CFA Level III · Private Markets Pathway

Private Real Estate Investments: formula sheet

Full chapter guide

Key formulas

Levered return (single period): Levered return = [Property return × Total assets − Interest rate × Debt] ÷ Equity; Equivalent to r_property + (Debt ÷ Equity) × (r_property − interest rate). Use when the question asks for the effect of leverage.
Loan-to-value (LTV): LTV = Loan amount ÷ Property value; Higher LTV means more leverage and more risk for the lender and the equity owner.
Debt service coverage ratio (DSCR): DSCR = Net operating income ÷ Debt service; Debt service is interest plus scheduled principal. Higher means more safety for the lender.
Net operating income (NOI): NOI = Rental income + other income − vacancy losses − operating expenses; Excludes depreciation, interest and income taxes.
Direct capitalization value: Value = NOI₁ ÷ Cap rate; Use forward (next-year) NOI if the cap rate is defined that way. Read the question to see which NOI is given.
Cap rate from a discount rate: Cap rate = Discount rate − Growth rate (constant growth in NOI); Holds for a perpetual, constant growth in NOI. A higher growth rate lowers the cap rate and raises value.
Cap rate from market evidence: Cap rate = NOI ÷ Sale price; Taken from comparable sold properties with similar risk, then applied to the subject property.
Net operating income: NOI = Potential gross income − Vacancy and collection loss + Other income − Operating expenses; NOI is before debt service, income taxes, depreciation and capital expenditures.
DCF value: Value = Σ [NOIₜ ÷ (1 + r)ᵗ] + Terminal value ÷ (1 + r)ⁿ; Terminal value is often NOIₙ₊₁ ÷ terminal cap rate, less selling costs if stated.
Cost approach value: Value = Land value + (Replacement cost new − Accumulated depreciation); Depreciation covers physical deterioration, functional obsolescence and external obsolescence.
Effective gross income (EGI): EGI = Potential gross income − Vacancy and collection losses + Other income; Start of the NOI build-up. Use the order the question gives.
Net operating income: NOI = EGI − Operating expenses; Exclude debt service, depreciation, income tax and capital expenditures.
Cap rate: Cap rate = NOI ÷ Property value; Use the NOI definition that matches the market comparables, usually first-year NOI.
Direct capitalization value: Value = NOI ÷ Cap rate; Use forward (next year's) NOI when the cap rate is based on forward NOI.
Cap rate and growth: Cap rate ≈ Discount rate − Growth rate of NOI; Holds approximately for a constant long-run growth rate.
Gross income multiplier: GIM = Value ÷ Gross income; Value = GIM × Gross income; Ignores expense and vacancy differences, so it is cruder than a cap rate.
Terminal value: Terminal value at year N = NOI in year N+1 ÷ Terminal cap rate; Net of selling costs if the question gives them.
DCF value: Value = Σ NOIt ÷ (1 + r)^t + Terminal value ÷ (1 + r)^N; r is the discount rate, not the cap rate.
Total return (single period): Total return = (NOI + (Ending value − Beginning value)) ÷ Beginning value; Equals income return plus capital return. Use consistent NOI and values for the same period. Ignores capital expenditures unless you adjust for them.
Income and capital return split: Income return = NOI ÷ Beginning value; Capital return = (Ending value − Beginning value) ÷ Beginning value; The two parts add to total return.
Appraisal smoothing model: Reported return(t) = α × True return(t) + (1 − α) × Reported return(t−1); α is between 0 and 1. A smaller α means more smoothing. Unsmoothing with this model is a possible exam application, but it is not confirmed whether a calculation will be tested, so learn the mechanics and understand what the model implies.
Unsmoothing (de-smoothing) a return series: True return(t) = [Reported return(t) − (1 − α) × Reported return(t−1)] ÷ α; Rearranged from the smoothing model. Unsmoothed volatility is higher than reported volatility when α < 1.
Effect of smoothing on standard deviation: σ(true) ≈ σ(reported) × √((2 − α) ÷ α), assuming serially uncorrelated true returns under the first-order smoothing model; Derived from σ²(reported) = α × σ²(true) ÷ (2 − α). Only an approximation. Equivalently, σ(reported) ÷ σ(true) = √(α ÷ (2 − α)), also approximate. The key point is the direction: unsmoothing raises volatility.
Loan-to-value: LTV = Loan amount ÷ Property value; Lower is safer for the lender. Use appraised value or purchase price as the question states.
Debt service coverage ratio: DSCR = NOI ÷ Debt service; Debt service = interest plus scheduled principal for the year. A DSCR below 1.0 means NOI cannot cover payments.
Debt yield: Debt yield = NOI ÷ Loan amount; Ignores interest rate and amortization, so loans can be compared on income alone.
Interest coverage ratio: Interest coverage = NOI ÷ Interest expense; Like DSCR but excludes principal repayment.
Levered return (one period): Equity return = [Property return × Total assets − Interest rate × Debt] ÷ Equity; Equivalent form: r_E = r_P + (D ÷ E) × (r_P − r_D), where r_P is the property return and r_D the cost of debt, both on the same basis.
Equity and debt link: Equity = Property value − Loan amount; Debt-to-equity D ÷ E = LTV ÷ (1 − LTV).
Loan amount from a DSCR limit: Maximum debt service = NOI ÷ Minimum DSCR; Then convert to a loan size using the payment factor or interest rate.
Total return of a property: Total return = income return + capital return; Income return = NOI ÷ beginning value. Capital return = (ending value − beginning value) ÷ beginning value. Before leverage and fees.
Equity return with leverage (approximate): Levered return ≈ unlevered return + (D ÷ E) × (unlevered return − cost of debt); D ÷ E is debt to equity. Leverage helps only when the unlevered return exceeds the cost of debt. Use the same pre-tax basis throughout.
Net return after fees: Net return = gross return − fees and costs (as a percent of the same base); Make sure fees are measured on the same base, such as invested equity, before subtracting.
Smoothing effect on volatility: Unsmoothed volatility > reported appraisal-based volatility; Rule of direction only. Unsmoothing raises estimated volatility and correlation with other assets.

Quick revision

NOI = effective gross income minus operating expenses, before debt service and income tax.
Value by direct capitalization = NOI ÷ cap rate.
Cap rate = NOI ÷ value, so a higher cap rate means a lower value for the same NOI.
Use forward-looking NOI consistently with how the cap rate was derived.
Main approaches: income, sales comparison and cost.
Cost approach is weakest for older or unusual properties.
Appraisal-based indexes are smoothed and lag, so volatility and correlations look too low.
Leverage magnifies gains and losses; it helps only if property return exceeds the cost of debt.
Lenders look at loan-to-value and debt service coverage.
Private real estate is illiquid, so check the client's liquidity needs first.
Real estate can diversify, but reported diversification is overstated by smoothing.
Tie every recommendation to objectives and constraints.

Common mistakes

Saying private real estate is diversified because it is a real asset. Fix: A single property is concentrated in one location and type. Diversification needs a pool of properties.
Treating appraisal-based returns as true market volatility. Fix: Say that appraisals lag and smooth returns, so risk is understated and correlations look lower than reality.
Subtracting debt service or depreciation when calculating NOI Fix: NOI is before debt service, income taxes, depreciation and capital expenditures. Only operating expenses and vacancy come off.
Using the wrong year's NOI with the cap rate Fix: Match NOI to the cap rate definition. If the cap rate is applied to forward NOI, use year-1 NOI.
Deducting interest, depreciation or income tax when computing NOI. Fix: NOI is before financing, depreciation and taxes. Deduct only operating expenses.
Using the discount rate as the cap rate. Fix: Cap rate ≈ discount rate − growth. They are equal only when NOI growth is zero.
Saying appraisal smoothing raises volatility. Fix: Smoothing lowers reported volatility and correlation. Unsmoothing raises them.
Treating repeat sales and hedonic indexes as the same thing. Fix: Repeat sales compares prices of the same property over time. Hedonic uses a regression across different properties with their characteristics.
Using total debt payments over several years, or monthly payments, against annual NOI. Fix: Convert to an annual figure first. Then divide annual NOI by annual debt service.
Using net income or cash flow after capital items instead of NOI in DSCR. Fix: Use NOI as given unless the question states another income measure.

Exam tips

Always tie each answer to a stated client objective or constraint; generic feature lists lose points.
For calculations, show the interest cost and equity base so a slip still shows method.
Know the trade-offs by form: direct (control, concentration, illiquid), pooled (diversified, fees), public (liquid, equity-like volatility).
Answer only as many points as asked; extra responses are not evaluated.
Read the command word. 'Calculate' needs a number; 'justify' needs a reason tied to the property and data.
In essays, a correct number typed on its own earns full credit for a calculation, so type the number and move on.
Check which NOI year and which cap rate the question uses before dividing.
When asked to choose an approach, name the property type and the data availability in one sentence each.