## Thursday, June 16, 2011

### Back to Basics: Simple vs. Compound Interest

PERSONAL FINANCE 101

Interest rates affect us mortals in different ways in different situations, depending on the role we take on as participants in financial markets. As investors and owners of debt securities like bonds, interest rates define how much periodic income we earn, in the form of coupon or interest payments. As borrowers and consumers, interest rates determine how much it costs us to use capital and enjoy goods and services now while delaying payment to some later date. At first glance, it seems simple enough: high interest rates attract investors while low interest rates spur borrowing. However, as we have seen in a previous post about add-on interest, the different ways that interest is presented or computed could often lead to confusion and inefficient economic decisions.

In this post we'll take a look at two more common forms of interest rate that many of us, unfortunately, still don't completely understand. Learning these basic concepts by heart will help us better evaluate the numerous economic choices that we encounter in real life and save or earn us a considerable amount of money in our lifetimes.

Simple vs. Compound Interest

With simple interest, interest is earned only on the original investment or principal. For example, if you invest 100,000 pesos at 5% simple interest per year, you will earn 5% of 100,000 or 5,000 in interest per year; at the end of five years, your investment will have grown to 125,000, or 25% more than your initial investment.

Year 1: 5% of 100,000 = 5,000 + 100,000 = 105,000
Year 2: 5% of 100,000 = 5,000 + 105,000 = 110,000
Year 3: 5% of 100,000 = 5,000 + 110,000 = 115,000
Year 4: 5% of 100,000 = 5,000 + 115,000 = 120,000
Year 5: 5% of 100,000 = 5,000 + 120,000 = 125,000

You might have already figured out the short cut in computing how much you'll have at the end of t years if you invest p pesos at r simple interest per year:

Total amount after t years = p × (1 + rt

And the total interest you will have earned is

Total interest in t years = prt

Compound interest pretty much runs along the same lines, with one very important difference: earned interest is added to the principal at the end of every period, leading to a higher interest in the next period. In other words, with compound interest, earnings are reinvested so that interest also earns interest, a process which is referred to as compounding. Using the same example above but this time with 5% compound interest per year, we get:

Year 1: 5% of 100,000 = 5,000 + 100,000 = 105,000
Year 2: 5% of 105,000 = 5,250 + 105,000 = 110,250
Year 3: 5% of 110,250 = 5,512 + 110,250 = 115,762
Year 4: 5% of 115,762 = 5,788 + 115,762 = 121,550
Year 5: 5% of 121,550 = 6,078 + 121,550 = 127,628

The results of the above computations show why it is important to distinguish between simple and compound interest: as investors, we earn more if interest on our investment is computed as compound interest than as simple interest, at 127,628 vs. 125,000.

Computing for the future value of an investment with compound interest is a tad more complex than what we had with simple interest. At a compound interest rate r for t years, a p peso investment will grow to:

Total amount after t years = p × (1 + r)^t

And the total interest you will have earned with compound interest is

Total interest in t years = p × (1 + r)^t - p

In many real-world applications like credit cards and bank deposits, interest is treated and computed as compound interest. However, there may be situations where unscrupulous individuals would try to pass off "5% interest per year" as either simple or compound interest, depending on which definition works best for them (and against you). That's why it's important to be sure of which kind of interest you are facing--simple or compound--by reading the fine print or by simply asking questions. Remember, knowing is half the battle. :)