IDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS


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1 IDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS Chrs Deeley* Last revsed: September 22, 200 * Chrs Deeley s a Senor Lecturer n the School of Accountng, Charles Sturt Unversty, Australa. Postal address: C/ School of Accountng, Charles Sturt Unversty, Locked Bag 588, Wagga Wagga, NSW 2678, Australa. Emal: Telephone: Fax: (02) (wthn Australa); (overseas) (02) (wthn Australa); (overseas)
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3 IDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS ABSTRACT Ths artcle examnes the conventonal method of solvng general annuty problems n whch general annutes are converted nto equvalent smple annutes, thereby enablng standard soluton routnes. It fnds that the method errs when the frequency of payments exceeds the frequency of nterest compoundng and nterest accrues lnearly between converson dates. It demonstrates, and advocates the adopton of, an alternatve method, whch acheves vald results n such crcumstances. Keywords: general annutes. JEL code: G300 INTRODUCTION We defne a general annuty as an annuty n whch the frequency of payments p dffers from the frequency of nterest compoundng m. For our present purposes we assume that p and m are ntegers, of whch ether s a factor of the other, obtaned from the same tme nterval; typcally, although not necessarly, one year. Gven these constrants, current technques for solvng general annuty problems typcally follow a method poneered by Hummell and Seebeck n the early 940s. General annutes are converted nto equvalent smple annutes, thereby enablng conventonal soluton routnes [Hummell & Seebeck, 948].
4 Identfcaton and correcton of a common error n general annuty calculatons In the preface to the frst edton of Mathematcs of Fnance the authors clam that ther book dffers wdely from exstng texts n the treatment of general annutes and that they have used ther advocated method n classroom teachng over the precedng eght years wth gratfyng success [Hummell & Seebeck, 948, v]. Bruck [949, 487] comments favourably on the technque, whch he descrbes as avodng a whole complex of complcated formulas. In the preface to the second edton of Mathematcs of Fnance, Hummell and Seebeck [956, v] clam that ther approach to solvng general annuty problems has been adopted by several other wrters. They also profess a belef, shared wth many colleagues throughout the country, that thers s the only teachable method of treatng general annutes. Hummell and Seebeck convert general annutes nto smple annutes by calculatng an equvalent payment per nterest compoundng perod. Contemporary fnance and fnancal mathematcs textbooks [see, for example, Mathematcs of Fnance by Knox, Zma & Brown, 999] may alternatvely calculate an equvalent nterest rate per payment perod. Ether way, the general annuty s converted nto a smple annuty, thereby enablng conventonal soluton routnes. Despte ther provenance and unchallenged acceptance, both methods conventonally used to convert general annutes nto smple annutes err when p > m and nterest accrues lnearly between converson dates. Ths artcle explans that error and how to avod t. 2
5 Identfcaton and correcton of a common error n general annuty calculatons IDENTIFICATION, EXPLICATION & CORRECTION OF THE ERROR Gven an ordnary general annuty and the constrants prevously stated, Equatons () and (2) ostensbly defne an equvalent nterest rate ' per payment perod and an equvalent payment P' per nterest compoundng perod. ' = ( + ) m/p () P' = P (2) ' where = nterest rate per compoundng perod P = payment. We can use ether equvalency to convert general annutes nto smple annutes and then use smple annuty soluton methods to solve general annuty problems. For example, Equatons (3A/B) and (4A/B) use respectvely ' and P' to defne the present and future values of an ordnary general annuty. PV = P ' n (3A) FV = P n ' (3B) PV = P' n (4A) FV = P' n (4B) where n = number of nterest compoundng perods. 3
6 Identfcaton and correcton of a common error n general annuty calculatons But f p > m and nterest accrues lnearly between converson dates, Equatons () and (2) are nvald. It follows that Equatons (3A/B) and (4A/B) are also nvald unless ' and P' are correctly redefned. In ths context t s more realstc to dentfy an equvalent payment per nterest compoundng perod that an equvalent nterest rate per payment perod. We therefore use Equaton (5) to provdes a vald defnton of P'. P' = Pa a a > (5) 2 2a where a = p/m = number of payments per nterest converson perod. When p > m, Equaton (2) gves a lower value of P' than Equaton (5). Equaton (6A) defnes the relatve dfference. = a a 0. 5 a a > (6A) The same relatve dfference apples to present and future values derved from the alternatve defntons of P'. The followng examples demonstrate how conventonal approaches to solvng general annuty problems err when p > m and nterest accrues lnearly between converson dates. We use the same examples to demonstrate the advocated alternatve approach, whch correctly solves these problems. Example An ordnary general annuty comprses $3,000 pad quarterly for ten years at an annual nterest rate of 8%, compounded annually. What s ts future value? 4
7 Identfcaton and correcton of a common error n general annuty calculatons Values of the relevant varables are therefore: P = $3,000 = 8% m = p = 4 a = p/m = 4/ = 4 n = 0 Equaton (5) correctly defnes the value of P' to be $2,360, as follows: P' = Pa a a > (5) 2a = $3, = $3, = $2,360 Table I confrms the foregong fgure by dentfyng and summng the nterest earned by each payment between converson dates. 3 5
8 Identfcaton and correcton of a common error n general annuty calculatons Table I: Summary of nterest earned by four quarterly payments of $3,000 at an effectve annual rate of 8% compounded annually Payment no. Amount Months of nterest Interest earned Algebrac notaton $ $3,000 9 P(3/a) 80 2 $3,000 6 P(2/a) 20 3 $3,000 3 P(/a) 60 4 $3, Total nterest earned 360 Equatons () and (2) erroneously defne the value of P' as $2,354.23, whch s approxmately % below the correct fgure, as follows: ' = ( + ) m/p () = (.08) /4 = % P' = P (2) ' = $3, = $2, Table II confrms the foregong fgure by dentfyng and summng the nterest ncorrectly assumed to be earned by each payment between converson dates. 6
9 Identfcaton and correcton of a common error n general annuty calculatons Table II: Summary of nterest earned by four quarterly payments of $3,000 at an effectve annual rate of 8% compounded quarterly Payment no. Amount Months of nterest Interest earned Algebrac notaton $ $3,000 9 P[( + ) 3 ] $3,000 6 P[( + ) 2 ] $3,000 3 P $3, Total nterest earned Equaton (6A) confrms a relatve dfference of %, as follows: = a a 0. 5 a = a > (6A) = = An error of less than 0.05% may seem nsgnfcant, but present and future values derved from the competng methods exhbt the same relatve error, the sgnfcance of whch s thereby amplfed. Usng the present example, we can confrm that ether of the conventonal soluton methods understates the future value by $83.6. FV = P n ' (3B) 7
10 Identfcaton and correcton of a common error n general annuty calculatons = $3, = $3, = $78, FV = P' n (4B) = $2, = $2, = $78, The correct equvalent annual payment of $2,360 correctly defnes a future value of $79,053.9, as follows: $2, = $79,053.9 Fnancal Calculator Soluton Fnancal calculators typcally compound nterest contnuously when solvng general annuty problems. Ths produces vald results when p < m, but not when p > m and nterest accrues lnearly between converson dates. For example, n the current case the Sharp EL738 calculator computes an ncorrect future value of $78,
11 Identfcaton and correcton of a common error n general annuty calculatons Example 2 A loan of $mllon s to be repad n equal monthly nstalments over four years. If the annual nterest rate s 0% compounded semannually, how much s the monthly repayment? Conventonal Soluton ' = ( + ) m/p () =.05 2/2 = % PV = P ' n (3A) P = PV ' n = $,000, = $,000, = $25, But f nterest accrues lnearly between converson dates, monthly repayments of $25, over four years at an annual nterest rate of 0% compounded semannually wll repay $93.64 more than $mllon, as shown by Equaton (7A) and confrmed n the Appendx to ths artcle. PV = P' n (4A) 9
12 Identfcaton and correcton of a common error n general annuty calculatons = Pa n a a > (7A) 2a = $25, = $25, = $,000,93.64 In ths example the conventonal soluton method has generated a postve error of %, as confrmed by Equaton (6B). a a 0.5 a = a > (6B) = = 0.05 = Advocated Soluton Equaton (7B) correctly defnes a monthly repayment of $25,260.70, as follows: PV = Pa n a a > (7A) 2a P = PV n a a a > (7B) 2a 0.05 = $,000,
13 Identfcaton and correcton of a common error n general annuty calculatons = $,000, = $25, Equaton (7A) and the Appendx confrm that monthly payments of $25, over four years at an annual nterest rate of 0% compounded semannually wll repay exactly $mllon; as follows: PV = Pa n a a > (7A) 2a = $25, = $25, = $,000,000 Calculator Soluton Appled to ths example, the Sharp EL738 calculator computes a monthly payment of $25,265.60, whch s ncorrect when nterest accrues lnearly between converson dates. ERROR ANALYSIS Examples and 2 confrm that, when p > m and nterest accrues lnearly between converson dates, the conventonal approach to solvng general annuty problems understates the present value of a gven payment stream and overstates the perodc amount requred to repay a gven loan
14 Identfcaton and correcton of a common error n general annuty calculatons Fgure llustrates the relatve magntude of that understatement of present and future values over a range of nterest rates and payments per nterest converson perod. % error 0.00% 0.05% 0.0% 0.5% p/m = 2 p/m = 3 p/m = % 0.25% 0.30% 0% 2% 4% 6% 8% 0% 2% 4% 6% 8% 20% Perodc nterest rate Fgure. Percentage errors produced by conventonal calculatons of the present and future values of general annutes when p > m and nterest accrues lnearly between converson dates Fgure confrms that the perodc nterest rate s the domnant factor n dstortng present and future values when conventonal soluton technques are appled to general annuty problems n whch p > m and nterest accrues lnearly between converson dates. Errors are nvarant to the term of an annuty. They can be nsgnfcant n ndvdual cases, but acqure sgnfcance through aggregaton. For example, a fnancal ntermedary could gan a sgnfcant advantage when loan repayments are structured as general annutes n whch p > m and perodc repayments are calculated n the conventonal manner. 2
15 Identfcaton and correcton of a common error n general annuty calculatons ADVOCACY FOR CHANGE Do these fndngs justfy changng the way we currently conceptualse and teach the solvng of general annuty problems when p > m? Before answerng that queston, we need to resolve a concomtant ssue: how does nterest actually accrue between converson dates? My own research confrms that nterest typcally accrues lnearly between converson dates, as depcted n Table I, rather than exponentally, as depcted n Table II and subsumed n fnancal calculators. That beng the case, the conventonal approach to solvng general annuty problems when p > m s founded upon a mstaken assumpton. I therefore advocate a shft towards realty n our teachng of general annutes. Mstaken assumptons that become known pretences are ndefensble bases for pedagogcal methodologes, however well entrenched. SUMMARY & CONCLUSION Ths artcle demonstrates how the conventonal approach to solvng general annuty problems s flawed when the frequency of payments exceeds the frequency of nterest compoundng and nterest accrues lnearly between converson dates. It dentfes the source of error and proposes an alternatve soluton technque, whch produces vald results. It fnds that the relatve errors generated by conventonal soluton methods are low (typcally less than 0.2%), but contends that the use of flawed routnes cannot be justfed when more robust alternatves are avalable. * * * * 3
16 Identfcaton and correcton of a common error n general annuty calculatons ENDNOTES When p > m and nterest accrues lnearly between converson dates, we need to defne ' as follows n order to valdate Equatons (3A/B): ' = a a 2a a > (8) where a = p/m However, we should not recommend ths approach, because, contrary to the mplcaton of Equatons (3A/B), nterest s not compounded between converson dates. Furthermore, the rate of nterest per payment perod s actually /a. Equaton (8) therefore defnes a spurous value of ', whch valdates Equatons (3A/B), but s unobservable. 2 We can adapt Equaton (5) to defne P' for general annutes n whch cash flows occur ether at the start or n the mddle of each payment perod, as follows: P' = Pa a a > (5A) 2a P' = Pa( + 0.5) a > (5B) 3 Appled to the same example, Equatons (5A) and (5B) defne equvalent payments per nterest converson perod of $2,600 and $2,480 respectvely. If payments occur at the begnnng of each quarter, the frst, second, thrd and fourth payments attract nterest of $240, $80, $20 and $60 respectvely. Smlarly, payments made n the mddle of each quarter earn nterest of $20, $50, $90 and $30 per successve payment. A contnuous payment stream acheves the same outcome. 4
17 Identfcaton and correcton of a common error n general annuty calculatons REFERENCES Bruck, R.H. Revew of Mathematcs of Fnance by PM Hummell and CL Seebeck, The Amercan Mathematcal Monthly, Vol. 56, No. 7 (August September, 949) Hummell P.M. and C.L. Seeback. Mathematcs of Fnance (New York, 948), McGrawHll. Hummell P.M. and C.L. Seeback. Mathematcs of Fnance (New York, 956), 2nd Edton, McGrawHll. Knox D.M., P. Zma and R.L. Brown. Mathematcs of Fnance (Sydney, 999), 2nd Edton, McGrawHll. * * * * 5
18 Identfcaton and correcton of a common error n general annuty calculatons APPENDIX: EXAMPLES OF LOAN REPAYMENT SCHEDULES I N P U T O U T P U T APR 0.00% m 2 p 2 EAR 0.25% Assumng contnuous compoundng Assumng lnear accrual of nterest between converson dates PMT Cl. bal. PMT Interest Cl. Bal. PMT Interest Cl. Bal. 0 ,000, ,000, ,000, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,
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