I came across a lecture notes on Ergodicity Economics: https://ergodicityeconomics.files.wordpress.com/2018/06/ergodicity_economics.pdf. It looks like the whole theory goes far beyond the article in the Nature Physics, we discussed in the previous post. Interestingly, Ole Peters applies his theory to find an optimal leverage which defines the portfolio with the highest time average growth rate. According to his calculations it is equal to:
We thought it might be a good idea to backtest a strategy on S&P 500 Total Return Index to see if the optimal leverage improves portfolio performance. An idea of a non-constant leverage to boost an investment performance is not a new one. There are multiple investment products such as Constant Proportion Portfolio Insurance (CPPI) products and volatility targeting funds. Those products are based upon different portfolio construction principles compare to the ones specified in ergodicity economics theory.
Unfortunately, the author does not specify the way he estimates an excess return of risky assets, which is µ and volatility of risky assets, which is σ in equation (316).
To estimate volatility we use an industry wide approach and calculate historical volatility for the last 500 trading days. For stock market returns, it is more tricky. There is no industry wide standard model to project future expected returns. We use three different indicators to project five-year returns: P/E ratio, Tobin’s Q and the Household Equity Share. P/E ratio and Tobin’s Q are well known measures used in valuations of companies, whereas the Household Equity Share is a relatively new predictor. It was introduced by Yang and Zhang in their paper: https://cba.lmu.edu/media/lmucollegeofbusinessadministration/responsivesite/research/Yang_Household%20equity%20share.pdf
The Household Equity Share represents retail investor positioning and according to Yang and Zhang it outperforms popular forecasters of market returns such as the cyclically adjusted price-earnings ratio, the equity share in new issuances, the consumption-wealth ratio, the term spread, and the Treasury bill yield.
To get an expected return on a five-year horizon we simply regress on a quarterly basis five-year rolling returns on our three indicators. i.e. to avoid forward data contamination we iteratively add an additional quarter of observation when it becomes available. The choice of a five-year horizon is somewhat arbitrary. Since we perform regressions on levels, we have an issue of nonstationary of our variables. Therefore, before we run regressions, we need to make sure that our indicators cointegrate with rolling returns. We have stronger cointegrating relationships between our indicators and five-year returns than between our indicators and ten-year returns. For that reason, a preference was made in favour of five-year return signals. The results of our historical back test are on the figure below. We assumed a zero bid/ask spread for a portfolio rebalancing and our funding rate is 3 month LIBOR. We further assumed, in line with what was described in the article, that in the case of a negative expected returns positions in the market are closed and all funds are allocated to cash, which also pays 3 month LIBOR.
Unfortunately, the author does not specify the way he estimates an excess return of risky assets, which is µ and volatility of risky assets, which is σ in equation (316).
To estimate volatility we use an industry wide approach and calculate historical volatility for the last 500 trading days. For stock market returns, it is more tricky. There is no industry wide standard model to project future expected returns. We use three different indicators to project five-year returns: P/E ratio, Tobin’s Q and the Household Equity Share. P/E ratio and Tobin’s Q are well known measures used in valuations of companies, whereas the Household Equity Share is a relatively new predictor. It was introduced by Yang and Zhang in their paper: https://cba.lmu.edu/media/lmucollegeofbusinessadministration/responsivesite/research/Yang_Household%20equity%20share.pdf
The Household Equity Share represents retail investor positioning and according to Yang and Zhang it outperforms popular forecasters of market returns such as the cyclically adjusted price-earnings ratio, the equity share in new issuances, the consumption-wealth ratio, the term spread, and the Treasury bill yield.
To get an expected return on a five-year horizon we simply regress on a quarterly basis five-year rolling returns on our three indicators. i.e. to avoid forward data contamination we iteratively add an additional quarter of observation when it becomes available. The choice of a five-year horizon is somewhat arbitrary. Since we perform regressions on levels, we have an issue of nonstationary of our variables. Therefore, before we run regressions, we need to make sure that our indicators cointegrate with rolling returns. We have stronger cointegrating relationships between our indicators and five-year returns than between our indicators and ten-year returns. For that reason, a preference was made in favour of five-year return signals. The results of our historical back test are on the figure below. We assumed a zero bid/ask spread for a portfolio rebalancing and our funding rate is 3 month LIBOR. We further assumed, in line with what was described in the article, that in the case of a negative expected returns positions in the market are closed and all funds are allocated to cash, which also pays 3 month LIBOR.
Source: Bloomberg, http://www.econ.yale.edu//~shiller/data.htm
In terms of an absolute performance of different indicators the results are drastically different. A portfolio which uses P/E ratio as a signal (represented by green line and right axis) significantly outperforms a portfolio based on Tobin’s Q and the one based on Household Equity Shares indicators. On the other hand, this portfolio has the sharpest (80%) drawdown. On a risk-adjusted basis, the performance of all portfolios is identical: the Sharpe ratio for all three portfolios is around 0.5.
How does that compare with the performance of a very naïve strategy that goes 100% long if expected return is positive and stays in cash when expected performance is negative? The results are on a figure below:
How does that compare with the performance of a very naïve strategy that goes 100% long if expected return is positive and stays in cash when expected performance is negative? The results are on a figure below:
Source: Bloomberg, http://www.econ.yale.edu//~shiller/data.htm
Although, this chart looks different compare to a previous one the risk-adjusted basis a performance of the naïve Long/Flat strategies is almost identical to the performance of portfolios with the leverage derived from ergodicity economics theory.
One can obviously argue that the indicators we used are not the perfect ones and one can perform much better if indicators were more precise. For that, reason we decided to test a performance of the optimal leverage strategy where long term return is known in advance. Specifically, we use 8% as a yearly expected total return rate which is roughly the rate of return of S&P 500 Total Return Index for the last 30 years. The results are on a figure below:
One can obviously argue that the indicators we used are not the perfect ones and one can perform much better if indicators were more precise. For that, reason we decided to test a performance of the optimal leverage strategy where long term return is known in advance. Specifically, we use 8% as a yearly expected total return rate which is roughly the rate of return of S&P 500 Total Return Index for the last 30 years. The results are on a figure below:
Source: Bloomberg
The portfolio had more than 90% drawdown and it took almost 20 years to get back to the High Water Mark. The Sharpe ratio for the strategy is 0.46. Both Sharpe ratio and drawdowns for this strategy are inferior to the Sharpe ratios and drawdowns for the strategies that used Tobin’s Q, P/E ratio and Household Equity Shares as indicators.
Curiously, upon further reading it appears that the author argues that an optimal leverage should always be equal to 1!
Curiously, upon further reading it appears that the author argues that an optimal leverage should always be equal to 1!
In our view, this statement is not entirely consistent with equation (316) and even more importantly, it contradicts to the volatility scaling technique, which is a founding block of modern risk management approach both on a buy-side and on a sale-side. To get things into perspective it would make sense to look at the performance on the Nikkei index on the figure below:
Despite negative interest rates, Abenomics and multiple rounds of quantitative easing the index today is almost 30% lower than it was back in 1989!
To conclude, we found very little evidence that ergodicity economics gives results consistent with the empirical market data. The optimal leverage recommended by the theory does not give much of an added value. It does not improve risk-adjusted return and significantly increases risk of potential drawdowns.
To conclude, we found very little evidence that ergodicity economics gives results consistent with the empirical market data. The optimal leverage recommended by the theory does not give much of an added value. It does not improve risk-adjusted return and significantly increases risk of potential drawdowns.
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