Quantitative Portfolio Management. Michael Isichenko

(1.3). Brokers provide services of trading “continuous futures,” or automatically rolled futures positions.

      

1.3 Linear vs log returns

       Given a list of consecutive daily portfolio pnls, compute, in linear time, its maximum drawdown.

      From a quant interview

      The linear return (1.1), also known as simple or accounting return, defines a daily portfolio pnl

through dollar position

:

      (1.8)

      Here boldface notation is used for vectors in the space of portfolio securities. For pnl computation, the linear returns are cross-sectionally additive with position weights. Risk factor models (Sec. 4.2) add more prominence to the cross-sectional linear algebra of simple returns.

      It is also convenient to use log returns

      (1.9)

      which, unlike the linear returns, are serially additive, for a fixed initial investment in one asset, across time periods. In quant research, both types of return are used interchangeably.

Over short-term horizons of order one day, stock returns are of order 1%, so the difference between the linear and the logarithmic return

(1.10)

is of order

, or a basis point (bps), which is in the ballpark of the return predictability (Sec. 2.3.3). The expectation, or forecast, of the log return (1.10) is

(1.11)

where

is the volatility (standard deviation) of the return. Due to the negative sign of the correction in (1.11), its effect can be meaningful even for a slightly non-dollar-neutral or volatility-exposed portfolio. Volatility is one of commonly used risk factors (Sec. 4.3).

      The difference between linear and log returns affects forecasting (Chapter 2), especially over longer horizons, because the operators of (linear) expectation and (concave) log do not commute. Even though statistical distribution of log returns may have better mathematical properties than those of linear returns, it is the linear return based pnl that is the target of portfolio optimization (Chapter 6). On the other hand, the log return plays a prominent role in the Kelly criterion (Sec. 6.9).