Introduction. Deriving the ADF For An Ordinary Annuity

Transcription

1 The Annuity Discount Factor (ADF): Generalization, Analysis of Special Cases, and Relationship to the Gordon Model and Fixed-Rate Loan Amortization Copyright 1993, by Jay B. Abrams, CPA, MBA Introduction Many of us have seen Annuity Discount Factors (ADFs) in present value tables after the present value of a single cash flow. The ADF has an imposing formula, [1-(1+r) n ]/r. I always studied it for 10 seconds, burped and backed off. I imagine many people have the same reaction, although perhaps you have better digestion than I have. In this article, I will do the following: (1) Derive the ADF. (a) Analyze the components of the ADF formula and relate them to the capitalization of perpetuities. (2) Generalize the ADF to handle stub years, growth, and flexible starting points. (3) Generalize the ADF to handle growth and flexible starting points (without stub years) and analyze special cases. (4) Show how ADFs relate to loan amortization My focus in this article is to provide an intuitive understanding of the family of ADF formulas. Additionally, to my knowledge, the generalization to handle growth and/or stub years has never been done before. Definitions: Deriving the ADF For An Ordinary Annuity ADF = Annuity Discount Factor NPV = Net Present Value r = Interest/Discount rate The ADF for a series of $1 payments received at the end of each period, discounted to NPV at rate, r, is: [1] ADF = (1+r) (1 + r) 2 (1 + r) 3 (1 + r) n Multiplying each side of the equation by 1/(1+r), we get: [2] 1 ADF = (1+r) (1 + r) 2 (1 + r) 3 (1 + r) n (1 + r) n+1 Subtracting [2] from [1] and noting that all the middle terms drop out, leaving only the leftmost term in [1] and the rightmost term in [2], 1

2 [3] ADF[1-1 ] = 1-1 (1+r) (1+r) (1 + r) n+1 [3a] ADF r = 1-1 (1+r) (1+r) (1 + r) n+1 Multiplying both sides of the equation by (1+r)/r, we come to: [3b] ADF = 1+r [ 1-1 ] r (1+r) (1 + r) n+1 Equation [3c]: The ADF For An Ordinary Annuity 1 [3c] ADF = 1 - (1 + r) n r r Equation [3c] is composed of two terms. The first term is the net present value (NPV) of a $1 annuity capitalized in perpetuity. For example, if r=10%, then the first term is $1/10% = $10. We can reorganize the equation as NPV r = Annual return, or $10 X 10% = $1 per year return forever. In other words, a prudent investor requiring a 10% return for an investment of this risk should be indifferent between receiving a $10 lump sum or $1 forever. The second term in [3c] is the NPV of the perpetuity of $1 per year starting at year n+1, as we will derive below in [4] through [6a]. Therefore the ADF for the ordinary annuity is the difference of the NPV of two perpetuities: from inception to infinity, less from n+1 to infinity = from inception to n. This formulation makes good, intuitive sense. Note that the formula for an ordinary annuity is usually shown in a more condensed form of: 1 ADF = 1 - (1 + r) n, r but for purposes of intuitively understanding what it means, it is better to show the two separate terms as in [3c]. This will also be true for other ADF formulas to follow in this article. I will always show them in their component terms to facilitate better understanding rather than show them in their most compact form. Now let's complete this portion of the analysis with the proof that the second term in [3c] is indeed the NPV of $1 from n+1 to infinity. [4] ADF = (1+r) n+1 (1 + r) n+2 (1 + r) n+3 (1 + r) I, where I = infinity Multiplying by 1/(1+r) gives us: [5] 1 ADF = (1+r) (1 + r) n+2 (1 + r) n+3 (1 + r) I Subtracting [6b] from [6a], and using the shortcut of bringing over the [1-(1/(1+r)] directly to the right hand side of the equation, we get: 2

3 [6] ADF = (1+r) 1 r (1+r) n+1 1 [6a] ADF = (1 + r) n, which is our proof. r Generalizing The ADF To Cash Flows With Stub Periods Background Of The Problem In this next section, we will develop a formula to handle annuities that have stub periods, constant growth in cash flows, and ones that start at any time, not just Year 1. I owe my thanks to my friend and colleague Robert Athy, J.D., CPA, who posed the following problem to me, which ultimately formed the basis of this entire article. Of course, my wife may have trouble forgiving him, because my obsession solving the problem kept me up late a few nights. Bob testified in a litigation, where an expert witness with a Ph.D. in Finance used a formula to calculate the NPV of an annuity1 of $10,000 per year, growing at 5.1% a year, with cash flows beginning 2.25 years from the valuation date and finishing years from the valuation date. Bob did his own calculation as an expanded schedule on a spreadsheet, and it did not match the Ph.D.'s answer. He asked me if I could figure out how the Ph.D. came to his answer. My recollection is that I never figured out his answer, but after quite some time banging my head against the wall, I derived the right answer in the form of a single, though complicated, equation. Derivation Using Mid-Year Cash Flows Let's begin with constructing a time line of the cash flows in Figure 1, using the following definitions: S = For end-of-year cash flow, S = the first cash flow. For midyear cash flows, S = the end of the first year's cash flow = 3.25 Years F1 = F2 = The "fiscal end" of the cash flows = Years (F is for "Finish") The end of the stub period = Years p = The proportion of a full year represented by the stub period = F2-F1 G = The annual growth rate in cash flows (or net income) 1 Relating to an income differential in a calculation of damages 3

4 Figure 1: Time Line of Cash Flows $1 (1+G) (1+G) 2 (1+G) 29 (1+G) 29.5 (F2-F1) $1 (1+G) (1+G) 2 (1+G) (F1-S) (1+G) F2-S (F2-F1) S S+1 S+2 F1 F2 The first cash flow differential occurs from Year 2.25 to We assume the cash flows would occur evenly throughout the year, which is tantamount to assuming all cash flows occur on average halfway through the year, i.e., at Year Therefore the first $1 cash flow has a net present value of 1/(1+r) Continuing in this fashion, the NPV of our series of cash flows would be: [7] ADF = 1 + (1+G) (1+G) F1-S + p(1+g) F2-S (1+r) S-.5 (1 + r) S+.5 (1 + r) F1-.5 (1 + r) F1+(F2-F1)/2 Multiplying both sides of the equation by (1+G)/(1+r), [8] (1+G) ADF2 = (1+G) (1+G) F1-S+1 + p(1+g) F2-S+1 (1+r) (1 + r) S+.5 (1 + r) F1+.5 (1 + r) F1+1+(F2-F1)/2 Subtracting [8] from [7], we get: [9] [1- (1+G)] ADF = 1 + p(1+g) F2-S - (1+G) F1-S+1 - p(1+g) F2-S+1 (1+r) (1+r) S-.5 (1 + r) F1+(F2-F1)/2 (1 + r) F1+.5 (1 + r) F1+1+(F2-F1)/2 The Left-Hand Side of the equation factors to ADF (r-g)/(1+r). Inverting (r-g)/(1+r) and moving it to the Right-Hand Side of the equation, we get: [10] ADF = (1+r) [ 1 + p(1+g) F2-S - (1+G) F1-S+1 - p(1+g) F2-S+1 ] (r-g) (1+r) S-.5 (1 + r) F1+(F2-F1)/2 (1 + r) F1+.5 (1 + r) F1+1+(F2-F1)/2 Cancelling the (1+r) and moving (r-g) below to a second denominator, we get: [10a] ADF = 1 + p(1+g) F2-S - (1+G) F1-S+1 - p(1+g) F2-S+1 (1+r) S-1.5 (1 + r) F1-1+(F2-F1)/2 (1 + r) F1-.5 (1 + r) F1+(F2-F1)/2 Combining the 2nd and 4th terms of [10a], we get: (r-g) [10b] ADF = 1 + p(1+g) F2-S [(1+r) - (1+G)] - (1+G) F1-S+1 (1+r) S-1.5 (1 + r) F1+(F2-F1)/2 (1 + r) F1-.5 (r-g) (r-g) (r-g) Call this term "C" 2 For lack of space, I omitted showing the 3rd and 4th terms of [7] and [8]. These terms drop out in subtracting [8] from [7]. 4

5 [10c] C = p(1+g) F2-S-1 (r-g) (1 + r) F1+(F2-F1)/2 (r-g) With (r-g) cancelling out in both numerator and denominator, substituting [10c] into [10b], substituting F2-F1 = p, and moving the term, C, to the end of the equation, we come to: Generalized Annuity Discount Factor With Stub Period--Mid Year Cash Flows [10d] ADF = 1 - (1+G) F1-S+1 + (F2-F1)(1+G) F2-S (1+r) S-1.5 (1 + r) F1-.5 (1 + r) F1+(F2-F1)/2 (r-g) (r-g) Call this Call this Call this term "C" term "A" term "B" Let's analyze the terms of [10d] to intuitively understand its meaning. "A" is a Gordon Model perpetuity with constant growth at G%, the cash flows of which begin at S-1 (remember that S is the end of the first year's cash flows) and continue forever, the proof of which will follow below in [10e] through [10h]. "B" is the perpetuity starting at F1+1 (proved in [10i] through [10l]), so that "A" - "B" covers the time period (S-1 to infinity) - (F1+1 to infinity) = S-1 to F1, which is the time period of the annuity. "C" is the singular cash flow that occurs at F2. It is not divided by (r-g) as the other terms are, because it is not a perpetuity or even a stream of cash flows. Therefore "A" - "B" + "C" is, indeed, logically and intuitively the ADF for a series of cash flows with a stub period. Proof of Interpretation of "A" in [10d] With the first cash flows occurring in the year from S-1 to S, we discount the first year's cash flow by (S-.5) years. [10e] ADF = 1 + (1+G) (1+G) I (1+r) S-.5 (1 + r) S+.5 (1 + r) I, where I = infinity Multiplying by (1+G)/(1+r) gives us: [10f] (1+G)ADF = (1+G) (1+G) I (1+r) (1 + r) S+.5 (1 + r) I Subtracting [10f] from [10e], and using the shortcut of bringing over the (1+r)/(r-G) directly to the right hand side of the equation, we get: [10g] ADF = (1+r) 1 (r-g) (1+r) S-.5, or 1 [10h] ADF = (1 + r) S-1.5, which is our proof. (r-g) Note that the ADF of a perpetuity, unlike a finite series of cash flows, is a single term, because the term on the extreme right is always to an infinite power. Therefore it drops out when multiplied by a ratio and subtracted from the original equation. Proof of Interpretation of "B" in [10d] 5

6 This is the perpetuity beginning at F1 plus one day. With the first year's cash flows occurring from F1 to (F1+1), we discount the first year's cash flows by (F1+.5) years. [10i] ADF = (1+G) F1-S+1 + (1+G) F1-S (1+G) I (1+r) F1+.5 (1+r) F1+1.5 (1 + r) I, where I = infinity Multiplying by (1+G)/(1+r) gives us: [10j] (1+G)ADF = (1+G) F1-S (1+G) I (1+r) (1 + r) F1+1.5 (1 + r) I Subtracting [10j] from [10i], and using the shortcut of bringing over the (1+r)/(r-G) directly to the right hand side of the equation, we get: [10k] ADF = (1+r) (1+G) F1-S+1 (r-g) (1+r) F1+.5, or (1+G) F1-S+1 [10l] ADF = (1 + r) F1-.5, which is our proof. (r-g) The procedure for deriving the End-of-Year Cash Flow ADF is identical to the Mid-Year, and I will not repeat it. The ADF is: Generalized Annuity Discount Factor With Stub Period--End-of-Year Cash Flows [10m] ADF = 1 - (1+G) F1-S+1 + (F2-F1) (1+G) F2-S (1+r) S-1 (1 + r) F1 (1 + r) F2 (r-g) (r-g) The individual terms in [10m] have the same meaning as in the Mid-Year Cash Flows of [10d]. The numerators are identical, while the denominators of the first two terms differ by one-half year more discounting. The difference of the discounting period in the third term, "C", is: F2-[F1+(F2-F1)/2] = F2-F1-(F2/2)+(F1/2) = (F2/2) - (F1/2) = (F2-F1)/2 years more than the Mid-Year convention. Examples of the Generalized Model Tables I-A and I-B show two examples of the model using the Mid-Year cash flows, and Table II is an example using End-of-Year cash flows. The top two-thirds of Table I-A is a detailed listing of the cash flows and their net present values each year. The first cash flows begin at Year 2.25 and continue to 3.25, with Year 2.75 as the midpoint from which we discount. Assumptions of the model appear at the bottom of Table I-A. With a first year cash flow of $10,000 and an annual growth rate of 5.1%, the cash flow differential in the year from 3.25 to 4.25 is $10,510, and so on for subsequent years. Using a 10% discount rate, the net present value of the entire series of cash flows is $129, Using Formula [10d] gives us a multiple of , which when multiplied by the initial cash flow of $10,000 yields the identical result as the longer method of scheduling all the cash flows. The section entitled Components of the Formula shows the individual terms in [10d], i.e., terms "A", "B", and "C". Term A, which is a perpetuity beginning at S-1 = 2.25 Years and ending at S = 3.25 Years, has a net present value of $172, From this we subtract "B", which is the perpetuity beginning from F1 plus one day, or Years in this example. The NPV of B is $44, Finally, we add the NPV of the stub period cash flow, i.e., the value of 6

7 the cash flow at Year 32.75, which is $ "A" - "B" + "C" = $129,687.26, which equals both the sum of the yearly NPVs of the full schedule cash flow and the singular amount determined by the formula. This equality demonstrates the accuracy of the individual terms of [10d]. Note that the stub period cash flow is slightly greater than 50% of the cash flow at Years. The reason is that first we add 5.1% growth to the Year cash flow to obtain the cash flow if this were a complete year; then we multiply this result by (F2-F1), which is the fraction of a year represented by the stub period. Since this stub period is Year Year 32.25, the stub period is exactly one-half of a year. We discount the stub period cash flow by its own halfway point (using a Mid-Year convention), which is Years. Table I-B is identical to I-A, with the exception of the starting time of the cash flows and the growth rate. The purpose of this table is to demonstrate that Formula [10d] works regardless of the starting period and the growth rate. Here we start the analysis 3.25 years prior to the valuation date, which would make S = Years. We then proceed in one year increments to Year 26.75, and finish off with the stub period at Note that the present value factors are greater than 1 for the first three cash flows, which means that we are charging positive interest for the cash flow years before the valuation date. Additionally, we used a negative 5.1% growth rate instead of the positive 5.1% in Table I-A to show the model works regardless of the sign of the growth rate. Table II is identical to I-A, except that here we are dealing with End-of-Year cash flows. This table proves Formula [10m]. The ADF With Growth But No Stub Period Now let us consider an ADF with growth, but without a stub period. We will term the last cash flow as period n. The series of cash flows would appear as follows, assuming End-of-Year cash flows for simplicity: [11] ADF = 1 + (1+G) (1+G) n-s (1+r) S (1 + r) S+1 (1 + r) n You can see the accuracy of the final cash flow being equal to (1+G) n-s = (1+G) = (1+G) 29 in Figure 1, replacing either F1 or F2 by n. Multiplying [11] by (1+G)/(1+r), we get: [12] (1+G) ADF = (1+G) (1+G) n-s + (1+G) n-s+1 (1+r) (1 + r) S+1 (1 + r) n (1+r) n+1 Subtracting [12] from [11], moving (1+G)/(1+r) to the right-hand side of the equation, we get: [12a] ADF = (1+r) [ 1 - (1+G) n-s+1 ] (r-g) (1+r) S (1+r) n+1 Cancelling out the (1+r), and moving the (r-g) below--as we did going from [10] to [10a], we come to a solution for this class of ADF: 7

8 ADF: Growth, No Stub Period, End-of-Year Cash Flows 1 - (1+G) n-s+1 [13] ADF = (1+r) S-1 (1 + r) n (r-g) (r-g) Let's analyze [13] to gain an intuitive understanding of the meaning of the formula. The first term is a Gordon Model perpetuity with its first cash flow in period S, discounted to net present value. We will prove this in [14] to [14d] below. The second term is a Gordon Model perpetuity beginning at n+1, which we prove below in [15] through [15d]. Therefore, [13], which is the difference of the two perpetuities, is logically and intuitively the ADF for the series of cash flows from periods S to n. Proof of Components of [13]: The net present value of a perpetuity growing at G% a year, first received at period S would look like: [14] ADF = 1 + (1+G) (1+G) I (1+r) S (1 + r) S+1 (1 + r) I, where I = infinity Multiplying both sides of [14] by (1+G)/(1+r), we get: [14a] (1+G) ADF = (1+G) (1+G) I (1+r) (1 + r) S+1 (1 + r) I, where I = infinity Subtracting [14a] from [14] and moving (1+r)/(r-g) to the left-hand side of the equation, we get: [14b] ADF = (1+r) 1 (r-g) (1+r) S = 1 [14c] ADF (1+r) S-1 (r-g), which proves our interpretation of the first term of [13] A perpetuity growing at G% a year, first received at period n+1, would look like: [15] ADF = (1+G) n-s+1 + (1+G) n-s (1+G) I (1+r) n+1 (1 + r) n+2 (1 + r) I, where I = infinity Multiplying [15] by (1+G)/(1+r), we get [15a] (1+G) ADF = (1+G) n-s (1+G) I (1+r) (1 + r) n+2 (1 + r) I Subtracting [15a] from [15], we get [15b] (r-g) ADF = (1+G) n-s+1 (1+r) (1 + r) n+1 [15c] ADF = (1+r) (1+G) n-s+1 (r-g) (1+r) n+1 8

9 (1+G) n-s+1 [15d] ADF = (1+r) n As a special case of [13], let G=0. [13] becomes: (r-g), which proves our interpretation of the second term of [13]. Special Case: G=0 1 - (1+0) n-s+1 [16] ADF = (1+r) S-1 (1 + r) n (r-0) (r-0), which reduces to: 1-1 [16a] ADF = (1+r) S-1 (1 + r) n r r ADF: No Growth, No Stub Period, End-of-Year Cash Flows Another Special Case: G=0, S=1 Finally we come to the most special case of an ordinary annuity, where there is no growth and the first cash flow occurs at time t = 1. We substitute S=1 in [16a] to get: 1 [16b] ADF = 1 - (1 + r) n r r The Classical Formula For An Ordinary Annuity Note that [16b] is identical to [3c]. Therefore, the formula for an ordinary annuity is merely a special case of the more general formula in [13]. One might think, as I originally did, that using [10d] as a general model with the restrictive assumption that F1 = F2 = n would suffice to handle annuities without a stub period. This reasonable hunch turns out to be incorrect. With no stub period, F1=F2. Let's call that last period n, so that we substitute n = F1 = F2 in [10d], remove the (F2-F1) from the numerator, and adjust the discounting to end-of-year cash flows to arrive at: [16c] ADF = 1 - (1+G) n-s+1 + (1+G) n-s (1+r) S-1 (1 + r) n (1 + r) n (r-g) (r-g) The first two terms of [16c] match [13]. The third term, which is the NPV of the nth period cash flow, doubles up on that period's cash flow, since it is already included in the second term. The third term renders [13] incorrect for cash flows without a stub period. Tables III-A and III-B Tables III-A and III-B demonstrate the use of the series of formulas for the no-stub period cases using end-of-year cash flows. Table III-A is identical to Table I-A, except for being an End-of-Year cash flow. Note that the formula calculates to the same amount--$122, as the expanded schedule and the Components of the Formula. 9

10 Table III-B is identical in structure to III-A. Here we use formula [16b], which is an ordinary annuity--being a special case of [13] starting at time 0 and with no growth. The results prove [16b]. The expanded series of cash flows totals $94,269.14, which comprises the upper 2/3 of the table. The complete formula calculates to a factor of , which when multiplied by the initial cash flow of $10,000, amounts to the same $94, The first term of [16b] is 1/r, which is 1/.1 = Multiplying this by the initial cash flow gives us a value of the $10,000 perpetuity with no growth of $100,000. From this we subtract the second term in [16b], or [1/(1+r) n ]/r, which is When multiplied by the $10,000 initial cash flow, this amounts to a $5, subtraction from the $100,000, coming to the net present value of $94, The $5, is the net present value of the $10,000 perpetuity starting from Year ADF: No Stub Period, Mid-Year Cash Flows We would derive this formula in exactly the same fashion as [13]. We would start with [11]; however the denominators, which are the time periods by which we discount the cash flows, would be one half-year less than those in [11]. Following the same methodology without going through all the steps, the solution for Mid-Year Cash Flows is: ADF: Growth, No Stub Period, Mid-Year Cash Flows [17] ADF = 1 - (1+G) n-s+1 (1+r) S-1.5 (1 + r) n-.5 (r-g) (r-g) Table III-C demonstrates the accuracy of [17]. Special Cases For Mid-Year Cash Flows Special Case #1--No Growth: G=0 [17a] ADF = 1-1 (1+r) S-1.5 (1 + r) n-.5 r r Special Case #2--No Growth/First Cash Flow at Year 1: G=0, S=1 Substituting S=1 into [17a], we get: [17b] ADF = 1-1 (1+r) -.5 (1 + r) n-.5 r r Table III-D demonstrates the accuracy of [17b]. Loan Amortization The amortization of loan principal in any time period is the NPV of the loan at the beginning of the period, less the NPV at the end of the period. While this is conceptually easy, this is a cumbersome procedure, and there is a better 10

11 way to make the calculation. First, let's develop some preliminary results that will lead us to a more efficient way to calculate loan amortization. Table IV Table IV is a loan amortization schedule that I have divided into three sections. Section I is a traditional amortization schedule for a $1 million loan at 10% for 5 years. The leftmost column is the row # in the spreadsheet, and the top row is the column designation. The row and column designations are there to make it easier to understand the formulas in the spreadsheet. Column A is the row number, and column B is the payment number. There are 60 months of the loan, hence 60 payments. Columns D and E are the interest and principal for the particular payment, while columns H and I are interest and principal cumulated in calendar year totals. Because the loan begins in March, the first year's totals in columns H and I are totals for the first 10 payments only. Column J is the present value factor (PVF) at 10%, and column K is the present value of each loan payment. Column L is the sum of the present values of the loan payments by calendar year. Let's look at Section II of Table IV. Here we calculate the net present value of each year's loan payment using [16a]. In column D, we are viewing the various years' cash flows from the inception of the loan, March 1, Note that these amounts exactly match those in column L of Section I, and the total is exactly $1 million, the amount of the loan, as it should be. This demonstrates the accuracy of [16a]. In column F, we are viewing the cash flows from January 1, Therefore, the 1993 cash flows drop out entirely, and the NPV of the cash flows increase relative to column E, because we discount the cash flows 10 months less. The difference between the sum of the NPVs from March 1, 1993 to January 1, 1994 is $1 million - $865,911 = $134,089, as seen in column F, row 85. We follow the same procedure each year to calculate the difference in the NPVs (columns G - K, row 85), and finally we come to a total of the reductions in NPV of $1 million, in column L, row 85. There are some significant numbers that repeat in southeasterly-sloped diagonals in Section II. The NPV $241,675 appears in cells F78, G79, H80, and I81. This means that the 1994 payments as seen from the beginning of 1994 have the same NPV as the 1995 payments as seen from the beginning of 1995, etc. through Similarly, the NPV of $218,767 repeats in cells F80, G80, and H81. The interpretation of this series is the same as before, except everything is moved back one year, i.e.: The 1995 payments as seen from the beginning of 1994 have the same NPV as the 1996 payments as seen from the beginning of 1995 and the 1997 payments as seen from the beginning of A Better Way To Calculate Loan Amortization This pattern gives us a clue to a more direct formula for loan amortization. As seen from the beginning of the loan, we have 60 payments of $21,247. In the first calendar year, 10 payments will be made, for a total of $212,470. At the end of the first year, which effectively is the same as January 1, 1994, there will remain 50 payments. The NPV of the final 50 payments discounted to January 1, 1994 is the same as the NPV of the first 50 payments discounted to March 1, 1993, because the entire time line will have shifted by 10 months and 10 payments. Therefore, the amortization of the loan can be represented by the NPV of the final 10 payments discounted to March 1, 1993, as that would comprise the only difference in the two series of cash flows as perceived from their different points in time. In other words, the reduction in NPV of the principal is the NPV of the opposite or "mirror-image" series of cash flows at the end of the loan, using formula [16a], which is the equation of an ADF with end-of-period cash flows and no growth in the cash flows, which is the typical case with a loan. In Table IV, Section III, we calculate the principal reduction using [18], which is the same as [16a] with special modifications of the starting and finishing periods in columns D and E, rows The spreadsheet formulas begin in column G of the same rows. Column D shows the first payment number of the loan in the calendar year, and 11

12 column E shows the last payment. Calendar 1993 starts with Payment #1 and finishes with Payment #10 of the loan. Calendar 1994 starts with Payment #11 and finishes with Payment #22, etc. Let's look at the 1993 cash flows in Row 92. The amortization of principal in 1993 is equal to the NPV of the last 10 payments of the loan. Letting n = the final payment period = 60, we want to calculate the NPV of payments 51-60, discounted to month 0. If we subtract the finishing month, 10, from n, we get: = 50, which is equal to the S-1 exponent in the first term of equation [16a]. The finishing point is n-(starting Month)+1. For 1993, this is = 60, which is the second term in [16a]. In other words, the formula to calculate the amortization of principal is: Formula For Loan Amortization 1-1 [18] Loan Amort ADF = (1+r) n-f (1 + r) n-s+1 r r We multiply the Loan Amortization ADF in [18] by the loan payment. The formulas in H in Section III are identical to [18], but require a little interpretation of the term, I, to see that. The term, I, is the monthly interest rate = 10%/12 months=.833%, which is equivalent to r in [18]. Again, formula [18] is the same as [16a], except that the amortization periods are the mirror image of the cash flows themselves, i.e., the amortization of the beginning periods are the same as the decrease in NPV of the ending cash flows, etc. Let's cycle through one more year using [18], as the concepts here are tricky, and I find it difficult to explain them in a simple, clear way. The amortization in 1994 is $176,309, per F93. Using [18], we calculate this as: 1-1 [18a] Loan Amort ADF = ( ) ( ) , or 1-1 [18b] Loan Amort ADF = ( ) 38 ( ) In 1993, we amortized the loan by calculating the ADF for the last 10 payments, months In 1994, we amortize the prior 12 payments, months Note that the principal amortization in column G, rows are equal to those in column H of Section I, which demonstrates the accuracy of [18]. Exponential Approximation of Loan Amortization This section of the paper is more philosophically and intellectually interesting than immediately practical. This section is quite optional, is not integral to understanding this article, and can be skipped. It is possible to approximate a loan amortization schedule with 99.9% accuracy using an exponential growth model. From an intellectual standpoint, it underscores the exponential nature of compound interest, which we can easily lose in the complexity and diversity of formulas comprising the ADF family. 12

13 The methodology to derive the formulas is very similar to the derivation of [13]. Definitions: P = Principal of the Loan P t = Principal amortization in month t, t=1,2,3,...,60 Y = Years of the Loan = 5 n = Number of compounding periods per year, assumed at 12 for monthly loans Mo = Number of months in the loan = ny = 12 * 5 = 60 r = Interest Rate = 10% A = Principal reduction in the 0th period of the exponential model We begin with an exponential growth model formula: [19] P t = Ae rt The total of all principal payments, P, would be: [20] P = Ae r/n + Ae 2r/n Ae Yr [21] P = A [e r/n + e 2r/n e Yr ] Multiplying "B" by e r/n, we get: Call this term "B" [22] e r/n B = e 2r/n e Yr + e [(ny+1)/n]r Subtracting [22] from the "B" portion of [21], we get: [23] B[1-e r/n ] = e r/n - e [(ny+1)/n]r, or: [24] B = e r/n - e [(ny+1)/y]r [1-e r/n ] Substituting into [24] into [21], we get: [25] P = A e r/n - e [(ny+1)/n]r [1-e r/n ] We can solve for A by inverting the equation as: [26] A = P [1-e r/n ] e r/n - e [(ny+1)/n]r Substituting [26] into [25] and [19], we get: Exponential Model For Principal Amortization in Month t [27] P t = P [1-e r/n ] e rt e r/n - e [(ny+1)/n]r 13

14 Table V Table V shows the exponential model in use for our familiar 5 year, 10% loan for $1 million. The first 7 columns are a traditional amortization schedule. P(0) = A, the starting principal according to [26], is $12,792, and appears at the bottom of the first page of Table V in the Assumptions section. Column (8) is the principal amortization calculated using [27]. Column (9) is a calculation of the formula error, which is the actual principal less the formula principal, or (6)-(8). Note that the formula error totals to zero, as it should. The error begins as a negative number and decreases until Month 33, when it becomes positive. This happens because the exponential approximation is too low in the beginning and is too high at the end. On average, it is exactly correct, because we have forced it to be so in [26]. Column (10) is a calculation of the percentage error, which is column (9)/(6). Column (11) is the absolute value of the error, or ABS(10). Taking the arithmetic mean of column (11) gives us the Mean Absolute Deviation of the formula error =.052%. In other words, the exponential formula is 99.5% correct as an approximation to the actual principal amortization. On the second page of Table V, we show the results for each year's amortization (in this model, we assumed the loan began January 1). The maximum error in any year is -$180 in the 5th year, which is 0.75% of the correct principal for the year. The formula for the yearly blocks is: [27] P s to f = P [1-e r/n ] e sr/n - e (f+1/n)r e r/n - e [(ny+1)/n]r 1-e r/n, where s and f are the starting and finishing months, respectively. The derivation of [27] is very similar to [13], but using the exponential model. We can approximate the NPV of the nominal loan payments by deriving the formulas as in [26] with the addition of dividing each term in the series by (1+r) t. The NPV equivalent of [26] becomes: [28] NPV = e r/n - e [(ny+1)/n]r A (1+r) [1-e r/n ] (1+r) [28] is reminiscent of [13]. (1+r) (ny+1)/n At this point, I cannot imagine a practical use for an exponential approximation when an exact formula for loan amortization is available. Nevertheless I include this section in this article because one never knows when the approximation formula may have use in the future, and as mentioned earlier, it does show how interest is an exponential phenomenon. Conclusion We can see that there is a family of Annuity Discount Factors (ADFs), from the simplest case of an ordinary annuity to the most complicated case of stub periods. The two most critical individual elements that determine which formula to use are whether the cash flow is Mid-Year vs. End-of-Year and whether there is a stub Period or not. The above two factors determine the appropriate model and formula to use, which is as follows: 14

15 Formula Cash Flow Type Stub Period No Stub Period Mid Year [10d] [17] End-of-Year [10m] [13] Loan Amortization NA [18] While this article consists of much tedious algebra, the focus has been on the intuitive explanation of each ADF. For cash flows without a stub period, the ADF is the difference of two Gordon Model perpetuities: the first term being the perpetuity starting from the beginning point of the cash flow, and the second one starting in the first period after the ending period of the ADF. For cash flows with a stub period, the preceding statement is true with the addition of a third term for the single cash flow of the stub period itself, discounted to NPV. Now you no longer need burp and back off when you encounter the need to use an Annuity Discount Factor. With this article tucked under your arm, you can now burp twice--and confidently go forward! 3 3 The author wishes to express that he has no current, nor contemplated interest in Alka Seltzer, Pepto Bismol, or any other product dealing with stomach upset--nor does any of his family or friends, to his knowledge. I'm just having some fun! 15

16 Table I-A ADF With Stub Period Mid-Year Formula Income/ PV Year Mid-Year Cash Flow Factor NPV , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Totals $129, Components of the Formula -(1+G)^(F1-S+1)/ Complete 1/(1+R)^(S-1.5) (1+R)^(F1-.5) (1+G)^(F2-S)(F2-F1)/ Formula /(R-G) /(R-G) (1+R)^[F1+(F2-F1)/2] Total Formula Multiple X Initial Cash Flow 10, , , , , NPV By Formula $129, $172, ($44,021.78) $ $129, Assumptions: S = Beginning Year-Cash Flows 3.25 F1= Ending Year Cash Flows-Regular Sequence F2= Ending Year Cash Flows-Stub Sequence R = Discount Rate 10.0% G = Growth Rate in Net Inc/Cash Flow 5.1% Beginning Year's Income $10,000

17 Table I-B ADF With Stub Period Mid-Year Formula Income/ PV Year Mid-Year Cash Flow Factor NPV (2.75) 10, , (1.75) 9, , (0.75) 9, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Totals $93, Components of the Formula -(1+G)^(F1-S+1)/ Complete 1/(1+R)^(S-1.5) (1+R)^(F1-.5) (1+G)^(F2-S)(F2-F1)/ Formula /(R-G) /(R-G) (1+R)^[F1+(F2-F1)/2] Total Formula Multiple X Initial Cash Flow 10, , , , , NPV By Formula $93, $94, ($1,128.37) $81.42 $93, Assumptions: S = Beginning Year-Cash Flows F1= Ending Year Cash Flows-Regular Sequence F2= Ending Year Cash Flows-Stub Sequence R = Discount Rate 10.0% G = Growth Rate in Net Inc/Cash Flow -5.1% Beginning Year's Income $10,000

18 Table II ADF With Stub Period End-of-Year Formula Income/ PV Year Cash Flow Factor NPV , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Totals $123, Components of the Formula -(1+G)^(F1-S+1) Complete 1/(1+R)^(S-1) (1+R)^F1 (1+G)^(F2-S)* Formula /(R-G) /(R-G) (F2-F1)/(1+R)^F2 Totals Formula Multiple X Initial Cash Flow 10, , , , , NPV By Formula $123, $164, ($41,973.12) $ $123, ASSUMPTIONS: S = Beginning Year-Cash Flows 3.25 F1= Ending Year Cash Flows-Regular Sequence F2= Ending Year Cash Flows-Stub Sequence R = Discount Rate 10.0% G = Growth Rate in Net Inc/Cash Flow 5.1% Beginning Year's Income $10,000

19 Table III-A ADF: No Stub Period End of Year Formula Income/ PV Year Cash Flow Factor NPV , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Totals $122, Components of the Formula Complete [1/(1+R)^(S-1)] -(1+G)^(N-S+1) Formula /(R-G) /(1+R)^N/(R-G) Total Formula Multiple X Initial Cash Flow 10, , , , NPV By Formula $122, $164, ($41,973.12) $122, ASSUMPTIONS: S = 1st Cash Flows 3.25 N = Ending Year Cash Flows R = Discount Rate 10.0% G = Growth Rate in Net Inc/Cash Flow 5.1% Beginning Year's Income $10,000

20 Table III-B ADF: No Stub Period, S=1, G=0 End of Year Formula Income/ PV Year Cash Flow Factor NPV , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Totals $94, Components of the Formula Complete [1/(1+R)^(S-1)] -(1+G)^(N-S+1) Formula /(R-G) /(1+R)^N/(R-G) Total Formula Multiple X Initial Cash Flow 10, , , , NPV By Formula $94, $100, ($5,730.86) $94, ASSUMPTIONS: S = 1st Cash Flows 1.00 N = Ending Year Cash Flows R = Discount Rate 10.0% G = Growth Rate in Net Inc/Cash Flow 0.0% Beginning Year's Income $10,000

21 Table III-C ADF: No Stub Period Mid-Year Formula Income/ PV Year Mid-Yr Cash Flow Factor NPV , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Totals $128, Components of the Formula Complete [1/(1+R)^(S-1.5)] -(1+G)^(N-S+1)/ Formula [17] /(R-G) (1+R)^(N-.5)/(R-G) Total Formula Multiple X Initial Cash Flow 10, , , , NPV By Formula $128, $172, ($44,021.78) $128, ASSUMPTIONS: S = 1st Cash Flows 3.25 N = Ending Year Cash Flows R = Discount Rate 10.0% G = Growth Rate in Net Inc/Cash Flow 5.1% Beginning Year's Income $10,000

22 Table III-D ADF: No Stub Period, S=1, G=0 Mid-Year Formula Income/ PV Year Mid-Yr Cash Flow Factor NPV , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Totals $98, Components of the Formula Complete [1/(1+R)^(S-1.5)] -(1+G)^(N-S+1)/ Formula [17] /(R-G) (1+R)^(N-.5)/(R-G) Total Formula Multiple X Initial Cash Flow 10, , , , NPV By Formula $98, $104, ($6,010.57) $98, ASSUMPTIONS: S = 1st Cash Flows 1.00 N = Ending Year Cash Flows R = Discount Rate 10.0% G = Growth Rate in Net Inc/Cash Flow 0.0% Beginning Year's Income $10,000

23 A B C D E F G H I J K L Table IV Amortization of Principal With Irregular Starting Point SECTION I: LOAN AMORTIZATION SCHEDULE Pmt NPV Annual Row # Date Pmt Int Prin Bal Int Prin PVF Pymt NPV /01/ /31/ /30/ /31/ /30/ /31/ /31/ /30/ /31/ /30/ /31/ /31/ /28/ /31/ /30/ /31/ /30/ /31/ /31/ /30/ /31/ /30/ /31/ /31/ /28/ /31/ /30/ /31/ /30/ /31/ /31/ /30/ /31/ /30/ /31/ /31/ /29/ /31/ /30/ /31/ /30/ /31/ /31/ /30/ /31/ /30/ /31/ /31/ /28/ /31/ /30/ /31/ /30/ /31/ /31/ /30/ /31/ /30/ /31/ /31/ /28/1998 Totals

24 A B C D E F G H I J K L Table IV (Cont'd) SECTION II: SCHEDULE OF NET PRESENT VALUES CALCULATED BY ADF FORMULA [16a] As Seen From The Beginning of Year Total NPV 1993 Payments NPV 1994 Payments NPV 1995 Payments NPV 1996 Payments NPV 1997 Payments NPV 1998 Payments NPV 1999 Payments Sum NPVs-All Pymts Reduction in NPV SECTION III: AMORTIZATION CALCULATED BY [18] Formulas For Principal Amortization, where: Start End Prin I=Monthly Interest=.833%, n=60 Months, Month Month Amort Pymt=$21,247/Month Calendar ($PYMT/$I)*((1/(1+$I)^($N-E92)-(1/(1+$I)^($N-D92+1)))) Calendar ($PYMT/$I)*((1/(1+$I)^($N-E93)-(1/(1+$I)^($N-D93+1)))) Calendar ($PYMT/$I)*((1/(1+$I)^($N-E94)-(1/(1+$I)^($N-D94+1)))) Calendar ($PYMT/$I)*((1/(1+$I)^($N-E95)-(1/(1+$I)^($N-D95+1)))) Calendar ($PYMT/$I)*((1/(1+$I)^($N-E96)-(1/(1+$I)^($N-D96+1)))) Calendar ($PYMT/$I)*((1/(1+$I)^($N-E97)-(1/(1+$I)^($N-D97+1)))) Total Assumptions: Prin $1,000,000 Int 10.0% Int-Mo 0.83% Years 5 Months = n 60 Pymt $21,247 Form-Prin $1,000,000 Start Month=S 3

25 Table V Monthly Loan Amortization Schedule--Exponential Model (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) =1/120 =4-5 = =9/6 =ABS(10) Total Formula Total Total Formula Mo Yrs=t Bal-Beg Pymt Int Prin Bal-End Prin Error % Err Abs(Err) Interest Principal Principal ,000,000 21,247 8,333 12, ,086 12, % 0.110% ,086 21,247 8,226 13, ,065 13, % 0.107% ,065 21,247 8,117 13, ,935 13, % 0.103% ,935 21,247 8,008 13, ,696 13, % 0.100% ,696 21,247 7,897 13, ,346 13, % 0.097% ,346 21,247 7,786 13, ,885 13, % 0.093% ,885 21,247 7,674 13, ,312 13, % 0.090% ,312 21,247 7,561 13, ,626 13, % 0.086% ,626 21,247 7,447 13, ,826 13, % 0.083% ,826 21,247 7,332 13, ,911 13, % 0.079% ,911 21,247 7,216 14, ,880 14, % 0.076% ,880 21,247 7,099 14, ,732 14, % 0.072% 92, , , ,732 21,247 6,981 14, ,466 14, % 0.069% ,466 21,247 6,862 14, ,081 14, % 0.066% ,081 21,247 6,742 14, ,576 14, % 0.062% ,576 21,247 6,621 14, ,951 14, % 0.059% ,951 21,247 6,500 14, ,203 14, % 0.055% ,203 21,247 6,377 14, ,333 14, % 0.052% ,333 21,247 6,253 14, ,339 14, % 0.048% ,339 21,247 6,128 15, ,220 15, % 0.045% ,220 21,247 6,002 15, ,974 15, % 0.041% ,974 21,247 5,875 15, ,602 15, % 0.038% ,602 21,247 5,747 15, ,102 15, % 0.034% ,102 21,247 5,618 15, ,472 15, % 0.031% 75, , , ,472 21,247 5,487 15, ,712 15, % 0.028% ,712 21,247 5,356 15, ,821 15, % 0.024% ,821 21,247 5,224 16, ,798 16, % 0.021% ,798 21,247 5,090 16, ,641 16, % 0.017% ,641 21,247 4,955 16, ,349 16, % 0.014% ,349 21,247 4,820 16, ,922 16, % 0.010% ,922 21,247 4,683 16, ,357 16, % 0.007% ,357 21,247 4,545 16, ,655 16, % 0.003% ,655 21,247 4,405 16, ,813 16, % 0.000% ,813 21,247 4,265 16, ,831 16, % 0.004% ,831 21,247 4,124 17, ,708 17, % 0.007% ,708 21,247 3,981 17, ,442 17, % 0.010% 56, , , ,442 21,247 3,837 17, ,032 17, % 0.014% ,032 21,247 3,692 17, ,476 17, % 0.017% ,476 21,247 3,546 17, ,775 17, % 0.021% ,775 21,247 3,398 17, ,926 17, % 0.024% ,926 21,247 3,249 17, ,928 18, % 0.028% ,928 21,247 3,099 18, ,781 18, % 0.031% ,781 21,247 2,948 18, ,482 18, % 0.035% ,482 21,247 2,796 18, ,031 18, % 0.038% ,031 21,247 2,642 18, ,425 18, % 0.042% ,425 21,247 2,487 18, ,665 18, % 0.045% ,665 21,247 2,331 18, ,749 18, % 0.048% ,749 21,247 2,173 19, ,675 19, % 0.052% 36, , , ,675 21,247 2,014 19, ,442 19, % 0.055% ,442 21,247 1,854 19, ,048 19, % 0.059% ,048 21,247 1,692 19, ,493 19, % 0.062% ,493 21,247 1,529 19, ,775 19, % 0.066% ,775 21,247 1,365 19, ,893 19, % 0.069% ,893 21,247 1,199 20, ,845 20, % 0.073% ,845 21,247 1,032 20, ,630 20, % 0.076% ,630 21, ,383 83,247 20, % 0.080% ,247 21, ,553 62,693 20, % 0.083% ,693 21, ,725 41,969 20, % 0.086% ,969 21, ,897 21,071 20, % 0.090% ,071 21, ,071 (0) 21, % 0.093% 13, , , Totals 1,000,000 1,000, % 1,000,000 1,000,000 ASSUMPTIONS: Mean Absolute Principal $1,000,000 Years 5 Deviation Interest 10.0% n=compound Per/Yr 12 P(0) By Formula $12,792 Months 60