4. Introduction to the present value of money Under the concepts of the time value of money, you can determine the current, or present, value of a cash receipt or payment that will occur at some specified time in the future, glven a specified rate of interest. This technique can be used to calculate the present value of a single or a series of future receipts or payments. Dakota and Gabriella are walking after class between the library and the best pizzeria near campus. They're discussing Dr. Johnson's latest financial management lecture, which addressed the concept of present value and the process for calculating it. In anticipation of tomorrow's quiz, they've decided to review their lecture notes and the textbook materials and then practice one or two problems. Complete the missing information in the conversation that follows GABRIELLAI SO, What is a present value, and why is it important to be able to calculate it? DAKOTA: According to De Johnson, an asset's present or receive in the future value is the current value of the cash flows that it will pay or discounted GABRIELLA: Waiti Can you give me an example of when compounded popriate to calculate a present value? DAKOTA: Sure, but it might make more sense for you to identity such a situation, 50, tell me in which of the following two scenarios you would use a present value calculation, and then explain why that is so Scenario 1: You would like to know how much you should place on deposit to have accumulated a certain amount of money by a specific future date. Scenario 2: You would like to know how much a given amount deposited today will grow into by a specific future date. GABRIELLA: Ummm. I think is the situation that requires the calculation of a present value. The reason is that the am scenario 2 pced on deposit is both and occurs DAKOTA: Very good so here! scenario 1 fuestion: How is the present value of a single amount calculated? GABRIELLA: It can be calculated by rearranging the formula that is used in the calculation of a future value. To see this DAKOTA: Wait,wait, wait. Could you show me what you mean by writing it down? Here is a sheet of paper, show me how to rearrange the future value formula to solve for a present value GABRIELLA: OK Dirst, let's write down the equation used to calculate a future value (PV) Scenario 1: You would like to know how much you should place on deposit to have accumulated a certain amount of money by a specific future date Scenario 2: You would like to know how much a given amount deposited today will grow into by a specific future date. GABRIELLA: Ummm. I think is the situation that requires the calculation of a present value. The reason is that the amount to be placed on deposit is both and occurs known DAKOTA: Very goodi so here's your next question: How is the pre r a single amount calculated? unknown GABRIELLA: It can be calculated by rearranging the formula that are the calculation of a future value. To see this DAKOTA: Wait,wait,waitCould you show me what you mean by writing it down? Here is a sheet of paper, show me how to rearrange the future value formula to solve for a present value GABRIELLA: OK, first, let's write down the equation used to calculate a future value (FV) Scenario 1: You would like to know how much you should place on deposit to have accumulated a certain amount of money by a specific future date Scenario 2: You would like to know how much a given amount deposited today will grow into by a specific future date. GABRIELLA: Ummm. I think is the situation that requires the calculation of a present value. The reason is that the amount to be placed on deposit is both and occurs DAKOTA: Very good so here's your next question: How is the present value of a single amoy now, or at the present time later, or in the future GABRIELLA: It can be calculated by rearranging the formula that is used in the calculation of errore DAKOTA: Walt, wait, wait. Could you show me what you mean by writing it down? Here is a sheet of paper show me how to rearrange the future value formula to solve for a present value GABRIELLA: OK, first, let's write down the equation used to calculate a future value (FV) The Calculation of a Future Value NO Future Value ge the equation to Isolate the present value (PV) term by dividing both sides of the equation by the be the equation to put the unknown variable, the PV term, on the left-hand side of the equation: Then, WE Principal FVX PV > (1+1) FVK 1+TY FVN ut PV PV So, does this make sense? We've rearranged the future value equation to solve for the present value. Also, notice that the present value interest factor is the of the future Value Interest factor. This means that you don't necessarily need two different interest factor tables for the single cash flow; you can make do using either simply the present value table or the future value table - so long as you use it correctly. The Calculation of a Future Value Next, let's rearrange the equat Annuity Interest Factor alue (PV) term by dividing both sides of the equation by the we'll simply rewrite the equatid Present Value able, the PV term, on the left-hand side of the equation: Then FVX- FVX PV X (1+1) (1 = PV PV FVN (14" +1) So, does this make sense? We've rearranged the future value equation to solve for the present value Also, notice that the present value Interest factor is the of the future value interest actor This means that you don't necessary need two different interest factor tables for the single cash flow you can make do using either simply the present value table or the ture value table 50 long as you use it correctly The calculation of a Future Value Then Next, let's rearrange the equation to isolate the present value (PV) term by Interest factor s of the equation by the we'll simply rewrite the equation to put the unknown variable, the PV term, Interest side of the equation: PV X (1+1) FV FVx X PY PV FVS * Try So, does this make sense? We've rearranged the future value equation to solve for the present value Also notice that the present value interest factor is the of the future value interest factor. This means that you don't nearly need two different interest factor tables for the single cash flow you can make do using either simply the present value table or the future value table so long as you use it correctly The Calculation of a Future Value Then, Next, let's rearrange the equation to Isolate the present value (PV) term by dividing both sides of the equation by the we'll simply rewrite the equation to put the unknown variable, the PU term, on the left-hand side of the equation PV term interest factor FVN PV X ( 11) FVXX - PV PV FVN So, does this make sense? We've rearranged the future value equation to solve for the present value Also, notice that the present value interest factor is the of the future Value interest factor This means that you don't necessarily need two different interest factor tables for the single cash flow, you can make do using either simply the present value table or the future value table -- so long as you use it correctly The calculation of a Future Value Then, Next, let's rearrange the equation to Isolate the present value (PV) term by dividing both sides of the equation by the we'll simply rewrite the equation to put the unknown variable, the PV term, on the left-hand side of the equation: PV X (1+1) FVN FVN PV PV FVxx 50, does this make sense we've rearranged the future value equation to solve for the present value Also, notice that the present value interest factor is the need two different interest factor tables for the single - so long as you use it correctly of the future value interest factor. This means that you don't necessary qan make do using either simply the present value table on the future valutable square inverse DAKOTA: So, do you think that we're ready to do a problem? GABRIELLA: Surel Here's one from our homework. You work and I'll work, and we'll see if our answers match. DAKOTA: OK. Let's get started. Homework Problem Sarah wants to reduce the cost of graduate school by starting a savings plan today. As a sophomore, she has estimated that she has three years to accumulate the $22,500 that she needs to help cover some of her projected expenses. The account she would open would earn 10% per year compounded annually. So how much would she have to deposit today to accumulate $22,500 in three years? Or, stated differently, what is the value of $22,5007 (Note: Round your answer to the nearest whole dollar) present future I think that Sarah wo deposit so that she would have the desired $22,500 at the end of three years Homework Problem Sarah wants to reduce the cost of graduate school by starting a savings plan today. As a sophomore, she has estimated that she has three years to accumulate the $22,500 that she needs to help cover some of her projected expenses. The account she would open would earn 10% per year compounded annually. So how much would she have to deposit today to accumulate $22,500 in three years? Or, stated differently, what is the value of $22,5007 (Note: Round your answer to the nearest whole dollar) I think that Sarah would have to deposit so that she would have the desired $22,500 at the end of three years. Is that what you got when you solved the $16,905 $18,849 $13,947 GABRIELLA: Yes it is I think we've got a good start on getting ready for Dr. Johnson's next quiz