Linear vs log returns

Also called linear or raw return is defined as: \[ \color{blue} { r_t = \frac{x_t - x_{t-1}}{x_{t-1}} = \frac{x_t}{x_{t-1}} - 1 } \] where:

• r_t - simple return for period t
• x_t - current value (price)
• x_t-1 - previous value (price)

Also called compound return is defined as: \[ \color{blue} { R_t = \ln({\frac{x_t}{x_{t-1}}}) = \ln({x_t}) - \ln{(x_{t-1})} } \] where:

• R_t - log return for period t
• ln - natural logarithm (base e - Euler's number)

Simple total return is defined as sum of individual returns for each period: \[ \color{blue} { str_t^n = {\sum_{t=1}^n{r_t}} } \] while simple average return is just the arithmetic mean: \[ \color{blue} { sar_t^n = \frac{\sum_{t=1}^n{r_t}}{n} } \] Example 1: let's take a stock A that is initially worth $100 and rises to $120 in t₁ and goes back to $100 in t₂

  a0, a1, a2 = 100, 120, 100
  r1_a = a1/a0 - 1
  print(r1_a)
  r2_a = a2/a1 - 1
  print(r2_a)
  str_a = r1_a + r2_a
  print(str_a)
  sar_a = tr / 2
  print(sar_a)

0.19999999999999996
-0.16666666666666663
0.033333333333333326
0.016666666666666663

Ooops! Something does not look right, total return is 3.33% even if we did not make any money (price falls back to $100).

Example 2: Let's take another stock B that was trading at $100 then $140 and $145.

  b0, b1, b2 = 100, 140, 145
  r1_b = b1/b0 - 1
  print(r1_b)
  r2_b = b2/b1 - 1
  print(r2_b)

0.3999999999999999
0.03571428571428581

Now, let's suppose that we have a portfolio with stock A (weight: 0.6) and stock B (weight: 0.4) and we want to calculate portfolio return for each period.

  r1 = 0.6 * r1_a + 0.4 * r1_b
  print(r1)
  r2 = 0.6 * r2_a + 0.4 * r2_b
  print(r2)

0.2693929340763724
-0.09510721979065809

Good! Return per each period seems correct given the weight of each stock.

Conclusions: simple returns

• ARE NOT additive over time
• ARE additive over assets

Compound total return is the sum of all log returns: \[ \color{blue} { ctr_t^n = {\sum_{t=1}^n{R_t}} } \] and for compound average return we need to use geometric mean because arithmetic mean does not account for compounding: \[ \color{blue} { car_t^n = ({\prod_{t=1}^n{(1 + R_t)}})^{\frac{1}{n}} - 1 } \]

Example: let's use the same example above and calculate the returns:

  from math import log
  r1_a = log(a1) - log(a0)
  print(r1_a)
  r2_a = log(a2) - log(a1)
  print(r2_a)
  ctr = r1_a + r2_a
  print(ctr)
  car = ((1+r1_a) * (1+r2_a))**(1/2) - 1
  print(car)

0.182321556793954
-0.182321556793954
0.0
-0.016761041288421596

Good! compound total return is 0% which is the correct result and compound average return seems to be correct and equal to simple average return 1.66%.

Example 2: same stock B as above

  r1_b = log(b1) - log(b0)
  print(r1_b)
  r2_b = log(b2) - log(b1)
  print(r2_b)

0.33647223662121206
0.03509131981127034

And portfolio return per each period:

  r1 = 0.6 * r1_a + 0.4 * r1_b
  print(r1)
  r2 = 0.6 * r2_a + 0.4 * r2_b
  print(r2)

0.24398182872485724
-0.09535640615186428

Ooops! quite a difference between simple (0.27) and compound (0.24) portfolio return for t₁ because when we add log returns we compound but in this case there is nothing to compound on, stock A and stock B are two different things which may/may not be correlated.

Conclusions: log returns

• ARE additive over time
• ARE NOT additive over assets

OK, both returns have pros/cons but which one is better and the answer is both, it depends on what kind of data you have and what you want to calculate.

A few more properties and intuition:

• log return as function of simple return

\[ \textcolor{black} { R_t = \ln({\frac{x_t}{x_{t-1}}}) \\ R_t = \ln({1 + \frac{x_t}{x_{t-1}}} - 1) } \] \[ \color{blue} { R_t = \ln(1 + r_t) } \]

• easy to calculate one given the other

\[ \textcolor{black} { e^{R_t} = e^{\ln(r_t + 1)} \\ e^{R_t} = r_t + 1 } \] \[ \color{blue} { r_t = e^{R_t} - 1 } \]

• aggregation (sum) is very efficient over log returns, a sub operation executed in O(1) \[ \textcolor{black} { R = \ln(\frac{x_f}{x_{f-1}}) + ... + \ln(\frac{x_1}{x_0}) \\ R = \ln(\frac{x_f}{x_{f-1}}\frac{x_{f-1}}{x_{f-2}}...\frac{x_2}{x_1}\frac{x_1}{x_0}) \\ R = \ln({\frac{x_f}{x_0}}) } \]

  \[ \color{blue} { R = ln(x_f) - ln(x_0) } \] where:

  • R - total return
  • x_f - final value (price)
  • x₀ - initial value (price)
• log return is faster to calculate because subtraction operation are numerically safe/faster than division
• log returns follow normal distribution since underlying prices are (hopefully) log-normal distributed (see Central Limit Theorem)
• simple returns are easier to reason about
• if you work with raw data (prices) use simple returns and arithmetic mean
• when you work with percentage/change in values use log returns
• if data series are volatile use log returns and geometric mean
• aggregate data over time use log returns
• aggregate data over different assets use simple returns