions. The diversification benefit is $DB = 1 - \sqrt{\rho + (1-\rho)/n}$. If the correlation is one, \(\rho = 1\), diversification is not possible as all assets behave similarly. In this case, the diversification benefit is zero, namely regardless of the number of positions, diversification cannot be achieved.

If the correlation is zero, the diversification benefit is $DB = 1 - \sqrt{1/n}$, namely the larger is the portfolio, the larger the diversification benefit.

The correlation can be negative. For 2 random variables, the correlation is between -1 and +1. But with more random variables, the correlations cannot be all -1 for all pairs of variables! For the present model with a constant correlation coefficient, it can be shown that the correlation coefficient is bounded with

\[\frac{1}{n-1} \leq \rho \leq 1\]

With the correlation at the lower bound, the volatility of the portfolio is null, and the diversification benefit is 1. This is the best that can be achieved. Alas, it is very difficult to find many assets with mutual correlation coefficients that are all negatives. Hence, this case is rather theoretical, and the realistic domain for this model is to consider positive correlations $\rho$.

For a correlation between 0 and 1, the term in $\rho$ in the square root is the component that cannot be diversified away. The second term given by $(1-\rho)/n$ is the part that is decreasing with increasing portfolio size. This simple model concludes that to have the largest diversification benefit, a portfolio should be constructed from a large number of positions with a low correlation $\rho$.

This simple model was constructed to make straightforward analytical computations. In general, the correlations on a typical stock market are in the range of 10 to 60%, and rarely null or negative. Then, the volatilities differ, and the position weights are different. But the conclusions of the simple model remain, namely diversified portfolios are made of a large number of positions with low correlations. And completely diversified portfolios cannot be constructed in practice, even with a very large number of positions.

Examples

=elDiversificationBenefit("volatility",C11:C50,D11:D50, "EUR", "10d", 95, "historicalInnovation",True)

The above example would return the diversification benefit for the volatility of a portfolio whose assets and quantities are in C11:C50 and D11:D50 respectively. The portfolio currency is EUR and the scenario type is historicalInnovation. The rest of the parameters are the default ones.

Syntax

elDiversificationBenefit(Measure, CalculationsLevel, AssetIDs[], Quantities[], Currency, Prices[], RiskHorizon, ConfidenceLevel, ScenarioType,Annualized)
Argument name Default Description
Measure   The name of the measure to be returned.
AssetIDs   Single or multiple asset identifiers (ISIN, FIGI, currency ISO, Edgelab ID).
Quantities   Asset quantities in the same orders as the specified asset ids.
Currency (Optional) local The ISO code of the reference currency for deriving the calculation results. Specify “local” to use the instruments native currency where appropriate. It is mandatory to specify a currency when the granularity is “portfolio”.
Prices (Optional) Edgelab prices You have the possibility to insert your own instrument prices. This parameter is optional and the function will take the Edgelab prices by default.
RiskHorizon (Optional) 10 The time interval over which the risks are estimated (i.e. between now and now + risk-horizon).
ConfidenceLevel (Optional) 0.95 The confidence level for the tail statistics estimators VaR and ES. It is number in percent between 85 and 99 [%].
ScenarioType (Optional) historicalInnovation How the possible scenarios for the asset prices are computed.
Annualized (Optional) true Whether the result should be expressed as an annualized figure. True or False.