EAA main formula
EAA controls the optimal portfolio asset allocation through an ingenious exponential scoring function of estimates for return (ri), volatility (vi) and index correlation (ci) as well as applying a portfolio concentration exponent: the non-negative elasticities wR, wV, wC respectively wS.
$\mathit{wi}\sim \mathit{zi}={\left(\frac{{\mathit{ri}}^{\mathit{wR}}\cdot {(1-\mathit{ci})}^{\mathit{wC}}}{{\mathit{vi}}^{\mathit{wV}}}\right)}^{\mathit{wS}}$ , if ri > 0 else wi = zi = 0, for i = 1 ... Nwhere for each asset i in an N-sized portfolio:
- wi is the normalized proportional optimal portfolio weight, where the summation of weights is equal to 100%
- zi is the generalized momentum score
- ri is the average return (total or excess*) calculated over the last 1, 3, 6 and 12 months
- vi is the volatility of total return measured over the last 12 months
- ci is the correlation of total returns with the equal weighted universe index measured over the last 12 months.
The four geometrical weights wR, wV, wC and wS are called "elasticities" due to their relative impact on the three terms (ri, vi, ci) of the EAA scoring function. Remember from math class:
- ${x}^{0.5}=\sqrt{x}$ and
- ${({x}^{0.5}\cdot y)}^{2}={x}^{(0.5\cdot 2)}\cdot {y}^{2}=x\cdot {y}^{2}$.
So when applying exponential values ranging between 1 to 0 the scoring effect is mitigated, while values ranging from 1 to 2 amplify the effect of the said term on the score. Note that with wS = 0 the EAA function will return zi = 1 for each and every asset, independent of ri, vi or ci (provided ri > 0). Put differently, with wS = 0 the asset allocation is equal weighted (apart from the safety net offered by a cash proxy fund, see below).
Different from FAA the proportionality with zi allows the weights wi to be not equal. Next to its exponential scoring function, EAA utilizes an optimal top quantile (TopN) of the portfolio size (N) and a C(r)ash Protection routine (CP) by allocating a proportional fraction of portfolio capital to a cash proxy fund (CPF) for every asset with non-positive return. In accordance with the concept of tactical asset allocation the portfolio is rebalanced at the end of each month.
During a stock market crash, like in 2008, the C(r)ash Protection kicks in. Note the unequal weights too (last column). |
Learning IS, testing OS
The authors have gone at great lengths in their quest for heuristically establishing the optimal formula settings for the EAA model. With a pool covering more than 100 years of monthly data, they were able to select the first 50 years as an exclusive learning or In-Sample period (IS), to avoid the risk of curve fitting on the subsequent 50 years of testing or Out-of-Sample period (OS). The five IS decades from 1914 through 1964 saw two World Wars, the Great Depression, various periods with increasing and decreasing interest rates and several inflationary regimes. Performance was optimized for three global, multi-asset universes consisting of N=7, N=15 and N=38 asset classes respectively.
Long story short: preliminary optimization sweeps over multi dimensions during IS pointed out the degrees of freedom for reaching optimal performance could be limited to only wS and wC, concurrently fixating wR = 1 and wV = 0. Note that with wV = 0 the volatility term becomes ${\mathrm{vi}}^{0}=1$, basically deactivating the volatility term in the equation. Note also that by preserving these two degrees of freedom, wC governs the intra-portfolio "hedging" by emphasizing lowly-correlated assets, while wS governs the degree of concentration of the portfolio weights.
Further testing proved the model could be kept even more constrained to merely two effective pairs of elasticities for wS and wC. On average these two pairs yield strong and stable IS performance. Additionally the two "golden pairs" allow for a substantial simplification of the EAA formula, resulting in quite elegant formulas.
Interestingly and contrary to common practice, the authors discovered that no more than 12 End-of-Month total return observations are sufficient for meaningful estimates for vi and ci.
Golden Models
The authors learned from their IS research two elasticity pairs with (close to) optimal IS performance on all three testing universes: the so-called "Golden Models".
Golden Defensive EAA: $\mathit{wi}\sim \mathit{zi}=\sqrt{(1-\mathit{ci})\cdot \mathit{ri}},$ if ri > 0 else wi = zi = 0In "defensive mode" the concentration elasticity wC = 1 puts emphasis on lowly-correlated assets (since -1 <= ci <= 1,
so 0 <= (1-ci) <= 2) through intra-portfolio hedging with the concentration elasticity wS = 0.5 mitigating both the effect of return and diversification at the same time.
Golden Offensive EAA: $\mathit{wi}\sim \mathit{zi}=(1-\mathit{ci})\cdot {\mathit{ri}}^{2},$ if ri > 0 else wi = zi = 0In "offensive mode" the impact of high returns is strongly amplified by the concentration elasticity wS = 2 while simultaneously preserving the intra-portfolio hedging effect through boosting the correlation elasticity wC = 0.5.
Equal Weight Models
As special cases of the EAA formula two equal weight models are appreciable too. Equal weighted allocations for the top portfolio quantile necessitates for adding a small positive number epsilon (eps = 1E-6 = 0.000001) to wS in the (simplified) EAA formula, so zi scores become almost identical, but are still discernible for ranking purposes. See example below.
The equal weighted hedged model is reached with wS = 0 and wC = 1:
Equal Weighted Hedged: $\mathit{wi}\sim \mathit{zi}={((1-\mathit{ci})\cdot \mathit{ri})}^{\mathit{eps}},$, if ri > 0 else wi = zi = 0The classical equal weighted return (only) model is reached with wS = wC = 0:
Equal Weighted Return: $\mathit{wi}\sim \mathit{zi}={\mathit{ri}}^{\mathit{eps}},$ if ri > 0 else wi = zi = 0
Stepwise summary
The below stepwise summary should crystallize the discussed EAA methodology (for the clarification remarks, an N=7 example universe is supposed). See the below screenshot examples too:
1. determine ri, vi and 1-ci
2. count the number of assets with ri <= 0;
(suppose 2 assets have non-positive ri, so count = 2)
3. allocate to CPF: count / N;
(suppose count = 2 and N = 7, then 2/7th or 28% of capital is allocated to the CPF)
4. calculate TopN = Min( 1 + roundup( sqrt( N ), rounddown( N / 2 ) );
(suppose N = 7, so TopN = 1 + roundup( 2.6 ) = 1 + 3 = 4, but rounddown( 7/2) is 3, so TopN becomes 3)
5. calculate zi using the EAA formula;
6. determine for TopN: wi ~ zi / ( sum zi of TopN ), where sum wi = 1 - count / N;
(suppose after CPF-allocation 100% - 28% = 72% of capital is still available for TopN, so wi = 0.72 * zi / sum zi of TopN )
When all portfolio assets register positive momentum, the CP-rule is not triggered (see column for avgRet% and Pos.Size%):
When one or more assets register non-positive momentum (ri <= 0), a corresponding fraction of portfolio weight is allocated to the cash proxy fund (here: $IEF). Note the CPF's zi score is not in the top quantile (here: TopN = 3), hence the yellow colored cell:
Importantly: while determining wi, remember the cash proxy fund is just an equal asset of the universe too, so it can receive an allocation share during the above step 6, provided the zi score of the CPF is among the TopN highest positive zi scores. In the below example weight is assigned to $IEF because of having the second highest zi score and due to the two assets with non-positive momentum:
Backtesting EAA over 20 years
The behavior of the different modes of the EAA model is demonstrated below with an N=7 global multi asset sample universe consisting of $MDY, $IEV, $EEM, $QQQ, $XLV, $IEF and $TLT. $IEF is also used as the cash proxy fund. ETF's are extended with their corresponding mutual funds (see here and here). Next to the four modes introduced above, performance is also compared with the equal weight (1/N) universe of the same 7 assets and with a buy and hold strategy for SPY. The backtest starts ultimo 1994 and ends ultimo 2014, covering 20 years with two secular stock market crashes and the subsequent recovery phases. Total return data (dividend adjusted closes) is from Yahoo! Finance.
During the first years until the start of the new millennium B&H SPY proved to be superior, but three years and a bear market later, the EAA model in all its four discussed modes, had shown themselves almost immune to the unwinding crash after the tech bubble burst. Note $QQQ is part of the portfolio universe too, loosing roughly 80% of its value in the course of these three years.
In the subsequent five years SPY experienced a strong recovery, doubling in nominal value. However, EAA in Offensive mode well-matched and redoubled too. In EW Return mode EAA performed even better, when measured in CAR.
When the housing bubble exploded in 2008 and stocks plunged like a waterfall, EAA in Offensive and EW Return modes collected some limited hits. At the same time the hedging effect of the Defensive and EW Hedged modes proved their value. Both modes were able to dodge most of the severe blows stocks experienced globally as shown in the performance of EW and SPY respectively.
Since the market bottomed early 2009, the spectacular recovery of EW and SPY was paralleled by the strong and steady performance of all four EAA modes.
To conclude the backtest review the below comparison table with several key performance indicators offers a more detailed analysis of all six portfolios during the investigated period (all rows are conditionally formated):
Rebalancing info
At the end of each month, the model provides the necessary information for reforming ones portfolio:
Final notes
- The backtest in this primer is performed using total returns for computing momentum values, contrary to the reported results achieved with risk free returns in the EAA-paper.
- In the context of the Cash Proxy rule for summing the number of assets with non-positive return, the CP-fund is excluded. This appears to be a small improvement of the EAA-model.
- For quants versed in R a post by Ilya Kipnis covering EAA is interesting: QuantStrat TradeR.
- Being able to contribute to the work of Wouter Keller and Adam Butler was a privilege and a great experience.
- EAA is also featured on AllocateSmartly.com. By signing up through this link, you support my work.
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