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Discrete vs. Continuously Compounded Returns

A Comparison

As discussed before, asset returns can be calculated in two primary ways: discrete (simple) returns and continuously compounded (logarithmic) returns. While both aim to measure the percentage change in an asset's value, they differ in their calculation and properties.

Key Differences

FeatureDiscrete Return (Simple Return)Continuously Compounded Return (Log Return)
FormulaR_t = (P_t - P_{t-1}) / P_{t-1}r_t = ln(P_t / P_{t-1})
Additivity over TimeNot directly additiveAdditive
SymmetryAsymmetricSymmetric
MagnitudeCan be larger than 1 (100%)Typically smaller in magnitude

Advantages of Continuously Compounded Returns

  1. Time Additivity: Continuously compounded returns are additive over time. This means that the return over multiple periods can be calculated by simply summing the returns of each individual period. This property is extremely useful for calculating returns over longer horizons.

    • r_{t, t+n} = r_t + r_{t+1} + ... + r_{t+n}

    This property does not hold for discrete returns. To calculate multi-period discrete returns, you need to compound the returns:

    • 1 + R_{t, t+n} = (1 + R_t) * (1 + R_{t+1}) * ... * (1 + R_{t+n})

  2. Symmetry: Continuously compounded returns are more symmetric than discrete returns. This means that an equal percentage increase and decrease in price will result in returns of approximately the same magnitude but with opposite signs. For example, if a stock goes from $100 to $110 (a 10% increase), the continuously compounded return is approximately 9.53%. If the stock then goes from $110 back to $100 (a 9.09% decrease), the continuously compounded return is approximately -9.53%. Discrete returns would be 10% and -9.09%, which are not symmetric. This symmetry can be beneficial in statistical modeling.

  3. Mathematical Convenience: Continuously compounded returns are often more convenient to work with in mathematical and statistical models. The logarithmic transformation often helps to stabilize variance and make the data more closely resemble a normal distribution, which is an assumption of many statistical tests.

  4. Approximation for Small Returns: For small return values (close to zero), the continuously compounded return is a good approximation of the discrete return. This is because ln(1 + x) ≈ x when x is close to zero.

When to Use Which?

  • Discrete Returns: Are generally easier to understand and interpret for single-period returns. They are also useful when dealing with situations where additivity is not a primary concern.
  • Continuously Compounded Returns: Are preferred when additivity over time is important, when working with statistical models, and when returns are relatively small. They are the standard in much of financial econometrics.

In conclusion, while both types of returns have their place, continuously compounded returns offer significant advantages in many financial applications due to their additivity, symmetry, and mathematical convenience.

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