Intrayear Cash Flow Patterns: A Simple Solution for an Unnecessary Appraisal Error


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1 Intrayear Cash Flow Patterns: A Smple Soluton for an Unnecessary Apprasal Error By C. Donald Wggns (Professor of Accountng and Fnance, the Unversty of North Florda), B. Perry Woodsde (Assocate Professor of Fnance, the College of Charleston), and Dlp D. Kare (Assocate Professor of Accountng and Fnance, the Unversty of North Florda) he Journal of Real Estate Apprasal and Economcs Wnter 1991 Introducton he apprasal and academc communtes have spent much tme and effort n recent years developng and refnng apprasal technques to make them as theoretcally correct and practcally applcable as possble. As a result, ncome apprasal technques such as dscounted cash flow and captalzaton of earnngs are commonly used n the apprasal of ncome producng real property and closely held busnesses. hese technques are theoretcally sound and have become the prmary valuaton methods for many apprasers. However, the hgh degree of dffculty of forecastng future revenues, expenses, profts and cash flows result n some unavodable applcaton problems. Actual results almost always devate from forecasts to some degree because of unforeseeable events and condtons, changng relatonshps between costs and revenues, changes n government polcy and other factors. Many of these problems are unavodable and the appraser s task s to lmt errors as much as possble through analyss. Whatever ther theoretcal soundness, there s one error bult nto most apprasal tools as they are commonly appled. hs error concerns the ntrayear tmng of cash flows and returns. Apprasal technques such as captalzaton of earnngs, Ellwood formulae and dscounted cash flow as they are most often appled nherently assume that ncome or cash flows occur at the end of each year. hs s obvously not realstc n the vast majorty of cases. he resultng apprased values may be sgnfcantly n error because of ths techncal assumpton mplct n the valuaton tool. he apprasal process s dffcult enough wthout havng known errors bult n an apprasal technque, especally f the errors are sgnfcant. hs artcle dscusses the problem and proposes a smple soluton. Assumptons Made n Income Apprasal Models he essence of the ncome approach to apprasal s that the value of a property reflects the present value of future benefts of property ownershp as measured by ncome or cash flows. he apprasal of an ncome producng property generally nvolves the use of an ncome valuaton model n addton to market determned multples of sales, earnngs or cash flow and cost approaches. here are several ncome approaches whch range from a smple net operatng ncome (NOI) captalzaton model to elaborate, comprehensve dscounted cash flow models. All ncome approaches requre determnaton of three crtcal factors whch determne the value of the property: return to the nvestor, tmng of the return, and rsk, whch determnes the requred yeld or dscount rate. he defnton of the return to the nvestor depends on the apprasal method employed. For example, the NOI captalzaton method uses operatng cash
2 flows pror to debt servce. Apprasals based on cash flows to equty use cash flows after debt servce. he dscount rate, also known as the yeld rate, reflects the nvestor s requred rate of return based on the rsk and type of property nterest beng valued. hs rate can be an all equty rate n whch the returns to be dscounted are aftertax cash flows to equty. Alternatvely, n net operatng ncome technques the yeld rate reflects returns to both debt and equty. he thrd factor, tmng of returns, takes nto account when cash flows and profts are receved and then adjusts to present value. he assumpton that ncome or cash flow wll be receved at specfc ponts n tme s bult nto all of the apprasal methods and t s ths assumpton that s a common source of error n apprasals. Some texts and artcles nclude dscussons of monthly or other cash flow patterns and some apprasers adjust for monthly cash flows (1, 487, 538)(2, 70). However, most sources and apprasals assume year end recepts. For example, he Apprasal of Real Estate contans an extensve dscusson of how mortgage equty can be used to determne captalzaton rates for ncome captalzaton models (1, ). All of the applcatons presented assume that the NOI s receved at the end of each year. Other authors also present valuaton models whch usually specfy annual returns and annual dscountng (2, )(3, )(4, 7384). he reason that property returns are measured on an annual bass s lkely due to conventon and the fact that t s smply more convenent to accumulate and project data on an annual bass. In realty, however, cash flows almost never are receved ether at the begnnng or at the end of the year but are spread out over the entre year. Many cash recepts, dsbursements and debt servce payments occur monthly due to normal bllng procedures, payment schedules, loan contracts and other agreement. For some propertes, cash flows may be even more frequent. Busness revenues normally are receved daly whle payroll and other cash expenses may be ncurred weekly, bweekly or n other patterns. Other expendtures, such as taxes, may be pad quarterly or annually. Realzable net benefts for nvestors, whether measured as cash flows to equty or some other measure, occur n the same pattern as the flow of recepts and expendtures. hus monthly cash flows approxmate realty much more closely than annual cash flows even though most apprasals assume yearly flows. Even f cash flows do not occur strctly n a monthly pattern, the monthly assumpton wll be accurate enough for use n almost all cases and s a substantal mprovement over the assumpton of year end recepts. Adjustment for Monthly Cash Flows Cash flows follow one of three patterns annutes, perpetutes and uneven. Every apprasal usng an ncome approach ncludes at least one of these and many nclude two or all three. Determnng the present value of any of these nvolves an assumpton as to when cash flows wll be receved wthn each year of the lfe of the apprased property. he problem of assumng year end rather than ntrayear cash flows apples to each pattern. Annutes
3 Although not a common stuaton, apprasals may nvolve the annuty cash flow patter n whch the annual cash flows are essentally level over the project s lfe. For example, a project may have cash flows of $15,000 per year for fve years: End of: Year 1 Year 2 Year 3 Year 4 Year 5 $15,000 $15,000 $15,000 $15,000 $15,000 If end of year cash flows are assumed as they are n most prnted present value tables, fnancal calculators and spreadsheets, the formula to value the annuty s: (1) 1 1 N (1 nom ) PVA( nom, N) = R X = R X PVIFA( nom, N) nom where: PVA( nom,n) R PVIFA ( nom,n) nom N = present value of the annuty, = annuty payment per year, = present value nterest factor for man yearly annuty, = nomnal annual nterest (or dscount) rate, and = number of years over whch the payments occur. Assumng a requred return of 12 percent results n a value of $54,072. However f the project really produces the cash flow monthly, the above formula msstates the pattern and value of the cash flows. he pattern s actually $1,250 per month for 60 months. Month 1 Month 2 Month 3 Month 4 Month 60 $1,250 $1,250 $1,250 $1,250 $1,250 o fnd the present value of ths stream, a monthly dscount rate must be appled. An annual effectve nterest rate of 12 percent compounded annually does not translate to a monthly rate of 1 percent. he formula used to fnd the monthly compounded rate mplct n a nomnal annual rate s: (2) where: m m =(1+ nom ) 1/12 1 = effectve monthly nterest rate. hus the monthly nterest rate mplct n an annual rate of 12 percent s: (3) m =(1.12) 1/12 1 = he annuty formula for monthly cash flows s: (4)
4 PVIFA( m 1 1 (1 ),12N) = m 12N m where: PVIFA( m,12n)= present value nterest factor for monthly annuty. Applyng ths monthly rate to the above example results n a present value of $56,985 as opposed to the $54,072 value calculated based on annual flows. hus, an appraser usng ths technque wth annual cash flows and an annual yeld rate has a bultn error n the fnal value of 5.4 percent. hs error s due solely to the mathematcal assumptons mplct n the present value equaton and not to normal forecastng dffcultes. In many cases, apprasers use prnted present value tables that assume annual cash flows. A table of adjustment factors can be derved to adjust these tables to the monthly cash flow assumpton usng the followng equaton: (5) PVIFA( AdjustmentFactor = PVIFA( m,12n) nom, N) he adjustment factors vary wth nterest rates but not wth the number of years and thus only a sngle adjustment factor s need for each nterest rate. he adjustment factors for annual nterest rates varyng from 8 percent to 30 percent appear n able 1. able 1 Annual Monthly Adjustment Factors Annual Interest Rate Adjustment Factor Annual Interest Rate Adjustment Factor 1% % he example gven above can be used to llustrate the use of the table. A present value of $54,072 was calculated for the fve year stream of annual cash flows of $15,000 usng a yeld rate of 12
5 percent. he adjustment factor n able 1 for a 12 percent return s Multplyng ths factor by $54,072 results n an adjusted present value of $56,985, whch s the same as the present value calculated above usng monthly cash flows. For apprasers usng fnancal calculators, spreadsheets or computerzed apprasal software packages wth annual cash flows, the adjustment s the same. Smply multply the fnal value by the adjustment factor n able 1 for the approprate nterest rate. Alternatvely, the problem can be solved by usng monthly cash flows and the monthly nterest rate derved from Equaton (2) n the calculator, spreadsheet or software, f the software wll allow t. Perpetutes A more common pattern of cash flows assumed n apprasals s a perpetuty whch s used n cases n whch the stream of benefts s expected to be level each year and contnue forever or ts practcal equvalent. hs pattern s the bass for the wdely used captalzaton methods of NOI, net ncome and cash flows. A perpetuty assumng year end flows would appear as: End of: Year 1 Year 2 Year 3 nfnty $15,000 $15,000 $15,000 $15,000 he formula for calculatng the present value of ths pattern s: (6) NetOperatngIncome( NOI ) Pr esentvalue = OverallCaptalzatonRate Assumng a captalzaton rate of 12 percent, the present value of the above perpetuty s $125,000. However, f monthly cash flows are used, the pattern s actually: End of Month Year 1 Year 2 Infnty January February January February $1,250 $1,250 $1,250 $1,250 $1,250 Usng a monthly captalzaton rate of percent as calculated n Equaton (2), the present value s actually $131,734 from the monthly adaptaton of Equaton (6): (6B) Pr esentvalue = $ $131,734 he factors from able 1 also can be used to adjust perpetutes from year end to monthly cash flows. In ths case, the value of $125,000 multpled by the 12 percent adjustment factor of yelds the same adjusted value of $131,734 calculated above. Uneven Cash Flows In many cases, cash flows vary from perod to perod. In these stuatons, each cash flow s treated ndependently and the appraser values a seres of ndvdual flows usng the followng equaton for each: (7)
6 Pr esentvalue CF = (1 nom ) t CF X (1 1 nom ) CF XPVIF $ ( nom, ) where: Present Value = present value of Year s cash flow, CF = year end cash flow n Year, and PVIF$ ( nom,) = present value nterest factor for year s cash flow. For example, assume an ncome producng property s projected to have the followng annual, end of year cash flows: Year 1 Year 2 Year 3 $9,000 $12,000 $18,000 Usng the same requred yeld rate of 12 percent and Equaton (7), the present value of ths cash flow stream s $30,414. If the addtonal assumpton s made that these cash flows occur monthly, an adjustment dentcal to those above can be made. he $30,414 present value s multpled by the factor for 12 percent to arrve at an adjusted value of $32,053. he accuracy of ths adjustment can be proven by calculatng present values of each year s cash flow separately usng the monthly recept assumpton. As an llustraton, consder the cash flows n Year 3 of $1,500 per month. o fnd the value today of these cash flows, two steps must be completed. Frst, the cash flows must be dscounted to the begnnng of Year 3 usng the annuty factor for twelve perods wth the monthly nterest rate of percent as calculated above usng Equaton (4). he resultng amount s then dscounted to the present for two years usng the annual nterest rate of 12 percent. he two year dscountng perod s used because of the fact that dscountng the monthly flows establshes the value as of the begnnng of Year 3 whch s the same as the end of Year 2. hs pont s two years n the future from today. he process s llustrated as: Step 1: Dscount Year 3 s monthly cash flows to the begnnng of the year: Year 3: January February March November December $1,500 $1,500 $1,500 $1,500 $1,500 $1,500 $16,937 Value at begnnng of Year 3 (end of Year 2) Step 2: Dscount ths value to today: $16,937 Value at begnnng of Year 3 (end of Year 2) Value oday $13,502 he process s repeated for each year s cash flow and the values for each year are summed to arrve at fnal total value. Mathematcally, the calculaton s: hs fnal value s the same as calculated above usng the adjustment factor from able 1.
7 Year Monthly Cash Flow % Monthly Present Value Annuty Factor 12 months Present Value at Begnnng of Year 12% Present Value Factor Present Value 1 $ $8, $8, , , , , , ,502 Fnal Value: $32,053 Reverson Values Many apprasal technques, especally dscounted cash flow and dscounted earnngs, nvolve forecastng ndvdual uneven annual returns for a number of years and captalzng the perpetuty at the end of ths tme to establsh a termnal or reverson value. If annual returns are used n ths process, the same tmng error occurs. hus, reverson values should be adjusted n the same manner as annual cash flows. MdYear Cash Flows A smple and approxmately correct alternatve to the monthly adjustment s to assume that all cash flows occur at mdyear. Under ths assumpton, the adjustment factor for all present values becomes: (8) ApproxmateAdjustmentFactor = 1 nom For example, usng ths equaton for 12 percent the adjustment factor s compared to the factor of n able 1. he resultng values would be approxmately 0.4 percent dfferent. he mdyear factor for 20 percent s compared to shown n able 1 or a dfference of 0.6 percent. hs observaton rases two mportant ponts. Frst, whle these dfferences are relatvely small, a bultn error of approxmately 0.5 percent s large enough that many apprasers may want to use the adjustments n able 1 n the nterest of accuracy. Secondly, the closeness of the mdyear adjustment to the more accurate factors presented n able 1 and ther substantal dfference from year end factors renforce the poston that an adjustment for ntrayear cash flow patterns should be made n order to avod errors. Summary As they are commonly descrbed n the lterature and appled n practce, apprasal technques have a bultn conceptual error. hs error occurs because of the clearly unrealstc assumpton that annual cash flows occur at the end of the respectve years of a property s or busness s lfe. hs artcle dscusses the problem as t relates to the varous cash flow patterns encountered n
8 ncome property and closely held busness apprasals and develops a theoretcally sound, smple adjustment to correct t. Don Wggns, D.B.A, ASA, CVA, CPA/ABV 904/ Don Wggns s presdent of both Busness Valuaton, Inc., a frm specalzng n busness valuatons, and Hertage Captal Group, Inc., a boutque nvestment bankng frm focusng on mddle market companes. Headquartered n Jacksonvlle, Florda, Wggns has over 30 years of experence advsng clents on a wde range of mergers and acqustons and fnance transactons, ncludng M&A, sales and dvesttures, captal placement, value enhancement, exst plannng and related busness owner transtons. He has advsed nternatonal and domestc companes and successfully led transactons n numerous sectors, ncludng busness servces, healthcare, dstrbuton and logstcs and manufacturng. Snce 1989, Busness Valuaton, Inc. has performed thousands of valuatons for professonal servce, as well as wholesale, retal and consumer product companes n a varety of ndustres ncludng healthcare, transportaton, logstcs and dstrbuton, manufacturng, technology and busness servces. Wth more than 30 years of experence, Hertage Captal Group, Inc. has earned a reputaton for negotatng hghly successful outcomes n both sellsde and buysde transactons, mergers and acqustons, value enhancement pror to ext or transton, captal placement, debt management and numerous other busness deals. Hertage s customers nclude hundreds of busness owners of mdszed companes n the southeastern Unted States and, as a foundngmember of M&A Internatonal, lead numerous, successful nternatonal transactons. he Hertage team s extensve transactonal and operatonal experence provdes a vtal perspectve on the most effectve means of maxmzng value as measured n the market by ether potental nvestors, partners or buyers. Wth a broad reach throughout the marketplace, Hertage Captal Group mantans a keen focus on the unque needs of owners of mddle market companes. Member FINRA/SIPC he world s leadng M&A Allance.
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