Average Return

Introduction

Average return is a fundamental concept in finance and investing, particularly significant in algorithmic trading (also known as “algotrading” or “automated trading”). It provides investors and traders with insights into the historical performance of a financial asset, portfolio, or strategy. The average return is a measure used to quantify the central tendency of returns over a specific period. This metric can be calculated in various ways, each here tailored to different analytical needs and scenarios.

Types of Average Return

Arithmetic Average Return

The arithmetic average return is the most straightforward calculation of average returns. It is computed by summing up the individual returns and then dividing by the number of periods. Formally, for a set of ( n ) returns ( R_1, R_2, \ldots, R_n ):

[ \text{Arithmetic Average Return} = \frac{1}{n} \sum_{i=1}^{n} R_i ]

This type of average is best used in short-term analysis where compounding effects are minimal. It provides a simple estimate of performance over multiple periods and is often used in backtesting trading strategies.

Geometric Average Return

The geometric average return, also known as the compounded annual growth rate (CAGR), is more suitable for long-term performance evaluation. It accounts for the effects of compounding over multiple periods. The formula for ( n ) returns ( R_1, R_2, \ldots, R_n ) is:

[ \text{Geometric Average Return} = \left( \prod_{i=1}^{n} (1 + R_i) \right)^\frac{1}{n} - 1 ]

This method is more accurate for assessing the performance sustainability of an asset or strategy over time as it smooths out volatility and highlights growth trends.

Modified Dietz Method

The Modified Dietz method is an approximation approach that incorporates external cash flows. It’s particularly useful in scenarios where a portfolio experiences significant cash inflows or outflows. The formula is:

[ \text{Modified Dietz Return} = \frac{EMV - BMV - C}{BMV + wC} ]

• ( EMV ) is the ending market value.
• ( BMV ) is the beginning market value.
• ( C ) is the net external cash flow.
• ( w ) is the weighting factor calculated as the fraction of the period the cash flow is present.

Applications in Algorithmic Trading

Algorithmic trading strategies deploy mathematical models and algorithms to execute trades at speeds and frequencies unachievable by human traders. Average return metrics form the backbone of evaluating, backtesting, and optimizing these strategies.

Backtesting

Backtesting involves running an algorithm on historical data to assess its viability. Average return types, particularly the arithmetic and geometric returns, are crucial to determining how well a strategy would have performed in the past, guiding future decision-making.

Risk Management

Returns analysis helps in refining the risk management elements of a trading algorithm. By knowing the average returns, traders can design strategies with acceptable risk levels and set appropriate thresholds for drawdowns, stop losses, and take-profits.

Performance Metrics

Average returns are also indispensable in constructing performance metrics, such as the Sharpe ratio, which adjusts returns by accounting for risk (standard deviation of returns).

[ \text{Sharpe Ratio} = \frac{R_p - R_f}{\sigma_p} ]

where ( R_p ) is the average return of the portfolio, ( R_f ) is the risk-free rate, and ( \sigma_p ) is the standard deviation of the portfolio returns.

Portfolio Optimization

In modern portfolio theory, average returns are central to constructing an optimal portfolio. The mean-variance optimization technique uses expected returns (geometric average) and covariance of asset returns to maximize returns for a given level of risk.

[ \text{Optimization Objective:} \quad \max_w \left( w^T \mu - [lambda](../l/lambda.html) w^T \Sigma w \right) ]

• ( w ) is the weights vector.
• ( \mu ) is the mean returns vector.
• ( \Sigma ) is the covariance matrix of returns.
• ( [lambda](../l/lambda.html) ) is the risk-aversion coefficient.

Major Institutions and Platforms

Bloomberg LP

Bloomberg’s Terminal (https://bloomberg.com/professional) provides traders with comprehensive datasets and tools to analyze average returns among other financial metrics. It supports both backtesting and real-time algorithmic trading analyses.

QuantConnect

QuantConnect (https://www.quantconnect.com/) is a prominent algorithmic trading platform offering a cloud-based backtesting and live trading environment. It leverages average return calculations in its performance module.

Interactive Brokers

Interactive Brokers (https://www.interactivebrokers.com) facilitate algorithmic traders with APIs and robust trading platforms. Traders rely on historical average returns to fine-tune their automated trading systems.

QuantInsti

QuantInsti (https://www.quantinsti.com/) is an educational institution offering training programs in algorithmic trading. It emphasizes the importance of average returns in strategizing and evaluating trading models.

TradeStation

TradeStation (https://www.tradestation.com/) offers tools for backtesting and developing custom trading algorithms. It includes features for calculating historical and average returns, crucial for strategy development.

Conclusion

Understanding and calculating average returns in various forms, such as arithmetic, geometric, and Modified Dietz, is imperative for investors and algorithmic traders. These calculations provide critical insights into the performance of financial assets and are instrumental in strategy development, risk management, portfolio optimization, and performance evaluation. The integration of average return metrics in prominent trading platforms and institutions substantiates their pivotal role in the algorithmic trading industry.