Symbols and Formulas

PCSYMBOL-G For Educational Purposes Only

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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 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

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 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 = Breakeven ratio × No. of units

Symbols and Formulas – 8 Appraisal Institute 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

ƒƒ If PV of future benefits > CO → NPV > 0 and IRR > Y

ƒƒ If PV of future benefits < CO → NPV < 0 and IRR < Y

ƒƒ 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