# Lesson 9 – Multipliers: Derivation and Valuation (The Income Approach to Value)

## Appraisal Training: Self-Paced Online Learning Session

### Multipliers – Derivation and Valuation

The investor decides to purchase property based on their anticipated income, expenses, and recapture. Appraisers use this direct relationship to derive multipliers and rates.

In Lessons 7 and 8, we discussed the use of anticipated income to derive income multipliers and the levels of the income stream to be processed to, in order to derive gross income multipliers and effective gross income multipliers.  The summary chart looked like this:

(PGI)
Anticipated Potential 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)

Where:

GIM
=
Gross Income Multiplier
EGIM
=
Effective Gross Income Multiplier

In this lesson, we will explain:

• How to develop a gross income and effective gross income multipliers.
• How to derive an opinion of value using the multipliers and applying the multiplier formula discussed in Lesson 8,
V = I × M.

Income may be capitalized by the use of gross income [GIM and EGIM], gross rent [GRM], or gross production [GPM] multipliers, derived by comparing sales prices of closely comparable properties (adjusted, if necessary, to cash equivalents) with their gross income, gross rents, or gross production.

Typically, the GIM [Gross Income Multiplier] considers gross annual income from all sources; other multipliers may sometimes be used, such as those using monthly income. The GRM [Gross Rent Multiplier] is the same concept, but uses only rental income and usually excludes income from other sources, such as billboards, laundry machines, parking, storage, and vending machines.

### Deriving Gross Income Multipliers

As the anticipated income stream is processed to different levels, various income multipliers and rates can be derived. Both multipliers and rates represent a relationship between a property's potential to produce an income stream and the value of that property.

Rule 8(h) states that income multipliers may be derived by comparing sale prices of closely comparable properties with their gross incomes, gross rents, or gross production. The investor's anticipated potential gross income is used to find a gross income multiplier by dividing the sale price, adjusted for cash equivalency, by the anticipated potential gross income.

The investor's anticipated potential gross income is used to find a Gross Income Multiplier (GIM). To derive a GIM, the sale price is divided by the anticipated potential gross income. This is expressed in the following formula:

Sale Price / Anticipated Potential Gross Income
= Gross Income Multiplier (GIM)
Processing the
Income Stream
PGI
-
V&CL
=
EGI
-
OE
=
NIBT
-
PT
=
NIBR

#### EXAMPLE 9-1: Deriving a Gross Income Multiplier (GIM)

Assume that a property recently changed ownership. The sale price, adjusted for cash equivalency, is \$850,000. The property is improved with a 20-unit apartment complex. Each unit is anticipated to rent for \$525 per month. The processing of the income stream and the derivation of the GIM are as follows:

1. Solve for PGI. Remember, PGI is always an annual number.

\$525 (per unit per month) × 20 (units) × 12 (months) = \$126,000 (PGI)

2. Solve for GIM using the formula: Sale Price / Anticipated PGI = GIM

\$850,000 (Sale Price) / \$126,000 (PGI) = 6.75 (GIM)

Deriving an Effective Gross Income Multiplier (EGIM) is done in the same manner as deriving a GIM; however, the income stream must be processed to the effective gross income level (EGI). EGI is the anticipated income from all rents after an allowance is made for vacancy and collection losses. Subtracting the anticipated vacancy and collection losses from the anticipated potential gross income produces an anticipated effective gross income. The sale price divided by the effective gross income equals the EGIM. This is expressed in the following formula:

Sale Price / Anticipated Eff. Gross Income
= Eff. Gross Income Multiplier (EGIM)
Processing the
Income Stream
GI
-
V&CL
=
EGI
-
OE
=
NIBT
-
PT
=
NIBR

#### EXAMPLE 9-2: Deriving an Effective Gross Income Multiplier (EGIM)

In the first example of the apartment complex, the buyers stated that they anticipated a loss of 5 percent rental income per year because of vacancy and collection losses. The income stream would be processed and an EGIM would be derived as follows:

1. Solve for EGI. Given a 5 percent vacancy and collection loss, subtract the value of 5 percent of the PGI from the PGI to arrive at an \$119,700 EGI.
Anticipated PGI
\$126,000
Anticipated V&CL (\$126,000 × 0.05)
-
6,300
Anticipated EGI
=
\$119,700
2. Solve for the EGIM using the formula: Sale Price / Anticipated EGI = EGIM

\$850,000 (Sale Price) / \$119,700 (Anticipated EGI) = 7.10 (EGIM)

When appraising smaller residential properties, appraisers commonly use a gross rent multiplier because such properties typically have no, or very little, non-rental income. Appraisers use GIM for properties, such as large apartment projects, that produce significant non-rental income attributable to the real property (e.g., income from parking, laundry facilities, or rental of storage areas). Gross production multipliers, also mentioned in Rule 8, are multipliers based on measures other than income. Appraisers seldom use them in real property appraisal.

#### Demonstration of Deriving a Multiplier

DEMONSTRATION 9-1: Deriving Gross Income Multiplier

In this demonstration, we will show you how to derive a gross income multiplier for comparable sales.

The subject property is a 10-unit commercial office building. It has a potential gross income of \$XXX.  There has been three recent sales of commercial office buildings. All of the sales are about the same age and condition as the subject property. They are also located in similar locations.

## Recent Sales of Commerical Office Buildings

Sale No. Sales Price Number of Units Anticipated Monthly Rent per Unit
1 \$2,300,000 15 \$1,100
2 \$1,950,000 13 \$1,050
3 \$2,100,000 13 \$1,200

The cost estimates for the subject property indicate that it would cost \$xxx to replace the existing improvements.

SOLUTION: In order to determine the gross income multiplier for each sale, you must first calculate the anticipated gross income for each property based on the number of units and monthly rent then annualize it. The gross income multiplier can then be calculated by dividing the sale price by the anticipated gross income.

## Sale Price ÷ Anticipated Gross Income = Gross Income Multiplier

Sale Number Number of Units Monthly Rent Number of Months Anticipated Gross Income
1 15 × \$1,100 × 12 = \$198,000
2 13 × \$1,050 × 12 = \$163,800
3 13 × \$1,200 × 12 = \$187,200

## Sale Price ÷ Anticipated Gross Income = Gross Income Multiplier

Sale Number Sale Price Gross Income GIM
1 \$2,300,000 ÷ \$198,000 = 11.616
2 \$1,950,000 ÷ \$163,800 = 11.905
3 \$2,100,000 ÷ \$187,200 = 11.218
DEMONSTRATION 9-2: Deriving Effective Gross Income Multiplier [EffGIM]

In this demonstration, we will show you how to derive an effective gross income multiplier (EGIM). The subject property is a 45 unit two-story apartment building; all units have two bedrooms and one bathroom. The subject property recently sold for \$2,750,000. Per returned income and expense questionnaire completed by the new owner, they purchased the property based on an anticipated monthly rental income of \$1,250 per unit; based trends in the current rental market, they also anticipated vacancy and collection loss of 7% per month. Derive an EGIM for the subject property.

Steps for deriving an EffGIM for the subject property:

1. Calculate the annualized anticipated gross income based on the anticipated monthly rent and the number of units. In this case:
Number of Units Monthly Rent Number of Months Anticipated Gross Income
45 × \$1,250 × 12 = \$675,000
2. Deduct the anticipated vacancy and collection loss from the anticipated gross income, to estimate the anticipated effective gross income.
Anticipated Gross Income Anticipated V&C loss V&C Loss Anticipated EffGI
\$675,000 × 7% = (\$47,250) = \$627,750
3. To derive the EffGIM for the subject property divide the subject's sales price by its indicated anticipated effective gross income.
Sale Price Anticipated EffGI EffGIM
\$2,750,000 ÷ \$627,750 = 4.381

In valuing the subject property three comparable apartment buildings were identified. Comparable properties are other apartments with all two bedroom and one bath units. The following table represents the buyers' anticipated income and vacancy and collection loss at the time of sale for each of the comparable properties.

Sale Number Sale Price Number of Units Monthly Rent per Unit Anticipated Vacancy and Collection Loss
1 \$3,000,000 50 \$1,200 6%
2 \$2,625,000 42 \$1,250 8%
3 \$2,668,000 46 \$1,175 5%

To derive an Effective Gross Income Multiplier (EffGIM) for each of the comparable properties:

1. Calculate each property's anticipated potential gross income.
Sale Number Number of Units Monthly Rent Number of Months Anticipated Gross Income
1 50 × \$1,200 × 12 = \$720,000
2 42 × \$1,250 × 12 = \$630,000
3 46 × \$1,175 × 12 = \$648,600
2. Calculate each property's anticipated effective gross income.
Sale Number Number of Units Monthly Rent Number of Months Anticipated Gross Income
1 50 × \$1,200 × 12 = \$720,000
2 42 × \$1,250 × 12 = \$630,000
3 46 × \$1,175 × 12 = \$648,600
Sale Number Anticipated Gross Income Anticipated V&C loss V&C Loss Anticipated EffGI
1 \$720,000 × 6% = (\$43,200) = \$676,800
2 \$630,000 × 8% = (\$50,400) = \$579,600
3 \$648,600 × 5% = (\$32,430) = \$616,170
3. Divide the property's sale price by the anticipated effective gross income.
Sale Number Sale Price Anticipated EffGI EffGIM
1 \$3,000,000 ÷ \$676,800 = 4.433
2 \$2,625,000 ÷ \$579,600 = 4.529
3 \$2,668,000 ÷ \$616,170 = 4.330
DEMONSTRATION 9-3: Deriving Effective Gross Income Multiplier [EffGIM] That Includes Personal Property

Sales of certain commercial property types may on occasion include the purchase of personal property that is necessary for the operation of the business and the generation of income by the real estate. Examples of such property types with on-going businesses are motels, hotels, restaurants, etc. For these commercial property types you can derive an EffGIM that includes the value of the personal property.

For example, you have a sale of a 100 room hotel for \$28,000,000. Included in the purchase price was \$5,600,000 of personal property. The hotel has a stabilized occupancy factor of 70% and an average daily room rate of \$125; the buyer expects the room rate and occupancy level will continue.

What is the indicated EffGIM for the total property, including the value of the personal property?

1. Calculate the subject property's anticipated gross income.
Number of Rooms Average Daily Rate Days in a Year Anticipated Gross Income
100 × \$125.00 × 365 = \$4,562,500
2. Calculate the subject property's anticipated effective gross income.
Anticipated Gross Income Occupancy Factor Anticipated EffGI
\$4,562,500 × 70% = \$3,193,750
3. Derive EGIM by dividing the sales price attributed to the real estate and personal property by the effective gross income.
Sale Price EffGI EffGIM
\$28,000,000 ÷ \$3,193,750 = 8.767

Remember: When you derive a multiplier from a sale that includes personal property, that multiplier must only be applied to other properties that have income generated from a similar mix of real property and personal property.

### Deriving Value from Multipliers

Income multipliers are very useful in the valuation of income-producing properties; however, they are very sensitive to variations in comparable characteristics, and therefore, must be used with care. The properties analyzed, from which the income multipliers were derived, must be closely comparable to the subject property. Properties that are very similar physically can have very different operating expense ratios, and therefore, not be comparable enough to the subject property to be used for valuation purposes. The properties must be comparable in remaining economic life, use, expense ratios, amount of vacancy, risk, tax rates, and other units of comparison. Vacancy and collection losses are less important when using EGIMs.

The income multiplier must be applied on the same basis in which it was derived.  In other words, an income multiplier based on potential gross income can only be applied to the potential gross income of the subject property. Likewise, an income multiplier based on effective gross income can only be applied to the effective gross income of the subject property.

The basic formulas for valuing property using income multipliers, as discussed in Lesson 6, is to multiply the potential gross income (PGI) by a GIM derived or to multiply the effective gross income (EGI) by an EGIM. Note that the market income derived from sales of comparable properties and not the buyer's anticipated income is used. Thus:

PGI
×
GIM
=
Value
EffGI
×
EffGIM
=
Value

#### EXAMPLE 9-3: Deriving Value from GIM

The subject property is an apartment complex with 15 two-bedroom units. A review of several comparable complexes indicates that the typical rent for apartment units similar to the subject complex is \$925 per month. A GIM of 7.0 is reported to be typical for the market place by area brokers. The subject property's indicated market value, using a GIM is as follows:

1. Calculate the PGI for the apartment complex.

\$925 (per unit per month) × 15 (units) × 12 (months) = \$166,500 (PGI)

Calculate the value of the subject using the formula: PGI × GIM = Value

\$166,500 (PGI) × 7.0 (GIM) = \$1,165,500 (Value)

GIMs are easily applied, but they should only be used when the comparable sales are very similar to the subject property. Thus, each comparable sale must be comparable to the subject property in terms of its income potential, expense ratios, location, land to building ratio, and physical characteristics.

When using a GIM, an appraiser capitalizes the market potential gross income of the subject property before allowing for property related expenses. Any variance in the relationship of net income to gross income, between the subject property and each comparable sale, will not be reflected in the resulting value estimate. Such a variance may distort the value estimate.

#### Demonstration of Deriving Value

DEMONSTRATION 9-4: Application of Gross Income Multiplier to Derive Value

In this demonstration, we will show you how to derive an estimate of value using a gross income multiplier. For purposes of this demonstration, the GIM will be given – it is 5.0.

The subject property is a 20 unit apartment building. A review of apartments that are in comparable locations to the subject, and whose units are comparable in size and amenities to the subject are renting for \$1,000 per month; which you determine to be economic rent for the subject property as well. Using a GIM of 5.0, what is the indicated value of the subject property?

SOLUTION:

1. First calculate the subject property's potential gross income using economic rents:
Number of Units Economic Rent per unit per month Number of Months in year PGI
20 × \$1,000 × 12 = \$240,000
2. Multiply the potential gross income by the market derived GIM of 5.0 to arrive at an indicated market value for the subject property.
PGI GIM Market Value
\$240,000 × 5.0 = \$1,200,000
DEMONSTRATION 9-5: Application of Gross Income Multiplier to Derive Value

In this demonstration, we will show you how to derive an estimate of value using gross income multipliers of comparable sales.

The property we are valuing is a non-franchise independently-owned 4,000 square foot fast food restaurant located on a one acre parcel. The following is sales and rent information from four other fast food restaurants.

Sale Number Sale Price Building Area Rent per square foot per month Remarks
1 \$1,050,000 4,000 square feet \$2.25 Similar location and parcel size
2 \$1,115,000 4,200 square feet \$2.25 Similar location and parcel size
3 \$950,000 4,500 square feet \$2.00 Inferior in location; smaller parcel size
4 \$1,950,000 5,250 square feet \$2.75 Superior location, larger parcel
1. To derive a GIM from each of the comparable sales, first determine each property's PGI:
Sale Number Building Area Rent per square feet per month Number of Months PGI
1 4,000 square feet × \$2.25 × 12 = \$108,000
2 4,200 square feet × \$2.25 × 12 = \$113,400
3 4,500 square feet × \$2.00 × 12 = \$108,000
4 5,250 square feet × \$2.75 × 12 = \$173,250
2. To derive a GIM divide the sale price of each comparable property by its indicated potential gross income:
Sale Number Sale Price PGI GIM
1 \$1,050,000 ÷ \$108,000 = 9.722
2 \$1,115,000 ÷ \$113,400 = 9.833
3 \$950,000 ÷ \$108,000 = 8.796
4 \$1,950,000 ÷ \$173,250 = 11.255
3. Determine what economic rent for the subject property is and calculate its potential gross income.
Sale Number Sale Price Building Area Rent per square feet per month Remarks
Subject N/A 4,000 square feet
1 \$1,050,000 4,000 square feet \$2.25 Similar location and parcel size
2 \$1,115,000 4,200 square feet \$2.25 Similar location and parcel size
3 \$950,000 4,500 square feet \$2.00 Inferior in location; smaller parcel size
4 \$1,950,000 5,250 square feet \$2.75 Superior location, larger parcel

Comparable sales number 1 and 2 are most similar to the subject in location, building area, and parcel size. Economic rent of \$2.25 per square foot is indicated for the subject property. Potential gross income for the subject is:

Building Area Rent per square feet per month Number of Months PGI
4,000 square feet × \$2.25 × 12 = \$108,000
4. Select an appropriate GIM and derive the subject property's value.
Sale Number Building Area PGI GIM Remarks
Subject 4,000 square feet \$108,000
1 4,000 square feet \$108,000 9.722 Similar location and parcel size
2 4,200 square feet \$113,400 9.833 Similar location and parcel size
3 4,500 square feet \$108,000 8.796 Inferior in location; smaller parcel size
4 5,250 square feet \$173,250 11.255 Superior location, larger parcel

Comparable sales number 1 and 2 are most similar to the subject with number 1 having the same building area as the subject as well being similar location and parcel size. A GIM of 9.722 is deemed appropriate.

PGI GIM Indicated Value
\$108,000 × 9.722 = \$1,050,000

### Summary

The lesson you just read discussed the derivation of multipliers and how to arrive at an estimate of value using multipliers. The next lesson will reiterate some important information about rates and factors and provide definitions for the various types of rates, some which were discussed in previous lessons and others that will be newly introduced.

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