Chris Conlan - Automated Trading with R: Quantitative Research and Platform Development

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Part 1
Problem Scope

Chris Conlan 2016

Chris Conlan Automated Trading with R 10.1007/978-1-4842-2178-5_1

1. Fundamentals of Automated Trading

Chris Conlan 1

(1)

Bethesda, Maryland, USA

Electronic supplementary material

The online version of this chapter (doi: 10.1007/978-1-4842-2178-5_1 ) contains supplementary material, which is available to authorized users.

The fundamental goal of trading is to maximize risk-adjusted return. When developing strategies, we will simulate trading performance in an attempt to maximize risk-adjusted return in simulation. There are many ways to measure risk-adjusted return. They involve examining the shape of the equity curve and the return series .

Equity Curve and Return Series

The equity curve is the trading account value plotted against time. It can otherwise be thought of as cash on hand plus the equity value of portfolio assets plotted against time. We want it to rise linearly if we trade with a uniform account size or exponentially if we reinvest gains. The return series is the list of returns on the account at each trading period. The return series depends only on which assets are traded when, not the trading account size, so it will be the same whether or not we reinvest gains.

Figure shows an example of an equity curve generated by a strategy that is long up to ten S&P 500 stocks at a time with a trading account of $10,000, trading once per day, without reinvesting gains. A gray reference line is plotted for an equivalent investment in the SPY S&P 500 ETF, a tradable fund that closely mimics the behavior of the S&P 500.

Figure 1-1.

Example equity curve

The return series is the portfolio gain or loss as a percentage of tradable capital at each trading period. Figure .

Figure 1-2.

Example return series

Characteristics of the Equity Curve

We will introduce some notation to study characteristics of the equity curve .

We define to be the dollar value of the portfolio before adjustment and to be the dollar value of the portfolio after adjustment for t in 0, 1, 2,..., T , where t= 0 represents the beginning of simulation and t=T represents the current time.

We assume that portfolio adjustments (or trades) happen instantaneously in time. The change in P from t 0 to t 1 represents change due to adjustment, while the change in P from ( t 1)1 to t 0 represents change due to movement of market prices of the assets in the portfolio. Chronologically, t evolves as , with transitions from t 0 to t 1 happening instantaneously when an algorithm automatically adjusts the portfolio.

We define C 0 as initial cash, and as uninvested cash at t 0 and t 1, and K t as trading costs incurred during instantaneous adjustment from t 0 to t 1. The equity curve at time t 0 is equal to the following:

Note that for t =0. Further, we note that the difference between and is the total of trading costs incurred during the adjustment period, from t 0 to t 1.

When we plot the equity curve and perform risk-return computations on it, we use only for t in 0, 1,..., T . The choice of over is intended to reflect the impact of commissions in the equity curve.

Characteristics of the Return Series

We define V t to be the tradable capital at time t 0. This is a value set by the trader. The total cash invested by the trader cannot exceed V t at any given time. We define t ( i 1) and t ( i 0) to be the times t 1 and t 0 at which trade i was initiated and exited, respectively. Trade i is considered to be active at time t if . We say that if i is active at t 1. We define j i as the asset initiated in trade i . Further, allow and to be subsettable by asset such that represents the value of asset j in the portfolio at time t 1.

If we make 15 trades in the instantaneous adjustment period occurring from t 0 to t 1, there will be 15 new i s subsettable to t for these transactions. This allows us to make infinitely many overlapping trades and describe them using our notation.

The tradable capital must meet the following condition for all t in 0, 1,..., T :

where V t is determined during or prior to ( t 1) based on information available at that time.

Verbally, this means the sum of the initial purchase prices of all active trades is less than or equal to the tradable capital. Note that there is no restriction regarding the relationship between V t and or . This is because and

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