Time Value of Money
Date: 03/20/2002 at 00:12:02 From: Daniel Wagner Subject: Time Value of Money Dear Dr. Math, My Dad sent me this math problom two weeks ago and I was not able to figure it out. He told me to get some help, but no one I called or sent it to can help. I called two engineers, one insurance company and one Professor at a College. After looking at it for a few days the Professor told me I needed to take it to people who actually do this sort of thing for a living. He called it the time value of money. Can you help? Please take a look. Capital Invested 5-17-00 Capital investment $24,000.00 6-26-00 Capital investment $10,000.00 8-24-00 Capital investment $30,000.00 10-05-00 Capital investment $ 7,500.00 2-21-01 Capital investment $ 2,500.00 Total Capital Invested $74,000.00 Capital Returned 3-15-01 Capital returned $37,500.00 8-18-01 Capital returned $36,000.00 1-26-02 Capital returned $ 500.00 Total Capital Returned $74,000.00 Profit on Investment 1-27-02 Profit on Investment $19,735.00 Determine the annual percentage rate of yield or return on this investment. Daniel Wagner
Date: 03/20/2002 at 10:09:50 From: Doctor Mitteldorf Subject: Re: Time Value of Money Dan - I'm surprised that none of the people you consulted could do this calculation for you. It is not difficult in principle, as mathematical problems go, but it requires enough calculation to be tedious without a computer. The principle is this: Create a "present value" function that multiplies each payment by exp(-rt), where r is an unknown rate and t is the date of the transaction, measured from any arbitrary date you want to define as t=0. (The zero-point date will cancel out of your calculation.) Investment payments count as positive amounts; returns from investment, (it doesn't matter if they are called principal or interest) count as negative. Now try different values for r, using a smart trial-and-error technique (e.g. Newton-Raphson) to home in on that value of r that makes the function equal to zero. The r that you find is your APR, defined as a true rate. For an APY, you can use the conversion exp(APR)-1 = APY. You can perform this calculation on a spreadsheet. Put your guess at a trial r into a cell of the spreadsheet, and put in all payments as positive or negative amounts next to their dates. In a third column, use an @ function to calculate the present value of each payment made on its particular date, using the trial date. At the bottom of the present values column, calculate the total present value as the @sum of the column. Now try different values in the rate cell, until you make the total present value come out equal to zero. Better yet, you can use the spreadsheet's "solve for" capability to automate this process. In one of my past lives, I wrote a financial calculator that was published and distributed commercially, mostly to lawyers and accountants. It does ROI problems, loan amortizations, APR calculations, and mortgage comparisons. It is conveniently set up so that you can solve for any one unknown, be it a rate or a payment or a date, just by leaving that cell blank and filling in all the others. The program, called Per%Sense, is now available free for download from my personal page at the Math Forum: http://www.mathforum.org/~josh/ http://www.mathforum.org/~josh/persense.zip Here is a screen shot from PerSense, displaying your computation: +--DEPOSITS-------------------+ Funds | Single Payments | IN +-Date---+--Amount---+Dollars-+ | 5/17/00| 24,000.00|CURRENT | | 6/26/00| 10,000.00|CURRENT | | 8/24/00| 30,000.00|CURRENT | |10/ 5/00| 7,500.00|CURRENT | | 2/21/01| 2,500.00|CURRENT | +--------+-----------+--------+ +--WITHDRAWALS----------------+ Funds | Single Payments | OUT +-Date---+--Amount---+Dollars-+ | 3/15/01| 37,500.00|CURRENT | | 8/18/01| 36,000.00|CURRENT | | 1/26/02| 500.00|CURRENT | | 1/27/02| 19,735.00|CURRENT | +--------+-----------+--------+ Interest +---------+ rate % | 24.1177| +---------+ Settings: 360 1950 - Doctor Mitteldorf, The Math Forum http://mathforum.org/dr.math/
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