Capitalization: Rate of Change & Discounting

Chapter: Capitalization: Rate of Change & Discounting

This chapter explores the dynamics of capitalization rates, specifically focusing on how anticipated changes in income and value, along with discounting principles, influence their calculation and application in real estate valuation. We will delve into various models that account for these factors, providing a comprehensive understanding of their scientific underpinnings and practical implications.

1. Understanding the Impact of Rate of Change

Real estate valuation often involves projecting future income streams. A critical aspect is anticipating how these income streams will change over time. These changes can be due to various factors like market trends, lease expirations, property condition, or economic cycles. Capitalization rates must reflect❓ these anticipated changes in order to provide accurate value estimates.

1.1. Straight-Line Change Model: A Simplified Approach

The straight-line capitalization procedure presents a basic model for understanding the change in income and value over time. In this approach, it’s assumed that the decrease or increase in income and value occurs evenly throughout the asset’s life.
* Concept: Assumes a constant, linear change in income and/or value over a specific period.
* Mathematical Representation:
* R = Y - (A / n)
* Where:
* R = Capitalization rate
* Y = Yield rate (required rate of return)
* A = Total relative change in value over n periods (expressed as a decimal, e.g., -0.25 for a 25% decline)
* n = Number of periods (typically years)
* Example:
* A property generates $19,000 income in the first year, and its value is expected to decline by 25% over 10 years. An investor desires a 12% yield.

*   R = 0.12 - (-0.25 / 10) = 0.12 + 0.025 = 0.145

*   Value = $19,000 / 0.145 = $131,034.48

• Relationship to Recapture: In the context of depreciating assets, the periodic rate of change can be interpreted as the recapture rate, and the reciprocal of this rate represents the economic life of the asset.

• Practical applications:

  • A leased fee interest where income is expected to decline predictably.
  • Situations where partial or complete return of invested capital occurs at resale.

• Limitations:

  • The straight-line premise is often unrealistic; market conditions rarely exhibit a consistent, linear trend.

1.2. Exponential-Curve (Constant-Ratio) Changes

This model assumes that both income and value change at a constant percentage rate each period. This is also known as the “frozen cap rate” pattern, due to the capitalization rate remaining constant.
* Concept: Income and value change at a consistent compound rate.
* Mathematical Representation:
* R = Y - CR
* Where:
* R = Capitalization rate
* Y = Yield rate (required rate of return)
* CR = Compound rate of change (expressed as a decimal; positive for growth, negative for decline)

• Example:

  • A property generates $50,000 net operating income (NOI) in the first year. Both NOI and value are expected to grow at 2% per year. An investor requires an 11% yield.

  • R = 0.11 - 0.02 = 0.09

  • Value = $50,000 / 0.09 = $555,555.56

• Derived Formulas:

  • Y = R + CR (Overall yield rate equals the overall capitalization rate plus the periodic adjustment)

1.3 Variable or Irregular Income and Value Changes

When income and value are not expected to follow a regular pattern of change, the present value of a property can be obtained by applying the standard discounting formula separately to each projected benefit, including the final reversion.

1.4 Level-Equivalent Income: Converting Variable Income into a Constant Stream

Converting a varying income stream to a level-equivalent income stream simplifies valuation using the traditional capitalization model. It involves finding a constant income amount that, when discounted at the appropriate yield rate, has the same present value as the projected, variable income stream.
* Concept: Transforming an irregular income pattern into a single, constant income figure that reflects the same present value.
* Methodology:
1. Calculate the present value (PV) of the projected cash flows (NOI) using the desired yield rate.
2. Calculate the annual payment (PMT) for an annuity with the total PV.

• Mathematical Representation:

  • NPV = Σ [CFt / (1 + r)^t] , where CF is the cash flow in period t, and r is the discount rate❓.
  • Level-Equivalent Income = NPV * (r / (1 - (1 + r)^-n))

• Example:

  • A property has the following projected net operating income (NOI) for the next five years: Year 1: $200,000, Year 2: $208,000, Year 3: $216,320, Year 4: $224,973, Year 5: $233,972. The required yield rate is 12%. The value is expected to increase 15% over a five-year projection period.

  • NPV = $774,096

  • Level-Equivalent Income = $774,096 x 0.277410 = $214,742
  • R = Y - Aa = 0.12 - 0.15(0.157410) = 0.096389
  • V=$214.742/0.096389 = $2,227,879

2. Discounting and Present Value

Discounting is a core principle in real estate valuation, reflecting the time value of money. A dollar received today is worth more than a dollar received in the future due to factors like inflation and the opportunity cost of capital. Discounting techniques are used to determine the present value of future income streams.

2.1. Fundamental Discounting Formula

The present value (PV) of a future cash flow (CF) is calculated as follows:

• PV = CF / (1 + r)^n

  • Where:
    • PV = Present value
    • CF = Cash flow (e.g., net operating income, reversion value)
    • r = Discount rate (required rate of return)
    • n = Number of periods (years) until the cash flow is received

2.2 Discounted Cash Flow (DCF) Analysis

DCF analysis is a sophisticated valuation method that projects all future cash flows associated with a property, including operating income and the reversion value (sale price at the end of the holding period), and then discounts them back to their present value using an appropriate discount rate.
* Process:
1. Project future cash flows over a defined holding period (e.g., 5-10 years).
2. Estimate the reversion value at the end of the holding period.
3. Discount each cash flow back to its present value using the appropriate discount rate.
4. Sum the present values of all cash flows, including the reversion value, to arrive at the property’s estimated value.

• Formula

  • PV = Σ [CFt / (1 + r)^t] + RV / (1 + r)^n
    • Where:
      • PV = Present value of the property
      • CFt = Cash flow in period t
      • r = Discount rate
      • t = Time period
      • RV = Reversion value (expected sale price at the end of the holding period)
      • n = Number of periods in the holding period

• Applicability: DCF analysis is particularly useful for:

  • Properties with irregular income streams.
  • Properties undergoing significant❓ changes (e.g., redevelopment, lease-up).
  • Complex investment scenarios where detailed analysis of cash flow timing is crucial.

• Critiques and Considerations:

  • DCF analysis relies heavily on projections, making it susceptible to errors if those projections are inaccurate.
  • Small changes in assumptions (e.g., growth rates, discount rates) can significantly impact the final value estimate.
  • However, it is still used by investors, particularly in regards to large, investment grade, multitenant properties.
  • The discount rates normally quoted by investors presume annual discounting in arrears. If an appraiser were to divide annual discount rates by 12 and analyze discounted cash flows on a monthly basis arguing that this is how the cash flows occur, those calculations would result in a higher value indication than the DCF analysis of annual cash flows.

3. Applying Discounting and Rate of Change: Practical Examples

3.1. Experiment: Sensitivity Analysis of Discount Rate

This experiment demonstrates the impact of varying the discount rate in a DCF analysis.

• Setup:

  • Create a simplified DCF model with projected cash flows for 5 years and an estimated reversion value.
  • Use a base discount rate (e.g., 10%).
  • Vary the discount rate by +/- 1% and +/- 2%.
  • Observe how the present value (property value) changes with each discount rate adjustment.

• Expected Outcome:

  • Higher discount rates will result in lower present values, and lower discount rates will result in higher present values. The magnitude of change will depend on the duration and magnitude of the projected cash flows.

• Interpretation: This experiment illustrates the sensitivity of property value to The discount rate.❓ A seemingly small change in the discount rate can have a significant impact on the overall valuation, highlighting the importance of carefully selecting and justifying the appropriate discount rate.

4. Investment Analysis Metrics

Discounted cash flow analysis techniques are often used to test the performance of real estate investments at a desired rate of return. Measures of investment performance include:
* Net present value
* Internal rate of return
* Payback period
* Profitability index (or benefit/cost ratio)
* Time-weighted rate

4.1. Net Present Value (NPV) and Internal Rate of Return (IRR)

Net present value (dollar reward) is the difference between the present value at a desired yield (discount) rate of all positive cash flows and the present value of all negative cash flows, or capital outlays.

The rate of discount that makes the net present value of an invest- ment equal zero is the internal rate of return. In other words, the IRR is the rate that discounts all returns from an investment, including returns from its termination, to a present value that is equal to the original investment.

• NPV = ∑ (Cash Flow / (1 + Discount Rate)^t) - Initial Investment
• IRR = Discount Rate where NPV = 0

5. Conclusion

Understanding the interplay between capitalization rates, rate of change, and discounting is fundamental to accurate real estate valuation. By mastering these concepts and the models presented in this chapter, appraisers can develop more robust and reliable value estimates, reflecting the complexities of the real estate market and the expectations of investors.