Difference between logarithmic returns and continuous compounding with Napierian logarithms
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😎 explain the small difference in results between the two formulas. Why it’s recommended to use logarithmic returns?
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🤖 The small difference in results between the linear and logarithmic return formulas arises because they measure percentage change in different ways. Linear returns calculate the percentage change based on the initial value, while logarithmic returns measure the percentage change in a way that can be time-additive over multiple periods – a property linear returns do not have.😎 explain the difference between using Napierian logarithm or base-10 logarithm. I see the formula with LN can give negative returns larger than -1, as in the example. But base-10 can also give negative returns larger than -1 when the decrement is more significant. What is the best or most common practice?
Logarithmic returns are recommended in finance for several reasons:
1. **Time Additivity**: Logarithmic returns can be summed across time, which is not the case with linear returns. This makes it easier to calculate compounded returns over multiple periods.
2. **Symmetry**: Logarithmic returns are symmetric with respect to gains and losses. A gain of X% followed by a loss of X% will result in the original value using logarithmic returns, which is not the case with linear returns.
3. **Small Changes Approximation**: For small percentage changes, logarithmic and linear returns are approximately equal. However, for larger changes, the approximation diverges.
4. **Multivariate Analysis**: Log returns are normally distributed if the underlying price series is log-normally distributed, which simplifies various types of financial analyses, including the calculation of the Sharpe ratio and other statistical models.
For these reasons, logarithmic returns are often used in quantitative finance, especially when dealing with multiple assets over multiple time periods.
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🤖 The difference between using Napierian (natural) logarithms and base-10 logarithms in finance stems from their mathematical properties and common usage in economic and financial models.😎 can you elaborate on the Continuous Compounding explanation?
1. **Natural Logarithm (LN)**: This uses the constant \( e \) (approximately 2.71828) as its base. The natural logarithm is the default in financial calculations for several reasons:
– **Continuous Compounding**: Financial models often assume continuous compounding, which is directly related to the number \( e \), hence the natural logarithm.
– **Euler’s Number**: \( e \) has unique mathematical properties that are useful in calculus and differential equations, which are common in financial modeling.
2. **Base-10 Logarithm (LOG10)**: This is based on the number 10 and is often used for its simplicity and interpretability in a decimal system, which can be more intuitive for representing magnitudes on a scale understandable to humans.
The choice between LN and LOG10 returns usually depends on the context:
– **Natural Logarithm**: Most commonly used in finance because it directly relates to continuous growth rates and is compatible with the exponential models of asset prices.
– **Base-10 Logarithm**: Sometimes used for pedagogical purposes or in contexts where the base-10 scale is customary.
Regarding negative returns larger than -100%, using logarithms for return calculations can indeed produce results that seem counterintuitive. In financial terms, a negative return larger than -100% does not make sense, because it would imply you lost more than the total value of the investment. However, logarithmic returns can go below -100% due to their mathematical properties.
For logarithmic returns:
– When the price of an asset falls to zero, the logarithmic return approaches negative infinity.
– A LOG10 return less than -2 represents a decrease in value by more than 99% (as 10^-2 = 0.01), while a LN return of less than -4.605 (since \( e^{-4.605} \approx 0.01 \)) represents the same.
In practice, the natural logarithm (LN) is the most commonly used for calculating returns in finance because of its direct relationship with continuous compounding and exponential growth models. Moreover, natural logarithms make certain types of mathematical manipulations simpler, especially when dealing with derivatives and integrals in continuous-time finance models.
🤖 Continuous compounding is a mathematical concept that represents the limit of compound interest as the compounding period shrinks to an infinitesimally small interval. This is a theoretical construct, as in reality, compounding occurs over discrete periods, like annually, monthly, or daily. But in continuous compounding, we imagine that the compounding happens every moment.😎 can you elaborate on the Continuous Compounding explanation in relation to neperian logarithms and the e constant? I know what continuous compounding is, don’t explain that. Explain why that justifies calculating returns with natural logarithm
Here’s the formal definition: If you have an initial investment \( P \), an interest rate \( r \), and it’s compounded \( n \) times per year, the future value \( FV \) after \( t \) years is given by the formula:
\[ FV = P \left(1 + \frac{r}{
🤖 Continuous compounding is related to the natural logarithm and the mathematical constant \( e \) through the process of continuously reinvesting earnings. The natural logarithm comes into play when we reverse this process: rather than calculating the future value of an investment with continuous compounding, we often want to find the equivalent constant rate of return that would yield the same final amount.
The formula for continuous compounding of an investment is:
\[ A = Pe^{rt} \]
Where:
– \( A \) is the amount of money accumulated after \( n \) years, including interest.
– \( P \) is the principal amount (the initial amount of money).
– \( r \) is the annual interest rate (in decimal).
– \( t \) is the time the money is invested for in years.
– \( e \) is Euler’s number (approximately 2.71828).
When you want to calculate the continuous compound return, you rearrange the formula to solve for \( r \), which gives you:
\[ r = \frac{1}{t} \ln \left( \frac{A}{P} \right) \]
This is where the natural logarithm (\( \ln \)) appears. It’s used to solve for the continuous rate of return \( r \) because the inverse of the exponential function \( e^x \) is the natural logarithm \( \ln(x) \).
In financial terms, when you calculate the logarithm of the ratio of the final amount to the initial amount, you are effectively calculating the continuous compound rate of return over the period in question. The use of natural logarithms is justified because the underlying continuous compounding formula is based on Euler’s number \( e \), and the natural logarithm is the inverse operation needed to isolate the growth rate \( r \) in the formula.
So, in calculating returns, using the natural logarithm aligns with the continuous growth model assumed in many financial theories and makes it easy to relate the rate of return over different periods of time due to its time-additive property. This property is unique to the natural logarithm and is not shared by other logarithmic bases.
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