Income Growth Models: Linear vs. Exponential

# Income Growth Models: Linear vs. Exponential

## Introduction

In income property valuation, accurately projecting <a data-bs-toggle="modal" data-bs-target="#questionModal-91814" role="button" aria-label="Open Question" class="keyword-wrapper question-trigger"><span class="keyword-container"><a data-bs-toggle="modal" data-bs-target="#questionModal-338828" role="button" aria-label="Open Question" class="keyword-wrapper question-trigger"><span class="keyword-container">future income streams</span><span class="flag-trigger">❓</span></a></span><span class="flag-trigger">❓</span></a> is critical for determining the present value of an investment. This chapter explores two fundamental models for projecting income growth: linear growth and exponential growth. Understanding the characteristics, applications, and limitations of each model is essential for sound valuation practices. We will delve into the underlying mathematical principles, examine practical examples, and discuss how to apply these models effectively.

## 1. Theoretical Foundation

### 1.1. Time Value of Money

Both linear and exponential growth models are built upon the core principle of the time value of money. This principle states that a dollar received today is worth more than a dollar received in the future due to its potential earning capacity.  Therefore, future income streams must be discounted back to their present value to reflect this difference. The discount rate, *i*, represents the required rate of return or opportunity cost of capital.

### 1.2. Present Value Concepts

The present value (PV) of a future cash flow is calculated using the following formula:

PV = CF / (1 + i)^n

Where:

*   PV = Present Value
*   CF = Cash Flow in year n
*   i = Discount Rate
*   n = Number of years in the future

### 1.3. Perpetuity

A perpetuity is an annuity that has no end date.  If the cash flow is constant each period, the PV can be simplified to:

PV = CF / i

## 2. Linear Growth Model (Straight-Line Change)

### 2.1. Definition

The linear growth model assumes that income increases by a fixed amount each period.  This model is appropriate for situations where income growth is expected to be consistent and predictable in absolute terms. It implies an arithmetic progression of income over time.

### 2.2. Mathematical Representation

Let:

*   I₀ = Initial income
*   h = Amount of change (increase) per period
*   n = Number of periods

Then, the income in period *n*, denoted as Iₙ, is:

Iₙ = I₀ + n * h

### 2.3. Present Value Calculation with Linear Growth

The present value of a series of linearly growing income streams can be calculated using a more complex present value formula. Based on the material provided, a relevant present value (PV) formula is:

PV = (1 − h * n) * an⎤ - h * (n - an⎤) / i

where `an⎤` stands for the present worth of a periodic payment, or `(1 - 1/(1+i)^n) / i`.

Note that this formula is suitable when the initial income is \\$1.

### 2.4. Practical Application and Example

Consider a property with an initial annual income (I₀) of \\$50,000, which is expected to increase by \\$2,000 per year (h) for the next 10 years (n).  The discount rate (i) is 8%.

1.  **Calculate the income for each year:**

    *   Year 1: \$50,000
    *   Year 2: \$52,000
    *   Year 3: \$54,000
    *   ...
    *   Year 10: \$68,000
2.  **Calculate the present value of each year’s income:**

    *   PV(Year 1) = \$50,000 / (1 + 0.08)^1 = \$46,296.30
    *   PV(Year 2) = \$52,000 / (1 + 0.08)^2 = \$44,683.70
    *   ...
    *   PV(Year 10) = \$68,000 / (1 + 0.08)^10 = \$31,493.74
3.  **Sum the present values:**

    *   Total PV = \$46,296.30 + \$44,683.70 + ... + \$31,493.74 = \$428,244.66 (using spreadsheet).

### 2.5. Experiment: Sensitivity Analysis

To illustrate the impact of the linear growth rate, consider the same property, and construct a table that shows the total PV calculated above based on different annual increases: \\$0, \\$1,000, \\$2,000, \\$3,000.

By plotting these points on a chart, you can visualize the relationship between the linear growth and present value.

### 2.6. Limitations

*   **<a data-bs-toggle="modal" data-bs-target="#questionModal-338824" role="button" aria-label="Open Question" class="keyword-wrapper question-trigger"><span class="keyword-container">unrealistic</span><span class="flag-trigger">❓</span></a> in the long run:** Linear growth cannot continue indefinitely as income may eventually reach unrealistic levels.
*   **Ignores market dynamics:** It does not account for external factors that influence income growth, such as inflation or changes in demand.
*   **Assumes constant change:** The model assumes that the fixed amount of increase is true every year.

## 3. Exponential Growth Model (Constant-Ratio Change)

### 3.1. Definition

The exponential growth model assumes that income increases by a constant percentage each period. This model is suitable for assets whose income changes in a compound rate per period. It reflects a compound progression of income over time.

### 3.2. Mathematical Representation

Let:

*   I₀ = Initial income
*   x = Ratio of change in the income for any period (growth rate as a decimal)
*   n = Number of periods

Then, the income in period *n*, denoted as Iₙ, is:

Iₙ = I₀ * (1 + x)^n

### 3.3. Present Value Calculation with Exponential Growth

The present value of a series of exponentially growing income streams can be calculated using the following formula (based on the provided excerpt):

PV = I₀ * [1 - ((1 + x) / (1 + i))^n] / (i - x)

This formula is valid when `i != x`.  If `i = x` then `PV = I₀ * n / (1 + i)`.

### 3.4. Practical Application and Example

Consider a property with an initial annual income (I₀) of \\$50,000, which is expected to grow at 3% per year (x) for the next 10 years (n).  The discount rate (i) is 8%.

1.  **Calculate the income for each year:**

    *   Year 1: \$50,000
    *   Year 2: \$50,000 * (1 + 0.03)^1 = \$51,500
    *   Year 3: \$50,000 * (1 + 0.03)^2 = \$53,045
    *   ...
    *   Year 10: \$50,000 * (1 + 0.03)^9 = \$64,770.79
2.  **Calculate the present value of each year’s income:**

    *   PV(Year 1) = \$50,000 / (1 + 0.08)^1 = \$46,296.30
    *   PV(Year 2) = \$51,500 / (1 + 0.08)^2 = \$44,327.78
    *   ...
    *   PV(Year 10) = \$64,770.79 / (1 + 0.08)^10 = \$30,010.74
3.  **Sum the present values:**

    *   Total PV = \$46,296.30 + \$44,327.78 + ... + \$30,010.74 = \$447,713.74 (using spreadsheet)

Alternatively, use the formula provided:

PV = 50000 * [1 - ((1 + 0.03) / (1 + 0.08))^10] / (0.08 - 0.03)
PV = 50000 * (1 - (1.03 / 1.08)^10) / 0.05
PV = 50000 * (1 - 0.6708) / 0.05
PV = 50000 * 0.3292 / 0.05
PV = 329,200

Note that this formula provides a different answer.  This is because this formula implicitly assumes income begins one period in the future. The first calculations are more comprehensive because they directly account for income streams each year.

### 3.5. Experiment: Impact of Growth Rate

Using the exponential growth formula, calculate the PV of the property for different exponential growth rates (0%, 2%, 4%, 6%) while holding other variables (I₀, i, n) constant.  Observe how the PV changes with different growth rates.  Plot the PV against growth rate to visualize the sensitivity.

### 3.6. Limitations

*   **May overestimate growth:** Constant percentage growth may not be sustainable in the long run due to market saturation, competition, or regulatory changes.
*   **Sensitivity to small changes:** Even small changes in the growth rate can significantly impact the present value, especially over longer periods.

## 4. Comparison and Selection Criteria

| Feature          | Linear Growth                                     | Exponential Growth                                 |
| ---------------- | ------------------------------------------------- | ------------------------------------------------- |
| Growth Pattern   | Constant amount per period                        | Constant percentage per period                       |
| Mathematical Complexity | Simpler calculation for individual years, more complex overall PV | Slightly more complex for individual years, overall PV has a more concise formula |
| Applicability     | Short-term, predictable, specific scenarios (e.g., lease step-ups) | Longer-term projections, general market trends       |
| Long-Term Realism | Less realistic, can lead to unrealistic income levels | More realistic, but must consider sustainability    |
| Sensitivity      | Less sensitive to changes in growth amount       | Highly sensitive to changes in growth rate          |

**Selection Criteria:**

*   **Time horizon:** For short-term projections with well-defined step-ups, linear growth may be appropriate. For longer-term projections, exponential growth is generally more realistic.
*   **Market conditions:** Consider the overall market trends and competitive landscape. If the market is experiencing steady, predictable growth, exponential growth might be suitable. If there are specific factors driving income changes, linear growth may be more appropriate.
*   **Data availability:** The model chosen should align with the available data. If reliable historical data on percentage income growth is available, exponential growth may be a better choice.

## 5. Advanced Considerations

### 5.1. Hybrid Models

In some cases, a combination of linear and exponential growth models may be appropriate. For example, income could grow linearly for a few years due to a specific lease agreement, and then switch to an exponential growth pattern to reflect broader market trends.

### 5.2. <a data-bs-toggle="modal" data-bs-target="#questionModal-338830" role="button" aria-label="Open Question" class="keyword-wrapper question-trigger"><span class="keyword-container">incorporating reversion value</span><span class="flag-trigger">❓</span></a>

Property valuation often involves estimating a reversion value, which represents the expected selling price of the property at the end of the projection period. The growth model used for income projections should also be considered when estimating the reversion value. For example, if income is projected to grow exponentially, the reversion value should also reflect this expected growth.

### 5.3. Discount Rate Selection

The discount rate (i) is a critical input in both linear and exponential growth models. It should reflect the risk and opportunity cost of capital for the specific property and market. Factors to consider when selecting the discount rate include:

*   Market interest rates
*   Property-specific risk factors (e.g., location, tenant quality, lease terms)
*   Investor expectations

### 5.4. Capitalization Rate (RO) Estimation

As demonstrated in the provided materials, the capitalization rate can be derived from income growth assumptions and the discount rate, especially when value changes occur (the reversion). The relationship is most directly given by:

RO = YO - CR (for exponential growth)

or

RO = YO - (∆ × 1⁄n) (for linear growth)

Where:

* RO = capitalization rate
* YO = yield rate
* CR = constant ratio (exponential growth rate)
* ∆ = Total property change
* n = Number of years

## 6. Conclusion

The choice between linear and exponential growth models depends on the specific characteristics of the income property, market conditions, and the appraiser's judgment. A thorough understanding of the underlying principles, mathematical representations, and limitations of each model is essential for accurate and reliable income property valuation. By carefully considering the factors discussed in this chapter, appraisers can select the most appropriate model and develop credible and defensible valuations.

## 7. Review Questions

1.  Explain the difference between linear and exponential growth.
2.  Under what circumstances is the linear growth model most appropriate?
3.  What are the limitations of the exponential growth model?
4.  How does the discount rate affect the present value calculation in both models?
5.  Describe a scenario where a hybrid model combining linear and exponential growth might be useful.
6.  How is the capitalization rate related to the discount rate and income growth assumptions?