Expert: Josh - 1/27/2011

Question
QUESTION: Is there another method that can be used to determine the annual percentage yield other than by the calculation (1 + r/n)^n - 1?

I would prefer something less complicated than the above calculation.

I thank you for your reply.

ANSWER: Kenneth,

Anything else that you can find must in the end reconcile with this expression, if this is the correct expression for the annual percentage yield.

Although (1+r/n)^n is, by definition, (1+r/n) multiplied by itself (n-1) times, in practice, people always find (1+r/n) first, then raise this to the power of n using a calculator. I do not think this is difficult to compute. Also, I cannot imagine anyone wanting to do this any other way.

I remember you had previously asked for the binomial expression of (1+r)^n. This involves expanding a compact expression and generally moves in the opposite direction to what you want to do (if finding a numerical answer for a fixed value of r and n is all you have in mind, unless you want to study the coefficients in the expanded series term-by-term).

---------- FOLLOW-UP ----------

QUESTION: Thanks for the reply!

Can you think of an "ad hoc" calculation just to see what is one possibility?

I thank you for your efforts!

Answer
I want to understand what you mean when you said "another method that can be used to DETERMINE the annual percentage yield other than by the calculation (1+r/n)^n - 1."

If "determine" here means "derive", then it might be possible to derive the expression in more ways than one (not that I remember any off the top of my head). If, on the other hand, you use "determine" to mean finding the numerical answer, then it means you have already accepted the legitimacy of the expression (1+r/n)^n - 1. You are then only interested in the arithmetic operations (a series of steps) that produce the answer, for some value of "r" and "n".

I will take the second interpretation here. The most computation intensive part obviously involves the (1+r/n)^n part. For illustration purpose, suppose r=0.12 and n=6. As I mentioned in the last reply, you can compute (1+r/n)^n recursively by multiplying the term inside the parenthesis by itself 5 terms.

Recursive approach:
step 1: (1+r/n) = 1+(0.12/6) = 1.02.
step 2: let X = 1.02
a) multiply by itself once, X*1.02 = 1.0404
b) multiply by 1.02 again (for the second time), 1.0404*1.02 = 1.061208
c) multiply by 1.02 again (for the third time), 1.061208*1.02 = 1.08243216
d) multiply by 1.02 again (for the fourth time), 1.08243216*1.02 = 1.1040808032
e) the final answer is (1+r/n)^6 = 1.1040808032*1.02 = 1.126162419264

Alternate approach:
step 1: (1+r/n) = 1+(0.12/6) = 1.02.
step 2: invoke the power function (1.02)^6 to obtain the answer (1.126162419264) directly.

Another approach:
step 1: (1+r/n) = 1+(0.12/6) = 1.02.
step 2: square this to get, say, y = 1.02*1.02 = 1.0404.
step 3: since (1+r/n)^6 = [(1+r/n)^2]^3 by the index law, final answer = 1.0404^3 =  1.126162419264. [You can do it the long way 1.0404*1.0404*1.0404 if you like]

People generally don't care what route one takes to compute the answer. The formula is all that really matters. Evaluating the answer is usually a simple matter with the power of today's computer.