<p>Usually one can think of <a href="https://en.wikipedia.org/wiki/Accumulation_function" rel="nofollow noreferrer">$a(t)$</a> as some form of interest. </p>

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

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

<p>$$A(t)= ka(t)$$ </p>

<p>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: </p>

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

CC BY-SA 3.0
2017-04-02T00:24:29.207Z
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?
WebUserLearner
CC BY-SA 3.0
2017-04-02T00:27:09.467Z
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
CC BY-SA 3.0
2017-04-02T00:28:31.793Z
So the _a_ has no meaning by itself? Like _k_ or _t_?
WebUserLearner
CC BY-SA 3.0
2017-04-02T00:32:58.647Z
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
CC BY-SA 3.0
2017-04-01T23:43:06.243Z
2017-04-02T00:25:03.953Z
2017-04-02T00:25:03.953Z
123
<p>A(t) represents the amount function, while a(t) represents the accumulation function.</p>
<p>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.</p>
<p>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.</p>

CC BY-SA 4.0
2021-09-19T03:40:32.817Z
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
Nader
CC BY-SA 4.0
2021-09-19T03:37:14.980Z
2021-09-19T03:37:14.980Z
Nader
<p>A number of interest functions, such as the accumulation function and the amount functions include an <em>A</em> (sometimes capitalized, sometimes not).  Could someone tell me what the <em>A</em> represents?  Here's an example:  <a href="https://i.sstatic.net/a8QMk.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/a8QMk.png" alt="enter image description here"></a>  </p>

<interest-rate>
What is the A in the Accumulation and Amount functions for the value of investments?
WebUserLearner
# What is the A in the Accumulation and Amount functions for the value of investments?

(Asked by: username_0 on 2017-04-01)

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: [![enter image description here](https://i.sstatic.net/a8QMk.png)](https://i.sstatic.net/a8QMk.png)

**Tags:** interest-rate

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## 2 answers

### Answer by username_1 on 2017-04-01. Score: 1 | Accepted answer

Usually one can think of [$a(t)$](https://en.wikipedia.org/wiki/Accumulation_function) 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:

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

> **Comments:**
> **username_0**: 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?  
> **username_1**: 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.  
> **username_0**: So the _a_ has no meaning by itself? Like _k_ or _t_?  
> **username_1**: 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.  

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### Answer by username_2 on 2021-09-19. Score: 0

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.

> **Comments:**
> **username_2**: 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  

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