Symbols ad Formulas PCSYBL-G
For Educatioal Purposes ly The materials preseted i this course represet the opiios ad views of the developers ad/or reviewers. Although these materials may have bee reviewed, the views ad opiios expressed herei are ot edorsed or approved by the Appraisal stitute as policy uless adopted by the Board of Directors pursuat to the Bylaws of the Appraisal stitute. While substatial care has bee take to provide accurate ad curret data ad iformatio, the Appraisal stitute does ot warrat the accuracy or timeliess of the data ad iformatio cotaied herei. Further, ay priciples ad coclusios preseted i this course are subject to court decisios ad to local, state, ad federal laws ad regulatios ad ay revisios of such laws ad regulatios. This course is preseted for educatioal ad iformatioal purposes oly with the uderstadig that the Appraisal stitute is ot egaged i rederig legal, accoutig, or other professioal advice or services. Nothig i these materials is to be costrued as the offerig of such advice or services. f expert advice or services are required, readers are resposible for obtaiig such advice or services from appropriate professioals. Nodiscrimiatio Policy The Appraisal stitute advocates equal opportuity ad odiscrimiatio i the appraisal professio ad coducts its activities i accordace with applicable federal, state, ad local laws. Copyright 2006, mior revisios 2018, Appraisal stitute. All rights reserved. No part of this publicatio may be reproduced, modified, re-writte, or distributed, electroically or by ay other meas, without the express writte permissio of the Appraisal stitute.
Symbols ad Formulas Symbols... 3 Summary of Basic Formulas... 6 Capitalizatio Selectio Tree... 15 Appraisal stitute Symbols ad Formulas 1
Symbols ad Formulas 2 Appraisal stitute
ariable, Expoet, ad Subscript Names Symbol ariable/expoet/subscript Descriptio a ariable Aualizer that adjusts to aual rate AL Subscript Assumed loa b ariable Balace B ariable Ratio of buildig value to total value B Subscript Buildig CF ariable Cash flow (icome or reversio) C ariable Cash outlay or capital outlay CP ariable Cosumer price idex CR ariable Compoud rate ariable Chage rate over holdig period DCR ariable Debt coverage ratio E Subscript Equity EG ariable Effective gross icome EG ariable Effective gross icome multiplier F ariable Factor or multiplier F ariable Future value G ariable Gross icome multiplier GR ariable Gross ret multiplier i ariable terest rate, discout rate ariable come B ariable Net icome to the buildig E ariable Net icome to the equity L ariable Net icome to the lad LF ariable Net icome to the leased fee LH ariable Net icome to the leasehold (or to the sadwich if there is a SLH) ariable Aual debt service, also called icome to the mortgage (or N) ariable Net operatig icome Appraisal stitute Symbols ad Formulas 3
Symbol ariable/expoet/subscript Descriptio SLH ariable Net icome to the subleasehold RR ariable teral rate of retur TA ariable stallmet to amortize e L ariable Ratio of lad value to total value L Subscript Lad LF Subscript Leased fee LH Subscript Leasehold (or sadwich if there is a SLH) ariable ariable Factor or multiplier Loa-to-value ratio Subscript ortgage RR ariable odified iteral rate of retur Subscript or Expoet Term (i periods) N Subscript or Expoet Term (i years) N ariable Net icome multiplier NR ariable Net icome ratio N Subscript New moey N (or ) ariable Net operatig icome NP ariable Net preset value Subscript verall or fee simple ER ariable peratig expese ratio PG ariable Potetial gross icome PG ariable Potetial gross icome multiplier PT ariable Level paymet P ariable Preset value R ariable Capitalizatio rate R B ariable Buildig capitalizatio rate R E ariable Pretax cash flow rate, also called equity capitalizatio rate R L ariable Lad capitalizatio rate R LF ariable Leased fee capitalizatio rate Symbols ad Formulas 4 Appraisal stitute
Symbol ariable/expoet/subscript Descriptio R LH ariable Leasehold capitalizatio rate (or sadwich cap rate if there is a SLH) R ariable ortgage capitalizatio rate R N ariable Termial capitalizatio rate R ariable verall capitalizatio rate R SLH ariable Subleasehold capitalizatio rate RR ariable Reivestmet rate 1/S ariable Sikig fud factor SLH Subscript Subleasehold T ariable Effective tax rate (icome tax) T ariable Teat improvemets ariable alue B ariable Buildig value E ariable Equity value L ariable Lad value LF ariable Leased fee value LH ariable Leasehold value (or sadwich value, if there is a SLH) ariable ortgage value (usually loa amout) N ariable Reversio value N E ariable Equity reversio value ariable verall or fee simple value SLH ariable Subleasehold value W Subscript Wraparoud loa Y ariable Yield or discout rate Y E ariable Equity yield rate Y LF ariable Leased fee yield rate Y LH ariable Leasehold yield rate (or sadwich yield rate if there is a SLH) Y ariable ortgage yield rate Y ariable Property or overall yield rate Y SLH ariable Subleasehold yield rate Appraisal stitute Symbols ad Formulas 5
Direct Capitalizatio: Factors To Extract Data from Comparable Sales To Apply to the Subject Property To Calculate come Geeral F= = F = F f = PG PG = PG = PG PG PG = PG f = EG EG = EG = EG EG EG = EG f = Ret ultiplier = = Ret ultiplier Ret = Ret ultiplier f = N = = N = N Direct Capitalizatio: Capitalizatio Rates Ca be subscripted for B, E, L, LF, LH,, N,, ad SLH Geeral To extract data from comparable sales To apply to the subject property To calculate icome To calculate reversio R= = R = R N+1 N = R N Symbols ad Formulas 6 Appraisal stitute
Direct Capitalizatio: Relatioship Formulas 1 R = N NR R = EG N = R 1 N = F NR = EG NR = R EG ortgage Formulas Loa to alue Defiitio: = = = Debt Coverage Ratio Defiitio: DCR = = DCR = DCR Uderwriter s ethod: R = R DCR R DCR = R Portio Paid ff: Balace outstadig Portio paid off = 1 rigial loa amout Appraisal stitute Symbols ad Formulas 7
Wraparoud ortgages: New moey amout ( N )= Total wraparoud ( W ) Assumed loa ( AL ) New moey cash flow = Wrap debt service Assumed loa debt service peratig Expese Formulas peratig expeses ER = EG ER = 1 NR NR = 1 ER peratig expeses + Breakeve ratio = PG acacy Formulas acat uits Physical vacacy rate = Total uits R acat square feet Physical vacacy rate = Total square feet Total ret loss due to vacat space Ecoomic vacacy rate = Total potetial gross reveue Breakeve Formulas peratig expeses + Breakeve ratio = PG Breakeve occupacy = Breakeve ratio No. of uits Symbols ad Formulas 8 Appraisal stitute
Property Tax Formulas Property taxes = Assessed value Tax rate Assessed value = arket value Assessmet ratio Actual taxes Effective tax rate = arket value ill rate Tax rate = 1,000 Effective tax rate = Assessmet ratio Tax rate Loaded capitalizatio rate = verall capitalizatio rate + Effective tax rate Note. These do ot apply i all jurisdictios. Please see course materials. Bad of vestmet ad Residual Formulas Lad ad Buildig = + L B = + L B R =L R +B R L B (R L R L) R B = B (R B R B) R L = L 1= B+ L B = 1 L L = 1 B = B B B o = B B B = = L L L = L L L = ortgage ad Equity = + E = + E R = R + (1 ) R ( R R ) R = E (1 ) E = o = = Usually 1, ot E, is used for the complemet of. Y Y + (1 - ) Y E, except uder special circumstaces. LF + LH, except uder special circumstaces. Appraisal stitute Symbols ad Formulas 9
Discouted Cash Flow Formulas F = P (1 + i) The base = (1 + i) P ordiary auity (1 + i) = P auity i advace 1 P of e factor = (1 + i) 1 1 1 1 P of e per period factor = + + +...+ 1 2 3 (1 + i) (1 + i) (1 + i) (1 + i) 1 2 3 = + + +...+ + 1 2 3 (1 + Y) (1 + Y) (1 + Y) (1 + Y) (1 + Y) Nomial aual rate Periodic rate i = Number of periods per year Effective aual iterest rate = (1 + i) z 1, where z = Number of compoudig periods per year P = P come + P Reversio E = P of equity icome + P of equity reversio Equity reversio = Property reversio ortgage balace N+1 N = R N Net Preset alue Formulas NP = P C f P of future beefits > C NP > 0 ad RR > Y f P of future beefits < C NP < 0 ad RR < Y f P of future beefits = C NP = 0 ad RR = Y (1 +Nomial rate) 1 +Real rate = (1 +Expected iflatio) (1 + Nomial rate) = (1 + Real rate) (1 + Expected rate of iflatio) Symbols ad Formulas 10 Appraisal stitute
odified RR Relatioships: f RR < RR RR < RR < RR f RR > RR RR > RR > RR f RR = RR RR = RR = RR U.S. come Tax Formulas PTCF (Pretax cash flow) ( E ) Tax ATCF (After-tax cash flow) Net operatig icome terest Depreciatio Taxable icome Tax rate Tax Pretax equity reversio Tax After-tax equity reversio Sale price Adjusted basis Capital gai Tax rate Tax Pretax RR After-tax RR T = Pretax RR Appraisal stitute Symbols ad Formulas 11
Property odel Formulas Ca be subscripted for B, E, L, LF, LH,,, ad SLH Geeral Premise: Perpetuity ( = 0) Premise: Level come (a = 1/S ) Premise: Same CR for come ad alue ( x a = CR) Solve for R R = Y a * R = Y R=Y 1/ S Solve for Y Y = R + a Y = R Y=R+ 1/ S or solve o fiacial calculator R = Y CR Y = R + CR Solve for value chage = Y R a = 0 Y R = 1 /S CR = Y R CR = F / P 1 or where = (1 + CR) N 1 or F P = P F P = P or = (1 + CR) N 1 Solve for F F = P F = P (1 + ) or solve o fiacial calculator F = P (1 + ) where = (1 + CR) N 1 * You will also see this formula writte these ways: R = Y a or R = Y ( )(a) These are all acceptable ways to write the product of two umbers. Symbols ad Formulas 12 Appraisal stitute
Cap ad Yield Rates: Sometimes Equal, Never the Same Capitalizatio Rate Yield Rate Abbreviatio R Y, i, or RR Syoyms* Cap rate Discout rate, iterest rate, iteral rate of retur, growth rate, compoud rate** Formula Where the upside potetial is reflected Aalogy Symbol for rate applicable to a property Syoyms for rate applicable to a property Symbol for a rate applicable to a mortgage Syoyms for a rate applicable to a mortgage Symbol for a rate applicable to equity Syoyms for a rate applicable to equity Solve for R R= the rate Sapshot i time aticipatig the plot R verall rate, overall cap rate R ortgage cap rate, aual loa costat, aual costat, aual mortgage costat R E Equity cap rate, equity divided rate, cash-ocash rate, cash throwoff rate, pretax cap rate Solve for Y CFj = j=1 (1 + Y) the cash flows Play the video to see the plot Y Property yield rate, iteral rate of retur i or Y ortgage yield rate, mortgage yield, mortgage iteral rate of retur, iterest rate Y E Equity yield rate, equity iteral rate of retur j * Note that fie distictios exist amog some of the syoyms. ** Growth ad compoud rates are ot yield rates, but all rates listed here behave accordig to the formula F = P (1+i), where i is the rate. Appraisal stitute Symbols ad Formulas 13
Six Fuctios of e Summary Name of Factor Syoyms Symbol From the HP-12C Formula Excel 1 Amout of e Future value of e, future worth of e, compoud iterest S 1 i P, solve for F (1+i) = F (i,, 0, 1) 2 Preset value of e Reversio factor 1/S 1 i F, solve for P 1 (1 ) + i = P (i,, 0, 1) 3 Preset value of e per period Level auity factor a 1 i PT, solve for P 1 1 (1 + i) i = P (i,, 1) 4 Amout of e per period Sikig fud accumulatio, future value of e per period S 1 i PT, solve for F (1 + i) 1 = F (i,, 1) i 5 Sikig fud factor Periodic paymet to grow to e 1/S or SFF 1 i F, solve for PT i (1 + i) 1 = PT (i,, 0, 1) 6 stallmet to amortize e Direct reductio or amortizig loa, partial paymet 1/a 1 i P, solve for PT 1 1 (1 + i) i = PT (i,, 1) Symbols ad Formulas 14 Appraisal stitute
Appraisal stitute Symbols ad Formulas 15
Symbols ad Formulas 16 Appraisal stitute