# What is the A in the Accumulation and Amount functions for the value of investments?

A number of interest functions, such as the accumulation function and the amount functions include an A (sometimes capitalized, sometimes not). Could someone tell me what the A represents? Here's an example: Usually one can think of $a(t)$ as some form of interest.

So, we can normalize some initial investment to 1 so that: $$a(0) = 1$$

letting $k$ denote some initial investment, we can have that the value of $k$ at time $t$ is give by your formula:

$$A(t)= ka(t)$$

So, for example, if you invest k=10, assume an interest rate $i=5$% and compounding interest, then ten periods after you invest your money, $k$ is worth:

$$A(10,k) = 10*(1+.05)^{10}$$

• Thanks, but my question was a little (or a lot) more basic. from your last illustration, it sounds like you're saying a is the final value, or what k is worth after applying the interest over time. Is that right? And if so, is there any difference between a capital A and lowercase a? Or is the difference just to show that you're referring to different values and not the same one appearing twice? --- And I noticed that you edited the last line of your formula to capitalize the a. Is that just formatting? Or would the formula have a different meaning without the correction? Apr 2 '17 at 0:24
• I made a typo that was likely confusing. I have fixed it. Capital A(t) is the value of your initial investment after $t$ periods whenever yo u fade interest rate $i$. Think of little a(t) then as some interest rule. If you click my hyperlink, you will visit a wiki on the subject that offers several examples of what a(t) might be. Another way to think of a(t) is the rule governing how your initial investment $k$ evolves over time.
– 123
Apr 2 '17 at 0:27
• So the a has no meaning by itself? Like k or t? Apr 2 '17 at 0:28
• It does have meaning. Its general definition is that it is the rule governing how the value of your initial investment evolves over time. However, the exact definition of $a(t)$ - compounding or simple interest rates, for example - is contextually specific.
– 123
Apr 2 '17 at 0:32

A(t) represents the amount function, while a(t) represents the accumulation function.

Definition 1.1.1: The accumulation function, denoted by a(t), gives the accumulated value at time t of an original investment of 1 at time 0.

Definition 1.1.2: The amount function, denoted by A(t), gives the accumulated value at time t of an original investment of k at time 0.

• A(t) = a(t), when k (your initial investment) equals 1; otherwise, A(t) = k * a(t), where k is your initial investment at time t = 0 Sep 19 at 3:40