11-02-17-Topic 8b

11-02-17-Topic 8b - Topic 8 Econ 154 R Boyer October 2005...

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Topic 8 Econ 154 R Boyer October 2005 The Notion of Present Value The idea of present value involves how to place a value at the present time on transactions (payments) that will not take place until the future. The simplest question might be, What value do you place on a payment of $100, that you will receive with certainty, one period from now? Notice that uncertainty and default are not the concerns we are getting at. We are just thinking about the difference in value you place on payments which you receive this period versus payments you receive in future periods. Let us state the obvious. A receipt of $100 a period (one year) from now is worth less than $100 today. The reason is that with $100 today you have the opportunity of putting those funds in the bank, where the interest paid is assumed to be equal to i, and then a year from now you have the $100 plus the interest you earned on that amount. Obviously this is better than just having $100, which is the situation which occurs when you have to wait for a year to get your money. This example suggests a solution to the present value question. The value today you attach to a future receipt of some quantity of dollars, is the amount of money you would have to put in the bank today in order to have the value in the account at the future date equal to the value of the payment you will receive at that time. So, for example, a receipt of $100 a year from now has a present value, if the rate of interest is .1, of $90.91. The reason is that $90.91 put in the bank today at a 10% rate of yield earns $9.09 in interest. The sum of the amount you put in the bank and the amount of interest earned after one year is exactly equal to $100. In the form of an equation, the present value, P, of a receipt of amount Q one year in the future with interest rates of i is equal to P = Q/(1+i) The future payment is said to be discounted at the rate i in order to determine its current value. Now the same argument can be made in order to determine that a receipt of funds Q which will occur n years in the future has a present value, P, of P = Q/(1+i) n The point is that funds left in the bank earn interest that is compounded. An account which has P in it initially and from which no funds are withdrawn, after one year will have P·(1+i) in it. After two years, since the interest from the first year also bears interest, the value of the account will be P·(1+i) 2 . And so forth for whatever value of n you want to consider.
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With this simple view of present value (sometimes called net present value or present discounted value), we can generate a number of inferences. It is immediately apparent that the present value of the receipt of two payments in the future is equal to the sum of the present values of those receipts. That this is true follows directly from our definition. The present value of the first receipt is the value we need in the bank today which would
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11-02-17-Topic 8b - Topic 8 Econ 154 R Boyer October 2005...

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