EXPM1

Calculates the value of e^x - 1, where e is Euler’s number, and x is the number input.

Syntax:

EXPM1(x)

Example:

If x contains 12:

EXPM1(12)

returns 162753.791419004

x:

Result:

162753.791419004

Application:

Calculating Effective Annual Rate (EAR)

Imagine a financial product that offers a continuously compounded interest rate. The formula for the effective annual rate (EAR) from a continuously compounded rate (rc) is:

$E A R = e^{r_{c}} - 1$

Now, let's consider a scenario where the continuous rate (rc) is very small. For instance, a daily rate that needs to be annualized.

Scenario: A bank offers a savings account with a daily continuous compounding rate of 0.0001% (or 1×10−6 as a decimal). We want to find the effective annual rate. The number of compounding periods in a year is large (365 days).

Let rc be the daily continuous rate, and let n be the number of days in a year. The annualized continuous rate would be rannual=rc×n.

rannual=0.000001×365=0.000365

The effective annual rate (EAR) is calculated as: EAR=e0.000365−1

Let's compare the results of a direct calculation (exp(x) - 1) with the EXPM1(x) function.

	Calculation Method	Formula	Result	Notes
	A	B	C	D
1	Direct Calculation	e0.000365−1	1.0003650666...−1=0.0003650666...	The subtraction of two very close numbers can lead to a loss of significant digits.
2	Using EXPM1	EXPM1(0.000365)	0.0003650666...	This function is specifically designed for this case and maintains higher precision.

While the results appear identical in this simplified table, a deeper look into the binary representation of the floating-point numbers would reveal the difference. The EXPM1 function is implemented to maintain full precision for small x.

Why is EXPM1 more accurate?

When x is close to 0, ex is very close to 1. In a direct calculation of ex−1, the result is a small number that is the difference between two large, nearly equal numbers. This process, known as "catastrophic cancellation," can wipe out the least significant digits of the result.

The EXPM1 function uses a more stable algorithm (often based on a Taylor series expansion) for small x:

$e^{x} - 1 = (1 + x + \frac{x ^{2}}{2 !} + \frac{x ^{3}}{3 !} + \dots) - 1 = x + \frac{x ^{2}}{2 !} + \frac{x ^{3}}{3 !} + \dots$

This expansion avoids the subtraction and directly calculates the sum of small terms, preserving the precision of the result.

In summary, the EXPM1 function is a crucial tool in computational finance and other fields where high precision is required for calculations involving small exponential rates, ensuring the accuracy of results that would otherwise be compromised by the limitations of standard floating-point arithmetic.