LOG1P

Application:

An application of the log1p function is in calculating the accurate percentage change of a stock price. When a stock price changes from $100.00 to $100.01, the percentage change is calculated as:

$\frac{100.01-100.00}{100.00}=\frac{0.01}{100}=0.0001\text{ or }0.01\%$

To get back the new value from the original value and the percentage change, we can use the formula:

New Value=Original Value×(1+percentage change)

So, 100.01=100×(1+0.0001)

This formula works well, but what if we want to calculate the natural logarithm of the ratio of the new value to the original value, which is a common calculation in financial modeling, especially for continuously compounded returns?

The formula for the continuously compounded return is:

$r=\ln \left(\frac{\text{New Value}}{\text{Original Value}}\right)=\ln \left(\frac{100.01}{100}\right)=\ln (1.0001)$

Now, let's see how log1p helps. The log1p function is defined as ln(1+x). In our case, the percentage change is x=0.0001. So, we can rewrite the formula as:

$r=\ln (1+0.0001)=\text{log1p}(0.0001)$

This is where the advantage of log1p becomes apparent. Due to the limitations of floating-point arithmetic in computers, when x is a very small number, the calculation of 1+x can lose precision. For example, if we were working with a very small change, like 1×10−16, adding it to 1 might result in 1 because of the way floating-point numbers are stored. The log1p function is specifically designed to calculate ln(1+x) accurately for very small values of x. It achieves this by using a different algorithm that avoids the intermediate addition of 1, thus preserving the precision of the small value x.

Here's a table comparing the direct calculation of ln(1+x) with log1p(x) for small values of x: