Quantitative Portfolio Management. Michael Isichenko
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(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
(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
is of order
where
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).
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