A simpler version of the compound interest formula is A = P( 1 + r)^{t} where A is the final balance, P is the principal, r is the annual interest rate compounded once per year, and t is the time in years.
The principal is the amount of money you deposit that you expect will grow over time.
Example #1
A businessperson invests 20000 dollars in a local bank paying 6% interest every year. How much money does the businessperson have in his account after 8 years?
Solution:
In this scenario, the interest rate is compounded or "calculated and added to the account" only once per year. Therefore, you can just use the formula A = P( 1 + r)^{t} to find the accumulated amount after 8 years.
A = P( 1 + r)^{t}
A = 20000( 1 + 6%)^{8}
A = 20000( 1 + 0.06)^{8}
A = 20000( 1.06)^{8}
A = 2000(1.5938480)
A = 31,876.96
Many banks though have plans in which interest is paid more than once a year. The number of interest periods is the number of times the interest is computed and paid per year.
If the interest is computed and added to the account twice a year, this means that the number of interest periods is 2.
If the interest is computed and added to the account quarterly or four times a year, this means that the number of interest periods is 4.
Suppose the interest rate is 6% per year and the number of interest periods is 4. Then, each time the interest is compounded, the bank will use 6% / 4 or 1.5%.
In general, if r is the yearly interest and n is the number of interest periods in a year, each time the interest is compounded, the bank will use r / n.
The number of payment periods will also change. In the simpler version of the formula shown above, the number of payment periods is t. And t is the number of years.
Suppose each year though the interest is compounded 4 times. After 8 years, the number of payment periods is 4 × 8 or 32.
The number of payment periods is the total number of times interest is added to the account. In this case, it was done 32 times.
In general, if n is the number of interest periods in a year and t is the number of years, then the number of payment periods is n × t.
Therefore, a more complete version of the compound interest formula is:
A = P( 1 + r / n)^{nt}
Example #2
Let us modify example #1 a little bit!
A businessperson invests 20000 dollars in a local bank paying 6% interest every year. The bank computes interest 4 times per year. How much money does the businessperson have in his account after 8 years?
Solution:
Now the interest rate is compounded or "calculated and added to the account" four times per year.
Therefore, you must use the formula A = P( 1 + r / n)^{nt }to find the accumulated amount after 8 years.
A = 20000( 1 + r / n)^{nt }
A = 20000( 1 + 6% / 4)^{4×}^{8 }
A = 20000( 1 + 1.5%)^{32}^{ }
A = 20000( 1 + 0.015)^{32}^{ }
A = 20000( 1.015)^{32}^{ }
A = 20000(1.61)
A = 32200
Notice that when the interest is paid 4 times a year, you end up with a little bit more money!
Sep 27, 22 08:34 AM