 Suppose you deposit $1,000 into a bank, and it earns interest at a rate of 5% per year. Two hours later, you withdraw the money. The accumulation function is 1.05 to power t, and two hours later will be... And again, to avoid round-off error, we'll use the exact values, so for every dollar that you deposit, the bank should give you... And $1,000 should give you... And so the bank should give you $1,000 and 1 cent. But it will probably only return your $1,000. And here's the issue. Time is a continuous quantity, so in theory, interest should accumulate continuously. But in practice, financial institutions treat time as a discrete quantity, and credit interest only at the end of a given time period. The end of the day, the end of the quarter, the end of the year, the end of the geological era. Suppose a bank pays interest at a rate i, but interest is computed and credited m times a year. We say the interest is compounded, or convertible, or payable m times a year. To compute the interest in each period, the amount of interest is computed using the nominal rate i, but since the year is broken into m time periods, the actual interest credited is divided by m. So in practice, an interest of i ms is paid at each period. We use the notation to indicate interest at a rate i pi 8 m times per year, and our accumulation function will then be where t is now measured in compounding periods. For example, suppose we deposit 1000 into an account turning 5% per year compounded quarterly, that's 4 times a year, and then find the amount at the end of the year. So we could write down our accumulation function, but remember, don't memorize formulas, understand concepts. Since interest of 5% is compounded quarterly, then 1 fourth of the crew of interest is paid each quarter. So the interest rate during the quarter is actually 5 fourths, 1.25%. So the accumulation function will be where t is the number of quarters. Now, one year is 4 quarters, so the number of quarters is 4 times the number of years. So after t equals 4 quarters, the dollar will grow too, and so $1000 will grow too. What about the effective annual rate? Since a dollar will grow to A of 4 in a year, the effective rate will be that amount, minus 1, or about 5.09%. Note that this is higher than the nominal rate. And in fact, we generally see a difference between the nominal and the effective rate. So the nominal rate is the rate that's actually used to compute interest. However, if the compounding period is different from the time unit, the effective rate will also be different. So 8% annual interest compounded annually, here the compounding period and the time units are the same, so the effective rate is also 8%. But if we have 8% annual interest compounded monthly, here the compounding period is different from the time unit, and so our effective rate is... Well, let's calculate it. Since there are 12 months in each year, during each period, 8%-12% interest is earned, and so our accumulation function will be where t is the number of months. Since one year is t equals 12 months, that a dollar will grow to, or about $459.39. Now, wait a minute... Remember, no computer was ever fired for making a mistake. So remember the percentage must be converted into a decimal to use in the accumulation function, and to convert a percentage into a decimal, divide by 100, or equivalently make the denominator 100 times larger. So the actual accumulation function will be, and in a year, a dollar will grow to, or about $1.08. So the effective interest rate is about 8.30%. Or let's have a great bank that offers you 10% interest compounded once per century. So over one century, that's a 100-year period, you would get paid 10% 100 times, so the rate is 1000% over a century. So our accumulation function will be, but now t is measured in centuries, and one year is one one-hundredth of a century. So in one year, $1 would grow to, and this gives us an effective interest rate of only 2.43%.