There is a special type of cash flow diagram that we can’t work with using the standard formula for calculating present value: the perpetual bond.

In this case, the number of compounding periods is infinite, because a perpetual bond makes payments forever (assuming that the bond issuer, a/k/a borrower, is still solvent!). Fortunately, this case allows us to use a much simpler formula:

In fact, the simplicity of this formula P = A/r may cause us to question whether it could really be true. To test whether the answer derived in the previous video makes any sense, we can consider the reciprocal problem, if we deposit a certain finite amount of money in a savings account at the present day, how much interest should we expect to be able to draw from that balance?

However, not all problems of time preference are *financial* problems. There are many contexts in which it is essential to understand the trade-offs between sacrifices made now for benefits that are reaped later that have little to do with borrowing and lending money. Extremely long-term problems, such as those associated with the environment or sustainability (e.g., global climate change) are not amenable to net-present-value analysis using exponential discounting.

As it turns out, exponential discounting is not the only sensible approach to understanding time preference. An alternative to exponential is *hyperbolic* discounting, which values the long-term more and the near-term less than exponential.

While the exponential discount rate is justified by the opportunity costs of compound interest, the hyperbolic discount rate is justified by human and animal behavior. That is, real human decision processes are better represented by the *hyperbolic* form.

Experiments like *The Marshmallow Test* exist in any number of human and animal decision problems. The next video uses the common example of the snooze button to demonstrate that people do not intuitively or heuristically exhibit time preferences that conform to exponential discount models.

###### As a consequence, if long-term environmental policies or investment must be justified exclusively by exponential discounting, it is likely that decisions will be made that we will eventually regret.

###### Related articles

- Prepared to wait? New research challenges the idea that we favour small rewards now over bigger later (bps-research-digest.blogspot.com)
- Why Is It So Hard To Occupy Yourself With Workouts? (explorebigworld.wordpress.com)
- Intro to Cash Flow, NPV, and IRR (cee300.com)
- The Science of Procrastination and How to Manage It, Animated (brainpickings.org)
- Central Bank of India to raise min 2 bln rupees via perp bonds (uk.reuters.com)
- Investors Get a ‘Perpetual’ Headache (blogs.wsj.com)
- The Irrational Consumer: Why Economics Is Dead Wrong About How We Make Choices (theatlantic.com)
- Don’t Repay the National Debt (slate.com)
- Asian Issuers Keep Tapping Perpetual Market (blogs.wsj.com)