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Q: Formula to determine DPW of declining income stream ( Answered,   0 Comments )
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 Subject: Formula to determine DPW of declining income stream Category: Business and Money > Finance Asked by: mceanswers-ga List Price: \$10.00 Posted: 12 Sep 2003 15:43 PDT Expires: 12 Oct 2003 15:43 PDT Question ID: 255233
 ```I begin with a stream of income delivered in one payment at the middle of each year. The income is declining at a constant rate (this number may vary from case to case, but for a given case it will be constant, e.g. 5%/year). I want a formula which, for varying decline rates, discount rates, and number of years for which payments are to be delivered, will tell me the present worth of said income stream. FOR EXAMPLE: Given: This year I will receive \$10,000. Next year I will receive \$9,000. In year three I will receive \$8,100, etc. (That is, income is declining at 10%/year.) This income will continue until there are no longer any payments (I know that this is not realistic, but I also understand that the DPW of such miniscule payments approaches zero). Question: What is the formula I would use to determine the DPW of this income stream at a 10% discount rate? at a 15% discount rate, etc? thanks, MCE```
 ```mceanswers, The present worth can be found by solving this problem as a geometric series The basic equation is this: Present Worth = a + ar + ar^2 + ar^3 + ... + ar^(n-1) + ... Where: a = Initial Sum, such as the \$10,000 in your example. r = (1 - Discount Rate) -> 10% discount becomes r = 90% n = number of payments, including the initial one For a value of "r" less than one, there is a simple formula that can tell you what the sum (the Discounted Present Worth) is: Sn = a * ( (1-r^n) / (1-r) ) Going with your example.... we start out with \$10,000, have a total term of 3 payments, and use a discount rate of 10%. Using the above equation, the value is: Sn = \$10,000 * 2.71 = \$27,100 Using common logic, like the initial example you used. Present Worth = \$10,000 + 0.9*\$10,000 + 0.9*\$9000 = \$27,100 I'm not sure what you are using this for, but this formula is useful for determining whether to make an investment based upon it's present earning power ("a" in this equation) and an external force like inflation (represented by "1-r"). Assuming that you have an infinite period of time in which to collect this income, the total present worth can be found by the simple formula: S = a / (1 - r) For the same numbers described above (\$10,000; 10%), the DPW is: \$100,000 However, if you are clever, you will notice that in the above equation you can simply substitute the raw discount rate for the value of "1 - r". So, just take the present income value (\$10,000) and divide it by the discount rate (10%) and you will get \$100,000. To tell you what you will have after a specified number of periods, use the first equation I wrote (Sn). Don't hesitate to ask any questions! krobert-ga``` Request for Answer Clarification by mceanswers-ga on 15 Sep 2003 09:29 PDT ```KRobert, I don't we've yet arrived at the answer. This should be an equation with 4 variables. We currently have just 3: a = Initial Sum, such as the \$10,000 in your example. r = (1 - Discount Rate) -> 10% discount becomes r = 90% n = number of payments, including the initial one We need to add a variable for the decline rate: d = annual percentage decline So, just to set up another example question, let's say that I am trying to determine the DPW for the following situation: a = \$30,000 r = 12% n = 17 d = 6.5% such that year 1 payment is \$30,000, yr 2 = 28,500, yr 3 = 26,227, etc out to 17 years. What is the 125DPW of this series of payments? thanks, MCE``` Request for Answer Clarification by mceanswers-ga on 15 Sep 2003 09:31 PDT ```KRobert, Please excuse the typos in the previous question. It should have read: I don't think we've yet arrived at the answer. This should be an equation with 4 variables. We currently have just 3: a = Initial Sum, such as the \$10,000 in your example. r = (1 - Discount Rate) -> 10% discount becomes r = 90% n = number of payments, including the initial one We need to add a variable for the decline rate: d = annual percentage decline So, just to set up another example question, let's say that I am trying to determine the DPW for the following situation: a = \$30,000 r = 12% n = 17 d = 6.5% such that year 1 payment is \$30,000, yr 2 = 28,500, yr 3 = 26,227, etc out to 17 years. What is the 12%DPW of this series of payments? thanks, MCE``` Clarification of Answer by krobert-ga on 15 Sep 2003 11:04 PDT ```mceanswers-ga, I want to make sure that I understand you.... This income stream is declining at 6.5% per year, but is also earning 12% per year? Is that right? krobert-ga``` Request for Answer Clarification by mceanswers-ga on 15 Sep 2003 12:21 PDT ```To clarify: This income stream is declining at 6.5% per year. What is the X DPW of this stream of income? (where X is a variable, in this case we shall say 12% per year) thanks, MCE``` Request for Answer Clarification by mceanswers-ga on 15 Sep 2003 12:24 PDT ```krobert-ga, Sorry, but to clarify again: This income stream is declining at 6.5% per year. What would I pay for this stream of income if I want to realize a 12% annualized rate of return, which, according to my understanding is the same as asking "what is the 12% DPW of this stream of income? thanks, MCE``` Clarification of Answer by krobert-ga on 15 Sep 2003 16:10 PDT ```mceanswers-ga, To be honest, I've never heard it quite put that way. One more question though, and we should have an answer: Let's take the one and two year models: If we have \$30,000 in year one, and you wanted to purchase this for an annual return of 12%, you would pay \$26,786. Does that agree with your calculation? For two years, we have income of \$30,000 in year one and \$28050 in year two (a 6.5% decline). So, we have the original amount, \$26,786, and a new amount \$22,361. So, you would be willing to pay \$49,147 for income of \$30,000 and \$28,050 (\$58,050 total), spaced one year apart, for a return of 12% per annum on your investment. (This assumes that you pay for the initial \$30,000 a year in advance). Is this correct, can you follow the logic? krobert-ga```