Compound interest (2024)

  • Future value
  • Present value
  • Effective Annual Yield
If you leave $500 in the bank at 4% interest for a year, you will have $520at the end of that year by the simple interest formula. Therefore if youleave the money in the bank for a second year, you should earn interest onthe $20 interest as well as the $500 original principal; $500×1.04 =$540.80, where the $.80 is the interrest on $20 interest for the first year. This process of paying interest on interest as well as principal is calledcompounding.

Future value

If interest is compounded annually, the formula for the amount to be repaid is:

A = P(1 + r)^t

where r is the annual interest rate and t is the number of years. Sometimesinterest is compounded more often than annually, For example, if 6% interest iscompounded four time per year (quarterly), then one receives 1.5% interest everythree months. The more general formula for the future value of a deposit withcompound intrest is:

A = P(1 + r/m)^(mt)

where m is the number of times the interest is compounded each year.

How much will $300 be worth in 2.5 years if the interest rate is 3% compoundedquarterly? A = $300×(1 + .03/4)^(4×2.5) = $323.27.

It is more difficult to solve for the interest rate that will produce a givenincrease than in the case of simple interst. It is also more difficult to solvefor the time required for a given increase, although this may be easily attainedby trial and error.

Exercise: How much will $250 dollars be worth in 5 years at 6% interestcompounded monthly? How long will be required for $250 to double to $500?

Present value (P)

The formula A = P(1 + r/m)^(mt) can be rewritten as:

P = A/((1 + r/m)^mt)

to get the present value, or how much you need to put in the bank now to have aspecified amount in the future. For example, if you want to give $200,000to your nephew in 21 years, how much must you deposit in the bank now at 5%compounded quarterly? P = $200,000/((1 + .05/4)^(4×21)) = $70,444.54.

Effective Annual Yield

Compounding increases the amount of interest one earns. Because the standard way toexpress interest rates is with the annual interest rate, the amount of interestwhich one earns with compounding is quantified as the Effective AnnualYield, which is the simple interest rate which produces the same yield for aone year period. This is computed as (1 + r/m)^m - 1. For example, 5% interestwith quarterly compounding has an effective annual yield of (1 + .05/4)^4 - 1 =.0509 or 5.09%. 18% compounded monthly has an effective annual yield of (1 +.18/12)^12 - 1 = .1956 = 19.56%.

CompetencyHow much money will one have in 7 years if he deposits $2000 inthe bank at 8% interest compounded monthly?
How much money must one deposit in the bank at 8% interest compounded monthly inorder to have $2000 seven years from now?
What is the effective annual yield of 8% interest compounded monthly?

Reflection:

Challenge:

April 2004

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campbell@math.uni.edu

Compound interest (2024)

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