Question

A formula exists for a monetary return on an investment of continuously compounded interest. If the interest is compounded only once a year, use the following formula. Suppose you have $100 to invest, but started investing only $50. What changes might increase your return on investment, if you plan to invest for 5 years?A = P(1 + r)n where A is the total amount, P is the principal invested, r is the annual interest rate, and n is the number of years

Answer 1

Your formula is missing the exponent sign "^", it should read: P(1+r)^n. Re: what changes would increase your return? - the compounding period (continuous compounding is higher than annual compounding), the higher "r" is the higher the return. The higher P is the higher the return - the beauty of compounding interest...interest paid on interest earned (already paid).
Example: Formula for annually compounded interest at 4%:
$50(1.04)^5 = $60.83
vs. if you invested all of the $100 now...
$100(1.04)^5 = $121.67
you have invested only $50 more, but you receive...
interest on the $50 = (60.83 - 50) = 10.83
interest on the $100 = (121.67 - 100) = 21.67
if you wait to invest the additional $50 you will lose the opportunity to receive interest on it, and interest on the interest paid each year during the 5 year period.

Above example with continuous compounding: Formula: P(e)^(r*t) where r= rate (here I use 4%) and t = time...."e" is a constant for continuous compounding, roughly equivalent to: 2.71828
$50(e)^(0.04*5) = $50(1.2214) = 61.07
$100(e)^(0.04*5) = $100(1.2214) = $122.14
you can see that with continuous compounding (vs. annual compounding) you earn more interest because interest is compounded more frequently (and that interest earns interest)...