


tadNPV in tadXL v3.0
tadIRR in tadXL v2.5
NPV may be best calculated using Excel spreadsheet program but the Excel NPV function has severe limitations as it only permits finding net present value of an ordinary annuity that makes end of period payments. That is all the native Excel financial function is capable of but as you may know finding net present value is such a complex task when you define the properties of the cash flows. Here the tadNPV function addresses all such limitations of Excel NPV function and offers a more robust and feature rich functionality that makes possible net present value for different scenarios. For example, you will be able to find NPV of a perpetuity, of a project, of a business or company valuation. These properties include but are not limited to use a schedule of discount rates where each of the cash flows is discounted at a different rate. This would be required for example in finding current value or market price of a coupon bearing bond that has a term structure of interest rates. A schedule of compounding frequencies of interest, a schedule of payment periods, growth and tax rates, possible hair cuts on income and permission to rig the discount rates.
Rates  5%  4%  3%  2%  1%  2%  3%  4% 
Growth  0%  0%  0%  0%  0%  0%  0%  0% 
Tax_Rates  30%  31%  32%  33%  34%  35%  36%  37% 
Cash_Flows  $(100)  $250  $350  $(450)  $550  $650  $750  $850 
Adjust_for_inflation  0  0  0  0  0  0  0  0 
Frequencies  4  365  24  365  INF  260  INF  5,200 
Types  1  0  1  0  0  1  0  0 
Compoundings  0.25  0.002739726  0.083333333  0.002739726  1  0.038461538  1  0.019230769 
Periods  0.25  0.002739726  0.083333333  0.002739726  1  0.038461538  1  0.019230769 
Concentrations  1  0.5  2  10  1  0.5  1  0.75 
Hair_Cuts  0%  20%  20%  20%  0%  20%  20%  20% 
Rate_Rigged_By  0%  15%  15%  15%  0%  15%  15%  15% 
Using tadXL function such as tadNPV finding net present value in Excel 2007, 2010 and 2013 becomes really easy. Let us now briefly look at this financial function in tadXL to find NPV using Excel:
Excel NPV function in tadXL v3.0
Excel NPV function in tadXL v2.5
 tadNPV ( discount rate, cash flows, type, compounding, period, concentration )
 tadNPVSchedule ( discount rates, cash flows, type, compounding, period, concentration )
 tadXNPV ( discount rate, cash flows, dates, compounding )
 tadXNPVSchedule ( discount rates, cash flows, dates, compounding )
 tadIncNPV ( discount rate, cash flows1, cash flows2, type, compounding, period, concentration )
 tadDecNPV ( discount rate, cash flows1, cash flows2, type, compounding, period, concentration )
NPV calculator(s) from FTWE 100
Here Abraham A presents you his personal collection of NPV calculators that are based on tadXL addon for Excel 2007, 2010 and 2013. All these financial calculators find net present value depending upon how much of information is available about the investment. For instance you will find NPV calculators that accept a single discount rate, and another that accepts a series of discount rates, others that accept a schedule of transactions dates and/or schedule of discount rates. Other calculators help find incremental and decremental net present value given that you have two set of cash flows for two investments.
tadNPV ( discount rate, cash flows, type, compounding, period, concentration )
If the cash flows were to be discounted using a single discount rate, then you should be using this NPV calculator. It also permits you to specify a number of compounding frequencies of interest, to use payment periods of different lengths, to select from a number of discounting conventions and type of annuity payments (whether start or end of period).
%
Data output
tadNPVSchedule ( discount rates, cash flows, type, compounding, period, concentration )
If the cash flows were to be discounted using a schedule of discount rates, then you should be using this NPV calculator. It allows you to specify a number of compounding frequencies of interest, to specify payment periods of different lengths, to choose from a number of discounting conventions and type of annuity payments (whether start or end of period).
Data output
tadXNPV ( discount rate, cash flows, dates, compounding )
If the cash flows were to be discounted using single discount rate but you have access to a schedule of transaction dates, then you should be using this NPV calculator. It permits the use of a number of compounding frequencies of interest.
%
Data output
tadXNPVSchedule ( discount rates, cash flows, dates, compounding )
If the cash flows were to be discounted using a schedule of discount rates and a schedule of transaction dates is available, then you should be using this NPV calculator. It permits the use of a number of compounding frequencies of interest.
Data output
tadIncNPV ( discount rate, cash flows1, cash flows2, type, compounding, period, concentration )
If you are analyzing two different investments with the same life (number of periods) then you would want to find the incremental net present value using this NPV calculator. It permits you to specify two set of cash flows along with options to use different compounding frequencies of interest, payment periods of different lengths, a choice of discounting convention.
%
Data output
tadDecNPV ( discount rate, cash flows1, cash flows2, type, compounding, period, concentration )
If you are analyzing two different investments with the same life (number of periods) then you would want to find the decremental net present value using this NPV calculator. It permits you to specify two set of cash flows along with options to use different compounding frequencies of interest, payment periods of different lengths, a choice of discounting convention.
%
Data output
NPV is the most commonly used DCF method emplyoed by an analyst to find an investor's ROI return on investment. The NPV definition states that it is the sum of all discounted cash flows (positive amounts and negative amounts). There are many other options that are left desired in net present value calculation which are explained herein. Here me Abraham A. will take you on a lengthy tutorial that will look at various aspects of net present value calculations. If you have around 40 minutes to spare then you will enjoy reading this superb article written for novices and professional analysts.
Hello reader, visitor to this site, please note that this web page on the topic of NPV is going through updates all this week. We will gradually add new content and replace it with existing one. Thanks for your understanding.
Although I previously stated two things, one that this page will get updates in all of this week starting on 10/14/2013 ending on 10/20/2013, and that the updates will gradually replace the old content. That said, as a writer, it takes time and effort to put together content and my guess on time was an underestimate. Thus for first I have removed all the older content from this page since the file size was getting really big as I was adding new updates exceeding 100K and slowing down the page. Now that the old content is totally gone, I have put in my latest updates and will keep on adding more content for as long as it takes to finish off this one of kind tutorial on net present value. Many thanks for your time in visiting us and making this site popular.
It has been almost a month since I started to rewrite my NPV tutorial or article as it turned out. But you wouldn't believe only 1/4th of the full tutorial has been compiled thus far. The introduction below shows my intent on the variety of topic I would include in this unmatched article on NPV. But I too am human and it takes imagination and knowledge of topics to be able to put together the events as they unfold. The whole article is written in form of Fifiction that is finance fiction, a new branch of writing style. The persons who are named in this article are fictional and any resemblence of these characters to someone living or deceased are purely coincidental. The author bears no responsibility for any offence that may be caused to the reader from the storyline presented herein.
From,
Abraham A.
update #1 as of 10/14/2013 6:30 AM EDT.
This will be a lengthy discussion on the topic of NPV  an acronym for net present value. In the simplest of term, an NPV is simply the sum of discounted cash flows (payments or receipts). But this rather trivial definition has hidden gems that I will explore in quite a detail before you get tired of reading all my bull**it. Here is what I do, I will begin with defining interest factors that are nothing more than finding present value of a $1 or a present value of series of periodic payments in amount of a $1. There will three basic present value interest factors, followed by extensions that will allow for growing and shrinking periodic payments in amount of $1 given a growth rate g. Here I will bring into discussion the element of intraperiod compounding of interest that will allow us to specify compounding frequency of interest other than the annual one. It will be explained how interest may be earned on savings or charged on loans on an ever small fraction of time leading to infinite or continuous compounding of interest. I will also divulge in to using a period other than a year for our periodic payments and will show how such periods affect the resultant net present value. Once you have understood, the basic elements that are required in discounting periodic cash flows, I will extend the discussion by including neverending stream of payments that go on forever and are referred to as a perpetuity. Here you will get to see how our prior discount factors for annuity payments may be extended to find present value of a perpetuity in amount of $1 or one that also grows by a growth or shrinks by a shrink rate that goes on forever. This will require us to find the limit of the two discount factors for annuity payments to get the results when N>∞. Before the discussion can be moved to finding net present value with payments in amounts different than $1, we will have a look at various 3rd party Excel financial functions that are part of tadXL addin that help you find all of the present value interest factors that we have thus far discussed. Although the actual tadXL addin is developed in C programming language and is distributed as an .XLL file, however I will present equivalent VBA code for these financial functions that will permit you to enter these functions in an Excel worksheet cells. A point to remember here is that the VBA version of tadXL function will be a stripped down version of the actual feature rich functions that are coded in tadXL using C programming langauge. Think of it this way, you will get something for free and hopefully you will spread the good word that would lead to an increased demand of the actual tadXL addin. Honestly speaking, I never say a lie am a very honest person, just like you I need money to keep my online business afloat and if you were one of my regular visitors you may have noticed that this site and it's sister concerns were taken offline at the end of July 2013 and I was barely able to bring this one back online but only with handful of the total content. I do not show adverts to my web vistors may it be from Google Adsense that I refer to as AdNonSense, over the years that I have been in this online business I have found that displaying adverts to web users may lead to unwanted, scam adverts for products and services that may cause our web visitors with loss of precious income that they have at present or have saved over their lifetime. My business model is simple, present to the user what she came looking for and offer such visitor with options to use online, windows based or Excel based financial software for a price that suits both of you and me. Alright, I think I have digressed from discussion of the original topic so let me revert back where we were. Once the discount factors are explained to you along with presentation of Excel VBA based financial code for tadXL functions, I will move ahead and begin to demonstrate finding net present values in a number of scenarios. This will include using closed form and open form formulas to find net present value of periodic payments in uniform amount, and ones that also grow or shrink by a growth rate or shrink rate. We will also look at calculating net present value when payments are not in uniform amount. The other elements of discussion will included using the same cash flows but then to use different compounding frequencies of interest and usage of period other than a year. The discussion will then be diverted to showing calculations for cash flows that may not be periodic and have a date schedule, or cash flows that are discounted using a rate schedule, and cash flows that are non periodic and are discounted using a rate schedule. We will also have a look at sort of annuity payments that increase or decrease by a constant amount and we will use short form formulas to find net present values for such sort of annuity payments. Here too, we will present you with Excel VBA code for financial functions from tadXL addin that will allow you to find net present value for various categories of annuity payments we have discussed including those that are in constant amount, or those that grow, or shrink by a rate and those that increase or decrease by a constant money amount. You will also be given VBA code for finding net present value when the cash flows are not necessarily periodic, and when cash flows are perioidic yet are discounted using a schedule of discount rates, and for annuities with non periodic cash flows that carry a schedule of discount rates. Once again as an alarm, you should not that all such Excel VBA code will be a stripped down version of the actual code that is more robust and feature rich as compared to the VBA code you will be given. I suppose I should end my introduction here and let you take a brief pause before letting you on a lengthy tutorial on net present value. Thanks for your time.
update #2 as of 10/14/2013 10:15 AM EDT.
The discount factor
As I first introduced the idea of discount factors that are key to finding net present value of a lump sum or series of periodic payments thus let me now move ahead and present you with various discount factors that are used in financial analysis to find NPV amongst other financial metrics. So what is this discount factor, the answer to this question may be put this way. Imagine you asked me for money now at time period t=0 and promised to payback this money to me at time period t=1. Obviously if I were to lend you the money now I will be letting go of the option of spending it myself and the only fair deal would suggest that I should charge you an amount $Y for borrowing an amount $X. In this case you are promising to pay me a $1 the $X at time period t=1, so I would have now determine the amount $Y that I will be willing to lend you now that when added interest to will equal $X in this case $1. This amount $Y is the loan amount and I can calculate it once you and me agree on a interest rate that you are willing to pay for the period t=0 till period t=1. Let us say you agree to pay me 10% interest for the duration of the loan, thus we will use the following math arrangement to find the loan amount that I should lend you at time period t=0.
PVIF  present value of a $1
Here we will find the present value of a $1 discounted at an interest rate i% for n time periods. This is our discount factor which when multiplied with the future value of $1 at time period t=n will give us the equivalent dollar amount at time period t=0.
PV = FV . PVIF( i%, n )
PVIF( i%, n ) = ( 1 + i )^{n}
PVIF example
In our loan arrangement between both you and me, it was agreed upon that an interest rate of 10% will be charged to you for borrowing an amount $X for time period t=1. Let us now put these numbers in to our PVIF formula to see what amount I must lend you at present to receive a time measured equivalent amount at time period t=1.
PVIF( 10%, 1 ) = ( 1 + 10% )^{1}
= ( 1.10 )^{1}
= 1 / 1.10
= 0.9091
Thus the following PV calculation shows the amount that I should lend you now to be able to receive an interest amount for 1 year at 10% interest rate.
PV = FV . PVIF( 10%, 1 )
= 1 x 0.9091
= $0.91
rounded to the nearest penny is the amount of money that you are borrowing now at time period t=0 and promising to pay back $1 a year from now. This means that you were charged an interest amount of $0.09 for the duration of the loan.
Intrayear compounding of interest
Our discussion on borrowing of money thus far assumed that interest is to be accrued at the end of the time period in this case a single year. What if I decided to charged interest on your borrowing at the end of each next 12 months. Obviously you would have to now pay me more interest as compared to the last agreement where interest was accumulated on borrowing at the end of each year. Let us now go back to the PVIF formula to see how this fits in to the picture.
PVIF( i%/c, n*c ) = ( 1 + i*c )^{n*c}
c = compounding frequency of interest
PVIF( 10%/12, 1*12 ) = ( 1 + 10%/12 )^{12}
=( 1 + 0.083334 )^{12}
=( 1.083334 )^{12}
=1/( 1.083334 )^{12}
=1/1.1047131
=0.9052
As you have just seen with monthly compounding of interest you get to receive a smaller amount of money from me now in amount of $0.9052 in contrast to $0.9090 which was the case with annual compounding of interest. This means that you are now paying a larger amount of interest on your borrowing which now amounts to $0.0948 this in contrast to an interest of $0.09 that was the case when compounding frequency of interest was agreed upon to be annual.
update #3 as of 10/14/2013 12:45 PM EDT.
But I want more money from you
Monthly compounding of interest resulted in me earning more money from you in form of interest and a lower amount of loan. But even though I am not greedy and am content with earning modest amount of money but what if I was like those who run Google, Yahoo or Microsoft you would want to see even more money than what their enterprises currently makes them. Ofcourse this too is possible when we increase the compounding frequency of interest, in our case let us agree upon the same loan arrangement but with weekly compounding of interest rather than the previously stated monthly compounding of interest. Follow the calculations that are listed after this paragraph to see what happens to the loan amount and the interest amount that now would have to pay under weekly compounding of interest. Keep in mind we assume that there are 52 weeks per year.
Weekly compounding of interest calculation
PVIF(i%/c, n*c) = (1 + i/c)^{n*c}
PVIF(10%/52, 1*52) = (1 + 10%/52)^{1*52}
= (1 + 10%/52)^{1*52}
= (1 + 0.0019231)^{52}
= (1.0019231)^{52}
= 1 / (1.0019231)^{52}
= 1 / 1.1051
= 0.9049
Now the amount of money I have to lend you is smaller than what it was with monthly compounding of interest, it turns out to $0.9049 and the amount of interest you have to pay goes up to $0.0951.
More money makes me happier
Not sure if money indeed buys happiness but it surely keeps away the heartache, and if I had a lust for more money I would take it one step further and start to compound interest on a daily basis. And as previously witnessed this will drive up the amount of interest you will have to repay and lower the amount of money I would have to lend you. Let us now see how much more money I can squeeze out of your wallet with the same loan agreement as discussed earlier.
Daily compounding of interest calculation
PVIF(i%/c, n*c) = (1 + i/c)^{n*c}
PVIF(10%/365, 1*365) = (1 + 10%/365)^{1*365}
= (1 + 10%/365)^{1*365}
= (1 + 0.0002739726)^{365}
= (1.0002739726)^{365}
= 1 / (1.0002739726)^{365}
= 1 / 1.10516
= 0.90485
You see that the amount of money I have to lend you at daily compouding of interest has gone down to $0.90485 and at the same time the amount of interest you must repay has gone up to $0.09515
But I want to earn interest while I sleep
Imagine that your investment earned you money on ever small fraction of time, or what is referred to as continuous compounding of interest also called infinite compounding of interest. This allows to max out my earnings from the loan money that I lend you. But just as I will be earning money while I sleep on the same token you will be paying interest while you sleep. I am not sure how the latter one would be possible unless you suffer from insomnia which will take your life if you did have this medical condition. But then who said lenders were saints, those who lend money have no heart if they do have one it is stone cold.
Infinite compounding of interest (continuous compounding)
If you noticed the trend in our previous examples that we were dividing the interest rate with larger compounding frequency and at the same time multiplying the number of periods with the same frequency. When N tends to ∞, the PVIF formula changes to the following form.
PVIF(i%/∞ , n*∞) = e^{in}
Here e is the mathematical constant called Euler's e with a value of 2.718281828459
The ultimate interest calculation
PVIF(10%/∞ , 1*∞) = e^{10%*1}
= e^{0.1*1}
= e^{0.1}
= 2.718281828459^{0.1}
= 1 / 2.718281828459^{0.1}
= 1 / 1.10517
= 0.904837
So limits have been maxed out and I would now have to lend you the least possible amount of money in exchange for a $1 at the end of the year and you are now obliged to pay the maximum possible interest amount of $0.095163
update #4 as of 10/15/2013 05:45 AM EDT.
Excel PVIF function
Now that you are learned about discount factors, I should now tell you about Excel PVIF function that you may use in a worksheet cell to perform the same discounting calculations as were shown to you. This financial function is called tadPVIF and it accepts the discount rate, the number of periods, the definition of the period such as year, month, or day and discounting convention.
Using tadPVIF function
This Excel PVIF function takes three values as follows:
=tadPVIF( rate, nper, compounding, period, distribution)

RATE
A value such as 10% or 0.10 may be used for interest rate. 
NPER
A value such as 10 or 24.5 may be used for number of periods. 
compounding
The value for compounding frequency of interest may be any value that you like. The annual compounding of interest is specified with a value of 1.
Any intrayear compounding of interest is denoted by a ratio of 1 over the compounding frequency such as 1/2 for semiannual compouding.
A quarterly compounding of interest is specifited by using a value of 3/12 or 1/4
A monthly compounding of interest is specified by using a value of 1/12
A weekly compounding of interest is specified by using a value of 1/52
A daily compounding of interest is specified by using a value of 1/365
An infinite compounding of interest is mentioned by using value of 0 
PERIOD = 1 for year
= 1/2 for half year
= 1/4 for quarter
= 1/12 for month
= 1/52 for week
= 1/365 for day
=tadPVIF(10%, 10, 1)
Where do I find tadPVIF, I don't see them in my Excel
tadPVIF series of financial functions for Excel 2007, 2010, and 2013 are available in Excel once you have installed the tadXL addin. This is a 3rd party library of 95 financial function that offer various Excel functions for financial analysis. You may go ahead and download tadXL now.
update #5 as of 10/16/2013 10:00 AM EDT.
Discount factors for an annuity
So far I have shown how one can discount a single payment or receipt that is due at some time into the future to find it's present value. But how about finding the present value of series of periodic payments in amount of $1. This is bit different than what we have done so far yet it is still related to the same discount factor. Let me explain to you what is going on here, imagine you have now borrowed $X from me at present and you wish to repay the loan amount in equivalent amount of $1 per period for N number of periods. For example you may wish to make twelve yearly payments at the end of each of the next 12 years in amount of $1 to pay off the loan amount of $X that you are borrowing from me at time period t=0. It seems that you are paying me a total of $12 spread throughout the 12 years at equal intervals. The loan arrangement is still on the same terms as before if you recall we agreed on an interest rate of 10% on borrowings. But this time the task becomes to find the current worth of your expected 12 payments in amount of $1 each. The sum of these 12 discounted payments in amount of $1 at an interest rate of 10% will equal the loan amount that I will be willing to lend you at present in return for 12 yearly repayments from you in amount of $1 each. In the following section, I will show you how we can use the concept of discount factor that you have learned thus far from this tutorial to find the present value of these periodic payments. Once this is explained and illustrated with example calculations, you will then be shown a simple closed form mathematical formula that will return the same results as the sum of 12 discounted payments in amount of $1. This formula is in fact the sum of first N terms of the geometric series ( an infinite series ), a topic that is usually taught in precalculus or Calculus I as a math course at colleges.
N  RATE  PMT  PVIF  PV 
1  0.1  1  (1+10%)^1  0.90909 
2  0.1  1  (1+10%)^2  0.82645 
3  0.1  1  (1+10%)^3  0.75131 
4  0.1  1  (1+10%)^4  0.68301 
5  0.1  1  (1+10%)^5  0.62092 
6  0.1  1  (1+10%)^6  0.56447 
7  0.1  1  (1+10%)^7  0.51316 
8  0.1  1  (1+10%)^8  0.46651 
9  0.1  1  (1+10%)^9  0.4241 
10  0.1  1  (1+10%)^10  0.38554 
11  0.1  1  (1+10%)^11  0.35049 
12  0.1  1  (1+10%)^12  0.31863 
NPV  6.81369 
update #6 as of 10/18/2013 06:00 AM EDT.
The table listed above this paragraph shows you how the sum of series of 12 payments in amount of $1 comes out to be $6.81369 when discounted using an interest rate of 10% compounded annually. In brief, I will now be willing to lend you an amount of $6.81369 in exchange for 12 annual payments from you in amount of $1 each. From your perspective, you would be obliged to pay me a cumulative interest in amount of $5.18631 for the loan you have taken from me. Later on, I will demonstrate how I may be able to squeeze more interest out of your pocket using intrayear compounding of interest that will lower the amount of money I have to lend you and at the same time permit me to collect more in interest.
Geometric series
If you look carefully at the table listed earlier, you would notice that we are looking at a sum of discounted cash flows where the discount factor takes the form of (1+i)^t for a particular time period. Assuming 1+i=x, we may rewrite it as following:
x^1 + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12
We can rearrange the series to write it with positive powers instead as follows:
(1/x) + (1/x)^2 + (1/x)^3 + (1/x)^4 + (1/x)^5 + (1/x)^6 + (1/x)^7 + (1/x)^8 + (1/x)^9 + (1/x)^10 + (1/x)^11 + (1/x)^12
This is referred to as a geometric series up to 12 terms, whereas a geometric series itself is an infinite series however the sum of it's first N terms is equal to the following formula:
PVIFA  an annuity discount factor
S = ( 1  x^n ) / (x1)
but since x = 1 + i, thus
PVIFA = [ 1  (1+i)^n ] / i
This is second of our discount factors this time it is applicable for annuity payments that are discounted at a discount rate i% for n periods. We will be commonly using this formula in our net present value calculations later on. We will also have a look at slightly altered form of PVIFA when annuity payments commence immediately.
Let us now confirm the results using this shortend formula for PVIFA rather than the lengthy sum of discounted payments in amount of $1. Recall our interest rate is 10% and there were 12 payments.
PVIFA calculation
PVIFA = [ 1  (1+10%)^12 ] / 10%
= [ 1  (1.10)^12 ] / 0.10
= [ 1  1/(1.10)^12 ] / 0.10
= [ 1  1/3.138428376721 ] / 0.10
= [ 1  0.31863081771 ] / 0.10
= 0.681369 / 0.10
= $6.81369
You have just seen how we have calculated the same present value of series of 12 periodic payments of $1 in amount of $6.81369.
update #7 as of 10/19/2013 00:15 AM EDT.
More in interest for less of a loan
As you would recall when we were dealing with a loan arrangement with a single repayment that we found that increasing the compounding frequency of interest led to a lower loan and an increase in interest repayment. Now we will try out the same calculations for an annuity payment and you will get to see how an increase in compounding frequency of interest will lead to a lower cost for me along with more income in terms of interest from you. Following is the tabulated data for the same loan arrangement where an interest rate of 10% applies for 12 monthly repayments of $1 each. This time I will charge you compound interest at the end of each month rather than at the end of each year.
N  RATE  PMT  PVIF  PV 
1  0.1  1  (1+10%/12)^12  0.9052 
2  0.1  1  (1+10%/12)^24  0.8194 
3  0.1  1  (1+10%/12)^36  0.7417 
4  0.1  1  (1+10%/12)^48  0.6714 
5  0.1  1  (1+10%/12)^60  0.6078 
6  0.1  1  (1+10%/12)^72  0.5502 
7  0.1  1  (1+10%/12)^84  0.498 
8  0.1  1  (1+10%/12)^96  0.4508 
9  0.1  1  (1+10%/12)^108  0.4081 
10  0.1  1  (1+10%/12)^120  0.3694 
11  0.1  1  (1+10%/12)^132  0.3344 
12  0.1  1  (1+10%/12)^144  0.3027 
NPV  6.6592 
As you can see the loan amount has come down to $6.6592 and you are now faced with paying me more in interest in amount of $5.3408. We have used the sum of discounted cash flows to find this answer, however we could have also used the PVIFA formula as discussed earlier to find the same amount. The following calculation shows us this in detail.
Annual Effective Yield
To be able to apply the monthly compounding of interest, we at first have to find the annualized effective rate to be used in our PVIFA calculations. To do this we will use the following formula:
AEY = (1 + i/c)^c  1
= (1 + 10%/12)^12  1
= (1.0083333)^12  1
= 1.104713067  1
= 0.104713067
AEY = 10.4713067%
PVIFA  Monthly compounding of interest
We will now find the net present value of 12 periodic payments of $1 discounted at 10% interest rate compounded monthly using the annual effective yield calculated as 10.4713067%. The remaining calculation will be just as it were before.
PVIFA = [ 1  (1+10.4713067%)^12 ] / 10.4713067%
= [ 1  (1.104713067)^12 ] / 0.104713067
= [ 1  1/(1.104713067)^12 ] / 0.104713067
= [ 1  1/3.303648968 ] / 0.104713067
= [ 1  0.302695598 ] / 0.104713067
= 0.697304402 / 0.104713067
= $6.6592
There you have seen the exact result from our PVIFA formula when an annual effective yield is used in case interest is compounded monthly. Thus using the PVIFA formula along with AEY produced the same results as was the case with sum of discouted cash flows in amount of $1.
update #8 as of 10/19/2013 01:30 AM EDT.
A little more money out of you won't hurt you, would it!
You have figured out by now that increasing the compounding frequency of interest makes it more difficult for you to payback the loan since the interest repayment goes up and the loan amount goes down. So you are in fact paying more money for less money, this is the beauty of charging compound interest where the lender always enjoys the drink on a beach with pretty girls in bikinis and bras where as you are obliged to pay for the lenders life style. This time, I will increase the compounding frequency to weekly to get more money from you. The results from the interest rate calculations in form of a table are given below followed by the use of PVIFA formula to get the same results.
N  RATE  PMT  PVIF  PV 
1  0.1  1  (1+10%/52)^52  0.9049 
2  0.1  1  (1+10%/52)^104  0.8189 
3  0.1  1  (1+10%/52)^156  0.741 
4  0.1  1  (1+10%/52)^208  0.6706 
5  0.1  1  (1+10%/52)^260  0.6068 
6  0.1  1  (1+10%/52)^312  0.5491 
7  0.1  1  (1+10%/52)^364  0.4969 
8  0.1  1  (1+10%/52)^416  0.4497 
9  0.1  1  (1+10%/52)^468  0.4069 
10  0.1  1  (1+10%/52)^520  0.3682 
11  0.1  1  (1+10%/52)^572  0.3332 
12  0.1  1  (1+10%/52)^624  0.3015 
NPV  6.6479 
See as compared to annual and monthly compounding now the loan amount is lowered to $6.6479 for which an interest is payable in amount of $5.3521. The previous tabular calculations are tedious so let me revert back to the shorter PVIFA formula to get the same numbers, and following the calculations for annualized efffective rate that is used for weekly compounding of interest in the PVIFA formula.
Annual Percentage Rate
We will now find the annualized percentage rate for weekly compounding of interest of 10% that is to be used in our PVIFA calculations. To do this we will use the following formula:
AEY = (1 + i/c)^c  1
= (1 + 10%/52)^52  1
= (1.001923)^52  1
= 1.105064793  1
= 0.105064793
AEY = 10.5064793%
PVIFA  Weekly compounding of interest
The interest rate of 10.5064793% which is the yield will now be inserted in to the PVIFA formula to get the net present value of 12 periodic payments in amount of $1 at an annual interest rate of 10% that is compounded weekly. The remaining calculation will be just as it were before.
PVIFA = [ 1  (1+10.5064793%)^12 ] / 10.5064793%
= [ 1  (1.105064793)^12 ] / 0.105064793
= [ 1  1/(1.105064793)^12 ] / 0.105064793
= [ 1  1/3.316293127 ] / 0.105064793
= [ 1  0.301541499 ] / 0.105064793
= 0.698458501 / 0.105064793
= $6.6479
The results from this NPV calculation do match those of $6.6479 that were derived with sum of discounted cash flows earlier in the listing.
update #9 as of 10/19/2013 04:30 AM EDT.
But I like to date two chicks at the same time
For me to be able to manage dating two pretty girls, it will be required of me to pay their bills that will now be double as compared to when I was dating a single broad. So where do I get this money from, obviously it would have to be borrowers like you from whom I can milk even more money as interest payment while lending a little less. My accountant said one way to do that would be to increase the compounding frequency of interest from weekly to daily which will increase the interest burden on the borrower. I like this accountant guy who is in his mid thirties and always carries his laptop with him to show me the interest calculation in an Excel spreadsheet. He came up with the following schedule of interest payments when I decided to charge you daily compounded interest. I think he deserves a raise which I will deal with a bit later as this is one more expense I would now have to worry about.
N  RATE  PMT  PVIF  PV 
1  0.1  1  (1+10%/365)^365  0.9048 
2  0.1  1  (1+10%/365)^730  0.8188 
3  0.1  1  (1+10%/365)^1095  0.7408 
4  0.1  1  (1+10%/365)^1460  0.6704 
5  0.1  1  (1+10%/365)^1825  0.6066 
6  0.1  1  (1+10%/365)^2190  0.5489 
7  0.1  1  (1+10%/365)^2555  0.4966 
8  0.1  1  (1+10%/365)^2920  0.4494 
9  0.1  1  (1+10%/365)^3285  0.4066 
10  0.1  1  (1+10%/365)^3650  0.3679 
11  0.1  1  (1+10%/365)^4015  0.3329 
12  0.1  1  (1+10%/365)^4380  0.3012 
NPV  6.6450 
The accountant was right, he just showed me that I would have to lend a lower amount of $6.6450 to you and in return get more interest from you in amount of $5.3550. One smart chap, he is indeed and then he showed me the notes from his notebook where he used the short formula for PVIFA to get the same results but he noted to do so would at first require to find the AEY for daily compounded annual interest rate of 10%. Here is what he had to say
AEY
A daily compounded interest rate would require finding the AEY using 10% as the annual rate and compounding frequency of 365. This rate will then be used as input for PVIFA formula to get the net present value of those 12 periodic payments in amount of $1 each.
AEY = (1 + i/c)^c  1
= (1 + 10%/365)^365  1
= (1.000274)^365  1
= 1.105155782  1
= 0.105155782
AEY = 10.5155782%
PVIFA  Daily compounding of interest
10.5155782% is the new annual effective yield that will give me the net present value of 12 periodic payments of $1 when discounted using daily compounding of interest and these calculations below were taken from my accountant's notebook.
PVIFA = [ 1  (1+10.5155782%)^12 ] / 10.5155782%
= [ 1  (1.105155782)^12 ] / 0.105155782
= [ 1  1/(1.105155782)^12 ] / 0.105155782
= [ 1  1/3.319571295 ] / 0.105155782
= [ 1  0.301243718 ] / 0.105155782
= 0.698756282 / 0.105155782
= $6.6450
I told you he is a genius since his use of AEY for daily compounded annual interest rate of 10% produced the exact same results as compared to detailed schedule of interest he had in his Excel workbook.
update #10 as of 10/19/2013 07:00 AM EDT.
One of my girlfriends called, she said she is pregnant
This one came as a shock to me, while I was getting ready to retire for the weekend, the Dutch blonde called and said she is expecting my baby!. Now other than paying the bills for two chicks I go out with, I would also have to worry about getting baby food in about mid June next year. I am already stretched out on my resources so this really came as a shock to my wallet. After the blonde hanged up the phone, I picked up my cell phone and rang the accountant and luckily he was still up at 10PM Saturday night. I told him of the news and asked him for advice, he said not to worry as he had figured out a way for me to earn even a larger chunk of money. I asked him how and from where and he said remember our debtor, the good old Schmuck who is borrowing money from us, we will have him pay the bills for the new arrival. I said to the accountant how would we do that, in reply he said we will charge our borrower the maximum amount of interest that is mathematically and legally possible by compounding interest on ever infinitesimal portion of time. It is called continuous compounding of interest whereby we will be lending him the least amount of money in exchange for maximum amount of interest out of his wits. I said thanks man, for a second I thought I was heading for a Chapter 11 bankruptcy proceeding. Well, I really have to give credit to my accountant for solving this monstrous problem that showed up from nowhere. By the early morning, my accountant send me the following schedule of interest rates that showed how a maximum amount of interest can be squeezed out of the pockets of my borrower in exchange for least amount of loan money.
N  RATE  PMT  PVIF  PV 
1  0.1  1  e^10%*1  0.9048 
2  0.1  1  e^10%*2  0.8187 
3  0.1  1  e^10%*3  0.7408 
4  0.1  1  e^10%*4  0.6703 
5  0.1  1  e^10%*5  0.6065 
6  0.1  1  e^10%*6  0.5488 
7  0.1  1  e^10%*7  0.4966 
8  0.1  1  e^10%*8  0.4493 
9  0.1  1  e^10%*9  0.4066 
10  0.1  1  e^10%*10  0.3679 
11  0.1  1  e^10%*11  0.3329 
12  0.1  1  e^10%*12  0.3012 
NPV  6.6445 
Would you believe if I told that now by compounding interest continuously I only have to lend an amount of $6.6445 for which I can earn a maximum allowed interest from him in amounts of $5.3555. I couldn't have figured this one out on my own, so I am thinking about making my accountant an equity partner in my firm. I am sure he would have more brilliant ideas as our business grows and attracts more customers like you who will be willing to pay unimaginable amount of interest on ever small amounts of principal amounts. Then the following notes were sent to me to prove that we can still use the shorter PVIFA formula to get the same principal amount.
APR
It is stated that an APR of annual percentage rate can be calculated by using the interest rate as a power to the Euler's e ( a constant value of 2.718281828 ) and then removing one from it to get the annualized rate required in PVIFA calculation listed below this.
AEY = e^i  1
= e^10%  1
= 2.718281828^0.1  1
= 1.105170918  1
= 0.105170918
AEY = 10.5170918%
PVIFA  Continuous compounding of interest
The annual percentage rate turns out to be 10.5170918% when an annual interest rate of 10% is compounded continuously. This rate is thus used as follows to find the least amount of money that I will be allowed to lend in exchange for 12 periodic payments of $1 that will earn me the most amount of interest that is possible under the State law.
PVIFA = [ 1  (1+10.5170918%)^12 ] / 10.5170918%
= [ 1  (1.105170918)^12 ] / 0.105170918
= [ 1  1/(1.105170918)^12 ] / 0.105170918
= [ 1  1/3.3201 ] / 0.105170918
= [ 1  0.3012 ] / 0.105170918
= 0.6988 / 0.105170918
= $6.6445
That's the story folks, we have just completed a whole cycle of interest rate calculations when loan repayment in made with periodic payments. And various frequencies of interest compounding are employed to get the maximum return for the buck.
update #11 as of 10/19/2013 09:30 AM EDT.
Excel PVIFA function
Now that you are learned about discount factors for annuity, I should now tell you about Excel PVIFA function that you may use in a worksheet cell to perform the same discounting calculations as were shown to you. This financial function is called tadPVIFA and it accepts the discount rate, the number of periods, the definition of the period such as year, month, or day and discounting convention.
Using tadPVIFA
Say you were to find PVIFA for i=10% and N=10 and compounding frequency of interest in annual thus you would use a value of 1 for compounding as follows:
=tadPVIFA(10%, 10, 1)
But now say you would like to use monthly compounding of interest for which you would have to use a value of 1/12 for compounding as follows
=tadPVIFA(10%, 10, 1/12)
And for continuous compounding of interest you may use a value of 0 such as
=tadPVIFA(10%, 10, 0)

RATE
A value such as 10% or 0.10 may be used for interest rate. 
NPER
A value such as 10 or 24.5 may be used for number of periods. 
compounding
The value for compounding frequency of interest may be any value that you like. The annual compounding of interest is specified with a value of 1.
Any intrayear compounding of interest is denoted by a ratio of 1 over the compounding frequency such as 1/2 for semiannual compouding.
A quarterly compounding of interest is specifited by using a value of 3/12 or 1/4
A monthly compounding of interest is specified by using a value of 1/12
A weekly compounding of interest is specified by using a value of 1/52
A daily compounding of interest is specified by using a value of 1/365
An infinite compounding of interest is mentioned by using value of 0 
PERIOD = 1 for year
= 1/2 for half year
= 1/4 for quarter
= 1/12 for month
= 1/52 for week
= 1/365 for day
Where do I find tadPVIFA, I don't see them in my Excel
tadPVIFA series of financial functions for Excel 2007, 2010, and 2013 are available in Excel once you have installed the tadXL addin. This is a 3rd party library of 95 financial function that offer various Excel functions for financial analysis. You may go ahead and download tadXL now.
update #12 as of 10/20/2013 04:30 AM EDT.
I want you to pay bills of my next many generations
Now that my Dutch girlfriend is expecting a baby in the Summer of 2014, I have been thinking about how my kid and his kid and their kids would survive without any recourse to income. This was quite troubling for me to figure out on my own that how can I ensure financial independence for my next generations. I looked around for answers even Googled it to no avail for a good reply but then the door bell rang and in walked my accountant who now owns part of my company as I had promised him. While having a cup of coffee together I posed the paradox to him to enquire if he has a solution to this financial problem that my children and their children would be facing. He said there is a solution to it as he recalled what his finance teacher had taught him while he was studying financial management. In his words, we could at present be able to secure the financial future of my progeny by demanding a never ending stream of loan repayments from our borrower ("you" the one who is reading this). I said to him who in their right mind be willing to pay forever. To which the accountant replied, that in lending and finance there is a special type of annuity that makes neverending payments and it is called a perpetuity. It is a common business practise to lend money and reap infinite benefits from it. My next logical question then was to be able to collect such payments that go on for ever I would have to loan a very very large sum of money to the borrower at present. He almost screamed in an negative and said NO NO NO. Then he went on to explain that there is a mathematical formula derived from PVIFA that will help me find the loan amount for which we will oblige the poor borrower ("you") to make annual payments in amount of $1 forever. Obviously you won't be alive forever so this means that my progeny will get payments from your progeny (the borrower) forever. It will be like masterslave relationship where your next many generations will be bonded in writing to pay my bills. I said to the accountant this sounds like what the Asian lenders (The Japs, the Chinese and the Singaporeans) are doing by buying the US Debt. The accountant said exactly my point. So I said why would the US Government allowed those foreign interests to put the US and our next many generations in a neverending bond? The accountant explained that it is the self interests of those who govern US and the same time pretend to speak at our behalf when there is nothing farthest from the truth. I said how did this all began, he said the Japs never forgotten or forgiven the US for dropping the nukes on Hiroshima and Nagasaki. They the Japs knew they can't fight the US militarily so they decided to launch a covert economic and financial war on the US. I said to him then what is the deal with Chinese to which the accountant referred me to the former President George Herbert Walker Bush Sr. who had spend considerable time in China in the 1970's and his personal relationship with the late Chairman Mao. He said George still has nostalgic memories of his days in China and it was his way of returning the favor to those at the helm of power at the Great Hall of China.
update #13 as of 10/20/2013 07:00 AM EDT.
Perpetual income formula
Later in the day the accountant sent me his notes as a MS Word file that showed me how he derived the net present value of never ending stream of payments from the PVIFA formula that he had shown us before. To understand how this is done, you would have to find the limit of the PVIF  the discount factor when the number of periods N>∞ as shown below:
PVIF = (1+i)^N
PVIF = 1/(1+i)^N
When N>∞ the denominator gets to become a very large number and the division with a numerator of 1 tends to 0.
PVIFA formula  perpetuity
PVIFA = [ 1  (1+i)^N ] / i
= [ 1  1/(1+i)N ] / i
= [ 1  1/∞ ] / i
= [ 1  0 ] / i
= 1 / i
There you have it folks, the net present value of a neverending stream of payments in amount of $1 discounted at an interest rate of i% that go on forever. And the accountant said this formula will give us the loan amount I have to lend the borrower ("you") at an interest rate of 10% to receive a neverending annual payment of $1. The accoutant said this is the way of ensuring the financial future of my next many generations while bonding the borrower ("you") in financial chains that is called a BOND.
NPV perpetuity
Now that I have the formula to find the loan amount that will pay infinite benefits, I was eager to find such amount at our agreed upon interest rate of 10% with you the borrower. See the calculations that follow:
PVIFA = 1/10%
=1/0.1
=$10
$10 of loan returns infinite repayments of $1
I told you before that this accountant guy is brilliant, all I have to do now is to lend you an amount of $10 at an interest rate of 10% for which you and your next many generations would be indebited to repay an annual payments of $1 forever. That solved the financial fiasco of my many generations and at the same time created a financial nightmare for your many next generations. Isn't this exactly what those Japs, Chinese and the Singaporeans have done to the many next generations of the US. A point to ponder!
update #14 as of 10/21/2013 06:00 AM EDT.
The Scandanavian chick called and said she has period so can't come over
Now that the Dutch blonde is pregnant and the Scandanavian broad has a PERIOD, I was left with no options but to spend the weekend on my own. Right after the Swedish chick hanged up the phone, I received a call from my borrower the good old Mr. Schmuck who said he wants to change the PERIOD associated with all of our previous loan arrangements. I didn't know what this was going to entail until I had spoken with my accountant Jimmy Bozo. So I asked my debtor for some time to which he replied that he will be sending me a complete list of interest schedules as prepared by his accountant Billy Moron. I said I be looking forward to your email. The next day I received the email from Mr Schmuck with the following list of interest schedule which required changing the frequency of loan repayments to monthly rather than the yearly payments as previously agreed. When I looked closer at the numbers in the Excel worksheet I noticed the net present value of these schedules was much higher than those we had agreed to in past, this meant I was now obliged to lend a larger sum of money to my debtor Mr. Schmuck. I didn't know what to make of it so I gave my accountant a call and asked him to come to offices on Monday and have a look at these numbers. For now you will notice that when period of payments is changed from a year to month that means the monthly loan repayments will pay out the loan by the end of the year and I would earn less in interest for a loan that is much higher in amount.
The notes sent to me from debtor
The following many tables were complied by Billy Moron the accountant working for my debtor good old Mr. Schmuck. I consulted my accountant Jimmy Bozo and he confirmed the results. Even though I would now have to make loans in higher amounts as compared to the last agreement that demanded annual repayments, my accountant assured me that we can still make more money out of Mr. Schmuck with a new agreement later down the lane. So I agreed to the new terms of the loan at an annual interest rate of 10% with 12 monthly repayments that allowed for various compounding frequencies of interest including annual, monthly, weekly, daily and continuous compounding of interest. The net present value calculations are computed as a sum of discounted payments in amount of $1. And the principle of using annualized rates for intraperiod compounding still apply as shown in the PVIF columns of the following interest schedules.
Monthly payments  annual compounding of interest
N  RATE  PMT  PVIF  PV 
1  0.1  1  (1+10%)^1/12  0.99209 
2  0.1  1  (1+10%)^2/12  0.98424 
3  0.1  1  (1+10%)^3/12  0.97645 
4  0.1  1  (1+10%)^4/12  0.96873 
5  0.1  1  (1+10%)^5/12  0.96107 
6  0.1  1  (1+10%)^6/12  0.95346 
7  0.1  1  (1+10%)^7/12  0.94592 
8  0.1  1  (1+10%)^8/12  0.93844 
9  0.1  1  (1+10%)^9/12  0.93101 
10  0.1  1  (1+10%)^10/12  0.92365 
11  0.1  1  (1+10%)^11/12  0.91634 
12  0.1  1  (1+10%)^12/12  0.90909 
NPV  11.40049 
Monthly payments  monthly compounding of interest
N  RATE  PMT  PVIF  PV 
1  0.1  1  (1+(1+10%/12)^12 1)^1/12  0.99174 
2  0.1  1  (1+(1+10%/12)^12 1)^2/12  0.98354 
3  0.1  1  (1+(1+10%/12)^12 1)^3/12  0.97541 
4  0.1  1  (1+(1+10%/12)^12 1)^4/12  0.96735 
5  0.1  1  (1+(1+10%/12)^12 1)^5/12  0.95936 
6  0.1  1  (1+(1+10%/12)^12 1)^6/12  0.95143 
7  0.1  1  (1+(1+10%/12)^12 1)^7/12  0.94356 
8  0.1  1  (1+(1+10%/12)^12 1)^8/12  0.93577 
9  0.1  1  (1+(1+10%/12)^12 1)^9/12  0.92803 
10  0.1  1  (1+(1+10%/12)^12 1)^10/12  0.92036 
11  0.1  1  (1+(1+10%/12)^12 1)^11/12  0.91276 
12  0.1  1  (1+(1+10%/12)^12 1)^12/12  0.90521 
NPV  11.37451 
Monthly payments  weekly compounding of interest
N  RATE  PMT  PVIF  PV 
1  0.1  1  (1+(1+10%/52)^52 1)^1/12  0.99171 
2  0.1  1  (1+(1+10%/52)^52 1)^2/12  0.98349 
3  0.1  1  (1+(1+10%/52)^52 1)^3/12  0.97533 
4  0.1  1  (1+(1+10%/52)^52 1)^4/12  0.96725 
5  0.1  1  (1+(1+10%/52)^52 1)^5/12  0.95923 
6  0.1  1  (1+(1+10%/52)^52 1)^6/12  0.95128 
7  0.1  1  (1+(1+10%/52)^52 1)^7/12  0.94339 
8  0.1  1  (1+(1+10%/52)^52 1)^8/12  0.93557 
9  0.1  1  (1+(1+10%/52)^52 1)^9/12  0.92781 
10  0.1  1  (1+(1+10%/52)^52 1)^10/12  0.92012 
11  0.1  1  (1+(1+10%/52)^52 1)^11/12  0.91249 
12  0.1  1  (1+(1+10%/52)^52 1)^12/12  0.90492 
NPV  11.37258 
Monthly payments  daily compounding of interest
N  RATE  PMT  PVIF  PV 
1  0.1  1  (1+(1+10%/365)^365 1)^1/12  0.9917 
2  0.1  1  (1+(1+10%/365)^365 1)^2/12  0.98347 
3  0.1  1  (1+(1+10%/365)^365 1)^3/12  0.97531 
4  0.1  1  (1+(1+10%/365)^365 1)^4/12  0.96722 
5  0.1  1  (1+(1+10%/365)^365 1)^5/12  0.95919 
6  0.1  1  (1+(1+10%/365)^365 1)^6/12  0.95124 
7  0.1  1  (1+(1+10%/365)^365 1)^7/12  0.94334 
8  0.1  1  (1+(1+10%/365)^365 1)^8/12  0.93552 
9  0.1  1  (1+(1+10%/365)^365 1)^9/12  0.92775 
10  0.1  1  (1+(1+10%/365)^365 1)^10/12  0.92005 
11  0.1  1  (1+(1+10%/365)^365 1)^11/12  0.91242 
12  0.1  1  (1+(1+10%/365)^365 1)^12/12  0.90485 
NPV  11.37208 
Monthly payments  infinite compounding of interest
N  RATE  PMT  PVIF  PV 
1  0.1  1  (e^10% 1)^1/12  0.9917 
2  0.1  1  (e^10% 1)^2/12  0.98347 
3  0.1  1  (e^10% 1)^3/12  0.97531 
4  0.1  1  (e^10% 1)^4/12  0.96722 
5  0.1  1  (e^10% 1)^5/12  0.95919 
6  0.1  1  (e^10% 1)^6/12  0.95123 
7  0.1  1  (e^10% 1)^7/12  0.94334 
8  0.1  1  (e^10% 1)^8/12  0.93551 
9  0.1  1  (e^10% 1)^9/12  0.92774 
10  0.1  1  (e^10% 1)^10/12  0.92004 
11  0.1  1  (e^10% 1)^11/12  0.91241 
12  0.1  1  (e^10% 1)^12/12  0.90484 
NPV  11.37199 
update #15 as of 10/21/2013 09:00 AM EDT.
I can't wait for the first payment, I want it now
Impatience has it's cost, what if I wanted my debtor to make the first repayment right now when I issue the loan. This sounds a bit unfair to be asking for the first payment immediately before the borrower has gotten a chance to earn a dime but the odds work out in his favor. How is that so you may ask? I would now have to lend him more money in terms of net present value of this new series of discounted payments of $1 as compared to when Mr. Schmuck made the first payment at the end of first year. You can confirm this from glancing over the interest rate schedule that follows this text. It would seem now that the first payment is left undiscounted and the last payment is discounted at time period t=11 as compared to t=12 for the case when he made end of year payments. This all leads to a net present value that is higher meaning I have to dole out a larger chunck of money to him as compared to the last agreement. Once you have gone through the interest rate schedule for this loan I would show you the shorter formula called PVIFAD that is an altered form of PVIFA formula that we have been using before.
N  RATE  PMT  PVIF  PV 
0  0.1  1  (1+10%)^0  1 
1  0.1  1  (1+10%)^1  0.90909 
2  0.1  1  (1+10%)^2  0.82645 
3  0.1  1  (1+10%)^3  0.75131 
4  0.1  1  (1+10%)^4  0.68301 
5  0.1  1  (1+10%)^5  0.62092 
6  0.1  1  (1+10%)^6  0.56447 
7  0.1  1  (1+10%)^7  0.51316 
8  0.1  1  (1+10%)^8  0.46651 
9  0.1  1  (1+10%)^9  0.4241 
10  0.1  1  (1+10%)^10  0.38554 
11  0.1  1  (1+10%)^11  0.35049 
NPV  7.49506 
PVIFAD formula  present value annuity due
PVIFAD = (1+i)[1  (1+i)^N] / i
As you can confirm the present value of annuity due is nothing more than the product of PVIFA and an extra interest factor which in this case is equal to (1+i). However do note that the interest factor for intrayear compounding will be different than this one as we will see a bit later in our calculations for monthly compounding of interest and PVIFAD calculations.
update #16 as of 10/22/2013 01:00 AM EDT.
The PVIFAD calculation
Let us now put in the numbers into our PVIFAD formula to check if this matches the results from sum of discounted cash flows as shown in the table above. Recall that our interest rate is 10% for 12 years yet this time the payments are made at the start of each year.
PVIFAD(i%,n) = (1+i)[ 1  (1+i)^N ]/i
= (1+10%)[ 1  (1+10%)^12 ]/10%
= (1.1)[ 1  (1.1)^12 ]/10%
= (1.1)[ 1  1/(1.1)^12 ]/10%
= (1.1)[ 1  1/3.138428376721 ]/10%
= (1.1)[ 1  0.318631 ]/10%
= (1.1)[ 0.6813692 ]/10%
= (1.1)[ 0.6813692 ]/0.1
= (1.1)6.813692
= 7.49506
There you have it the same loan amount is returned by the PVIFAD formula as was calculated using sum of discounted cash flows that were shown in a table earlier. Thus it is shown that PVIFAD is nothing more than the product of an interest factor (1+i) with the PVIFA result.
update #17 as of 10/22/2013 03:00 AM EDT.
Monthly payments beginning immediately with monthly compounding of interest
We can now go ahead with checking the net present value for monthly payments that begin at time period t=0. Meaning that the moment the borrower has received the loan, he has to make an immediate loan repayment of $1. This would result in a payment without any interest due on it since no time has lapsed between issue of the loan and the first payment. These interest calculations are at display in the table given below where interest rate is compounded per month for the monthly payments. A bit later we will confirm these results with the use of PVIFAD formula that was presented to you a bit earlier.
N  RATE  PMT  PVIF  PV 
0  0.1  1  (1+10%/12)^0  1 
1  0.1  1  (1+10%/12)^12  0.90521 
2  0.1  1  (1+10%/12)^24  0.81941 
3  0.1  1  (1+10%/12)^36  0.74174 
4  0.1  1  (1+10%/12)^48  0.67143 
5  0.1  1  (1+10%/12)^60  0.60779 
6  0.1  1  (1+10%/12)^72  0.55018 
7  0.1  1  (1+10%/12)^84  0.49803 
8  0.1  1  (1+10%/12)^96  0.45082 
9  0.1  1  (1+10%/12)^108  0.40809 
10  0.1  1  (1+10%/12)^120  0.36941 
11  0.1  1  (1+10%/12)^132  0.33439 
NPV  7.3565 
Loan amount formula  monthly payments  monthly interest
Here I will show you the PVIFAD calculations for the same loan for which a net present value of $7.3565 was calculated using the sum of discounted cash flows. The interest rate is still quoted at 10% per annum with monthly compounding. There are 12 loan repayments in total the first one due at the time of loan issue.
The annualized interest rate
Before we can make use of PVIFAD formula to find net present value of series of monthly payments commencing immediately, we have to at first find the annualized interest rate as follows
AEY = (1 + 10%/12)^12  1
= (1 + 0.008333333)^12  1
= (1.008333333)^12  1
= 1.104713067  1
= 0.104713067
= 10.4713067%
PVIFAD  monthly compounding of interest
PVIFAD(i%,n) = (1+i)[ 1  (1+i)^N ]/i
= (1+10.4713067%)[ 1  (1+10.4713067%)^12 ]/10.4713067%
= (1.104713067)[ 1  (1.104713067)^12 ]/10.4713067%
= (1.104713067)[ 1  1/(1.104713067)^12 ]/10.4713067%
= (1.104713067)[ 1  1/3.303648968 ]/10.4713067%
= (1.104713067)[ 1  0.302695598 ]/10.4713067%
= (1.104713067)[ 0.697304402 ]/10.4713067%
= (1.104713067)[ 0.697304402 ]/0.104713067
= (1.104713067)6.659191818
= 7.3565
update #18 as of 10/23/2013 02:00 AM EDT.
Excel PVIFAD function
Now that you are learned about discount factors for annuity, I should now tell you about Excel PVIFAD function that you may use in a worksheet cell to perform the same discounting calculations as were shown to you. This financial function is called tadPVIFAD and it accepts the discount rate, the number of periods, the definition of the period such as year, month, or day and discounting convention.
Using tadPVIFAD
Say you were to find PVIFAD for i=10% and N=10 and compounding frequency of interest in annual thus you would use a value of 1 for compounding as follows:
=tadPVIFAD(10%, 10, 1)
But now say you would like to use monthly compounding of interest for which you would have to use a value of 1/12 for compounding as follows
=tadPVIFAD(10%, 10, 1/12)
And for continuous compounding of interest you may use a value of 0 such as
=tadPVIFAD(10%, 10, 0)

RATE
A value such as 10% or 0.10 may be used for interest rate. 
NPER
A value such as 10 or 24.5 may be used for number of periods. 
compounding
The value for compounding frequency of interest may be any value that you like. The annual compounding of interest is specified with a value of 1.
Any intrayear compounding of interest is denoted by a ratio of 1 over the compounding frequency such as 1/2 for semiannual compouding.
A quarterly compounding of interest is specifited by using a value of 3/12 or 1/4
A monthly compounding of interest is specified by using a value of 1/12
A weekly compounding of interest is specified by using a value of 1/52
A daily compounding of interest is specified by using a value of 1/365
An infinite compounding of interest is mentioned by using value of 0 
PERIOD = 1 for year
= 1/2 for half year
= 1/4 for quarter
= 1/12 for month
= 1/52 for week
= 1/365 for day
Where do I find tadPVIFAD, I don't see them in my Excel
tadPVIFAD series of financial functions for Excel 2007, 2010, and 2013 are available in Excel once you have installed the tadXL addin. This is a 3rd party library of 95 financial function that offer various Excel functions for financial analysis. You may go ahead and download tadXL now.
update #19 as of 10/27/2013 09:00 AM EDT.
My Secretary of State Ms. Busty Babe called and said Tax Attorney Joe TriSexual is here from TriState Area
Just before I was to take off for the lunch after a busy morning at my Midtown Manhattan office, that the phone rang and it was no other than our Secretary Ms Busty Babe. She said that our Taxattorney Mr Joe TriSexual from TriState area wants to see you. I said to send him to the cafeteria where I will join him for a lunch. This was the first time I was to meet Joe and I found him in the cafeteria lounge sitting idly. I asked why do you refer to it as TriSexual to which Joe responded that his business is a limited partnership with Jack, Jill and Chris the odd one LLP. I suppose it all make sense now, so my next question to Joe was what brings him here. He said taht the IRS has caught up to your shady loan deals with Mr. Schmuck after a lengthy undercover investigation and that Uncle Sam is looking for it's share of the deal. I said how much of a smacker do I have to pay the IRS on my loan deals with Mr. Schmuck. Joe said, it is inflation adjusted percentage of the gross revenue and it will take out a big piece of your income pie. He looked at his notes and said we are looking at a marginal rate of 20% per annual income. After we had lunch, I promised to Joe that I will get back to him about my tax obligations once I had discussed this with my accountant Jimmy Bozo. At that point we parted our ways and I called Busty Babe to tell her I was quitting for the day and to remind Jimmy to come see me first thing next morning at my Midtown office.
A tax expense paid with a loan repayment that grows by a exponetial growth rate
Jimmy was awaiting my arrival at my offices the next morning as I walked in with soaked clothes as there had been heavy pours since late last night. I told Jimmy about my meeting with Joe from TriState and informed him of the IRS demands for a tax payment on my loan deals with Mr. Schmuck. Jimmy looked outside the window facing Broadway and gazed at me and said I got it!. I said what is it. He said since the tax rate is 20% of the gross income, we can oblige our debtor Mr. Schmuck to pay for inflation adjusted loan repayments meaning the $1 repayment will be adjusted for inflation. I said what does it mean, to which Jimmy said the he will calculate the expected inflation over the next twelve years and include this as a growth rate for the annual loan repayments. This will result in a larger income per $1 we will be lending to Mr. Schmuck. Jimmy Bozo promised to send me the detailed interest schedule for this inflation adjusted loan repayments along with detailed description of how such payments and it's net present value is calculated. For this I and you will have to wait for Jimmy Bozo's reply.
update #20 as of 10/27/2013 10:30 AM EDT.
Net Present Value  Growing Annuity of $1
As promised by Jimmy Bozo, I received a schedule of interest rates that showed how net present value may be calculated when periodic payments in amount of $1 are adjusted for inflation using a growth rate of 5%, recall our loan agreement still states an interest rate of 10%. This growth rate is also called a GRADIENT as explained by Jimmy in his notes. The following table shows how each periodic payment grows at 5% and at the same time is discounted at 10%. The sum all such compounded and discounted periodic payment results in a net present value of a growing annuity. Later in his notes, Jimmy showed me a closed form formula called PVIFGA that may be used to get the same results as is the case with sum of such payments.
N  RATE  GROWTH  PVIF  PV 
1  0.1  0.05  (1+5%)^0 (1+10%)^1  0.9091 
2  0.1  0.05  (1+5%)^1 (1+10%)^2  0.8678 
3  0.1  0.05  (1+5%)^2 (1+10%)^3  0.8283 
4  0.1  0.05  (1+5%)^3 (1+10%)^4  0.7907 
5  0.1  0.05  (1+5%)^4 (1+10%)^5  0.7547 
6  0.1  0.05  (1+5%)^5 (1+10%)^6  0.7204 
7  0.1  0.05  (1+5%)^6 (1+10%)^7  0.6877 
8  0.1  0.05  (1+5%)^7 (1+10%)^8  0.6564 
9  0.1  0.05  (1+5%)^8 (1+10%)^9  0.6266 
10  0.1  0.05  (1+5%)^9 (1+10%)^10  0.5981 
11  0.1  0.05  (1+5%)^10 (1+10%)^11  0.5709 
12  0.1  0.05  (1+5%)^11 (1+10%)^12  0.545 
NPV  8.5557 
PVIFGA  closed form formula
PVIFGA = [ 1  {(1+g)^N (1+i)^N} ] / (ig)
Jimmy Bozo showed me this formula that returns the present value of growing annuity compounded at g% and discounted at i% for N number of periods.
And following are the net present value calculations using this PVIFGA formula:
PVIFGA = [ 1  {(1+5%)^12 (1+10)^12} ] / (10%5%)
= [ 1  {(1+5%)^12 (1+10)^12} ] / 5%
= [ 1  1.795856326 * 0.318630818 ] / 5%
= [ 1  0.57221517 ] / 5%
= 0.42778483 / 0.05
= 8.5557
update #21 as of 10/28/2013 01:30 AM EDT.
Present value growing annuity  monthly compounding of interest
Jimmy explained that we will be able to get more money in interest from Mr. Schmuck if we were to compound the inflation and discount the interest using monthly compounding of interest. This will make possible a lower loan amount that I have to lend to the debtor and at the same time make possible earning in excess of what was previously possible using the annual compounding of inflation and interest rate. Then the following interest schedule was sent to me by Jimmy Bozo that detailed the sum of compounded cum discounted periodic payments that grow over each time period.
N  RATE  GROWTH  PVIF  PV 
1  0.1  0.05  (1+5%/12)^0 (1+10%/12)^12  0.9052 
2  0.1  0.05  (1+5%/12)^12 (1+10%/12)^24  0.8613 
3  0.1  0.05  (1+5%/12)^24 (1+10%/12)^36  0.8196 
4  0.1  0.05  (1+5%/12)^36 (1+10%/12)^48  0.7798 
5  0.1  0.05  (1+5%/12)^48 (1+10%/12)^60  0.742 
6  0.1  0.05  (1+5%/12)^60 (1+10%/12)^72  0.7061 
7  0.1  0.05  (1+5%/12)^72 (1+10%/12)^84  0.6718 
8  0.1  0.05  (1+5%/12)^84 (1+10%/12)^96  0.6393 
9  0.1  0.05  (1+5%/12)^96 (1+10%/12)^108  0.6083 
10  0.1  0.05  (1+5%/12)^108 (1+10%/12)^120  0.5788 
11  0.1  0.05  (1+5%/12)^120 (1+10%/12)^132  0.5507 
12  0.1  0.05  (1+5%/12)^132 (1+10%/12)^144  0.524 
NPV  8.3871 
PVIFGA for monthly interest compounding
But Jimmy stated we can get the same results for net present value of growing annuity compounded at monthly rate by using the PVIFGA formula but it would require making use of annualized rate in the PVIFGA calculation. Here is what Jimmy showed to me in terms of the math required to solve this problem.
Compound Annual Growth Rate  Monthly interest
AEY = (1+5%/12)^121
= (1+0.004166667)^121
= (1.004166667)^121
= 1.0511618981
= 0.051161898
= 5.1161898%
Compound Annual Interest Rate  Monthly interest
AEY = (1+10%/12)^121
= (1+0.008333333)^121
= (1.008333333)^121
= 1.1047130671
= 0.104713067
= 10.4713067%
PVIFGA Calculation for monthly compounded rate
PVIFGA = [ 1  { (1+5.1161898%)^12 / (1+10.4713067%)^12 } ] / ( 10.4713067%  5.1161898% )
= [ 1  { (1.051161898)^12 / (1.104713067)^12 } ] / 0.05355117
= [ 1  { 1.819848874 / 3.303648968 } ] / 0.05355117
= [ 1  0.550860243 ] / 0.05355117
= 0.449139757 / 0.05355117
= 8.3871
update #22 as of 10/28/2013 02:30 AM EDT.
Present value growing annuity  weekly compounding of interest
But as it was shown to me in earlier interest calculations when the annuity payments were in constant amount that if the compounding frequency were to be increased that led to a lower loan amount and a higher interest earnings. This time too, my accountant friend J. Bozo explained it will be possible to lend even lower amount by increasing the compounding frequency of growth and interest to weekly. As a proof of this claim, Jimmy included the following interest schedule in his MS Word file that does confirm that the principal amount has been reduced in comparison to when interest and growth were compounded monthly. You can skim through the following table to see it for yourself.
N  RATE  GROWTH  PVIF  PV 
1  0.1  0.05  (1+5%/52)^0 (1+10%/52)^52  0.9049 
2  0.1  0.05  (1+5%/52)^52 (1+10%/52)^104  0.8609 
3  0.1  0.05  (1+5%/52)^104 (1+10%/52)^156  0.8189 
4  0.1  0.05  (1+5%/52)^156 (1+10%/52)^208  0.779 
5  0.1  0.05  (1+5%/52)^208 (1+10%/52)^260  0.7411 
6  0.1  0.05  (1+5%/52)^260 (1+10%/52)^312  0.705 
7  0.1  0.05  (1+5%/52)^312 (1+10%/52)^364  0.6707 
8  0.1  0.05  (1+5%/52)^364 (1+10%/52)^416  0.638 
9  0.1  0.05  (1+5%/52)^416 (1+10%/52)^468  0.6069 
10  0.1  0.05  (1+5%/52)^468 (1+10%/52)^520  0.5774 
11  0.1  0.05  (1+5%/52)^520 (1+10%/52)^572  0.5493 
12  0.1  0.05  (1+5%/52)^572 (1+10%/52)^624  0.5225 
NPV  8.3746 
PVIFGA for weekly interest compounding
Jimmy added some extra notes in his MS Word file that showed how we may still be able to find the net present value of growing annuity that uses weekly compounded rate with the help of the PVIFGA formula. This would at first require us to find the annualized growth and interest rates as shown below.
Compound Annual Growth Rate  Weekly interest
AEY = (1+5%/52)^521
= (1+0.000961538)^521
= (1.000961538)^521
= 1.0512458421
= 0.051245842
= 5.1245842%
Compound Annual Interest Rate  Weekly interest
AEY = (1+10%/52)^521
= (1+0.001923077)^521
= (1.001923077)^521
= 1.1050647931
= 0.105064793
= 10.5064793%
PVIFGA Calculation for weekly compounded rate
PVIFGA = [ 1  { (1+5.1245842%)^12 / (1+10.5064793%)^12 } ] / ( 10.5064793%  5.1245842% )
= [ 1  { (1.051245842)^12 / (1.105064793)^12 } ] / 0.053818951
= [ 1  { 1.821593602 / 3.316293127 } ] / 0.053818951
= [ 1  0.549286065 ] / 0.053818951
= 0.450713935 / 0.053818951
= 8.3746
update #23 as of 10/28/2013 03:30 AM EDT.
Present value growing annuity  daily compounding of interest
But greed has no limit, even though I would be happy with compounding interest on a weekly basis but as Jimmy suggested that we can save on our loan amount and earn more of interest if we were to compound growth and interest on a daily basis. As a proof of this statement the following schedule was appended to the notes sent to me by Jimmy Bozo. And just as before this too highlights the saving in lending and exceed in profits in terms of interest on daily basis. Have a look for yourself at the following table.
N  RATE  GROWTH  PVIF  PV 
1  0.1  0.05  (1+5%/365)^0 (1+10%/365)^365  0.9048 
2  0.1  0.05  (1+5%/365)^365 (1+10%/365)^730  0.8607 
3  0.1  0.05  (1+5%/365)^730 (1+10%/365)^1095  0.8188 
4  0.1  0.05  (1+5%/365)^1095 (1+10%/365)^1460  0.7788 
5  0.1  0.05  (1+5%/365)^1460 (1+10%/365)^1825  0.7409 
6  0.1  0.05  (1+5%/365)^1825 (1+10%/365)^2190  0.7047 
7  0.1  0.05  (1+5%/365)^2190 (1+10%/365)^2555  0.6704 
8  0.1  0.05  (1+5%/365)^2555 (1+10%/365)^2920  0.6377 
9  0.1  0.05  (1+5%/365)^2920 (1+10%/365)^3285  0.6066 
10  0.1  0.05  (1+5%/365)^3285 (1+10%/365)^3650  0.577 
11  0.1  0.05  (1+5%/365)^3650 (1+10%/365)^4015  0.5489 
12  0.1  0.05  (1+5%/365)^4015 (1+10%/365)^4380  0.5221 
NPV  8.3714 
PVIFGA for daily interest compounding
The same net present value of growing annuity payments given daily compounding frequency of interest rate and inflation is also possible when we make use of the shorter PVIFGA formula. But to do this, we must at first find the annualized interest and inflation rates as shown in the following calculations of NPV of growing annuity payments in amount of $1.
Compound Annual Growth Rate  Daily interest
AEY = (1+5%/365)^3651
= (1+0.000136986)^3651
= (1.000136986)^3651
= 1.0512674961
= 0.051267496
= 5.1267496%
Compound Annual Interest Rate  Daily interest
AEY = (1+10%/365)^3651
= (1+0.000273973)^3651
= (1.000273973)^3651
= 1.1051557821
= 0.105155782
= 10.5155782%
PVIFGA Calculation for daily compounded rate
PVIFGA = [ 1  { (1+5.1267496%)^12 / (1+10.5155782%)^12 } ] / ( 10.5155782%  5.1267496% )
= [ 1  { (1.051267496)^12 / (1.105155782)^12 } ] / 0.053888285
= [ 1  { 1.822043927 / 3.319571295 } ] / 0.053888285
= [ 1  0.548879288 ] / 0.053888285
= 0.451120712 / 0.053888285
= 8.3714
update #24 as of 10/28/2013 04:30 AM EDT.
Present value growing annuity  infinite compounding of interest
But just as some drug addict said that Heroin gets you the ultimate high, Bozo said we could get the ultimate earnings from Mr. Schmuck if we were to compound interest on ever continuous basis thus allowing for lending the least amount of money. Wow, I said Jimmy you must be a genius. To which Jimmy showed a smirk on his face and said see the following interest rate schedule to check how this is made possible. And indeed if you look at the NPV column at the bottom of the table it does show the least amount of net value I have to lend to Mr. Schmuck in return for maximum possible interest.
N  RATE  GROWTH  PVIF  PV 
1  0.1  0.05  e^5%*0 e^10%*1  0.9048 
2  0.1  0.05  e^5%*1 e^10%*2  0.8607 
3  0.1  0.05  e^5%*2 e^10%*3  0.8187 
4  0.1  0.05  e^5%*3 e^10%*4  0.7788 
5  0.1  0.05  e^5%*4 e^10%*5  0.7408 
6  0.1  0.05  e^5%*5 e^10%*6  0.7047 
7  0.1  0.05  e^5%*6 e^10%*7  0.6703 
8  0.1  0.05  e^5%*7 e^10%*8  0.6376 
9  0.1  0.05  e^5%*8 e^10%*9  0.6065 
10  0.1  0.05  e^5%*9 e^10%*10  0.5769 
11  0.1  0.05  e^5%*10 e^10%*11  0.5488 
12  0.1  0.05  e^5%*11 e^10%*12  0.522 
NPV  8.3709 
PVIFGA for infinite interest compounding
The infinitely compounded net present value of annuity payments that grow by a growth rate and discounted at an interest rate is also possible when we make use of the PVIFGA formula. For this purpose, you have to at first find the annualized growth and interest rates as shown below.
Compound Annual Growth Rate  Infinite interest
AEY = e^5%  1
= (2.718281828459)^0.05  1
= 1.051271  1
= 0.051271
= 5.1271%
Compound Annual Interest Rate  Infinite interest
AEY = e^10%  1
= (2.718281828459)^0.10  1
= 1.105171  1
= 0.105171
= 10.5171%
PVIFGA Calculation for infinite compounded rate
PVIFGA = [ 1  { (1+5.1271%)^12 / (1+10.5171%)^12 } ] / ( 10.5171%  5.1271% )
= [ 1  { (1.051271)^12 / (1.105171)^12 } ] / 0.053899822
= [ 1  { 1.8221188 / 3.320116923 } ] / 0.053899822
= [ 1  0.548811636 ] / 0.053899822
= 0.451188364 / 0.053899822
= 8.3709
update #25 as of 10/30/2013 04:00 AM EDT.
Abe, this is Busty Babe calling to let you know that Oh Boy! George STRAIGHT is here from IRS
Mom I am coming out of closet. You know what Mom! I am STRAIGHT. Oh, honey I know how hard it must have been to come out openly STRAIGHT with all the peer pressure not to mention the constantly reinforced messages on popular TV and movies about GAY being trendy and sheikh. It really takes courage and tell you what my son, I support your openness in coming out STRAIGHT out of the box. But wait mom, my friend Matt had to make even a tougher decision when he came out of closet and told his mom that "He is White". OMG Abe, I can imagine how difficult that decision would have been for Matt to openly come out of closet and tell the world that he is white. Abe, this reminds me of the struggles of US whites in the early 21st century when we had to fight for our rights in a society and government dominated by colored people. I recall the famous million white man march led by our Reverend David Duke and his deliverance of the famous "Lo! I have a NIGHTMARE speech" in the shadows of Lincoln Memorial in Washington D.C.
I apologize to you the reader for digressing away from the topic. Right after Busty Babe annouced the arrival of George Straight from IRS, George walked in to my office carrying a bunch of files. He sat on the chair right in front of my desk and said, Abe those at IRS offices want you to come out straight as we have found that you are rigging the inflation rate that you are charging your debtor Mr. Schmuck. That scared the living daylights out of me and I tried to hide my confusion and asked George for more details. He said IRS is demanding that you only charge a GROWTH RATE (inflation rate) that is the same as your INTEREST RATE for the calculation of net present value of growing annuity in amounts of $1. In reply I told George I would only be able to reply back to the IRS once I have spoken to my confidant Jimmy Bozo. George was pleased with my reply and left me his email address for a reply as soon as I have spoken to J. Bozo. It was Friday afternoon and Jimmy was not expected at the Offices till Monday so I asked Busty Babe to forward the minutes of my meeting with George Straight to Jimmy Bozo and to ask Jimmy for a solution on Monday. For the reply from Jimmy, me and you would have to wait till the weekend is over.
update #26 as of 10/30/2013 07:30 AM EDT.
Number of payments discounted once will get you the NPV of growing annuity when i=g
I ran into Jimmy Bozo in the lobby of my office building on Monday. He looked at me and said Abe, I got the solution to the distressing problem created by George Straight from the IRS offices. I said to Abe, please go right ahead and explain all this nonsense to me before it drives me nuts and bolts. Jimmy said let's head to your office and we discuss this in detail. Once we arrived at the office, Jimmy turned on his Notebook PC and showed me his MS Powerpoint presentation on NPV calculations for a growing annuity of $1 when interest rate i is the same as the growth rate g. Jimmy said, first I will show you the closed form formula for these net present value calculations and later I will illustrate to you how I arrived at this formula using a schedule of interest rates where i=g.
PVIFGA formula  discount rate equals growth
PVIFGA = N (1+i)^1
Looking at this new PVIFGA formula, I said to Jimmy it would seem that net present value of a growing annuity in amount of $1 when i=g is nothing more than number of payments discounted at a single interest factor. Jimmy noded in an affirmative action like the one needed to fix the wrongs done by Founding fathers to help advance those in society who have been historically disadvantaged. I said Jimmy this sounds like taking advantage of our debtor Mr. Schmuck and the IRS at the same time. Jimmy showed a smirk on his face and said you got it BOSS is never wrong.
NPV of a growing annuity when growth rate is the same as interest rate
Jimmy said let me now show you how I derived this new formula of PVIFGA for growing annuities' net present value when the interest rate is the same as the growth rate. He said, look at the following table and keep an eye of the PVIF column. Recall that in past when calculating PVIFGA using the sum of discounted cash flows, we compounded each $1 payment at a number of period that was right below the time period at which we discounted the same $1 payment. It looked like this (1+g)^(t1) (1+i)^(t) and when i=g it becomes this (1+i)^(t1) (1+i)^(t) or this when simplified (1+i)^(t1t) and this gets us this (1+i)^1.
From here you can see that PVIF values for all of the time periods is (1+i)^1 and this leads to the closed form of N (1+i)^1 as the present value of growing annuities when discount rate equals the growth rate. Go ahead and look deeply into the following interest schedule.
N  GROWTH  RATE  PVIF  PVIF  
1  0.1  0.1  (1+10%)^1+0  (1+10%)^1  
2  0.1  0.1  (1+10%)^2+1  (1+10%)^1  
3  0.1  0.1  (1+10%)^3+2  (1+10%)^1  
4  0.1  0.1  (1+10%)^4+3  (1+10%)^1  
5  0.1  0.1  (1+10%)^5+4  (1+10%)^1  
6  0.1  0.1  (1+10%)^6+5  (1+10%)^1  
7  0.1  0.1  (1+10%)^7+6  (1+10%)^1  
8  0.1  0.1  (1+10%)^8+7  (1+10%)^1  
9  0.1  0.1  (1+10%)^9+8  (1+10%)^1  
10  0.1  0.1  (1+10%)^10+9  (1+10%)^1  
11  0.1  0.1  (1+10%)^11+10  (1+10%)^1  
12  0.1  0.1  (1+10%)^12+11  (1+10%)^1  
NPV  12 * (1+10%)^1 
update #27 as of 11/05/2013 05:30 AM EDT.
Perpetual price of a Barrel of Poison  the Arab Oil
Recall the example presented earlier where it was shown how a once economic superpower like United States has been put in economic bondage by those who dispise the very values that our nation was founded on. Now let me turn to how other players have emerged on the scene who have used their Oil Wealth to put the US in an economic bondage from which there seems no escape. Likes of those Texans who control the Oil trade has put the American lifes in peril when we had to send our forces to liberate Kuwait from Iraqi forces in the 1st Gulf War. We were told that it was in US interests to liberate Kuwait and safeguard the dictators in neighbouring Saudi Arabia. But what in reality are those interests, this question begs to be asked. In brief, the Oil traders in Texas wear two hats: A businessman's and one that of a politician. Defending those Oil fields in Middle East are of utmost importance to the Oil companies in Texas and the commodity traders on future exchanges that fix the price of a barrel of Oil that is more like a poison. Those Kings, Shieks and Fiefs in Middle East have money to burn literally but they ain't that stupid somebody told them they can earn perpetual income from their PetroDollars by investing the surplus in US financial system. Thus there aren't many large US enterprises where you won't find the investment of Arabs even media enterprises in US are financed with Oil money of those Middle Eastern dictators. The next time you take your kids to the Galleria Mall on Main Street USA, it would be noteworthy to note that shops selling designer clothing pays dividends to their investors in Middle East. Such is the cycle of financial dependence that our men and women have to sent to those lands to liberate those lands where the poor neighbors try to take over the Oil pits in the deserts of Arabia. I suppose this is what you would call the Arabian Knights. Long story short, this cycle of enslavement leads me to explain how I can earn ever more money from Mr. Schmuck and next many generations by demanding never ending payments in amounts of $1 that now grow each year by a growth rate. You have probably figured by now that to understand the math we would have call our friend Jimmy Bozo. And surely Jimmy will be back in the office tomorrow so he will enlighten you and me with financial math formulas and interest schedules that would make all of this possible.
for update #28 , please stay tuned...