May 2, 2023

The PAA (Protective Asset Allocation) portfolio is a Global Tactical Asset Allocation model using a multi-asset universe. It can be easily implemented through ETFs. The strategy uses an innovative dual momentum framework and originated in the paper of Keller and Keung in early 2016. We revisit its performance seven years after publication and still find it to be a very interesting alternative to the traditional 60/40 stock/bond portfolio. However, we also discovered slight hints that the authors may have overfitted to the data period used in their study and put forward a more robust hyper-parameter setting.

Protective Asset Allocation is a dynamic ETF asset allocation model published by the renowned Keller and Keung in early 2016. They consider their model to be a **tactical variation of the traditional 60/40 stock/bond portfolio** where the optimal mix is determined by a simple** dual momentum framework**. This way, they hope to guarantee a low risk/absolute return portfolio that packs a vigorous 'crash protection'. Since it's been seven years since its publication, we reevaluate their PAA model to assess how its reported performance measures have held up. We're particularly interested in how overfitted (or tuned) the final model is on the period considered in the study. In short, we investigate the following:

- Is PAAs strategy performance in line with what the authors reported for the period spanning 01/1970 to 01/2015? We evaluate its performance on the period spanning 01/2016 to 04/2023.
- How sensitive is the model to its hyper-parameter configurations? A larger sensitivity could imply that the strategy is more prone to overfit on specific periods.
- Is the strategy always better than the 60/40 benchmark?
- What does the PAA portfolio look like today? (April 2023)

Every month, the model allocates all capital between a **risky **and a **safe portfolio **and repeats the process below:

- The
**risky portfolio**is based on price momentum. Every month, the risky portfolio selects - from a multi-asset universe of 12 assets (which we call N) - the top n assets with the highest momentum (relative momentum). The multi-asset universe can be found in the section Data below. From the top n momentum assets, only those assets with positive absolute momentum (which we call n_{+ }) are selected. Finally, an equally weighted risky portfolio is constructed which thus consists of n_{+}assets. - Momentum is computed as price divided by its moving average of the last
*12*months. - The
**safe portfolio**only consists of one asset: intermediate-term treasury bonds. - Every month a portion of capital is allocated to the safe portfolio using a formula that takes into account both the number of positive momentum assets (n
_{+})and a risk-aversion coefficient called A. The higher the number of positive momentum (n_{+}) assets in a given month, the lower the allocation to the safe portfolio. The higher the risk-aversion coefficient (A), the higher the allocation to the safe portfolio. The introduction of a risk-averse coefficient to determine the safe allocation, makes it possible to scale total risk towards certain desired risk-profiles. The risk-aversion coefficient can vary between 0 and 3. The exact formula = (N - n+) / (N - x), with x = A * (N / 4). - After the allocation to the safe portfolio is computed, the remaining portion of capital is allocated to the risky portfolio.

We use index price data available on Bloomberg to match the proposed asset universe of ETFs by the authors. Our data starts in 1970. This mapping is not entirely accurate so our results might diverge slightly from the original paper.

The following lists the proposed universe of risky assets (N): We both show the representative ETF for the assets as the index we used for bac testing purposes:

- US equities:
**SPY, QQQ, IWM**(MSCI USA USD Net return Index, Nasdaq Index, Russel 2000 Index) - Developed International Market equities:
**VGK, EWJ**(MSCI Europe USD net return index, MSCI Japan USD net return Index) - Emerging Market equities:
**EEM**(MSCI Emerging Markets USD net return Index) - Commodities:
**GSG**(Bloomberg Total Return Commodity Index) - US Real Estate:
**IYR**(Dow Jones REITs Net Return Index) - Gold:
**GLD**(Gold spot) - Bonds:
**HYG, LQD, TLT**(Bloomberg Barclays USD Corporate High Yield, USD Corporate Investment Grade, Bloomberg Barclays USD Long Treasury)

The safe asset used is **SHY** or **IEF**. (Bloomberg Barclays Treasury USD Index)

Crunching Numbers

We performed our backtest using vectorbtpro (a fantastic python package). We made sure not to introduce data leakage by delaying allocation changes one timestep after signal.

The paper suggests that PAA with max six risky assets (top n = 6) and a higher risk aversion (A=2) performs best. We evaluated this PAA(2) model for the same period as the paper (1970-2015) and for the period after publication (2016-4/2023) to assess whether the reported performance remained consistent. The results are outlined in the following table. Note that the 60/40 benchmark consists of a static allocation to 60% US stocks (MSCI USA USD Net return Index) and 40% bonds (Bloomberg Barclays Treasury USD Index).

We find indeed that the PAA portfolio performs much better than the benchmark on all measures during the period considered in the paper. However, the period after publication paints a different story. Although PAA remained less volatile with a smaller max drawdown, it was unable to beat the benchmark on return-based measures. This of course begs the question: was the PAA(2) model overfitted to the data period used in the study? Let's look at it more closely.

Studies often report optimal sets of model hyper-parameters values, implying these will continue to produce strong performance after publication. However, maybe those values just happen to work well on the specific subset of data it was tested on. For this reason, it's often a good exercise to look at how sensitive a model is to small changes in its hyper-parameters. If performance fluctuates wildly for small perturbations, the model could be overfit to a specific period.

PAA comes with an array of hyper-parameters. We list the main ones:

- Number of risky assets to select from the risky universe (
**top n**) - Risk aversion parameter that controls how aggressively we shift to the risk-off portfolio (
**A**) - The lookback period for the simple moving average
- The rebalancing frequency

For this exercise we only look at fluctuations in performance for combinations of **top n** and **A**. The lookback period of 12 months to measure momentum and monthly rebalancing frequency have a long and proven track record both in academics and practice (see e.g. Quantitative Momentum by Jack Vogel and Wesley Gray). Sharpe ratios for the different strategies obtained during years 1970 to 2015 are outlined in the following heatmap:

Sharpe ratios seem quite robust to small changes in values for risk aversion and size of the risky portfolio. There are no isolated cases of abnormal high Sharpe ratios and the performances slightly trend in function of the parameter changes. That's a good sign. Other than that, we notice that performance increases for larger risky universes (diversification is a well-established free lunch after all), and decreases for larger values of A (risk aversion). For large values of A, the model shifts large amounts (if not all) capital into the safe portfolio which only consists of one asset (SHY/IEF). Lower performance may therefore be explained by having a less diversified portfolio on average.

Performance fluctuations from 2016 to 4/2023 paint a different picture:

It turns out the hyper-parameter configuration for PAA(2) wasn't that robust after all (as we already saw in the previous section). Lower values for risk aversion were much more robust, which makes us belief that the strategy should be run with A=0 instead. This is hindsight of course, but might be interesting going forward. Part of the reason why performance deteriorated so badly over this period was the use of a single bond asset to run towards in times of turmoil. In risk-off scenarios the portfolio is incredibly lacking in diversification and the bond asset performed poorly in this particular period. Perhaps the authors did not account for an inflationary regime with rising interest rates. Running with lower values of A partially alleviates this issue.

Note that we also ruled out that the degradation was not caused out by a disproportionate amount of time spent in a risk-off setting or that the number of positive momentum assets were lower than usual.

The next table revisits the performance metrics from before with PAA(0) included:

PAA Versus 60/40: Is It Always Better?

Many studies typically show the equity curve from their strategy beating that of a classical benchmark (like the 60/40 portfolio).

Let's do the same for PAA(0) and PAA(2) versus the 60/40 portfolio for the entire period of 1970 - April 2023.

Amazing. If an investor would've used PAA starting in 1970, she would have ended up performing much better than the 60/40 benchmark. But that's all this graph really tells. By now, we know that an investor who started in 2016 with the PAA(2) model would not have done much better than the 60/40 portfolio. That's sadly not something that can be inferred from just looking at the graph. Luckily we have a neat little trick for that: we show the value of the model portfolios divided by that of the 60/40 portfolio:

Each time the strategy value increases faster than that of the benchmark, a positive trend shows in the graph above. It's now much more obvious to see when PAA outperforms the 60/40 portfolio. Surprisingly, it turns out that the benchmark has been recovering lost ground on PAA since 2009. Note that this only looks at absolute returns and does not consider risk measures such as drawdowns and volatility.

Today, PAA(2) remains in a risk-off mode with 83% of capital allocated to the safe asset. PAA(0) is less risk averse allocating only 41% to the safe asset. Compared to the past, PAA is relatively bearish as more defensive allocations happen only in 40% of the times.

The following pie charts show today's portfolios:

The PAA portfolio remains an interesting alternative to the 60/40 portfolio. However, we did find that PAA(2) as proposed by the original authors was somewhat overfitted to the study period (1970-2015). PAA(0) seemed much more robust and we would likely prefer this model configuration going forward.

Advantages:

- Straight-forward allocation method using a diverse risky asset universe.
- Easy to implement and low maintenance.
- Beats the 60/40 portfolio based on volatility and drawdown measures, especially during turbulent times.
- The PAA(0) model was able to beat 60/40 even in the last seven years.

Disadvantages:

- Only uses one asset (intermediate bonds) as safe haven. In certain configurations this leads to an increasingly concentrated (non-diversified) portfolio.
- The method seems slightly overfitted as it has a hard time accommodating inflationary regimes with rising yields.
- PAA does not always perform better than the 60/40 benchmark.