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1 A Time Line In financial matters the timing of cash flows has important consequences. Here we develop a tool called a time line to help visualize situations.

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2 0 1 2 3 …n

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3 End of Period Convention You will notice on the previous slide a time line that starts with the number zero. Then I have marked off equal time periods (usually months or years) and called the last period the general time period n. In specific situations n will be made clear. As an example, if you win the Powerball lottery and take your payments over the next 30 years, n would be 30 years. In financial matters, time consideration usually follows the end of period convention. This means that we start at time zero and think about the end of the first period, the end of the second period and so on. So, the 1 on the time line means the end of the first period. (We could use beginning of year convention and our line would be labeled starting with a 1 for the beginning of the first period.) This end of period convention is similar to how we consider our age. When born, we are zero. At the end of the first year we become 1.

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4 Interest and Compounding In financial matters, it is customary to incorporate interest growth when considering money values over time. As an example of this, you often hear the phrase a dollar today is worth more than a dollar next year. Here is the logic of the phrase (I hope you understand the logic, but you do not have to agree or like the logic). If you have a dollar today you could put it into an interest earning account and have more than a dollar next year (by next year I mean the end of the first year from right now, today.) The interest rate helps determine what you have at the end of the first year. Compounding of interest means that in any one period if interest is earned and kept in the interest earning account, then in future periods not only will the original funds earn interest, but interest earned from previous periods will also earn interest.

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5 Compound and Simple Interest In the history of the world compound interest was not always the standard. Simple interest used to rule the day. This meant that if you held funds in an account for many periods only the original amount would earn interest each period. Somewhere along the line compound interest took over. In essence, this means that an original sum of money will be worth more after many periods under compounding because the interest, as well as the initial amount (sometimes called the principal amount), can earn interest.

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6 Years and Periods On the time line I mentioned the end of each period. The compounding time period is what really defines the period. If interest is compounded once a year, then the periods on the time line represent years. But, if interest is compounded monthly (like with mortgages and credit cards), then the periods on the time line represent months. A 30 year mortgage on a home would really have us look at 360 months (n on the time line would be 360.)

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7 Yearly Interest Rate and Period Interest Rate Interest rates are typically mentioned in yearly terms and less explicitly the compounding period is mentioned. For example, on a credit card the rate might be 12% per year, but the fine print says you are charged interest each month. This suggests a monthly compounding period. If compounding is only once a year, called compounded annually, then the time period of interest is years and the rate each period is the annual rate. If compounding is monthly, then the time period is months and the interest rate per month is the annual rate divided by 12. Sometimes interest is compounded quarterly. In general, what is the interest rate per quarter? It would be the annual rate divided by 4.

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8 Getting Money and Giving Money In our financial lives we sometimes receive money and we sometimes payout, or disburse, money. (We say disburse because the thing you and I call a purse used to be called a burse. Back in the wild west days when you would pay for something before you would reach into the purse you would say to the shop keeper, “I am going to get de money out of dis burse.” That way the shopkeeper would not think you were getting out your smith and wesson. – OK, so I should have taken a break before I typed this!) On our timeline from earlier, we will put vertical lines of various heights to represent money flows at various times. We will have receipts rise above the line and disbursements go down from the line. Let’s see an example on the next screen.

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9 0 1 2 3 …n 100 150 Say you put $100 into a stock and three years later you sold the stock and received $150. Note when you bought the stock and paid out $100 I have a line down at time zero and I labeled it 100. At the end of 3 periods I have a line rising to 150. By using the time line we have a visual summary of the cash flows in the problem

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10 Well, here we have just considered some basic ideas that have to do with financial matters. We will build on these ideas as we consider single payment ideas, uniform series ideas and other ideas to come.

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Multiple Cash Flows –Future Value Example

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