Overview
This study guide covers the essential mathematical and appraisal concepts tested on the Real Estate Broker Exam. Topics include commission calculations, the three approaches to value, depreciation methods, income capitalization, and property tax math. Mastering these formulas and principles is critical for both the exam and professional practice.
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The Golden Rule of Real Estate Math
Before diving in, memorize this framework. Almost every math problem uses one of these three relationships:
$$\text{Part} = \text{Total} \times \text{Rate}$$
$$\text{Rate} = \text{Part} \div \text{Total}$$
$$\text{Total} = \text{Part} \div \text{Rate}$$
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Commission & Price Calculations
Key Concepts
Commission problems test your ability to work forward (finding commission from sale price) and backward (finding sale price from net proceeds or commission earned).
Core Formulas
| Goal | Formula |
|------|---------|
| Find total commission | Sale Price × Commission Rate |
| Find agent's share | Total Commission × Agent's Split % |
| Find sale price from commission earned | Commission Earned ÷ Commission Rate |
| Find sale price from net proceeds | Net Proceeds ÷ (1 − Commission Rate) |
Worked Examples
Forward Commission:
• Property: $425,000 × 6% = $25,500 total commission
• Listing agent at 60%: $25,500 × 0.60 = $15,300
Backward from Net Proceeds:
• Seller wants to net $280,000 after 7% commission
• $280,000 ÷ (1 − 0.07) = $280,000 ÷ 0.93 = $301,075
Backward from Commission Earned:
• Agent earned $9,450 at 6% rate
• $9,450 ÷ 0.06 = $157,500 sale price
Key Terms
• Commission Rate – Percentage of sale price paid to the brokerage
• Net Proceeds – What the seller receives after commission is deducted
• Commission Split – Division of total commission between broker and agent
• Complement of the Rate – (1 − rate); used in net proceeds problems
Watch Out For ⚠️
• Commission is always calculated on the SALE price, not the list price. If a property lists for $350,000 but sells at 95%, calculate the commission on $332,500.
• The net proceeds formula uses the complement (1 − rate), not the rate itself. A common mistake is dividing by the rate directly.
• Multi-step splits: Read carefully — problems may involve a broker-to-agent split AND a listing-to-buyer-agent split.
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Appraisal: The Three Approaches to Value
Overview of the Three Approaches
| Approach | Best Used For | Core Concept |
|----------|--------------|--------------|
| Sales Comparison | Residential properties | Adjust comparable sales to match subject |
| Cost Approach | New construction, special-use properties | Land + Depreciated Cost of Improvements |
| Income Capitalization | Income-producing properties | Convert NOI into value |
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Appraisal Principles
Key Terms & Definitions
• Substitution – A buyer will pay no more than the cost to acquire an equally desirable substitute. (Foundation of all three approaches)
• Conformity – Maximum value is achieved when a property is similar to its neighbors (homogeneous neighborhood)
• Progression – A lower-value property gains value when surrounded by higher-value properties
• Regression – A higher-value property loses value when surrounded by lower-value properties
• Contribution – Value of a component = what it adds to total value, not its actual cost
• Anticipation – Value is based on the expectation of future benefits
• Supply & Demand – Value is influenced by the balance of available properties vs. buyer demand
• Highest & Best Use – The legally permissible, physically possible, financially feasible, and maximally productive use of a property
Watch Out For ⚠️
• Progression vs. Regression: These are opposites. Progression = benefit from being near higher-value properties. Regression = harm from being near lower-value properties.
• Contribution ≠ Cost: Adding a $30,000 pool may only contribute $15,000 in value — the principle of contribution says the value added is what matters.
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Sales Comparison Approach
The Adjustment Rule
> "CBS" — Comparable Better, Subtract; Comparable Worse, Add
| Comparable vs. Subject | Direction of Adjustment |
|------------------------|------------------------|
| Comparable is superior (has feature subject lacks) | Subtract from comp's price |
| Comparable is inferior (lacks feature subject has) | Add to comp's price |
Worked Example
• Comparable sold for $310,000 but lacks a garage the subject has (valued at $18,000)
• Comparable is inferior → Add: $310,000 + $18,000 = $328,000 adjusted price
Key Terms
• Subject Property – The property being appraised
• Comparable (Comp) – A recently sold, similar property used for comparison
• Adjustment – Dollar amount added or subtracted to account for differences
• Adjusted Sale Price – The comp's price after all adjustments
Watch Out For ⚠️
• Adjustments are always made to the COMPARABLE, never to the subject property.
• Think of it this way: "What would the comp have sold for if it were identical to the subject?"
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Income Capitalization Approach
Core Formulas
$$\text{NOI} = \text{Effective Gross Income} - \text{Operating Expenses}$$
$$\text{Effective Gross Income} = \text{Gross Income} \times (1 - \text{Vacancy Rate})$$
$$\text{Value} = \text{NOI} \div \text{Cap Rate}$$
$$\text{Cap Rate} = \text{NOI} \div \text{Value}$$
$$\text{NOI} = \text{Value} \times \text{Cap Rate}$$
IRV Triangle
Use this memory device — cover the variable you want to find:
```
[ I ]
[ R V ]
```
• I = Income (NOI)
• R = Rate (Cap Rate)
• V = Value
Worked Examples
Finding NOI:
• Gross income: $95,000
• 20% vacancy: $95,000 × 0.80 = $76,000 EGI
• Operating expenses: $28,000
• NOI = $76,000 − $28,000 = $48,000
Finding Value:
• NOI: $54,000 ÷ Cap Rate: 9% = $600,000
Finding Cap Rate:
• NOI: $60,000 ÷ Value: $750,000 = 8%
Gross Rent Multiplier (GRM)
$$\text{GRM} = \text{Sale Price} \div \text{Gross Annual Rent}$$
$$\text{Value} = \text{GRM} \times \text{Gross Annual Rent}$$
Example:
• 12 units × $900/month × 12 months = $129,600 annual rent
• $129,600 × GRM of 110 = $1,188,000
Key Terms
• Gross Income – Total potential rental income at 100% occupancy
• Vacancy & Credit Loss – Allowance for empty units and uncollected rent
• Effective Gross Income (EGI) – Gross income minus vacancy and credit loss
• Net Operating Income (NOI) – EGI minus operating expenses (does NOT include mortgage payments)
• Capitalization Rate (Cap Rate) – Rate of return used to convert income into value
• Gross Rent Multiplier (GRM) – Quick valuation tool; ratio of price to gross rent
Watch Out For ⚠️
• NOI does NOT deduct mortgage payments (debt service). Operating expenses only.
• Higher cap rates = lower value (and indicate higher risk). Lower cap rates = higher value.
• GRM is a quick estimate only — it ignores operating expenses, making it less accurate than full capitalization.
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Cost Approach & Depreciation
The Cost Approach Formula
$$\text{Indicated Value} = \text{Land Value} + (\text{Reproduction/Replacement Cost} - \text{Accrued Depreciation})$$
> Important: Land is never depreciated. Only improvements depreciate.
Straight-Line Depreciation
$$\text{Annual Depreciation} = \frac{\text{Improvement Cost}}{\text{Economic Life (years)}}$$
$$\text{Accrued Depreciation} = \text{Annual Depreciation} \times \text{Effective Age}$$
Example:
• Building cost: $280,000 | Economic life: 50 years | Age: 10 years
• Annual: $280,000 ÷ 50 = $5,600/year
• Accrued: $5,600 × 10 = $56,000
Three Categories of Depreciation
| Type | Description | Example |
|------|-------------|---------|
| Physical Deterioration | Wear and tear from use and age | Worn roof, cracked foundation |
| Functional Obsolescence | Outdated design or features | One bathroom in a 5-bedroom home |
| External (Economic) Obsolescence | Outside forces reducing value | New highway built nearby |
Curable vs. Incurable Depreciation
• Curable – Cost to fix ≤ Value added (makes economic sense to repair)
• Incurable – Cost to fix > Value added (not economically feasible to repair)
Worked Example (Full Cost Approach)
• Land value: $75,000
• Reproduction cost: $310,000
• Accrued depreciation: $42,000
• Value = $75,000 + ($310,000 − $42,000) = $343,000
Key Terms
• Reproduction Cost – Cost to build an exact replica using same materials
• Replacement Cost – Cost to build a functionally equivalent building using current materials
• Economic Life – Total estimated useful life of the improvement
• Effective Age – Age based on condition, which may differ from actual/chronological age
• Accrued Depreciation – Total depreciation from all causes to date
Watch Out For ⚠️
• External obsolescence is always incurable — you cannot fix what's outside the property.
• Know the difference between reproduction (exact copy) and replacement (functional equivalent).
• Effective age ≠ actual age. A well-maintained 20-year-old building may have an effective age of 10 years.
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Property & Investment Math
Property Tax Calculations
$$\text{Assessed Value} = \text{Market Value} \times \text{Assessment Ratio}$$
$$\text{Annual Tax} = \text{Assessed Value} \times \text{Tax Rate}$$
Tax Rate Conversions:
• Per $100: Divide assessed value by 100, then multiply by rate
• Per $1,000 (mills): Multiply assessed value by (mills ÷ 1,000) or by the decimal equivalent
| Expression | Equivalent |
|-----------|-----------|
| 1 mill | $0.001 per dollar of AV |
| 1 mill | $1 per $1,000 of AV |
| 35 mills | $0.035 per dollar of AV |
Example:
• Market value: $420,000 × 60% = $252,000 assessed value
• 35 mills = $252,000 × 0.035 = $8,820 annual tax
Area & Acreage Calculations
$$\text{Area} = \text{Length} \times \text{Width (in square feet)}$$
$$\text{Acres} = \text{Square Footage} \div 43,560$$
Key Conversion: 1 acre = 43,560 square feet
Example:
• 660 ft × 1,320 ft = 871,200 sq ft
• 871,200 ÷ 43,560 = 20 acres
Appreciation/Return
$$\text{Appreciation \%} = \frac{\text{Profit}}{\text{Original Price}} \times 100$$
Example:
• Bought $500,000, sold $575,000
• $75,000 ÷ $500,000 = 15% appreciation
Key Terms
• Assessed Value – Value assigned by tax assessor (often a % of market value)
• Assessment Ratio – Percentage of market value used for tax purposes
• Mill – Unit of tax rate equal to $0.001 or $1 per $1,000 of assessed value
• Millage Rate – Tax rate expressed in mills
Watch Out For ⚠️
• Always calculate assessed value FIRST before applying the tax rate.
• Mills must be converted to decimals: 35 mills = 0.035 (divide mills by 1,000).
• Area formulas require consistent units. Convert everything to feet before calculating.
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Master Formula Reference Sheet
| Calculation | Formula |
|------------|---------|
| Total Commission | Sale Price × Rate |
| Sale Price (from net) | Net Proceeds ÷ (1 − Rate) |
| Sale Price (from commission) | Commission ÷ Rate |
| NOI | EGI − Operating Expenses |
| EGI | Gross Income × (1 − Vacancy Rate) |
| Property Value (income) | NOI ÷ Cap Rate |
| Cap Rate | NOI ÷ Value |
| GRM | Sale Price ÷ Gross Annual Rent |
| Annual Depreciation | Cost ÷ Economic Life |
| Accrued Depreciation | Annual Depreciation × Effective Age |
| Cost Approach Value | Land + (Repro Cost − Accrued Depreciation) |
| Assessed Value | Market Value × Assessment Ratio |
| Property Tax | Assessed Value × Tax Rate |
| Appreciation % | Profit ÷ Original Price |
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Quick Review Checklist
• [ ] I can calculate total commission and agent splits in multi-step problems
• [ ] I can work backward from net proceeds to find required sale price using the complement formula
• [ ] I can work backward from commission earned to find the sale price
• [ ] I know when to use each of the three appraisal approaches
• [ ] I can apply the CBS rule for sales comparison adjustments (Comparable Better = Subtract)
• [ ] I understand the IRV triangle and can solve for any missing variable
• [ ] I can calculate NOI step-by-step: Gross Income → EGI → NOI
• [ ] I know the difference between cap rate and GRM, and the limitations of each
• [ ] I can compute straight-line depreciation (annual and accrued)
• [ ] I can identify the three types of depreciation and distinguish curable from incurable
• [ ] I can complete a full Cost Approach problem (Land + Cost − Depreciation)
• [ ] I can convert mills to decimal form and calculate property taxes
• [ ] I remember that land is NEVER depreciated
• [ ] I know that 1 acre = 43,560 square feet
• [ ] I understand the appraisal principles (substitution, conformity, progression, regression, contribution)