Research

**Outline****Install the latest version****Import required libraries****Partitioning volatility into regimes****Decomposition of the portfolio return distribution****Decomposition of the benchmark return distribution****A portfolio "robustness" score****Constructing the score****Examples of the robustness score in action****Conclusion and further research directions**

The essential idea is that decomposing the return distribution of a portfolio into separate distributions based upon market regimes (as defined using a volatility indicator, such as the VIX index, and changes thereof) may elicit a stronger understanding of the drivers of performance, and furthermore, give some indication as to how robust, and hence well constructed the portfolio is. For example, if the returns are broadly normal in times of market calm (with the expectation of future calmness), but strongly left skewed and platykurtic (fat-tailed) in times of turbulence (with the expectation of future turbulence), this may be indicative of a portfolio whose performance is overly driven by market exposure or being short vol.

Intuitively, it seems clear that the notion of robustness is connected to that of antifragility popularised by Nassim Taleb [1]. A simple, universal measure of financial robustness allows one to objectively rate any portfolio in terms of its invariance to different market regimes. It is a useful complement to the traditional suite of risk and performance metrics (average returns, volatility, VaR, Sharpe ratio, drawdowns, diversification ratio and factor exposures).

This will be explored in further detail in this notebook. However, before we begin, it is necessary to ensure we have the latest version of ** Tau**,

[1] Antifragile, Nassim Nicholas Taleb, 2012, Penguin Books

For comparison, we repeat the analysis with the portfolio benchmark, the Morningstar Asset Allocation TR Index:

We summarise the return distributions of the multi-asset portfolio and benchmark index for side-by-side comparison:

We have partitioned the economic climate into four regimes based on investors' sentiment (VIX > 20 implies strong uncertainty/turbulence) and their short-term outlook (change in VIX over the last 10 days > 0 implies becoming more uncertain/turbulent, < 0 implies a return to calm is expected). We can then analyse how a portfolio performs in each regime by looking at the conditional distribution of returns.

Our idea is that in order to be robust, a portfolio needs to be impervious to regime changes. The **ALPIMA** MASTR strategy is robust because it rebalances monthly to ensure it remains optimally diversified through regime changes, so the return distributions are closer to invariant (the four curves have a similar shape to the bars, which is the overall distribution) than one which is less well balanced. This is the case with the first multi-asset portfolio we analysed (MASTR).

When a portfolio has return distributions which look markedly different in different regimes, in particular exhibiting clear skewness and/or fat-tails, this implies that it is not so robust to regime changes, and likely to exhibit poor performance in one of the regimes, e.g., the negative skew of returns in the high VIX, rising regime in the second (benchmark) portfolio we analysed.

Our idea is to construct a normalised robustness score by comparing the return distribution across different volatility regimes from times of "calm" (VIX low and falling) to the distribution in times of turbulence (VIX high and rising). The claim is that a robust portfolio would have a less extreme difference in these distributions than a poorly constructed one, which was overly exposed to market events.

We formally estimate the probability distributions of the returns in the required regimes by appealing to kernel density estimation (KDE) under a standard gaussian kernel, whose bandwidth is estimated via maximum likelihood estimation. Full details of the KDE construction can be found in our *Volatility Surface Construction* notebook (available upon request, please contact **info@alpima.net**).

These various distributions are compared using a variation on the *Kullback–Leibler divergence*, which can be interpreted as the average difference in the number of bits required for encoding samples of one probability distribution using a code optimised for the other probability distribution and is related to the Shannon entropy.

We construct a quantitative measure of portfolio robustness, by applying a scaled non-linear transformation to this divergence to form an invariance measure. The **ALPIMA Robustness Indicator™** is then constructed by bracketing portfolios into ranks by this invariance measure. A portfolio with an **ALPIMA Robustness Indicator™** of rating **A** is very robust, whilst one which receives a rating of **G** is less robust. No value is assigned if there is insufficient data to calculate a meaningful decomposition of distributions (currently we require at least 18 years of data to ensure a number of major business/economic/market cycles have been captured).

It should be noted that a strong **ALPIMA Robustness Indicator™** does not necessarily signify a superior investment in terms of risk adjusted returns. It is simply a measure of how much the investment returns will likely change under a change of volatility regime. As such, it is another useful tool for modern investors to add to their arsenal; a lens under which to view their portfolio, which when augmented with other measures available on the ** ALPIMA** platform enables them to gain a rather complete picture of their portfolio characteristics and performance.

To illustrate the **ALPIMA Robustness Indicator™** in action, we calculated its value for a range of portfolios and instruments, including the multi-asset portfolio above (**ALPIMA MASTR**) along with its multi-asset benchmark (Morningstar Multi Asset TR Index), a standard 60/40 equity/fixed income portfolio, the equity market itself (S&P 500 Index) and a range of other markets and instruments.

The results are in the following table:

Below are a few things to note from this table:

- Robustness is clearly distinct from risk-adjusted performance. Assets like Gold and Nikkei are robust, i.e., relatively invariant to market regimes, but have relatively low Sharpe ratios on their own;
- Somewhat surprisingly, during the testing period, the robustness of the classic 60/40 portfolio is not very different from that of the S&P 500 itself. This is due to the fact that, as illustrated in the figure below, adding Treasuries to the S&P 500 does not increase the relative stability of the distributions between calm and turbulent volatility regimes (and therefore does not lead to an increase in the
**ALPIMA Robustness Indicator™**), even if adding Treasuries significantly increases the Sharpe ratio of the basket; - Achieving higher regime invariance requires a dynamic allocation methodology (as opposed to a "static" fixed-weight scheme such as 60/40) in order to adapt to changing market conditions. The allocation engine of the
**ALPIMA MASTR**strategy, whose aim is to maximise portfolio diversification is a good example. The fact that it rebalances monthly to remain optimally diversified is a plausible explanation for why it scores highly on both the**ALPIMA Robustness Indicator™**and Sharpe ratio dimensions.

The relationship between the **ALPIMA Robustness Indicator™** and the Sharpe ratio is highlighted in the following figure:

In this piece we introduced the **ALPIMA Robustness Indicator™**, a new measure of market regime invariance for the return distribution of any portfolio. Simply put, the higher the **ALPIMA Robustness Indicator™**, the more impervious is the portfolio's return distribution to changes in market regimes.

Simple extensions of this idea include using different measures to characterise the difference between the distributions, for example, the total variation, Wasserstein or Hellinger distances. Fixed portfolios or instruments could also be used to bound the measure at extremes to known values, enabling us to construct a linear measure.

The introduction of this framework for quantifying robustness gives rise to the possibility of constructing portfolios where the allocation could be directed by maximising the portfolio's robustness indicator itself, either alone or alongside other metrics such as portfolio diversification or inverse volatility. As this is a notion of particular pertinence in the current economic climate, we will be revisiting this shortly in a forthcoming notebook.

Finally, it should be noted that the **ALPIMA Robustness Indicator™**, as we have defined it here, is essentially a measure of distributional invariance. It measures the constancy of a strategy's return distribution in different market regimes, which sheds interesting light to a portfolio's behaviour. The higher the indicator, the more invariant the expected return distribution is in different market regimes. This being said, it is a symmetric measure of deviation, similar in that respect to a second order moment such as variance. It does not distinguish between distributional differences which serve to increase the portfolio returns (cf., Sharpe vs Sortino ratio). It is quite possible for portfolios to exhibit superior performance in turbulent markets to that in calmer ones, e.g., rolling long puts. For such a portfolio, the **ALPIMA Robustness Indicator™** would be low, but this could indicate a positive portfolio feature, i.e., *antifragility*. Accounting for this positive deviation requires the introduction of higher order measures, which when used in combination with the **ALPIMA Robustness Indicator™** would give a very clear and complete picture of the portfolio characteristics and behaviour under market regimes of interest. This is a topic of current research, which will be published shortly.

This piece is a good example of how ** ALPIMA** notebooks coupled with our powerful API,

Please contact us at **info@alpima.net** for more information on this analysis, or if you would like to explore how we can help you and your team re-think portfolio design and asset allocation for a new world.

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