How to measure portfolio performance
To evaluate and compare different strategies or to improve an existing strategy, we need metrics that reflect their performance with respect to our objectives. In investment and trading, the most common objectives are the return and the risk of the investment portfolio.
Typically, these metrics are compared to a benchmark that represents alternative investment opportunities, such as a summary of the investment universe like the S&P 500 for US equities or the risk-free interest rate for fixed income assets.
There are several metrics to evaluate these objectives. In this section, we will review the most common measures for comparing portfolio results. These measures will be useful when we look at different approaches to optimize portfolio performance, simulate the interaction of a strategy with the market using Zipline, and compute relevant performance metrics using the pyfolio library in later sections.
We'll use some simple notation: let R be the time series of one-period simple portfolio returns, R=(r1, ..., rT), from dates 1 to T, and Rf =(rf1, ..., rfT) be the matching time series of risk-free rates, so that Re=R-Rf =(r1-rf1,..., rT-rfT) is the excess return.
Capturing risk-return trade-offs in a single number
The return and risk objectives imply a trade-off: taking more risk may yield higher returns in some circumstances, but also implies greater downside. To compare how different strategies navigate this trade-off, ratios that compute a measure of return per unit of risk are very popular. We'll discuss the Sharpe ratio and the information ratio in turn.
The Sharpe ratio
The ex ante Sharpe ratio (SR) compares the portfolio's expected excess return to the volatility of this excess return, measured by its standard deviation. It measures the compensation as the average excess return per unit of risk taken:
Expected returns and volatilities are not observable, but can be estimated as follows from historical data:
Unless the risk-free rate is volatile (as in emerging markets), the standard deviation of excess and raw returns will be similar.
For independently and identically distributed (IID) returns, the distribution of the SR estimator for tests of statistical significance follows from the application of the central limit theorem (CLT), according to large-sample statistical theory, to 
and 
. The CLT implies that sums of IID random variables like and 
converge to the normal distribution.
When you need to compare SR for different frequencies, say for monthly and annual data, you can multiply the higher frequency SR by the square root of the number of the corresponding period contained in the lower frequency. To convert a monthly SR into an annual SR, multiply by 
, and from daily to monthly multiply by .
However, financial returns often violate the IID assumption. Andrew Lo has derived the necessary adjustments to the distribution and the time aggregation for returns that are stationary but autocorrelated. This is important because the time-series properties of investment strategies (for example, mean reversion, momentum, and other forms of serial correlation) can have a non-trivial impact on the SR estimator itself, especially when annualizing the SR from higher-frequency data (Lo, 2002).
The information ratio
The information ratio (IR) is similar to the Sharpe ratio but uses a benchmark rather than the risk-free rate. The benchmark is usually chosen to represent the available investment universe such as the S&P 500 for a portfolio on large-cap US equities.
Hence, the IR measures the excess return of the portfolio, also called alpha, relative to the tracking error, which is the deviation of the portfolio returns from the benchmark returns, that is:
The IR has also been used to explain how excess returns depend on a manager's skill and the nature of her strategy, as we will see next.
The fundamental law of active management
"Diversification is protection against ignorance. It makes little sense if you know what you are doing."
– Warren Buffet
It's a curious fact that Renaissance Technologies (RenTec), the top-performing quant fund founded by Jim Simons, which we mentioned in Chapter 1, Machine Learning for Trading – From Idea to Execution, has produced similar returns as Warren Buffet, despite extremely different approaches. Warren Buffet's investment firm Berkshire Hathaway holds some 100-150 stocks for fairly long periods, whereas RenTec may execute 100,000 trades per day. How can we compare these distinct strategies?
A high IR reflects an attractive out-performance of the benchmark relative to the additional risk taken. The Fundamental Law of Active Management explains how such a result can be achieved: it approximates the IR as the product of the information coefficient (IC) and the breadth of the strategy.
As discussed in the previous chapter, the IC measures the rank correlation between return forecasts, like those implied by an alpha factor, and the actual forward returns. Hence, it is a measure of the forecasting skill of the manager. The breadth of the strategy is measured by the independent number of bets (that is, trades) an investor makes in a given time period, and thus represents the ability to apply the forecasting skills.
The Fundamental Law states that the IR, also known as the appraisal risk (Treynor and Black), is the product of both values. In other words, it summarizes the importance to play both often (high breadth) and to play well (high IC):
This framework has been extended to include the transfer coefficient (TC) to reflect portfolio constraints as an additional factor (for example, on short-selling) that may limit the information ratio below a level otherwise achievable given IC or strategy breadth. The TC proxies the efficiency with which the manager translates insights into portfolio bets: if there are no constraints, the TC would simply equal one; but if the manager does not short stocks even though forecasts suggests they should, the TC will be less than one and reduce the IC (Clarke et al., 2002).
The Fundamental Law is important because it highlights the key drivers of outperformance: both accurate predictions and the ability to make independent forecasts and act on these forecasts matter.
In practice, managers with a broad set of investment decisions can achieve significant risk-adjusted excess returns with information coefficients between 0.05 and 0.15, as illustrated by the following simulation:
Figure 5.1: Information ratios for different values of breadth and information coefficient
In practice, estimating the breadth of a strategy is difficult, given the cross-sectional and time-series correlation among forecasts. You should view the Fundamental Law and its extensions as a useful analytical framework for thinking about how to improve your risk-adjusted portfolio performance. We'll look at techniques for doing so in practice next.