Dependencies and aggregations

 Wide diversification is only required when investors do not understand what they are doing. –

Warren Buffett

Nevertheless in the insurance industry the benefits of diversification are significant: the QIS5 capital requirement decreased by 35.1% due to diversification. Diversification is one of the most difficult topics as it needs to fulfil a wide spectrum of requirements:

• Mathematical: correlations need to be consistent: e.g. once the correlation between A&B, and B&C is defined, this defines the correlation between A&C;
• Data: correlations need to withstand validation tests which are mainly based on performing historical data analysis;
• Economic rationale: dependencies need to be backed up by economic rationale.

Several layers of diversification are possible within an insurance company (the following list is from complex to easy):

• Geographical diversification: the interaction of different parts within an international insurance risk group, e.g. the UK and US business unit.
• Intra risk: between risk classes. For example the interaction between US market and US life risks.
• Inter risk class: between risks within a risk class, for example within market risk (equity and property risk), or life risk (mortality, longevity).
• Inter risk: within a risk driver, for example the interaction within US equity between different US stocks. Most insurance companies won’t go into this level of detail but instead apply a benchmark to model these risks (e.g. S&P500).
• Individual risk drivers: for example between UK equity and US property risk. In practice, this is the lowest level of detail and usually the hardest to define: as the risks are very specific it will be difficult to collect sufficient data to justify extreme events.

Although agreeing on the level of diversification is doable, there is the risk of taking excess diversification benefits or double counting benefits during the aggregation process due to several layers in organisation. 

Aggregation mechanisms

The most common method in the insurance industry for describing dependencies is a Pearson correlation. Pearson correlation is a measure of linear dependency which is less suitable for describing non linear dependencies: for example the function y = x² is fully dependent, however it has a Pearson correlation of zero implying no dependency. Therefore the term dependency is a more general term than (linear) correlation which also contains non-linear relationships.

An alternative to the Pearson correlation is rank correlation. This method tries to deal with the shortcoming of linear dependency by ordering each variable in the sample: when calculating a rank correlation only the degree of ordering of the sample matters. For example, the rank correlation for a polynomial (which is fully dependent) remains 1 but the linear correlation decreases as shown in the table below:

Despite this, also rank correlations have difficulties with describing non linear events (e.g. also the rank correlation of y =x² equals 0).

Current techniques

Correlations are included in modelling in mainly 2 ways:

• Variance covariance methodology: used in ICA and the Solvency II Standard Formula aggregation. First the capital requirement per individual risk driver is calculated. Thereafter a correlation matrix is used to aggregate all individual capital requirements.

Variance covariance is easy to implement and to explain, however due to lack of tail data it is difficult to derive sensible correlations. Furthermore, the correlations in this method are independent on the economic cycle and hence don’t reflect any tail behaviour. Lastly, this method does not cope well with non-linearities such as options in insurance contracts.

• Monte Carlo: dependency is included in the simulations. A number of runs are performed and a scenario giving the 1 in x capital requirement is selected. The main advantage of Monte Carlo is that it can deal with non-linearities. The disadvantage of Monte Carlo is that one has to take the average of scenarios around the 1 in x scenario to avoid sampling error and this causes  the runtime to increase substantially (required to get tail numbers; more averaging required when there are less runs).

Challenges

Although these current methods are strongly embedded in the industry, they have some drawbacks:

• Positive semi definite (PSD) requirement (mathematical requirement): because of the requirement of making the correlation matrix PSD, manual adjustments might be required on top of the empirically derived correlations. Some dependencies might not feel sensible after these adjustments and difficult to justify.
• Granularity (mathematical requirement): higher numbers of risk drivers require a larger correlation matrix and make the PSD requirement more difficult to fulfil. Also, when the level of granularity increases diversification benefits increase and will be harder to justify. Lastly, more granularity makes it more challenging to backup dependencies by empirical data.
• Allocation of diversification benefits to group and business units: possible methods are proportional, and based on marginal contribution. However sampling error can de-stabilise the allocation mechanism.
• Data limitations (data requirement): little data might be available or there are stale prices (due to lack of updated data, prices remain the same over several time periods) for tail events. For example, in many countries property data is not stock traded or only provided on a quarterly or semi-annual basis leading to a sample of insufficient size.
• Changes in the data set over time (economic rationale): regime shifts (e.g. US inflation policy before and after 1982) make the choice of an appropriate time horizon to determine correlations tricky. One can attribute more weight to more recent data however this can over-estimate higher correlations if the recent history was a recession and under-estimate when the recent history was a boom. When assigning more weights to recent data, pro-cyclicality increases which is not a desirable effect.

Tail dependencies

Dependencies behave differently in the tail of the distribution than the body. Copulas (a copula is a generalised dependency structure) can deal with tail behaviour. The most used copula in the industry is the Gaussian copula, mainly as it is the easiest to implement: the variance covariance is an example of a Gaussian copula. However, the Gaussian copula does not model tail dependency (as it has a tail dependency of zero) or asymmetries. Other copulas (e.g. t-copula, Clayton) can have tail dependency or asymmetry.

The figures below show that a t-Copula with limited degrees of freedom has tail dependency (which is removed once the degrees of freedom increase when it becomes a Gaussian Copula):

However, copulas are less intuitive and the parameterisation is not straightforward.

Left tail and right tail dependency

While for some stresses (interest rate risk, inflation) we might expect a symmetrical distribution, this is not the case for dependencies that tend to differ between positive and negative stress levels and hence behave differently in the left and right tail. The scatter plot below shows that downside risks are more correlated than upside risks:

It therefore appears diversification is not there when it is needed most.

The effect of this is that it is not always clear if increasing the number of counterparties (e.g. asset classes, policyholders) decreases the overall risk. The American subprime market consisted of mortgages from people of different backgrounds. However it turned out that most securitised schemes were vulnerable to the same underlying economic factors therefore reducing the actual diversification effect.

In most calibrations we assume that equity and concentration risk decrease when we increase the number of companies we invest in therefore reducing the unsystematic/company specific risk component. However, do diversification benefits still exist in extreme scenarios?

Case study

Estimating correlation matrices from past data is a notoriously difficult exercise, even ignoring the impact of tail dependency. We present a simple example which demonstrates this.

The specific example is portfolio diversification where there are a range of possible assets. This example has its roots in the ground-breaking paper by Markowitz (1952). The paper set the framework for modern portfolio theory.

In the version we consider, investors can invest in up to N risky assets. We make a number of assumptions:

1. We assume that the return on the assets are joint-normally distributed;
2. There are no taxes or transaction costs;
3. Securities can be bought or short sold in any quantity without restriction; and
4. All investors have the same time horizon.

Under these assumptions, the set of possible expected returns is bounded by a hyperbola in (return, standard deviation of return) space.

Suppose that we also assume that there is a unique risk free rate at which investors can borrow or lend money. This is illustrated in the figure below.

The unique market portfolio is the point of tangency of the hyperbola and the straight line. Investing in different proportions of the risk free asset and the market portfolio achieves returns and standard deviations of returns along the straight line, with higher returns to the right of the market portfolio being achieved by borrowing money to invest in the market portfolio.

The market portfolio has the highest “Sharpe ratio”, which is a risk-adjusted measure of return.

The matrix of the correlations between asset returns and the expected asset returns are the key in determining the market portfolio. However, these cannot be known with certainty and can only be estimated from past data. One way of testing this is comparing the performance of an estimated market portfolio with an alternative investment approach.

The alternative we consider is to invest 1/N of the portfolio in each of the N assets. We assume that the portfolio is rebalanced monthly to keep to the 1/N rule.

The 1/N strategy has a long history. Rabbi Issac bar Aha gave the earliest explanation of this strategy that we are aware of, in about 4BC:

“One should always divide his wealth into three parts: A third in land, a third in merchandise and a third ready to hand.”

There are two possible approaches to comparing the strategies. One is empirical testing, using actual past data. The other approach is to use simulated data, and assume all the assumption we outlined above are true. With either approach, in-sample data is used to calibrate the Markowitz approach, and out-of-sample data is then used for testing.

With empirical data, the 1/N strategy generally outperforms performs the Markowitz strategy. The following chart shows the relative Sharpe ratio of the 1/N strategy and the Markowitz strategy across 6 different sets of data:

For all 5 out of the 6 data sets, the 1/N strategy outperforms the Markowitz strategy. In the sixth data set, the Markowitz approach marginally outperforms the 1/N strategy. This investigation is described in greater detail in the paper by De Miguel, Garlappi and Uppal.  The following table shows the data used:

With simulated data, the Markowitz approach may be expected to perform better. There are features of real world assets returns, such as fat tails, which are not present in the assumptions underpinning the Markowitz approach. However, it turns out that the 1/N strategy still performs remarkably well on the simulated data. One way of measuring this is how long the time series of past data of monthly returns needs to be for the Markowitz approach to be expected to outperform the 1/N strategy. With 10 assets, the Markowitz approach is expected to outperform with 6000 months’ worth of past data. However, with 25 assets 6000 months’ worth of data is not adequate and the 1/N strategy is expected to outperform. Again, the paper by De Miguel, Garlappi and Uppal has further details.

One of the key issues in the case study is that estimating large correlation matrices is extremely difficult. The amount of data required for an estimated correlation matrix to converge to the true correlation matrix can be unrealistic. The estimate matrices are quite unstable over time, and the portfolio chosen can materially change given a small change in the matrix.

Conclusion

The current industry dependency methods are easy to implement and to explain. However, it appears to be problematic to fulfil both economic rationale, empirical data, and mathematical requirements. Associative measures based on historical data are prone to miss out on events that haven’t happened in the past. The level of data required for a robust calibration may be too large to be realistic. Also, standard correlation methodologies do not take into account tail behaviour which is the main focus of capital modelling as we focus on extreme events. Therefore, new ideas, like empirical copulas, are necessary to come to a better assessment of tail events.

References:

• De Miguel, Garlappi and Uppal, Optimal versus naïve diversification: How inefficient is the 1/n portfolio strategy?, Review of Financial Studies, May 2009.
• Markowitz, Portfolio Selection, The Journal of Finance, March 1952
• Shaw, Smith, Spivak, Measurement and Modelling of Dependencies in Economic Capital, 10 May 2012
• Steven Verschuren, Copula-GARCH Models to Estimate Capital Requirements for Pension Funds, AENORM 74, February 2012
• Stephane Loisel, Pierre Arnal, and Romain Durand, Correlation crises in insurance and finance, and the need for dynamic risk maps in ORSA, 15 July 2010

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This article was written by Matthew Cocke, Servaas Houben, and Elliot Varnell and published in AENORM vol. 20 (76) in August 2012. The original article can be found on pages 21 to 25 of the following link http://issuu.com/vsae/docs/definitief_76/1

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