Expert: Abe Mantell - 8/30/2010

Question
QUESTION: Hello:

If the principal earns compound interest for 4 years. expanded calculation is as follows:

P(P   rP/)^4 = (P   rP)  r(P   rP)  r[(P   rP) r(P   rP)]  r[(P   rP)  r(P   rP)  r[(P   rP) r(P   rP)]]

If the principal is compounded twice a year for two years, would the following expaned calculation be as follows?

P(P   rP/2)^2*2 = P(P   rP/2)^4 = (P   rP/2)  r(P   rP/2)  r[(P   rP/2)  r(P   rP/2)]  r[(P   rP/2)  r(P   rP/2)  r[(P   rP/2)  r(P   rP/2)]]

I thank you for your reply.



ANSWER: Do not raise the principal (P) to a power...it would just be:
P(1   r/2)^(2*2)=P(1 r/2)^4

Abe


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

QUESTION: Hello:

I want to thank you for your reply.

Would the expanded calculation then be as follows compounded twice a year for two years?

P(1   r/2)^(2*2) = (P   r/2*P)^4 = (P   r/2*P)  r(P   r/2*P)  r[(P   r/2*P)  r(P   r/2*P)]  r[(P   r/2*P)  r(P   r/2*P)  r[(P   r/2*P)  r(P   r/2*P)]]

I thank you for your follow-up reply.  

Answer
At the end of:
1st half year: P (r/2)P
2nd half year (or 1 yr): P (r/2)P   (r/2)[P (r/2)P]
3rd half year (or 1.5 yrs):
P (r/2)P   (r/2)[P (r/2)P] (r/2)[P (r/2)P (r/2)[P (r/2)P]]
4th half year (or 2 yrs):
P (r/2)P   (r/2)[P (r/2)P] (r/2)[P (r/2)P (r/2)[P (r/2)P]]
 (r/2)[P (r/2)P   (r/2)[P (r/2)P] (r/2)[P (r/2)P (r/2)[P (r/2)P]]]

To make it look cleaner:
1st half year: P1=P (r/2)P=P(1 r/2)
2nd half year (or 1 yr): P2=P1 (r/2)P1=P1(1 r/2)=P(1 r/2)^2
3rd half year (or 1.5 yrs): P3=P2 (r/2)P2=P2(1 r/2)=P(1 r/2)^3
4th half year (or 2 yrs): P4=P3 (r/2)P3=P3(1 r/2)=P(1 r/2)^4

I hope this clarifies it for you now.

Abe