Finance and Infinite Series

Infinite series, though seemingly abstract mathematical concepts, find surprisingly practical applications within the realm of finance. They are particularly useful for valuing streams of future cash flows, especially when those cash flows are expected to continue indefinitely.

One of the most common applications is in the valuation of perpetuities. A perpetuity is an annuity that continues forever, paying out a fixed amount at regular intervals. Think of it as an investment that provides an income stream in perpetuity, such as certain types of preferred stock. The present value (PV) of a perpetuity can be calculated using the formula derived from the sum of an infinite geometric series:

PV = C / r

Where:

• PV = Present Value of the perpetuity
• C = Cash flow per period (the periodic payment)
• r = Discount rate (the rate of return required on the investment)

This formula is based on the assumption that the cash flows (C) are constant and the discount rate (r) remains stable over time. The logic stems from discounting each future cash flow back to its present value and then summing them all up. The resulting infinite geometric series converges to the simple formula above.

Another area where infinite series are beneficial is in valuing growing perpetuities. A growing perpetuity is similar to a regular perpetuity, except that the cash flows are expected to grow at a constant rate (g) indefinitely. The formula for the present value of a growing perpetuity is:

PV = C / (r – g)

Where:

• PV = Present Value of the growing perpetuity
• C = Cash flow in the next period
• r = Discount rate
• g = Growth rate of the cash flows

This formula is valid only when the growth rate (g) is less than the discount rate (r). If the growth rate exceeds the discount rate, the series diverges, and the present value becomes infinitely large (or undefined in a practical financial sense). The condition (r > g) ensures that the present values of the future cash flows become progressively smaller, allowing the series to converge to a finite value.

Beyond simple perpetuities, infinite series can be employed to model more complex financial scenarios. For instance, certain dividend discount models (DDM) used to value stocks can incorporate assumptions about constant growth in dividends that extend infinitely into the future. While the long-term sustainability of such growth is debatable in reality, the infinite series provides a theoretical framework for estimating the intrinsic value of a company.

It’s crucial to acknowledge the limitations. Applying infinite series requires careful consideration of the underlying assumptions, especially regarding the constancy of cash flows, growth rates, and discount rates over the very long term. Financial markets are inherently unpredictable, and these assumptions may not hold in reality. However, as a foundational tool, infinite series offer a powerful way to conceptualize and evaluate investments with long-term income streams.