The formula for converting a continuously compounded rate to a periodically compounded rate is
Rc = m(eRc/m - 1)
Rc = continuously compounded interest rate, which is 3.75% in this question.
Rm = periodically compounded interest rate, compounded m times per year.
m = compounding times per year, which in this case is 2 for semiannual compounding.
e = Euler's number, a constant with a value of roughly 2.71828.
----- SOLUTION -----
Plugging in the values produces Rc = 2(e.0375/2 -1) = 3.7854%
----- EXCEL STEPS -----
In Excel, the function exp() raises e to a power. In this case, eRc/m is handled as exp(.0375/2), which resolves to 1.0189.... Note that all the terms in the power component (in this case, Rc/m) must be within the parentheses following the letters "exp".
The full equation in Excel is =2*(exp(.0375/2)-1)
----- CALCULATOR STEPS -----
Steps for a scientific calculator vary by calculator. For the CalcTastic smartphone app in scientific mode, eRc/m is handled by pressing the ex button (via SHIFT, ln), then typing .0375/2) . To finish the problem, type -1=, then *2=.
For other scientific calculators, such as the standard Windows calculator in scientific mode, use a combination of the e and xy buttons: e, xy, (.0375/2)-1=, *2=.
----- REVERSE -----
To convert from a periodically compounded rate to a continuously compounded rate, use
Rc = m ln (1 + Rm/m)
(Note, ln denotes the natural log, or loge(x).)
Plugging in the values produces Rc = 2 ln (1 + .037854/2) = 3.75%
In Excel, the full equation is =2*ln(1+.037854/2).
On the CalcTastic smartphone app, press the ln button, then 1+.037854/2)*2=. On the standard Windows calculator in scientific mode, type .037854/2+1, ln, *2=
----- PRACTICAL APPLICATION -----
Converting between periodic compounding and continuous compounding is an essential skill for working with derivative securities. Underlying assets, such as bonds or loans, commonly use periodic compounding, but derivatives commonly use continuous compounding. The use of very frequent compounding is necessary so that the precise time periods associated with a derivative security can be accounted for, and fortunately, formulas with continuous compounding can be less cumbersome than those with periodic compounding. For practical purposes, continuous compounding produces largely the same result as daily compounding.
----- ALGEBRA REFRESHER -----
Formulas with continuous compounding require some familiarity with e and logarithms. e represents the maximum possible result of continuous compounding at 100% during one period. e is an irrational constant equal to roughly 2.71828, where “irrational” simply means that there are an infinite number of digits after the decimal place, as with pi (3.14159…)
(Note, for daily compounding on a 100% rate, (1+r/n)^n = (1+100%/365)^365 = 2.714, which is very close to e's value of 2.718....)
The inverse of e is the natural log and is denoted by ln.
A further refresher on the mechanics of working with exponents and logarithms may be helpful for understanding the derivation of many of the formulas used with derivative securities. The essentials above, though, may suffice for many practical applications.
----- LEARN MORE -----
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