 The compounding period determines when interest is credited. So compounded daily, interest would be paid every day. Compounded monthly, interest would be paid every month. Compounded centenially, interest would be paid every century. So how much would you get if you withdrew in the middle of a day, month, or century? To answer this question, we introduce the greatest integer function, which is the greatest integer less than or equal to x. So for example, let's find the greatest integer 1.98, 5, and 17 and 11-12. So 1.98 is the greatest integer less than or equal to 1.98. Well, that's going to be 1. The greatest integer less than or equal to 5 is 5. And since 17-12 is not quite 18, then the greatest integer less than or equal to will be 17. The greatest integer function is a way to simplify computations regarding compound interest. So suppose the bank pays compound interest at rate i per time unit. Then the accumulation function will be a of t equals 1 plus i raised to the power of the greatest integer less than t. For example, suppose a bank pays interest at an annual rate of 3% compounded. You deposit 1,000 on January 1st, find the amount on December 31st, and the amount on January 1st of the next year will assume a non-leap year. So our accumulation function will be... Now from January 1st to December 31st is one day less than one year, so it's 364 days. And this is 364-365 years, so we want to find a of this value. Since the greatest integer less than this fraction is zero, we have, and so 1,000 will become 1,000. Now if we wait one more day from January 1st to January 1st of the next year, is t equal to one year, so our accumulation function will be, and since the greatest integer less than or equal to one is one, we find, and so 1,000 will become 1,030. And notice that if you withdraw even one second before t equals one, you won't get any interest at all. And this is sometimes referred to as the penalty for early withdrawal. You don't get the interest you should have earned. Now let's consider another situation. Suppose interest is credited m times per time unit. The bank would credit im, that's im's, the interest each time. So the accumulation function should be where we retain the original time units. While this is what we should do, often we use the compounding interval as a new time unit, and this allows us to write our accumulation function as. So for example suppose a bank pays 8% interest compounded monthly, you deposit 1,000 on January 1st, how much can you withdraw on February 28th? Since the interest rate is annual but interest is compounded monthly, then our interest per compounding unit is 8% divided by 12, which will be, since the interest is compounded monthly, we'll measure t in months. So the accumulation function is, but remember, if it's not written down, it didn't happen. Because we're measuring t in months, we'll want to indicate that when we give our accumulation function. From January 1st to February 28th is t equals some, some number of months. Let's figure that out. Now from January 1 to February 1 is one month. And from February 1 to February 28th is 27 days. Since February has 28 days, usually, this is 27, 28 months. So from January 1st to February 28th is one and 27, 28 months. So we'll evaluate our accumulation function, and so the amount will be approximately. Now if interest isn't credited until a full compounding period passes, this motivates certain investment strategies and discourages others. For example, if interest is credited annually, investors are discouraged from withdrawing money before one year has passed, and that's because if you deposit $1,000, you'll only get back $1,000 if you withdraw any time before t equals one. At t equals one, you'll be able to withdraw $1,000 plus interest, but nothing more until t equals two. And so this means that investors are motivated to withdraw money the instant interest is credited. And what this means is that banks would have a long period of time where nobody was in, and then everybody would want all their money at once. So why not pay out interest every minute, or every second, or every shake? A shake is one 100 millionth of a second. So let's do some calculus. Suppose the annual interest rate is i compounded m times per year. The effective annual rate as m goes to infinity will be the limit, which turns out to be e to the i. So if interest at rate i is compounded continuously, then our accumulation function will be where t is measured in the same time units as the interest rate. In other words, we've gotten rid of that compounding factor. So suppose you deposit $100 in the Hilbert Bank, which pays 4% annual interest compounded continuously. Let's find the amount you'll have after 23 days. The accumulation function for 4% annual interest compounded continuously will be where t is in years. Now, 23 days is 23, 365 years, so we'll evaluate. And so $100 would become