Need help understanding concept of Compound Interest.

Question

Need help understanding concept of Compound Interest.

A bank offers two types of interest accounts The first account receives 6% compounded monthly. The second account receives 5% interest compounded continuously. What account has more and why?

Answer 1

 

Continuously Compounded Interest is a great thing when you are earning it! Continuously compounded interest means that your principal is constantly earning interest and the interest keeps earning on the interest earned!

 

   The formula for compounded interest is ... A = Pr^rt

 

                  A = amount accumulated

                  P = Principal

                  e = The mathematical constant e

                  r = rate

                  t = time, in years

 

   The formula for interest compounded monthly is ... A = P(1 + r/n)^nt

 

                  A = amount accumulated

                  P = Principal

                  r = rate

                  n = number of compounds, per year ... for monthly, n = 12

                  t = number of years 

 

You can plug in values for both to see the difference in the amounts.  

 

In this example, I will use 5000 for the principal, and 5 years for the time

 

For 6% compounded monthly

 

      A = 5000(1 - .06/12)²⁽⁵⁾

         = 5000(1 -.005)10

         = 5000(.995)¹⁰

         = (5000).951

         = 4755.55

 

For 5% compounded continously,

 

     A = 5000e^(.05)(5)

        = 5000e^.25

        = 6420.13

 

So even though the rate us lower, the compouding continuously yields the most income

Answer 2

The first account receives 6% annually but it is compounded monthly, or 12 times a year. So, assuming a principal amount of $1 accruing interest over 1 year,  the interest accrued would be

 

1*(1+.06/12)¹² = 1.0617

 

**(Using the compound interest formula P*(1+r/n)^n*t where r is the annual rate, n is the number of times it is compounded in 1 year, and t is the number of years (1 in this example). In this example we divide by 12 since the rate (6%) is compounded 12 times in 1 year)**

 

Continuous compounding follows a different formula Pe^rt, where e represents an irrational number (not a variable), r is still the annual rate and t is the number of years.

 

1*e^.06 = 1.0618