Unfolding the Enigmas: Exploring Quirks of Net Present Value (NPV)

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Net Present Value (NPV) is a cornerstone in actuarial science and financial decision-making. It's a tool for valuing future cash flows in today's terms.

But beneath its straightforward facade lie some interesting quirks that as an actuarial professional, you might find particularly fascinating.

Let's delve deeper into the Perpetuity Conundrum and Mirrlees' Paradox, two intriguing phenomena inherent to NPV.

The Perpetuity Conundrum

In the world of actuarial science, perpetuity represents an endless stream of cash flows. The present value of a perpetuity is calculated as the Cash Flow divided by the Discount Rate.

Things get more interesting when we consider a "growing perpetuity" where cash flows grow at a constant rate.

If the growth rate surpasses the discount rate, we encounter the Perpetuity Conundrum. Here, the present value of perpetuity skyrockets to infinity, a compelling mathematical phenomenon that we actuaries find intriguing.

This scenario has practical implications for high-growth companies or assets whose perceived value can seem infinite.

Here's a simple example. Let's consider a cash flow of $100 per year, growing at a rate of 3% annually. If the discount rate is 2%, then the NPV of this growing perpetuity will be $100 / (0.02 - 0.03) = -$10,000. Yes, you read that right, a negative value implying infinity! This conundrum, where the NPV becomes infinite if the growth rate exceeds the discount rate, adds a fascinating layer of complexity to the basic principles of financial valuation.

The Perpetuity Conundrum also serves as a cautionary reminder for actuaries to be judicious in assigning growth and discount rates, as erroneous estimates can lead to significant overvaluations.

Mirrlees' Paradox

Another fascinating paradox is named after the Nobel laureate James Mirrlees.

This paradox indicates that an investor can sometimes enhance NPV by deliberately delaying some cash flows.

On the surface, this seems to defy the fundamental principle of the Time Value of Money, which states that money received sooner is more valuable than money received later.

But, the paradox emerges when we consider the multidimensionality of NPV that accounts for the amount, timing of cash flows, and the discount rate.

Suppose you have a project with expected cash inflows of $5000 in Year 1 and $6000 in Year 2. The discount rate is 10%. The NPV of this project will be around $9572. Now, suppose you delay the Year 1 cash inflow to Year 2. If you can invest the delayed $5000 at an interest rate of 12%, the inflow in Year 2 will be $5600. Now, the NPV will be around $9715, an increase despite the delay

The Mirrlees' Paradox highlights the strategic side of actuarial practice. It exemplifies how actuaries can optimise their financial decisions by considering the timing and opportunity cost of capital, in addition to the cash flow amount.

Conclusion

As an actuary, we can appreciate the nuances of tools like NPV.

Intricacies such as the Perpetuity Conundrum and Mirrlees' Paradox shed light on the fascinating interplay between mathematical theory and actuarial practice.