Surprisingly, in the literature there are only few effective formulae for asset allocation. They are based on the asset types and, in theory, they should define investors’ risk appetite. For instance, a large exposure in stocks should define aggressive investment style in comparison to investing in bond market. While, in general, this approach is true and intuitive, it lacks a quantitative consistency and is based only on common sense assumptions.
Let’s start with a short review of publicly available methods. Vanguard, Corporate Finance Institute provide a set of simple, clear and common sense recipes distinguishing between bonds and stocks with a time comparison. It states asset classes with associated risks and rewards within. Investopedia defines asset allocation as a trade-off between reward, risk appetite and time horizon. While horizon factor is said to be covered by asset classes, the risk vs reward is an art of balance. As it sounds legitimate, investors often fail to estimate future rewards and risks. Moreover, given all instruments in the portfolio are already selected for its potential value, the asset allocation comes down only to risk reduction. Hence, this should be a reviewed definition. Markowitz’s Modern Portfolio Theory defines the mathematical relationship between risk (volatility) and reward (returns) resulting in the Nobel Prize. In fact, his work is still popular today for its elegance and simplicity among scholars, but in real life application turns out to be unstable. The optimal portfolio with minimized variance has very unstable proportions in time and assumptions of variance and returns are very unrealistic. Next, at investor.gov, U.S. Securities and Exchange Commission studies the literature providing an extra look at diversification and rebalancing. Again, diversification is covered by asset classes stressing that even within each class there are multiple options. Although rebalancing is a legitimate point in terms of asset allocation, we will drop this topic from the scope. Finally, Almog & Shmueli (2019) discussed Structural Entropy with application to risk monitoring (VaR) making use of clustering of correlation matrix. We recommend this study to the reader as an interesting approach.
A good asset allocation should consider certain properties:
- Scalable: more independent assets invested should lead to better allocation measure resulting in less variance in value;
- Dependency fragile: measure should be penalized by correlation of assets to the benchmark;
- Proportion fragile: unlike number of assets, the measure should be sensitive how equally the investments are allocated;
- Interpretable: we can easily compare various portfolios against each other and decide which portfolio is allocated more effectively;
- Hedge effective: considering long and short positions the measure should consider directions of investments;
- Effective: good allocation measure should result in smaller volatility of the portfolio. This, we will check below;
- Simple: equation is easy to understand, implement and computationally cheap.
In the article, we will briefly discuss theory for the entropy equation. Next we will run a sample simulation of trades and prices shocks (random walk), calculate portfolios, entropies, correlations and asset allocation measures. We will verify how it reflects in future volatility (shocks) of portfolios. We will run the simulations for cryptocurrencies only for arbitrary defined time window. Finally, we will provide visualizations to verify results.
Hypothesis: Decreased allocation measure leads to the increased shocks of the portfolio in the future.
Let’s define [v1,v2,…,vn] to be a vector of volumes invested at nth asset class. The volume always remains positive regardless the position we hold (long or short). Further on, we can define proportions vector [p1,p2,…,pn] from volumes by a simple standardisation: Having the we can define the entropy value of the following proportions:
Entropy is a measure of dispersion and “information” in random distribution. By subtracting distribution sample with proportions of investments we can see how well assets are diversified, E:[p1,p2,…,pn]→[0,∞]. This means equally distributed assets would maximise the value and an entire investment only in one asset would lead to entropy value 0. Due to a logarithmic component the following measure ranges from 0 to inf but in practice this value should remain small.
Let C define the correlation matrix. Given investment direction vector D=[d1,d2,…,dn]. Now we want to inverse correlation parameters how independent they are by adjusting correlation matrix:
Let’s consider the kth element denotes the benchmark index we can extract correlation vector for the kth row: Having linear dependency we achieve transitory relationship between assets the correlation to benchmark is sufficient for further definitions. Hence, we assume adjusted correlations assume noise assigned to each instrument.
Asset Allocation Proposed Measure
Finally, we can define asset allocation measure A:
where is an entropy of the portfolio which increases along with more equal distribution of volumes, and is an absolute value sum of direction-adjusted correlations to the benchmark, which, in this case, denotes accumulated noise.
Logarithmic component scales down the value of total noise so can be a part of equation operating on similar numeric levels. Also, the following formula is indirectly penalised by the asset correlation to the index. Due to logarithmic component following measure ranges from 0 to ∞ but in practice this value should remain small, e.g. for 0 value: ln(100)≈4.6. We also penalise poorly hedged positions that should result with a higher volatility (when A is small). Following measure should satisfy necessary properties of asset allocation measure mentioned at the beginning.
Visit Quant at Risk to learn about the simulation and application process: https://quantatrisk.com/2022/07/14/effective-allocation-measure-with-entropy-application-for-correlated-crypto-assets/.
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