One metric that is often used when talking about the performance of a trading strategy is the Sharpe ratio: the excess return above the risk-free rate per unit of risk (or volatility) that a strategy delivers. This metric is often subject to inflation, in the literal sense. Let’s deflate it somewhat.
To simplify the following analysis, we assume a risk-free rate of 0% to begin with, which makes the Sharpe ratio equivalent to the information ratio (“IR”), the strategy’s return divided by its volatility. While quick to calculate, the IR might not be the most intuitive to understand. We can reframe it, however, in terms of a simpler measure that also crops up in other fields such as sports and betting: the hit ratio. This is the ratio of correct bets to all bets taken, or the percentage of trades you got right. For example, you went long, the market went up; you went short, the market went down. You said heads/tails, and the coin landed heads/tails. If your hit ratio is 60%, you are right 6 out of 10 times. Can we link the hit ratio to the IR? A high hit ratio should give us a high IR! Going with the simplest possible model for our random outcome (winning the bet), we settle on the binomial distribution. Let’s say we are right with a probability p on each bet. When we win, we make $1; otherwise, we lose $11. We play repeatedly, N times.
Our expected return over those N bets is:
![\begin{equation*} \mathbb{E}(R)= N [p \cdot 1 + (1-p) \cdot (-1)] = N (2p - 1) \end{equation*}](https://www.grahamcapital.com/wp-content/ql-cache/quicklatex.com-ddfe3b30baf25721728034ee3fde71ef_l3.png "Rendered by QuickLaTeX.com")
The variance of our return is written down quickly2:
![\begin{equation*} \mathrm{Var}(R)= N \left(\left[p \cdot 1^2 + (1-p) \cdot (-1)^2\right] - (2p-1)^2\right) = 4Np(1 - p) \end{equation*}](https://www.grahamcapital.com/wp-content/ql-cache/quicklatex.com-ea8d60147fd6c70b948266dc3aafe63a_l3.png "Rendered by QuickLaTeX.com")
Now the IR we are interested in is:
Typically, we talk about annualized IRs, this would mean N = 252 (business days). We can now just plug values of p into this expression, and get this:
| 𝑝 | 0.50 | 0.52 | 0.55 | 0.60 |
| IR | 0 | 0.64 | 1.60 | 3.24 |
So, an IR of 0.64 (quite reasonable for classic macro strategies) corresponds to a hit ratio of 0.52. Conversely, a hit ratio of 0.55 – not that much better than a coin toss – would give an IR of 1.6! And mythical IRs above 3 can be reached with a “modest” hit ratio of 0.6. Note that a hit ratio of 0.5 gives a zero IR, as expected.
Another way of increasing the IR is to trade faster. Say we fix our hit ratio at 0.52 and trade D times a day. Our annualized IR would be:
| D | 2 | 4 | 8 | 16 |
| IR | 0.90 | 1.27 | 1.80 | 2.54 |
One does not have to do high frequency trading to get to (gross) IRs well above 2! Where is the caveat? There are various costs associated with trading, and these become substantial when trading faster, bringing down the net IR, as outlined in the Transaction Costs research note (Graham Capital Management, 2017). In addition, there may be capacity issues at faster execution horizons, limiting the feasible trading size.
Now that we have gained some intuition for the relationship between hit ratio and IR it is time to bring back our risk-free rate! For years, in a close to zero rate environment, the distinction between IR and Sharpe ratio was of little practical importance. With rates back at substantial non-zero levels, the difference between these two ratios is worth illuminating. Denoting our risk- free rate as r, the trading strategy’s return as s, and the trading strategy’s volatility as σ, we write down our two earlier definitions:
Clearly, to achieve a non-negative or non-zero Sharpe, a strategy needs to clear the risk-free rate hurdle first. In a higher interest rate environment, this hurdle becomes more challenging. But there is an upside! Many macro strategies (especially in the futures space) make use of leverage, in that not all capital has to be deployed to trade. Given the generally liquid nature of the underlying markets and instruments traded, macro strategies typically do not borrow money or rely on external funding, which is a form of leverage often employed by other hedge fund styles. Instead, only some margin m has to be posted. Any unencumbered cash (1 – m) can be invested at the risk-free rate. We can thus write down an expression for our more realistic levered Sharpe ratio:
Fixing our hit ratio and thus IR, say at 0.52 and 0.64, respectively, as well as setting our volatility at σ = 10%, we can solve for s and tabulate values for SRlev for varying levels of r and m.
| Margin m | Risk-free rate r | ||||||
|---|---|---|---|---|---|---|---|
| 0% | 1% | 2% | 3% | 4% | 5% | 6% | |
| 10% | 0.64 | 0.63 | 0.62 | 0.61 | 0.60 | 0.59 | 0.58 |
| 25% | 0.64 | 0.61 | 0.59 | 0.56 | 0.54 | 0.51 | 0.49 |
| 50% | 0.64 | 0.59 | 0.54 | 0.49 | 0.44 | 0.39 | 0.34 |
| 100% | 0.64 | 0.54 | 0.44 | 0.34 | 0.24 | 0.14 | 0.04 |
Table 1: Levered Sharpe ratio SRlev for ranges of r and m when p = 0.52 ⇔ IR = 0.64.
These values make sense: when the risk-free rate is zero, it does not matter how large our margin requirement is, as we won’t be able to earn a risk-free return on the unencumbered cash. And our levered Sharpe is largest when we have a small margin requirement, as we can put a lot of remaining cash to good use. A margin requirement of 10% is quite realistic for a futures trading strategy, while many single name equity strategies require full allocation of capital (m = 100%), or borrowing of cash to finance the strategy, which lowers the final Sharpe. For completeness, we show levered Sharpe ratios when p = 0.55 and IR = 1.60. While a strategy with low margin requirement can deliver an effective Sharpe well above 1.0 when the hit ratio is 0.55, a strategy that requires full capital allocation is much more sensitive to the level of the risk-free rate, as it cannot earn cash to offset the risk-free rate hurdle in the Sharpe ratio definition.
| Margin m | Risk-free rate r | ||||||
|---|---|---|---|---|---|---|---|
| 0% | 1% | 2% | 3% | 4% | 5% | 6% | |
| 10% | 1.60 | 1.59 | 1.58 | 1.57 | 1.56 | 1.55 | 1.54 |
| 25% | 1.60 | 1.57 | 1.55 | 1.52 | 1.50 | 1.47 | 1.45 |
| 50% | 1.60 | 1.55 | 1.50 | 1.45 | 1.40 | 1.35 | 1.30 |
| 100% | 1.60 | 1.50 | 1.40 | 1.30 | 1.20 | 1.10 | 1.00 |
Table 2: Levered Sharpe ratio SRlev for ranges of r and m when p = 0.55 ⇔ IR = 1.60.
Higher rates can thus act as a tailwind for macro. You are not only getting the alpha of the strategy, you are able to compound it with the return on the unencumbered cash, achieving greater portfolio efficiency over time. Coupling that with uncorrelated returns to public and private market exposures, macro-oriented strategies can deliver true alpha to a portfolio – and here consistency is key!
1Note that in practice, profits and losses are not fixed amounts, and outcomes may not be entirely independent. However, these factors do not alter the overall conclusion of this analysis.
2We use the following identity:
References
Ed Tricker, Saurabh Srivastava, and Marci Mitchell. Transaction Costs. Research Note, Graham Capital Management, July 2017.
DISCLOSURE
This presentation includes statements that may constitute forward-looking statements. These statements may be identified by words such as “expects,” “looks forward to,” “anticipates,” “intends,” “plans,” “believes,” “seeks,” “estimates,” “will,” “project” or words of similar meaning. In addition, our representatives may from time to time make oral forward-looking statements. Such statements are based on the current expectations and certain assumptions of Graham Capital Management’s (“GCM”) management, and are, therefore, subject to certain risks and uncertainties. A variety of factors, many of which are beyond GCM’s control, affect the operations, performance, business strategy and results of the accounts that it manages and could cause the actual results, performance or achievements of such accounts to be materially different from any future results, performance or achievements that may be expressed or implied by such forward-looking statements or anticipated on the basis of historical trends.
This document is not a private offering memorandum and does not constitute an offer to sell, nor is it a solicitation of an offer to buy, any security. The views expressed herein are exclusively those of the authors and do not necessarily represent the views of Graham Capital Management. The information contained herein is not intended to provide accounting, legal, or tax advice and should not be relied on for investment decision making.
Tables, charts and commentary contained in this document have been prepared on a best efforts basis by Graham using sources it believes to be reliable although it does not guarantee the accuracy of the information on account of possible errors or omissions in the constituent data or calculations. No part of this document may be divulged to any other person, distributed, resold and/or reproduced without the prior written permission of GCM.
