Accumulation function, n-th year interest factor and effective interest factor

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 $A(1)$?

$$ A(1) = A(0) \times ? $$

With our concrete numbers:

$$ 110 = 100 \times ? $$

By very simple algebra we just divide both sides by $A(0)$ to obtain that growth factor:

$$ \frac{A(1)}{A(0)} = \frac{110}{100} = 1.10. $$

If I consider $A(2)$ and ask for the growth factor starting at the beginning $A(0) = 100$, then you will do:

$$ \frac{A(2)}{A(0)} = \frac{121}{100} = 1.21. $$

That growth factor represents the accumulation function. It quantifies how much I changed compared to the beginning.

What we just computed, the growth factor from one period to another, is more precisely called the period growth factor.

Connecting to the accumulation function to amount function

We can now introduce a more general notation. Let $a(t)$ represent the accumulation function, which tells us the growth factor from the initial time to time $t$.

For our amount function $A(t)$, we can express any value as:

By definition, $a(0) = 1$ since there’s no growth at the starting point, which you can verify from the formula: $A(0) = A(0) \times a(0)$ implies $a(0) = 1$.

Using our specific values from the beginning, we have $A(0) = 100$ and $A(1) = 110$, so:

$$ a(1) = \frac{A(1)}{A(0)} = \frac{110}{100} = 1.10. $$

Similarly, since $A(2) = 121$:

$$ a(2) = \frac{A(2)}{A(0)} = \frac{121}{100} = 1.21. $$

And for $A(3) = 133.1$:

$$ a(3) = \frac{A(3)}{A(0)} = \frac{133.1}{100} = 1.331. $$

Notice that these accumulation values tell us the cumulative growth: after one period we have 1.10 times our starting amount, after two periods we have 1.21 times our starting amount, and so on. The accumulation function $a(t)$ captures the total multiplicative change from the beginning to any point in time.

But recall that we computed the growth from $t=0$ to $t=1$, which was:

$$ \frac{A(1)}{A(0)} = \frac{110}{100} = 1.10. $$

With the general formula above for the amount function in terms of the accumulation function, we can re-express this growth factor between periods. Since $A(t) = A(0) \times a(t)$, we have:

$$ \frac{A(1)}{A(0)} = \frac{A(0) \times a(1)}{A(0) \times a(0)} = \frac{a(1)}{a(0)} = \frac{1.10}{1.00} = 1.10. $$

Notice how $A(0)$ cancels out, leaving only a ratio of accumulation functions! This ratio represents the growth specifically during the period from time 0 to time 1.

The n-th Year Interest Factor

This concept generalizes beautifully. More generally, for any period $t$, the growth factor from time $t-1$ to time $t$ is called the n-th year interest factor (also called the period interest factor), denoted $(1 + i_t)$:

Using our example, the interest factor for the first period would be:

$$ (1 + i_1) = \frac{a(1)}{a(0)} = \frac{1.10}{1.00} = 1.10. $$

We started with an amount at $t=0$ of 100 and then we went to an amount of 110. Note that 110 is the new amount, not the increase itself. To find the actual increase, we need to subtract our starting value of 100, which gives us 10.

So what’s the actual percent increase? In terms of ratios:

$$ \frac{A(1) - A(0)}{A(0)} = \frac{110 - 100}{100} = \frac{10}{100} = 0.10. $$

This number that’s analogous to the percent increase is called the effective interest rate. It’s effective because it takes into account what you started with and computes a more meaningful measure of change. This is the $i_t$ in our interest factor formula above.

The Effective Interest Rate

The effective interest rate for period $t$ is defined as:

Alternatively, we can express this in terms of the change between consecutive periods:

$$ i_t = \frac{a(t) - a(t-1)}{a(t-1)}. $$

This alternative form is useful because it directly shows the change in accumulation relative to where we started the period. To see how we arrive at this, we simply manipulate the definition above:

$$ i_t = \frac{a(t)}{a(t-1)} - 1 = \frac{a(t)}{a(t-1)} - \frac{a(t-1)}{a(t-1)} = \frac{a(t) - a(t-1)}{a(t-1)}. $$

This shows that the effective interest rate is simply the fractional change in the accumulation function from one period to the next. Returning to our concrete example from the beginning:

$$ i_1 = \frac{a(1) - a(0)}{a(0)} = \frac{1.10 - 1.00}{1.00} = 0.10 = 10\%. $$

Summary

We can now see the complete picture:

$$ \begin{array}{|c|c|c|c|c|} \hline \text{Time} & \text{Amount } A(t) & \text{Accumulation } a(t) & \text{Interest Factor } (1+i_t) & \text{Effective Rate(\%) } i_t \\ \hline t=0 & 100 & 1.00 & — & — \\ t=1 & 110 & 1.10 & 1.10 & 10\% \\ t=2 & 121 & 1.21 & 1.10 & 10\% \\ t=3 & 133.1 & 1.331 & 1.10 & 10\% \\ t=4 & 146.41 & 1.4641 & 1.10 & 10\% \\ \hline \end{array} $$

The accumulation function $a(t)$ tells us total growth from the beginning, while the effective interest rate $i_t$ tells us the growth during each individual period. Both perspectives are essential for understanding how investments grow over time.