Perpetuity

Overview

Engineering reference data for Perpetuity in miscellaneous.

Key Formulas

Unit Conversion

$y = x \cdot k$

Multiply by conversion factor.

Linear Interpolation

$y = y_1 + \frac{(x - x_1)(y_2 - y_1)}{x_2 - x_1}$

Estimate between two known points.

Percentage

$p = \frac{\text{part}}{\text{whole}} \times 100\%$

Part as fraction of whole.

Variables

Examples

Basic Perpetuity Example

Consider an asset that generates a fixed annual cash flow of $100 forever. With a discount rate of 10%, the present value is calculated as:

$P = \frac{F}{i} = \frac{100}{0.10} = 1000$

This value represents the fair price of the perpetuity.

Growing Perpetuity Example

For a growth-oriented asset, such as a company with a first-year net income of $200 and a constant growth rate of 3% annually, assuming a 10% discount rate, the present value is:

$P = \frac{F}{i - g} = \frac{200}{0.10 - 0.03} = \frac{200}{0.07} \approx 2857.14$

Note: The growing perpetuity formula requires $i > g$ to ensure the present value is finite and positive.

Key Considerations

• Perpetuities model infinite-life assets like endowments, trusts, and resource fields.
• The basic formula $P = F / i$ derives from summing the infinite series $\sum_{n=1}^{\infty} \frac{F}{(1+i)^n}$.
• For growing perpetuities, the growth rate $g$ must remain constant and less than $i$ to avoid divergence.
• Real-world applications often involve adjusting $F$ for inflation or using risk-adjusted discount rates $i$.

References

• Original Source