Sunday, April 10, 2011

about Roots and Exponents...

Sometimes when we are calculating returns, we need to know the meanings of roots and exponents. This may be especially true when we have to enter things on the calculator, like CIPM candidates are required to do on exams.

In mathematics, the n-th root of a number x is a number r which, when raised to the power of n, equals to x. For example, the 3rd root of 343 is 7. This means that 7 times 7 times 7 = 343.

Exponentiation is a mathematical operation, expressed as a raised to the n-th power, involving two numbers, the base a and the exponent n. When n is a positive integer, exponentiation corresponds to repeated multiplication. Using the prior example, 7 raised to the third power is 343.

Roots are special cases of exponentiation, where the exponent is a fraction. That is to say, the nth root of x is the same thing as raising x to the 1/n power.

For example, the third root (aka the 3rd root or cube root) of 343 is the same thing as raising 343 to the 1/3 power.

Why is this important? On some calculators, you can look at the math from either a root-taking point of view or an exponentiation point of view. But, for the calculators that are allowed for the CIPM exams, you must do things by taking an exponentation point of view, because there is no button that corresponds to taking "the x-th root of y."

For example, let's say that you had to calculate an IRR, and your calculator result comes out as a quarterly number but the answer requires an annual return. Suppose the quarterly return was 5.5%. In order to obtain the appropriate annual return, you must do the following steps:

1. make the return a decimal: 0.055
2. create the corresponding wealth relative by adding 1: this makes 1.055
3. raise 1.055 to the 4th power: this yields 1.2388
4. subtract 1: this yields 0.2388
5. make the decimal number a percentage: thus, the final answer is 23.88%

If you were thinking about this from a root-taking standpoint, would say that the return represents 1/4 of a year (i.e., 0.25 years), thus you could take the .25th root of 1.055, which is equivalent to raising 1.055 to the 4th power.

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