# Lesson 11 – Derivation of Overall Rates (OARs) from Sales and by Band of Investment (The Income Approach to Value)

## Appraisal Training: Self-Paced Online Learning Session

In Lessons 8 and 10, we discussed some of the formulas for deriving an opinion of value. In this lesson, we will demonstrate how to derive overall rates by two different methods.

Rule 8(g) provides two methods of developing overall rates:

1. By comparing the net incomes that could reasonably have been anticipated from recently sold comparable properties with their sales prices, … (the market-derived rate).
2. By deriving a weighted average of the capitalization rates for debt and for equity capital … (the band-of-investment method).

This lesson discusses the following:

• Deriving Overall Rates from Sales
• Deriving Overall Rates by the Band-of-Investment Method

### Deriving Overall Rates from Market Sales of Comparable Properties

Rule 8(g)(1) states that a capitalization rate may be developed:

By comparing the net incomes that could reasonably have been anticipated from recently sold comparable properties with their sales prices, adjusted, if necessary, to cash equivalents (the market-derived rate). This method of deriving a capitalization rate is preferred when the required sales prices and incomes are available.

In much the same manner as processing the income stream to derive a multiplier, appraisers must use the anticipated income stream to derive an OverAll Rate [OAR]. When deriving a gross income multiplier, the income is processed to the level of anticipated gross income, to anticipated effective gross income when deriving an effective gross income multiplier. When deriving an OAR, the appraiser goes a step further: expenses are deducted from the effective gross income and the income is processed down to the level of Net Income Before deducting for Recapture [NIBR].

Subtracting the anticipated vacancy and collection losses and anticipated operating expenses from the anticipated gross income produces the anticipated Net Income Before deducting recapture and Taxes [NIBT]. Investors typically anticipate having to pay property taxes. They are a necessary expense to the property owner, and therefore, should be deducted as an operating expense when developing the OAR. NIBT less the anticipated property taxes produces NIBR.

The steps used in processing the income stream to value a property are summarized as follows:

(GI)
Anticipated Gross Income
[Process to this level to derive GIM]
(minus)
(V&CL)
Anticipated Vacancy and Collection Losses
(equals)
(EGI)
Anticipated Effective Gross Income
[Process to this level to derive EGIM]
(minus)
(OE)
Anticipated Operating Expenses
(equals)
(NIBT)
Anticipated Net Income Before deducting Recapture & Taxes
(minus)
(PT)
Anticipated Property Taxes
(equals)
(NIBR)
Anticipated Net Income Before deducting Recapture
[Process to this level to derive Rates]
(minus)
Anticipated Allowance for Recapture
(equals)
Anticipated Net Income (or Anticipated Yield Income)
Processing the
Income Stream
GI
V&CL
=
EGI
OE
=
NIBT
PT
=
NIBR

The anticipated NIBR divided by the sale price produces a market-derived OAR. An OAR is an income rate for a total property interest that reflects the relationship between a single year's net operating income expectancy and the total property value. The OAR makes no distinction between recapture (return OF an investment) and yield (return ON an investment). The basic formula for deriving an OAR is:

Anticipated NIBR / Sale Price
= OAR

#### EXAMPLE 11–1: Deriving Overall Rates by Market Derivation

A 20-unit apartment complex recently sold for \$850,000. The current contracted rent is \$525 per unit. The buyers stated that they anticipate a 5 percent vacancy and collection loss. The buyer also stated that they anticipate the following annual expenses:

Maintenance and Repair
\$12,750
Insurance
\$4,500
Utilities
\$4,500
Swimming Pool Contract
\$1,500
Management
\$6,300
Apartment Owners Assoc. Dues
\$1,200
Chamber of Commerce Dues
\$1,250
Property Taxes
\$8,500
Income Taxes
\$20,000
TOTAL EXPENSES
\$60,500

The apartment owner's association dues, chamber of commerce dues, and income taxes are excluded from being a part of the operating expenses. (Operating expenses were discussed in Lesson 7, Processing the Income Stream). The association dues, chamber of commerce dues, and income taxes are expenses that are not necessary to maintain the income stream. In addition, Property Tax Rule 8 specifically excludes income taxes as an operating expense in the valuation process. However, property tax is an operating expense and may be deducted from the anticipated income stream in the process to derive OAR. For property tax purposes, however, it is recommended that the deduction be performed as a separate step from deduction of the other operating expenses. For the derivation of an OAR, the allowable operating expenses from the above expense roll, excluding property taxes, is \$29,550.

1. The processing of the income stream is as follows:
Anticipated GI
\$126,000
Anticipated V&CL (\$126,000 × 0.05)
\$6,300
Anticipated EGI
=
\$119,700
Anticipated Expenses
Maintenance & Repair \$12,750
Insurance \$4,500
Utilities \$4,500
Swimming Pool Contract \$1,500
Management \$6,300
\$29,550
Anticipated NIBT
\$90,150
Anticipated Property Taxes
\$8,500
Anticipated NIBR
=
\$81,650
2. The derivation of the OAR is as follows:
Anticipated NIBR / Sale Price
= OAR
\$81,650 (Anticipated NIBR) / \$850,000 (Sale Price)
= 0.096 or 9.6% (OAR)

### Deriving Overall Rates by Band-of-Investment

Rule 8(g)(2) provides that a capitalization rate may also be developed by deriving a weighted average of the capitalization rates for debt and equity, with weights based on the typical loan-to-value ratio for the property being appraised. This methodology is called the band of investment method.

The band-of-investment method is a technique in which the capitalization rates attributable to components of a capital investment are weighted and combined to derive a weighted average rate attributable to the overall investment. The rate developed is a weighted average between the mortgage investment (the lender's component) and the equity investment (the buyer's component). The property interests are separated into bands of investment based upon the financial investment of each participant in a transfer. The band-of-investment method is most useful when market investors are primarily concerned with equity capitalization rates.

Because OARs reflect the complete cash flow requirements of an investment, the components should reflect the complete cash flow requirements for each segment of capital. The cash flow requirement for the debt segment is simply the total requirement to service the debt, including the principal payments as well as the interest. The cash flow rate to the debt component [Rm] is the mortgage constant [ƒ] based on the effective interest rate and the term of the loan. The equity requirement should reflect the cash flow rate that will satisfy an equity investor (Re) in the current market.

The debt capitalization rate [Rm] is the ratio of the annual debt service to the principal amount of a loan. In real estate appraisal, it is called the mortgage constant [ƒ or MC]. The mortgage constant is a function of the interest rate, the frequency of amortization, and the term of the loan. The mortgage constant was discussed in detail in the Time Value of Money - Six Functions of a Dollar, offered by the Board of Equalization at: http://www.boe.ca.gov/info/tvm/, and in Lesson 5 of this learning session. As previously discussed elsewhere, the factors for the mortgage constant for monthly payment, fully-amortizing, loans can be found in column 7 of Assessors' Handbook Section 505, Capitalization Formulas and Tables. For interest‑only loans, the interest rate is the Mortgage Constant; for annual payment, fully-amortizing, loans, the annual Periodic Repayment factor, column 6 of Assessors' Handbook Section 505, Capitalization Formulas and Tables, is the Mortgage Constant.

The equity capitalization rate [RE] is the ratio between the buyer's anticipated pre-tax cash flow to equity and the equity investment in the property. The pre-tax cash flow to equity is the buyer's anticipated NIBR less the annual debt service. The equity investment is the total property value less the outstanding loan balance. Equity capitalization rates can be derived from comparable sales or by surveying investors.

The formula for developing an OAR using the band of investment can be expressed as:

Capital Source Formula
Equity Component: + (1-M) × Re =
OAR or Ro
Where:
M
=
The loan ratio of the sale price. It is the percent the debt component (the portion borrowed) is of the overall investment. When the percent borrowed of the sale price (M) is not known, then M may be derived by using the following formula:
M =
Loan Amount / Sale Price
(1-M)
=
The cash down of the sale price. It is the percent the equity component (the down payment) is of the overall investment. When the percent down of the sale price (1-M) is not known, then 1-M may be derived by using the following formula:
1 - M =
Down Payment ÷ Sale Price
Rm or ƒ
=
Debt Capitalization Rate. It is also referred to as the mortgage constant [MC or ƒ]. It is the ratio of the annual debt service to the principal amount of a loan.
Re
=
Equity Capitalization Rate. It is also referred to as cash flow rate. It is the ratio of annual pre-tax cash flow expectancy to the amount of equity investment.
Ro
=
OAR

#### EXAMPLE 11–2: Deriving Overall Rates by the Band of Investment Method

The buyers of the apartment complex stated that they paid \$125,000 as a down payment of the \$850,000 sale price. This equates to 14.7 percent of the total purchase price (calculated by dividing \$125,000 by \$850,000). The remainder of the purchase price was financed by a new loan bearing an 8 percent interest rate, compounded monthly. The term of the loan is 25 years. Typically, people who invest in apartment complexes expect a 10 percent return on and of their equity investments. The OAR developed by market derived band-of-investment method would be as follows:
Capital Source Weighting Rm or Re Weighted Amount
Debt Component: 85.30% × 0.092618 [25 yrs @ 8%] [Rm] = 0.079003
Equity Component: 14.70% × 0.100000 [Re] = 0.014700
WEIGHTED OAR (Ro) = 0.093703, or 9.4%

If there is more than one mortgage or trust deed, the formula for developing an OAR using the band of investment, as shown above, can be expanded thusly:

Capital Source Formula
1st Trust Deed: M1 × ƒ1 +
2nd Trust Deed: M2 × ƒ2 +
additional Trust Deeds: M × ƒ +
Down Payment (1−(M1+M2+M) × Re =
OAR or Ro

Where the Trust Deeds percentages, M1, M2, M, etc. (with their corresponding rates or mortgage constants, ƒ1, ƒ2, ƒ, etc.) make up the Debt Component, and the Down Payment (and corresponding cash flow rate Re) is the Equity Component.

Re may be determined by interviewing investors of different types of property. If the Re is not known (for example, when there are few or no sales), the Re can be calculated by dividing the pre-tax cash flow of a property by the equity investment. Pre-tax cash flow (equity income) is the portion of net operating income that remains after total mortgage debt service is paid but before ordinary income tax on operations is deducted. Equity Income is calculated by subtracting the annual debt service from the NIBR. The debt service is calculated by multiplying the loan amount by the mortgage constant. The mortgage constant is based on the effective interest rate and the term of the loan. The equity income is converted to Re by dividing the equity income by the equity investment. This can be summarized using the following formulas:

Re = Equity Income ÷ Equity Investment
Equity Income defined below
Equity Income = NIBR − Debt Service
Debt Service defined below
Debt Service = Loan Value × Mortgage Constant [ƒ or MC]

So: Re = [NIBR − (Loan Value × MC)] ÷ Equity Investment

Where:

Equity Income
=
The portion of net operating income that remains after total mortgage debt service is paid but before ordinary income tax on operations is deducted.
Equity Investment
=
The amount of down payment made against the sale price.
Debt Service
=
The amount paid in principal and interest on a loan over a period of time. In general, that period is one year.

Alternatively, Re can also be derived using the following:

GI
Gross Income
(minus)
(V&CL)
Vacancy and Collection
equals
EG
Effective Gross Income
(minus)
(OE)
Operating Expenses
equals
NIBT
Net Income Before Recapture & Taxes
(minus)
(PT)
Property Taxes
equals
NIBR
Net Income Before Recapture
(minus)
Debt Service (Loan Value × MC)
equals
Equity Income
Re =
Equity Income / Equity Investment

#### EXAMPLE 11–3: Deriving Overall Rates by the Band of Investment Method

A property owner recently refinanced his retail store. The bank valued the property at \$1,500,000, and they approved a loan of 75 percent of the appraised value. The effective interest rate is 8.5 percent. The loan is to be paid in full over a 20-year term.

A new lease was negotiated six months ago. The contract rent is \$9,500 per month. Historically, the operating expenses have consumed approximately 30 percent of the annual effective gross income. A vacancy and collection allowance of 7 percent would be typical. The annual property tax is \$22,500. The OAR developed by market derived band-of-investment method would be as follows:

Using the formula:

Capital Source Formula
Debt Component: M × Rm +
Equity Component: (1-M) × Re =
OAR or Ro
Where:
M
=
Percent of the Loan or Mortgage, that is, the Loan Amount divided by the Sales Price or Property Value
Rm
=
The cash flow the lender is receiving, as a percentage of the loan amount – usually the mortgage constant, on a fully-amortized, level payment, loan, or the interest rate, on an interest only loan
Re
=
The cash flow the investor is receiving, as a percentage of the investment, usually the down payment.

We discover we have not been given the Re. As such, we will need to use the following formula to derive Re before we can continue.

Re = Equity Income ÷ Equity Investment
Equity Income defined below
Equity Income = NIBR − Debt Service
Debt Service defined below
Debt Service = Loan Value × Mortgage Constant [ƒ or MC]

Before we work all three lines to determine Re, we examine if we have enough information to process the first line. If not, we determine if we have enough information to process the second line. If not, we start with the third line. For this example, we do not have enough information to process the first or second line, so we will start with the third line.

Starting with the third equation, we need to derive the Debt Service.

Debt Service
=
Loan Value × Mortgage Constant (20 yrs @ 8.5%)
Debt Service
=
\$1,125,000 × 0.1041388
Debt Service
=
\$117,156

Next, we need to derive Equity Income. To do so, we will have to process the income stream to NIBR. The income stream is processed as follows:

Anticipated GI
\$114,000
Anticipated V&CL (\$114,000 × 0.07)
\$7,980
Anticipated EGI
=
\$106,020
Anticipated Operating Expenses
\$31,806
Anticipated NIBT
=
\$74,214
Anticipated Property Taxes
\$22,500
Anticipated NIBR
=
\$51,714

Now we can solve for Equity Income.

Equity Income
=
NIBR – Debt Service
Equity Income
=
\$51,714 – \$117,156 (derived above)
Equity Income
=
(\$65,442)

In this example, the Re is a negative amount. This is not unusual. To increase leverage and increase profits through mortgage financing, many investors incur debt deliberately and seek maximum yields on minimum down payments. The Re can be determined by dividing the equity income by the value of the property held in equity. In this illustration, the value of the property held in equity is \$375,000 (\$1,500,000 × 25 percent).

Now that we have the equity income, we can finally we can derive the Re.

Re
=
Equity Income ÷ Equity Investment
Re
=
(\$65,442) ÷ \$375,000 (down payment)
Re
=
(0.174512) − 17.4512%

The OAR can be calculated by band-of-investment method as follows:

Capital Source Weighting Rm & Re Weighted Amount
Debt Component: 75% × 0.1041388 [20 yrs @8.5%] = 0.078104
Equity Component: 25% × (0.174512) = (0.043628)
WEIGHTED OAR (Ro) = 0.034476, or 3.4%

### Summary

The lesson you just read explained how to derive overall rates using two methods, by using data from comparable sales and the band-of-investment. The next lesson will address valuation using overall rates.

Note: Before proceeding on to the next lesson, be sure to complete the exercises for this lesson.