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Symbols and Formulas Symbols and Formulas PCSYMBOL-G For Educational Purposes Only The materials presented in this course represent the opinions and views of the developers and/or reviewers. Although these materials may have been reviewed, the views and opinions expressed herein are not endorsed or approved by the Appraisal Institute as policy unless adopted by the Board of Directors pursuant to the Bylaws of the Appraisal Institute. While substantial care has been taken to provide accurate and current data and information, the Appraisal Institute does not warrant the accuracy or timeliness of the data and information contained herein. Further, any principles and conclusions presented in this course are subject to court decisions and to local, state, and federal laws and regulations and any revisions of such laws and regulations. This course is presented for educational and informational purposes only with the understanding that the Appraisal Institute is not engaged in rendering legal, accounting, or other professional advice or services. Nothing in these materials is to be construed as the offering of such advice or services. If expert advice or services are required, readers are responsible for obtaining such advice or services from appropriate professionals. Nondiscrimination Policy The Appraisal Institute advocates equal opportunity and nondiscrimination in the appraisal profession and conducts its activities in accordance with applicable federal, state, and local laws. Copyright © 2006, minor revisions 2018, Appraisal Institute. All rights reserved. No part of this publication may be reproduced, modified, re-written, or distributed, electronically or by any other means, without the express written permission of the Appraisal Institute. Symbols and Formulas Symbols ................................................................................................................... 3 Summary of Basic Formulas ...................................................................................... 6 Capitalization Selection Tree ................................................................................... 15 Appraisal Institute Symbols and Formulas – 1 Symbols and Formulas – 2 Appraisal Institute Variable, Exponent, and Subscript Names Symbol Variable/Exponent/Subscript Description a Variable Annualizer that adjusts ∆ to annual rate AL Subscript Assumed loan b Variable Balance B Variable Ratio of building value to total value B Subscript Building CF Variable Cash flow (income or reversion) CO Variable Cash outlay or capital outlay CPI Variable Consumer price index CR Variable Compound rate ∆ Variable Change rate over holding period DCR Variable Debt coverage ratio E Subscript Equity EGI Variable Effective gross income EGIM Variable Effective gross income multiplier F Variable Factor or multiplier FV Variable Future value GIM Variable Gross income multiplier GRM Variable Gross rent multiplier i Variable Interest rate, discount rate I Variable Income IB Variable Net income to the building IE Variable Net income to the equity IL Variable Net income to the land ILF Variable Net income to the leased fee ILH Variable Net income to the leasehold (or to the sandwich if there is an SLH) IM Variable Annual debt service, also called income to the mortgage IO (or NOI) Variable Net operating income Appraisal Institute Symbols and Formulas – 3 Symbol Variable/Exponent/Subscript Description ISLH Variable Net income to the subleasehold IRR Variable Internal rate of return ITAO Variable Installment to amortize One L Variable Ratio of land value to total value L Subscript Land LF Subscript Leased fee LH Subscript Leasehold (or sandwich if there is an SLH) Variable Factor or multiplier M Variable Loan-to-value ratio M Subscript Mortgage MIRR Variable Modified internal rate of return n Subscript or Exponent Term (in periods) N Subscript or Exponent Term (in years) NIM Variable Net income multiplier NIR Variable Net income ratio NM Subscript New money NOI (or IO) Variable Net operating income NPV Variable Net present value O Subscript Overall or fee simple OER Variable Operating expense ratio PGI Variable Potential gross income PGIM Variable Potential gross income multiplier PMT Variable Level payment PV Variable Present value R Variable Capitalization rate RB Variable Building capitalization rate RE Variable Pretax cash flow rate, also called equity capitalization rate RL Variable Land capitalization rate RLF Variable Leased fee capitalization rate Symbols and Formulas – 4 Appraisal Institute Symbol Variable/Exponent/Subscript Description RLH Variable Leasehold capitalization rate (or sandwich cap rate if there is an SLH) RM Variable Mortgage capitalization rate RN Variable Terminal capitalization rate RO Variable Overall capitalization rate RSLH Variable Subleasehold capitalization rate RR Variable Reinvestment rate 1/Sn Variable Sinking fund factor SLH Subscript Subleasehold T Variable Effective tax rate (income tax) TI Variable Tenant improvements V Variable Value VB Variable Building value VE Variable Equity value VL Variable Land value VLF Variable Leased fee value VLH Variable Leasehold value (or sandwich value, if there is an SLH) VM Variable Mortgage value (usually loan amount) VN Variable Reversion value VNE Variable Equity reversion value VO Variable Overall or fee simple value VSLH Variable Subleasehold value W Subscript Wraparound loan Y Variable Yield or discount rate YE Variable Equity yield rate YLF Variable Leased fee yield rate YLH Variable Leasehold yield rate (or sandwich yield rate if there is an SLH) YM Variable Mortgage yield rate YO Variable Property or overall yield rate YSLH Variable Subleasehold yield rate Appraisal Institute Symbols and Formulas – 5 Direct Capitalization: Factors To Extract Data from To Apply to the To Calculate Income Comparable Sales Subject Property V V General F= V =I×F I= I F VO VO If I = PGI PGIM = VO = PGI × PGIM PGI = PGI PGIM VO VO If I = EGI EGIM = VO = EGI × EGIM EGI = EGI EGIM If I = VO VO Multiplier = V=O Rent × Multiplier Rent = Rent Rent Multiplier V V NIM = O O If I = IO VOO = I × NIM I=O IO NIM Direct Capitalization: Capitalization Rates Can be subscripted for B, E, L, LF, LH, M, N, O, and SLH General I To extract data from comparable sales R= V I To apply to the subject property V= R To calculate income I=V×R IN+1 To calculate reversion VN = RN Symbols and Formulas – 6 Appraisal Institute Direct Capitalization: Relationship Formulas 1 1 IO R=O NIR = NIM NIM = EGI RO NIR R=O NIR = R ×EGIM EGIM NIM = F O Mortgage Formulas Loan to Value Definition: VMO =V ×M V M= M VO V V= M O M Debt Coverage Ratio Definition: IO DCR = IM I I= O M DCR IOM = DCR×I Underwriter’s Method: ROM = M×R ×DCR R DCR = O M×RM Portion Paid Off: Balance outstanding Portion paid off = 1 − Original loan amount Appraisal Institute Symbols and Formulas – 7 Wraparound Mortgages: New money amount (VNM)= Total wraparound (VW) – Assumed loan (VAL) New money cash flow = Wrap debt service – Assumed loan debt service Operating Expense Formulas Operating expenses OER = EGI OER =1 − NIR NIR =1 − OER Operating expenses + I Breakeven ratio = M PGI Vacancy Formulas Vacant units Physical vacancy rate = Total units OR Vacant square feet Physical vacancy rate = Total square feet Total rent loss due to vacant space Economic vacancy rate = Total potential gross revenue Breakeven Formulas Operating expenses + IM Breakeven ratio = PGI Breakeven occupancy = Breakeven ratio × No. of units Symbols and Formulas – 8 Appraisal Institute Property Tax Formulas Property taxes = Assessed value × Tax rate Assessed value = Market value × Assessment ratio Actual taxes Effective tax rate = Market value Mill rate Tax rate = 1,000 Effective tax rate = Assessment ratio × Tax rate Loaded capitalization rate = Overall capitalization rate + Effective tax rate Note. These do not apply in all jurisdictions. Please see course materials. Band of Investment and Residual Formulas Land and Building Mortgage and Equity V =V +V O LB VO =V ME +V I =I +I O LB IO =I ME +I R =L×R +B×R OLB ROM =M×R +(1− M ) ×R E (R− L× R ) OL ()ROM− M×R R=B R= B E (1− M ) (R− B × R ) R= OB L L VMO= MV × VM Vo = 1=BL+ VBO= BV × VLO= LV × M BL=1 − V V VM B L M = Vo = VO = LB=1 − B L VO V V B = B L = L V V Usually 1 – M, not E, is used for the O O complement of M. YO ≠ M × YM + (1 -– M) × YE, except under special circumstances. VO ≠ VLF + VLH, except under special circumstances. Appraisal Institute Symbols and Formulas – 9 Discounted Cash Flow Formulas FV = PV×(1 + i )n The base = (1 + i) PV ordinary annuity × (1 + i) = PV annuity in advance 1 PV of One factor = (1 +i )n 111 1 PV of One per period factor = + + +...+ (1+)iii123 (1+) (1+) (1+) in II I IV V =12 + +3 +...+nn + (1+)YYY123 (1+) (1+) (1+) YYnn (1+) Nominal annual rate Periodic rate i = Number of periods per year Effective annual interest rate = (1 + i)z − 1, where z = Number of compounding periods per year PV = PVIncome + PVReversion VE = PV of equity income + PV of equity reversion Equity reversion = Property reversion – Mortgage balance IN+1 V=N RN Net Present Value Formulas NPV = PV− CO If PV of future benefits > CO → NPV > 0 and IRR > Y If PV of future benefits < CO → NPV < 0 and IRR < Y If PV of future benefits = CO → NPV = 0 and IRR = Y (1 +Nominal rate) 1 +Real rate = (1 +Expected inflation) (1 + Nominal rate) = (1 + Real rate) (1 + Expected rate of inflation) Symbols and Formulas – 10 Appraisal Institute Modified IRR Relationships: If RR < IRR → RR < MIRR < IRR If RR > IRR → RR > MIRR > IRR If RR = IRR → RR = MIRR = IRR U.S. Income
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