When we look at data, we are often not interested in every individual fluctuation. We want to see the bigger picture: is this growing? Is this declining? Where is it heading? The moving average is one of the simplest tools we have to answer that question. Before we define it formally, let us build the intuition from the ground up.

Consider those three data points: 10, 20, 10

20 is like a bump or big increase in our data. Now average those three values.

Real-World Analogies

Think about how you pay for services in everyday life.

A post-paid phone plan is a perfect analogy for an annuity immediate. You make calls throughout the month, and only at the end of the period does the carrier tally up your usage and send you a bill. You consume first, pay later.

Conversely, a prepaid internet plan mirrors an annuity due. You load credit before the service begins. If you want internet access for the month, you pay upfront

This article introduces the accumulation function and effective interest rates through concrete examples, showing how we measure investment growth over time. Breaking down total growth into period-by-period factors allows us to better understand how an investment performs and compare growth rates across different time periods.

Starting with Numbers

Let me define a simple amount function at different time periods:

$$A(0) = 100, \quad A(1) = 110, \quad A(2) = 121, \quad A(3) = 133.1, \quad A(4) = 146.41$$

What’s the growth factor between $t=0$ and $t=1$? What number do you need to multiply $A(0)$ by to get to $A(1)$?