# STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION

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1 Page 1 STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION C. Alan Blaylock, Hendeson State Univesity ABSTRACT This pape pesents an intuitive appoach to deiving annuity fomulas fo classoom use and attempts to assess student esponse to such pesentation in a pinciples of finance couse by obseving student behavio and suveying student sentiment. Afte pesenting thee deivations of an annuity fomula, a quiz was administeed to detemine the coect use of a paticula deivation of the student s choosing. The deivation used and its coect use by students is epoted. In addition, a suvey is conducted to assess student sentiment. Implications fo instuctos ae also discussed. INTRODUCTION AND LITERATURE REVIEW This pape attempts to assess student eaction to teaching the deivation of annuity fomulas in a pinciples of finance couse. The eseach methods include obseving student behavio and suveying student sentiment. The esults pesented in this pape allow the instucto some insight into student peceptions of how annuity fomulas should be taught. Many students find the use of mathematics in any discipline somewhat difficult. This is indeed tue fo finance couses as well, especially fo the finance pinciples couse that is usually equied of all business majos. Some instuctos allow the use of fomula sheets duing testing. Fo those students that ae not given such luxuy, memoizing and compehending mathematical fomulas can be a daunting task. Many students distinguish between mateial that is moe concept and that which equies the use of fomulas. They fail to ealize that the two ae insepaable; fomulas ae simply concepts witten in mathematical tems. If students wee allowed to see the easoning behind the fomulas, they may bette be able to undestand and use those fomulas. Time Value of Money (TVM), the valuing of cash flows though time, is a majo topic in the teaching of finance. Of the vaious fomulas available to adjust dolla values, those involving the pesent and futue value of annuities may be egaded as pehaps the most difficult to compehend in the pinciples couse. This may be due to the fact that they ae lengthy and thei logical deivation is not so easily gasped in the way they ae usually pesented. Most copoate finance texts wite the Pesent Value of an Annuity (PVA) fomula using a constant ate fo each payment peiod in the fom Jounal of Economics and Economic Education Reseach, Volume 15, Numbe 3, 2014

2 Page 2 1 PVA PMT 1 1 (1) whee PVA is the value of the annuity one peiod befoe the fist payment, PMT is the annuity payment at the end of each peiod, is the ate of change in dolla values pe peiod, and n is the numbe of peiods. Some othe finance texts use othe foms of the fomula, some of which will be developed shotly. A common fom of the Futue Value of an Annuity fomula is FVA PMT 1 1 (2) To a student with a limited math backgound, these fomulas can look ovewhelming in thei concept and size. In an age without computes Sheitt (1944) suggests limiting the numbe of fomulas, esticting the numbe of symbols acoss all fomulas, and not equiing that fomulas be developed. To futhe assist student leaning he gaphically connects each fomula to a time line diagam. In ode to avoid efeing to a geometic pogession to explain annuity fomulas Watson (1936) instead points out that the FVA fomula is the compound inteest of \$1 divided by the inteest ate pe peiod and that the PVA fomula is the compound discount on \$1 divided by the inteest ate pe peiod. In ode to help students match a set of cash flows with an appopiate valuation fomula Skinne (1994) begins with a geometic pogession and deives a geneal souce fomula that can, by imposing diffeent assumptions elated to a set of cash flows, be used to calculate the pesent value of an odinay annuity a pepetuity a gowing pepetuity a gowing annuity a gowing annuity with constant ates of gowth. Using this fomula with vaying sets of assumptions, establishes the citical link between the stuctue of the cash flow to be valued and the appopiate model to be used (Skinne, 1994, p. 87). Most intoductoy finance texts simply omit any explanation of the annuity fomulas. A few, howeve, do pesent to vaying degees the concept that an annuity is a piece of a pepetuity, and then use that concept to deive the annuity fomulas. Notable texts that offe such pesentations eithe in the main text, footnotes, o as web extensions o chaptes include Bealey, Myes, and Allen (2014), Bigham and Ehhadt (2014), Ross, Westefield, and Jaffe (2013), and Welch (2009). Although seemingly ovewhelming to a student in intoductoy finance the fomulas actually epesent this quite simple concept of valuing an annuity as a pat of a pepetuity. Jounal of Economics and Economic Education Reseach, Volume 15, Numbe 3, 2014

3 Page 3 The connection an annuity has with a pepetuity is not easily seen in the often used foms of annuity fomulas. Howeve, assuming that students easily accept the pepetuity fomula they may be bette pepaed to accept annuity fomulas if they see that such fomulas can deived fom the simple pepetuity fomula. Newcombe et al. (2009) cite seveal studies in which elementay school childen wee bette able to detemine solutions to faction poblems when they wee able to connect such poblems to elated infomal expeiences. Newcombe et al. also cite eseach that shows that confidence in one s math ability inceases pefomance in science, technology, engineeing, and mathematics (STEM) leaning activities. Beginning with the simple pepetuity fomula may povide students with familia mateial that can be used to build confidence to use the moe difficult annuity fomulas. The majo question this pape asks is, How do students espond to a pesentation of the deivation of an annuity fomula fom the pepetuity fomula? This pape uses the method of obseving student behavio and suveying student sentiment. Fist, students ae pesented with the deivation of the annuity fomulas that is simila to Bealey, Myes, and Allen (2014), Bigham and Ehhadt (2014), Ross, Westefield, and Jaffe (2013), and Welch (2009). Thee common foms of each of the fomulas ae given, with each fom epesenting egessive closeness to the simple concept fom which it was deived. As will be shown shotly, the fist fom is witten in a way that bette eflects the concept of its deivation (the most intuitive fom). The subsequent two foms, each a deivation of the pevious fom, pogessively distance themselves fom the initial simple deivation (less intuitive foms). Afte pesenting the thee foms, students ae given a TVM quiz which equies them to wite the fomulas they used. Howeve, although some students may use one fom, they still may have benefited fom being exposed to othe foms. This cannot be detected by obsevation. Theefoe, students ae asked if they did eceive some benefit fom being taught how the annuity fomulas wee deived. Both negative and positive eactions ae anticipated. Although some students may benefit, othes may deem such deivations too complex, deep, and confusing. By evealing the benefits and costs involved though student usage and peceptions, the esults pesented in this pape allow the TVM instucto some insight into how the annuity fomulas should be taught. Such analysis of students esponse to fomula deivation is lacking in the liteatue. DERIVING THE ANNUITY FORMULAS Befoe the esults ae epoted, the deivation of the PVA fomula into thee foms is given as a sample of what students wee pesented. One of the simplest fomulas in finance is that of the Pesent Value of a Pepetuity (PVP). The PVP fomula is PVP PMT (3) whee PMT is the pepetual peiodic cash flow, and is the discount pecentage ate. The pesentation pesumes students undestand this simple fomula. The simple concept in deiving the pesent value of an annuity esults fom the fact that the annuity is simply a pat of a pepetuity; the only diffeence between a pepetuity and an annuity is that a pepetuity has an infinite steam of payments wheeas an annuity has a finite steam of payments. The object of deiving the annuity fomula fom the pepetuity fomula ests in Jounal of Economic and Economic Education Reseach, Volume 15, Numbe 3, 2014

4 Page 4 subtacting the pat of the pepetuity that is not needed in ode to leave the pat of the pepetuity that is needed, the annuity. This is illustated in Figue 1. Pepetuity B is subtacted fom Pepetuity A to esult in Annuity Z. Since the pat of Pepetuity A that is not needed extends into infinity the subtacted pat is, itself, is an annuity (Pepetuity B). The emaining payments of Pepetuity A (that no longe extend into infinity) would fom the desied annuity, Annuity Z. Of couse, in ode to subtact the value of the futue pepetuity fom the value of the pesent pepetuity, the two values must be at the same time peiod. The two values must be adjusted to the desied time peiod, and the esulting diffeence would be the value of the annuity at that time peiod. This pocess is illustated in moe detail in Figue 2 by deiving the pesent value of a foupeiod annuity (value at t = 0; n = 4). The value of a pepetuity at peiod 0 is calculated. The value of the fifth and emaining payments needs to be subtacted in ode to etain and eveal the value of the fou-peiod annuity. This is done by fist calculating the value at peiod 4 (t + n) of those unwanted payments. Since the unwanted payments also fom a pepetuity, this esults in the value of that pepetuity at peiod 4. Since the value of the annuity is to be calculated at peiod 0, the value of the unwanted payments, PVP4, is adjusted to peiod 0 using the simple pesent value fomula (which would have been peviously pesented in class). The value at peiod 0 of the desied annuity can then be found by subtacting PVP4 fom PVP0. Thus the Pesent Value of an Annuity (PVA) can be calculated as PVA PMT PMT O moe geneally 1 1 (4) PVA PMT PMT 1 1 (5a) whee n is the numbe of payment peiods poceeding (to the futue of) time peiod t. This is the fist fom of the PVP fomula and will be designated as Fom A. The concept of how the Jounal of Economics and Economic Education Reseach, Volume 15, Numbe 3, 2014

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7 Page 7 annuity fomula is deived is best seen in this fom. Reaanging Fom A esults in foms B and C, espectively, FORM B: PVA PMT (5b) FORM C: PVA 1 PMT 1 1 (5c) Notice that Equation 5c (Fom C) is the same as Equation 1 pesented ealie. A simila pesentation is given to deive thee foms of the Futue Value of an Annuity (FVA) which ae not pesented hee. METHODOLOGY Students wee fist pesented the deivations of the futue and pesent value annuity fomulas. Students wee not equied to deive the annuity fomulas at any time duing the semeste. Within a week of when the annuity fomula deivations wee taught, students wee given a pop quiz to test thei ability to calculate: (1) pesent value of an annuity (PVA), (2) futue value of an annuity (FVA), (3) the payment of an annuity given its pesent value (PMT), (4) pesent value of a pepetuity (PVP). Having the students calculate the payment fo an annuity is included to assess student eaction of having to manipulate any given fom to solve fo a missing vaiable. Although students could use a financial calculato, they wee equied to wite the fomula used in solving the poblems. Hypothesis tests ae used to detemine if any significant diffeence exists between the following: 1. The popotion of students who ecalled a given fomula and those who did not. 2. Of those who could ecall a given fomula, the popotion of students who ecalled a paticula fom of that fomula and those who ecalled anothe fom. 3. Of those who could ecall a given fomula, the popotion of students who calculated the coect answe and those who could not. 4. Of those who could ecall a given fom of a fomula, the popotion of students who calculated the coect answe and ecalled a paticula fom of a fomula and those who calculated the coect answe and ecalled anothe fom. Although students may use one of the thee foms, they still may have benefited by being exposed to the two othe foms. Fo instance, a student may use fom C, but the student may be bette able to use that fom in calculating the coect answe because they wee exposed to foms A and B and to thei deivations. Theefoe, a suvey of student sentiment was conducted. Within the last two weeks of the semeste, students wee asked to (1) wite the PVA and FVA fomulas Jounal of Economic and Economic Education Reseach, Volume 15, Numbe 3, 2014

8 Page 8 in the fom of thei choosing togethe with the PVP fomula, and (2) state in witing if exposue to the deivation of the annuity fomulas benefited them in thei classoom pefomance. Chi-squae and elated hypothesis tests ae used to detemine significance in egad to ecalling a paticula fomula, a paticula fom of a fomula, and if the student esponded with a positive, negative, o neutal sentiment to the annuity fomula deivations. RESULTS Table 1 pesents the esults of the pop quiz. A total of 86 students took the quiz. Of those students, a total of 77 (90%) wee able to coectly wite the PVP fomula, moe than the numbe who could coectly wite the PVA o FVA fomulas. This may be a somewhat obvious indication that the PVP fomula is much easie to compehend than the PVA o FVA fomulas. Pat B of Table 1 shows that of those who emembeed the fomulas, 90% actually coectly calculated the answe to the FVA poblem compaed to only 85% fo the PVA poblem. Only 71% who emembeed the PVA fomula could use it to solve fo the annuity payment. Also, 95% of those who emembeed the PVP fomula could use it to coectly answe the PVP poblem. The heat of this study is to assess student peceptions of the deivations of the annuity fomulas. Pat A of Table 1 epots the numbe (and pecentage) of students who coectly emembeed each of the thee foms. Pat B of Table 1, eflecting the infomation pesented in Pat A, epots the numbe (and pecentage) of students answeing the poblems coectly using each of the thee foms. The majoity of students favoed Fom C, followed by Fom A and then Fom B in all thee types of annuity poblems. The popotion of students who used Fom B in computing the PVA (15%) is significantly less than those who used eithe Fom A (38%) o Fom C (48%). No significant diffeence is found between the popotion of students who used Fom A and those who used Fom C. Howeve, when used to compute the FVA o the payment fo the PVA, Fom C is used significantly moe than eithe of the othe two foms. Table 1C shows that while students who used Fom A wee the least able to calculate the coect answes fo the PVA poblem, no significant diffeence exists in the coect use of each fom. Although a significant popotion of students used Fom C in the FVA poblem, the popotion of students coectly using eithe Fom A o Fom B was significantly geate than those using Fom C. In othe wods, moe students used Fom C, but a lage popotion of students used Foms A and B coectly. The fom that povided the best pefomance in calculating the payment fo the PVA poblem was Fom C, followed closely by Fom B. This seems easonable, since using Fom C o Fom B involves less manipulation of the PVA fomula to isolate the annuity payment (PMT). Jounal of Economics and Economic Education Reseach, Volume 15, Numbe 3, 2014

9 Page 9 Table 1 Quiz Results A1. PVA FVA PMT fo PVA PVP Total Attempts Total coect fomulas / total attempts (%) a : 61/86 (71%)*** 62/86 (72%)*** 59/86 (69%)*** 77/86 (90%)*** A2. PVA FVA PMT fo PVA PVP Total coect foms / total coect fomulas (%): Fom A 23/61 (38%)*** 16/62 (26%)*** 17/59 (29%)*** - Fom B 9/61 (15%)*** 10/62 (16%)*** 10/59 (17%)*** - Fom C 29/61 (48%)*** 36/62 (58%)*** 32/59 (54%)*** - Pai-wise Compaisons b : Fom A vs. Fom B 38% >15%*** 26% > 16%*** 29% > 17%*** - Fom A vs. Fom C 38% < 48%*** 26% < 58%*** 29% < 54%*** - Fom B vs. Fom C 15% < 48%*** 16% < 58%*** 17% < 54%*** - B1. PVA FVA PMT fo PVA PVP Total coect poblems / total fomulas (%) a : 52/61 (85%)*** 56/62 (90%)*** 42/59 (71%)*** 73/77 (95%)*** B2. PVA FVA PMT fo PVA PVP Total coect poblems / total coect fom (%): Fom A 18/23 (78%)*** 16/16 (100%)** 7/17 (41%)*** - Fom B 8/9 (89%)*** 10/10 (100%)** 8/10 (80%)*** - Fom C 26/29 (90%)*** 30/36 (83%)** 27/32 (84%)*** - Pai-wise Compaisons b : Fom A vs. Fom B 78% < 89%*** 100% 100%*** 41% < 80%*** - Fom A vs. Fom C 78% < 90%*** 100% >83%*** 41% < 84%*** - Fom B vs. Fom C 89% < 90%*** 100% > 83%*** 80% < 84%*** - ***, **, * indicate significance at the 1, 5, and 10 pecent, espectively a Chi-Squae test is used b Maascuillo pocedue is used Table 2 epots students ecall of the fomulas and thei peceptions of how the fomulas wee pesented in the classoom. Students wee asked if they benefited fom the annuity fomula deivation lectue and wee also asked the wite down the fomulas fo PVA, FVA, and PVP. Seventy-seven students wee suveyed in total. A lage numbe of students (24 out of 77, o 31%) did not state thei peceptions while othes gave eithe an unclea o neutal esponse. The vast majoity had a favoable impession of the classoom pesentation; 47 students (61%) stated that they wee positively affected fom having the deivation of the fomulas pesented to them in class. Only 6 students (8%) stated that they wee negatively affected. Jounal of Economic and Economic Education Reseach, Volume 15, Numbe 3, 2014

10 Page 10 Table 2 Suvey Results A. Positive Negative N/A* Total Total Response 47/77 (61%) 6/77 (8%) 24/77 (31%) 77 Pai-wise Compaisons b : Positive vs. Negative 61% > 8%*** Positive vs. N/A 61% > 31%*** Negative vs. N/A 8% < 31%*** B1. Positive Negative N/A* Total PVA: Total Fomulas Recalled 30/47 (64%) 1/6 (17%) 15/24 (63%) 46/77(60%) Pai-wise Compaisons b : Positive vs. Negative 64% > 17%*** Positive vs. N/A 64% > 63%*** Negative vs. N/A 17% < 63%*** B2. Positive Negative N/A* Total PVA: (Total coect fomulas / total esponse) Fom A 8/30 (27%) 0/1 (0%) 2/15 (13%) 10/46 (22%) Fom B 6/30 (20%) 0/1 (0%) 0/15 (0%) 6/46 (13%) Fom C 16/30 (53%) 1/1 (100%) 13/15 (87%) 30/46 (65%) Pai-wise Compaisons b : Fom A vs. Fom B 27% > 20%*** 0% 0%*** 13% > 0%*** 22% > 13%*** Fom A vs. Fom C 27% < 53%*** 0% < 100%*** 13% < 87%*** 22% < 65%*** Fom B vs. Fom C 20% < 53%*** 0% < 100%*** 0% < 87%*** 13% < 65%*** C1. Positive Negative N/A* Total FVA: Total Fomulas Recalled 26/47 (55%) 1/6 (17%) 11/24 (46%) 38/77 (49%) Pai-wise Compaisons b : Positive vs. Negative 55% > 17%*** Positive vs. N/A 55% > 46%*** Negative vs. N/A 17% < 46%*** C2. Positive Negative N/A* Total FVA: (Total coect fomulas / total esponse) Fom A 8/26 (31%) 0/1 (0%) 1/11 (9%) 9/39 (23%) Fom B 6/26 (23%) 0/1 (0%) 1/11 (9%) 7/39 (18%) Fom C 12/26 (46%) 1/1 (100%) 9/11 (82%) 23/39 (59%) Pai-wise Compaisons b : Fom A vs. Fom B 31% > 23%*** 0% 0%*** 9% 9%*** 23% > 18%*** Fom A vs. Fom C 31% < 46%*** 0% < 100%*** 9% < 82%*** 23% < 59%*** Fom B vs. Fom C 23% < 46%*** 0% < 100%*** 9% < 82%*** 18% < 59%*** D. Positive Negative N/A* Total PVP: Total Fomulas Recalled 39/47 (83%) 1/6 (17%) 4/24 (17%) 44/77 (57%) Pai-wise Compaisons b : Positive vs. Negative 83% > 17%*** Positive vs. N/A 83% > 17%*** Negative vs. N/A 17% 17%*** * Includes no esponse to sentiment o those esponses that wee eithe unclea o neutal in egads to sentiment ***, **, * indicate significance at the 1, 5, and 10 pecent, espectively a Chi-Squae test is used b Maascuillo pocedue is used Table 2 also shows that those students who wee positively affected by the deivation pesentation wee bette able to ecall fomulas than those students who wee negatively affected. Jounal of Economics and Economic Education Reseach, Volume 15, Numbe 3, 2014

11 Page 11 Sixty-fou pecent of the positively affected students wee able to ecall the PVA fomula (in vaious foms) compaed to only 17% of those who wee negatively affected. The patten holds fo the FVA (PVP) fomula with 55% (83%) of the positively affected students able to ecall the FVA (PVP) fomula compaed to only 17% (17%) fo those negatively affected. Those who liked the pesentation and wee able to ecall the PVA fomula pefeed Fom C (Table 2.B2), but no significant diffeence exists between the foms fo the FVA fomula (Table 2.C2). CONCLUSION AND IMPLICATIONS This pape attempts to measue student esponse to teaching the deivation of annuity fomulas in the classoom by obseving fomula usage as well as suveying student sentiment. Oveall, pesenting annuity fomula deivations seems to benefit students. Student usage of all thee foms indicates that some benefit is associated with pesenting all thee foms although, depending on the type of poblem pesented, some foms ae used moe than othes and the usage of some foms esults in bette pefomance. Also, those students who wee positively affected by the deivation pesentation ae bette able to coectly ecall fomulas than those negatively affected. This esult may be intepeted in a vaiety of ways. On the one hand, since those who wee not able to coectly ecall the fomulas in any fom stated they wee negatively affected, an assumption may be dawn that only pooe pefoming students dislike the deivation pesentation. In othe wods, only the bad students do not appeciate good teaching. On the othe hand, one would expect that those who wee negatively affected would not be able to ecall the fomulas. Othe factos that may affect an instucto s decision to include the deivation lectue include the amount of class time devoted to deiving annuity fomulas as well as the benefits and costs of using the vaious foms thoughout the semeste. Afte pesenting the fomula deivations, an instucto may illustate how to calculate annuity poblems with eithe all thee foms o with only one fom. Fo instance, an instucto may show how to calculate the pesent value of a bond s coupons (which fom an annuity) with each of the thee foms o with only Fom C as is commonly the case. Of couse, using all thee foms is time consuming, and may also confuse a few students. With the inceasing technology of online teaching platfoms, an altenative appoach would be to pesent the deivation of the fomulas in video o PowePoint fomat as supplemental mateial via the intenet. In summay, deiving the FVA and PVA annuity fomulas esults in thee common foms of each of the fomulas. Results fom a quiz equiing the use of one of the thee foms fo each fomula eveals that some foms ae used moe than othes. The fom that povides bette pefomance depends on the type of poblem pesented. In addition, the majoity of students indicated a positive sentiment towad pesenting the deivation of annuity fomulas in the classoom while a small minoity disliked the deivation pesentation. The esults pesented in this pape allow the instucto some insight into student peceptions of how annuity fomulas should be taught. Jounal of Economic and Economic Education Reseach, Volume 15, Numbe 3, 2014

12 Page 12 REFERENCES Bealey, R., S. Myes, and F. Allen (2014). Pinciples of Copoate Finance (Eleventh Edition). McGaw Hill Iwin. Bigham, E.F. and M.C. Ehhadt (2014). Financial Management: Theoy and Pactice (Fouteenth Edition). South- Westen. Newcombe, N.S., N.Ambady, J. Eccles, L. Gomez, D. Klah, M. Linn, K. Mille, and K. Mix (2009). Psychology s Role in Mathematics and Science Education. Ameican Psychologist, 64(6), Ross, S., R. Westefield, and J. Jaffe (2013). Copoate Finance (Tenth Edition). McGaw-Hill Iwin. Sheitt, L.W. (1944). Simplifying the Pesentation of Compound Inteest Fomulas Accounting Review, 19(3), Skinne, F. S. (1994). An Integative Appoach to Teaching Valuation Equations. Financial Pactice and Education, 4(1), Watson, J. D. (1936). Explaining Annuity Fomulas. Accounting Review, 11(4), Welch, Ivo. (2009). Copoate Finance: An Intoduction. Pentice Hall. Jounal of Economics and Economic Education Reseach, Volume 15, Numbe 3, 2014

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