## What Is Ad Infinitum?

Ad infinitum is a Latin phrase meaning "to infinity"—in other words, forever. In finance, the term is associated with a perpetuity, in which the payments derived from an asset at fixed intervals are assumed to go on forever and ever, or ad infinitum. Contrary to conventional wisdom, some financial products have no maturity or expiration date. Thus, they're called perpetuities, and certain strings of cash flows are said to carry-on ad infinitum.

### Key Takeaways

- Some types of financial products do not mature or expire.
- Ad infinitum is a Latin phrase meaning "to infinity," which helps to characterize non-maturing products or infinite streams of cash flows.
- Ad infinitum can be another phrase used in business to refer to a perpetuity.

## Understanding Ad Infinitum

Payments received ad infinitum can go on for a lifetime and beyond. It is important to realize though that, because of the time value of money, the present value (i.e., the value today) of those payments very far off in the future (say, 50 years from now) is negligible. Thus the present value of an ordinary annuity (i.e., one with a fixed end) of 50 years is not very much less than that of a perpetuity whose payments go on ad infinitum.

There are slightly different financial calculations used for calculating the present value of an ordinary annuity and the present value of a perpetuity annuity.

More generally, the expression ad infinitum means "continue forever, without limit." So this definition can be adapted to a number of financial contexts characterized by a non-terminating process, a non-terminating repeating process, or a set of instructions to be repeated "forever." In mathematics, a sequence such as this is not uncommon. Thus, it's a natural extension to finance as markets have increasingly grown more computationally heavy, especially in the area of economics. Investment strategies utilizing algorithms, machine learning, and big data regularly turn to infinite loops which carry-on ad infinitum.

Within finance, and especially in the sale and marketing of investment products, ad infinitum can again come up but in a sarcastic or cynical context. For instance, many strategists who crave information but only get more of the same often lament the market hyper-analyzes certain trends 'ad infinitum.' For example, it wouldn't be uncommon to hear someone remark how certain cable news outlets are "talking about ETFs ad infinitum."

## Examples of Ad Infinitum

The fields of mathematics and finance contain many examples of sequences that carry on ad infinitum. For example, the digits that form pi continue ad infinitum. The common numerical series beginning “1, 2, 3, 4...” carries on ad infinitum. The perimeter of a fractal can be iterated ad infinitum. Insurance companies sell annuity products that offer ad infinitum cash flows for investors.

The concept of ad infinitum also appears in other disciplines, such as in the discussion of the problem of infinite regress in epistemology. Classically, this problem is illustrated with the saying “It’s turtle’s all the way down,” which alludes to the mythological belief in a World Turtle that carries the world on its back, and is in turn standing on the back of another, larger turtle, which is in turn standing on the back of another, even larger, turtle, and so on.

The children’s song, “The Song That Never Ends,” from the television program *Lamb-Chop's Play Along, *is another example of an infinitely iterative phenomenon. Much to the chagrin of parents all over the English-speaking world, “The Song That Never Ends” consists of a single verse repeated ad infinitum.