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 Tax Formulas
PTCF (Pretax cash flow) (IE)
− Tax
ATCF (After-tax cash flow)
Net operating income
− Interest
− Depreciation
Taxable income
× Tax rate
Tax
Pretax equity reversion
− Tax
After-tax equity reversion
Sale price
− Adjusted basis
Capital gain
× Tax rate
Tax
Pretax IRR− After-tax IRR T = Pretax IRR
Appraisal Institute Symbols and Formulas – 11 Property Model Formulas
Can be subscripted for B, E, L, LF, LH, M, O, and SLH
Premise: Premise: Premise: Same CR for General Perpetuity Level Income Income and Value (∆ = 0) (a = 1/Sn) (∆ x a = CR)
Solve for R R = Y – ∆ × a* R = Y R=Y−∆×1/ S R = Y – CR n
Solve for Y Y = R + ∆ × a Y = R Y=R+∆×1/ S Y = R + CR n or solve on financial calculator
Solve for YR− ∆ = 0 YR− CR = Y – R value change ∆= ∆ = a 1 /S n − n CR=1 FV/ PV or where ∆ = (1 + CR)N – 1 FV− PV ∆ = or PV FV− PV or ∆ = PV ∆ = (1 + CR)N – 1
Solve for FV FV = PV FV = PV × (1 + ∆) FV = PV × (1 + ∆) or solve on where financial calculator ∆ = (1 + CR)N – 1
* You will also see this formula written these ways:
R = Y – ∆a or R = Y – (∆)(a)
These are all acceptable ways to write the product of two numbers.
Symbols and Formulas – 12 Appraisal Institute Cap and Yield Rates: Sometimes Equal, Never the Same
Capitalization Rate Yield Rate
Abbreviation R Y, i, or IRR
Synonyms* Cap rate Discount rate, interest rate, internal rate of return, growth rate, compound rate**
Formula Solve for R Solve for Y
I n CF R= V = j V ∑ j=1 (1 +Y ) j
Where the upside In the rate In the cash flows potential is reflected
Analogy Snapshot in time Play the video to see the plot anticipating the plot
Symbol for rate RO YO applicable to a property
Synonyms for rate Overall rate, overall cap Property yield rate, internal applicable to a property rate rate of return
Symbol for a rate RM i or YM applicable to a mortgage
Synonyms for a rate Mortgage cap rate, Mortgage yield rate, applicable to a mortgage annual loan constant, mortgage yield, mortgage annual constant, annual internal rate of return, mortgage constant interest rate
Symbol for a rate RE YE applicable to equity
Synonyms for a rate Equity cap rate, equity Equity yield rate, equity applicable to equity dividend rate, cash-on- internal rate of return cash rate, cash throw- off rate, pretax cap rate
* Note that fine distinctions exist among some of the synonyms. ** Growth and compound rates are not yield rates, but all rates listed here behave according to the formula FV = PV × (1+i)n, where i is the rate.
Appraisal Institute Symbols and Formulas – 13 Six Functions of One Summary
Name of From the Synonyms Symbol Formula Excel Factor HP-12C
1 Amount of Future value Sn 1 in PV, (1+i)n = FV (i, n, 0, –1) One of One, future solve for FV worth of One, compound interest
2 Present Reversion 1/Sn 1 in FV, 1 = PV (i, n, 0, –1) value of factor solve for PV (1+ i )n One
3 Present Level annuity 1 in PMT, = PV (i, n, –1) an 1 value of factor solve for PV 1 − n (1+ i ) One per period i
n 4 Amount of Sinking fund Sn 1 in PMT, (1+−i ) 1 = FV (i, n, –1) One per accumulation, solve for FV i period future value of One per period
5 Sinking Periodic 1/Sn 1 in FV, i = PMT (i, n, 0, –1) fund factor payment to or solve for (1+−i )n 1 grow to One SFF PMT
6 Installment Direct 1/an 1 in PV, i = PMT (i, n, –1) to amortize reduction or solve for 1 1 − One amortizing PMT n (1+ i ) loan, partial payment
Symbols and Formulas – 14 Appraisal Institute Appraisal Institute Symbols and Formulas – 15 Symbols and Formulas – 16 Appraisal Institute