Abstract

Traders in the financial world are assessed by the amount of money they make and, increasingly, by the amount of money they make per unit of risk taken, a measure known as the Sharpe Ratio. Little is known about the average Sharpe Ratio among traders, but the Efficient Market Hypothesis suggests that traders, like asset managers, should not outperform the broad market. Here we report the findings of a study conducted in the City of London which shows that a population of experienced traders attain Sharpe Ratios significantly higher than the broad market. To explain this anomaly we examine a surrogate marker of prenatal androgen exposure, the second-to-fourth finger length ratio (2D4D), which has previously been identified as predicting a trader’s long term profitability. We find that it predicts the amount of risk taken by traders but not their Sharpe Ratios. We do, however, find that the traders’ Sharpe Ratios increase markedly with the number of years they have traded, a result suggesting that learning plays a role in increasing the returns of traders. Our findings present anomalous data for the Efficient Markets Hypothesis.

Introduction

Knowing that a trader has made $10 million tells us little about the skill involved in making this money unless we also know how much risk was taken. If this trader could have, with equal probability, lost $100 million then we would have to conclude that the gain of $10 million was merely a bit of luck, even dumb luck. To control for this possibility, trading managers and investors increasingly look at the returns made on capital (in excess of the risk free rate) and then divide by the standard deviation of these returns, giving them a measure known as the Sharpe Ratio [1]. The Sharpe Ratio plays an important role in Modern Portfolio Theory [2]. [3]. and in the influential Efficient Market Hypothesis [4]. [5]. [6] .

According to this hypothesis market prices provide the best estimate of asset values because they incorporate all available information. Prices change with new information, which by its nature arrives unexpectedly, making the markets random and preventing anyone from consistently outperforming a broad market such as the S&P index or the Dax, the German stock index. The Efficient Market Hypothesis (EMH) does not deny that investors can, through asset allocation decisions, increase the return on their capital, but they can do so only by increasing the amount of risk taken, risk in this case being defined as the standard deviation of returns. If we plot the possible rates of return against the levels of risk needed to achieve them, we should, according to EMH (or at least the Capital Asset Pricing Model, which is the testable prediction of EMH) find a curve that is linear with a slope equal to the Sharpe Ratio of the broad market. EMH implies, therefore, that one can increase one’s returns but one cannot systematically increase one’s Sharpe Ratio above that of the broad market. We tested this hypothesis by examining the Sharpe Ratios of a group of traders.

To do so we analysed profit and loss (P&L) statements over a 20 month period, between 2005 and 2007, for 53 traders on a trading floor in the City of London. These traders are all male and at the beginning of the study had an average age of 29 years. They engage in what is variously called ‘noise’ or ‘high frequency’ trading, meaning they buy and sell futures contracts on a range of underlying assets, mostly bonds and equities, with the occasional position in currencies or commodities, and hold their positions for short periods of time, usually seconds or minutes. Their trading is proprietary, meaning they trade for their own accounts and do not make markets for clients; they do not therefore benefit from the bid-offer spread on market-making, a lucrative source of profit for what are called ‘flow’ traders at investment banks; nor do they receive fees or commissions of any kind. Lastly, they do not receive a year-end bonus based in part on the performance of the firm as a whole. Their P&L thus derives purely from their trading skill, and this in turn determines their income. Selection acts quickly in this trading environment. Money losing traders do not last long; but the more successful traders on the floor can earn over $10million per year [7] .

We calculated monthly Sharpe Ratios for this cohort of traders as well as the broad market. To calculate a Sharpe Ratio one normally calculates the return on invested capital, subtracts the risk-free rate of interest to give the investment’s excess return over the risk free rate, and divides by the standard deviation of the returns (Note S1 ). However, the calculation of a Sharpe Ratio for leveraged traders, like high frequency traders, is slightly different: high frequency traders rarely position trades overnight so do not need to post capital, making it difficult to calculate their rate of return. This does not mean they do not need capital: they must have on hand capital enough to post margin on positions and to cover any trading losses. But that capital, while untouched, is invested in liquid deposits or government bonds and earns on average a rate of interest close to the risk free rate, meaning the return on this capital less the risk free rate nets close to zero [8]. Calculating the Sharpe Ratio therefore reduces to taking mean trading P&L and dividing by the standard deviation of P&L.

Results

We began by plotting the traders’ P&L against their risk, i.e. the standard deviation of their P&L. We found a curve that is, in accordance with the predictions of EMH, linear ( Figure 1 ). We then calculated the average monthly Sharpe Ratios for the 53 traders and found that it was 0.70.

Trader Sharpe Ratios compared to Dax.

We were concerned, however, that this test did not take into account the fact that the Sharpe Ratio of the Dax is itself a random variable which we observed at a value of 0.534 but which could, over the study period, have been higher or lower. To control for the stochastic nature of the Dax’s Sharpe we compared it to the trader Sharpes by means of a bootstrap test. This test consists of the following procedure: we use the observed time series of Dax monthly returns over the period to create new time series of returns by resampling. For each of these new time series we compute a Sharpe Ratio. We can in this way create a distribution of Sharpes by resampling the observed monthly returns of the Dax (Note S1 ). We can similarly calculate new Sharpes for the traders by resampling their monthly returns. The bootstrap test repeats this process and measures how often the observed Dax Sharpe takes on higher values than the average traders’ Sharpe. Our bootstrap tests confirmed our first estimates, that the Sharpe Ratios of the beginner traders were not significantly different from that of the Dax (p=0.72, n=27) but that the Sharpes of experienced traders were significantly higher (p=0.032, n=26). Running the bootstrap test for the Dax Sharpe denominated in Sterling gives, as expected, even more significant results (p=0.001, n=27).

In the present study we looked into the possibility that higher pre-natal androgen exposure improved the traders’ profitability through increased Sharpe Ratios. To conduct this and subsequent analyses the P&L and Sharpe Ratio data were Box-Cox transformed to correct for a pronounced right skew. The correlation between the traders’ 2D4D and their Sharpe Ratios, however, while displaying the expected sign, only approached significance at the 5% level (n=47, r=−0.26, p=0.08). This finding suggested to us that androgens may have their main effect through the amount of risk taken by traders rather than through the amount of money made per unit of risk. We therefore employed robust regression to determine the effect on traders’ P&L of i) the standard deviation of their P&L; ii) years of experience; and iii) 2D4D. We found that the regression was significant (F test, p<0.00001) and displayed high explanatory power (R 2 =0.94). The risk variable was highly significant (p<0.001); as was years of experience (p=0.004); but 2D4D was not (p=0.911). 2D4D was, however, correlated with risk in a simple regression (n=47, r=−.43, p=0.001. Figure 3 ). The low 2D4D traders are more profitable and survive longer in the markets, as was previously reported, but we now find the effect is largely mediated through a higher tolerance for risk.

2D4D vs risk.