# The relationship between trading frequency and achievable alpha

#### Published in Automated Trader Magazine Issue 42 Q1 2017

## Some of the most successful investors have long holding periods and investors are often advised to "buy and hold". We investigate how the average holding period of an optimal trading strategy relates to the alpha it can generate. Is longer really better?

A common belief amongst investors is that long-term investing is preferable to short-term trading. The great Warren Buffett, whose favourite holding period is "forever", is often cited as evidence to support this point of view.

On the other hand, it is fairly obvious that increasing trading frequency, at least hypothetically, enables capturing more variance and should therefore be more profitable overall.

In this article, we first show the difficulty of generating significant alpha at low trading frequencies. After that, we quantify the relationship between trading frequency and maximum theoretically achievable alpha.

#### Figure 01: Maximum return strategy with 7 trades on the S&P 500 (since 1871)

#### Figure 02: Maximum alpha strategy with 6 trades on the detrended S&P 500 (since 1871)

### THE MAXIMUM RETURN STRATEGY AT A LOW TRADING FREQUENCY

We will start our analysis at a very low trading frequency, with an average period of 25 years between two trades (corresponding to so-called 'secular' bull/bear markets). Between 1871 and 2016 (145 years), this average trading period would have allowed about six trades.

Figure 01 shows the maximum return directional strategy (long or short 100% assets) that can be generated on the S&P 500 with seven trades (green = 100% long position, red = 100% short position). Here, we look at excess returns (i.e. total returns including dividends above the risk-free rate). Note that we use seven trades instead of six, as an odd number of trades yields the optimal strategy in this case. (We will later consider a maximum alpha strategy for which an even number of trades is optimal instead.)

This strategy would have generated a compounded annualized excess return of 9.08%, with a volatility of 13.99%, yielding a Sharpe ratio of 0.64. This is quite underwhelming for a strategy that was short during the three largest market crashes over the past 150 years and long otherwise. Additionally, it is worth noting that the maximum return strategy would have been long 87.8% of the time, thereby capturing a sizable amount of equity risk premium.

### THE MAXIMUM ALPHA STRATEGY AT A LOW TRADING FREQUENCY

As its title indicates, this paper is about alpha rather than total return. Therefore, to remove the equity risk premium from the S&P 500, we detrend it by subtracting its average return from each observation. This enables us to define the 'maximum alpha strategy', which has a neutral long-term exposure to the S&P 500 and goes long (short) during periods when the S&P 500 outperforms (underperforms) its long-term trend, as shown in Figure 02.

#### Figure 03: Shiller's CAPE ratio (4 cycles in 135 years)

The maximum alpha strategy with six trades would have generated an annualized alpha of 6.79%, with a volatility of 14.10%. Despite having a completely unrealistic knowledge of all highs and lows, this strategy would have generated a Sharpe ratio of 0.48.

On an alpha generation basis, it is structurally impossible for an investor (or an indicator) with a secular viewpoint of the S&P 500 to outperform this Sharpe ratio. Moreover, a reallife secular investor would obviously capture a small fraction of alpha generated by the maximum alpha strategy.

This demonstrates the difficulty of generating significant riskadjusted alpha at very low trading frequencies. That said, one could state that very few investors try to time the market at such a low trading frequency (one change of mind every 25 years). Fair enough.

#### Figure 04: Maximum alpha strategy with 12, 24 and 48 trades

However, investors often pay a lot of attention to indicators that have very long cycles (valuation ratios, credit cycle indicators, 10-year return forecasting models etc.).

Our point is that these long cycle indicators are structurally incapable of producing significant alpha, regardless of how we use them. (Not to mention the fact that, even on an insample basis, they cannot be validated statistically due to the low number of trades they would generate.)

Consequently, these indicators should rank very low on an investor's watch list. The popular Shiller CAPE ratio (Figure 03), which experienced only four cycles during the past 135 years, is a typical example of a long cycle indicator.

In our view, instead of being a cause for alarm every time it reaches a new local high, such an indicator should (at best) be seen as a tiny breeze that might slow down future equity returns.

#### Figure 05: Power law relationship between trading frequency and Sharpe ratio

### THE MAXIMUM ALPHA STRATEGY AT HIGHER TRADING FREQUENCIES

If we progressively double the number of trades to 12, 24 and 48, we observe a regular increase in the Sharpe ratio of the maximum alpha strategy. This is illustrated in Figure 04.

Beyond a certain trading frequency, we can observe a power law relationship between the number of trades and the Sharpe ratio of the maximum alpha strategy (Figure 05). In other words: on the S&P 500, doubling trading frequency increases the maximum achievable alpha of a market timer by around 36% (= 20.44 - 1).

The fact that we observe a power law relationship is not surprising. We can link this to the concept of fractal dimension, first introduced by Benoit Mandelbrot in his 1967 paper, "How long is the coast of Britain? Statistical self-similarity and fractional dimension".

This seminal paper showed how the length of Britain's coastline changes according to the scale at which it is measured (see Figure 06).

The measured length of Britain's coastline is a function of the measurement scale and can be approximated by the following power law

with a constant and the fractal dimension of Britain's coastline (around 1.25).

Britain's coastline and the S&P 500 are fractal objects, meaning that they have irregular shapes at all scales of measurement. To a certain extent, we can consider that the maximum alpha strategy (which buys lows and sells highs) measures the 'length' of the asset's trajectory at different scales. Consequently, it is logical to observe a power law relationship between trading frequency and maximum achievable alpha.

Note that the exponent in the power law between trading frequency and maximum achievable alpha is, in fact, an estimator of the asset's Hurst exponent. Financial assets generally have a Hurst exponent value lower than 0.5, indicating a tendency for them to mean revert in the long term.

#### Figure 06: The length of Britain's coastline as a function of the measure scale used

### CONCLUSION

Before jumping to the conclusion that high frequency is necessarily better, one needs to be reminded that there are all sorts of factors that will differentiate the actual outcome from the hypothetical optimal outcome. Trading successfully at high frequencies faces a number of obstacles:

• Decreased signal-to-noise ratios

• Increased market impact

• Increased transaction costs.

Especially at sufficiently high frequencies the concept of trying to follow a time series with an optimal timing strategy becomes obviously absurd. There is a clear limit to how much we can expect to improve by simply increasing the frequency by which we alter position signs. Nevertheless it is interesting to see how performance behaves on the longer time frames.