<p>Usually, you want to calculate the PV as of one period prior to the first cashflow (like a bond price).   Whereas, usually you want to calculate the FV coincident with the last cashflow (no economic purpose beyond that).  Does this explain it?</p>

CC BY-SA 4.0
2021-10-17T15:19:51.363Z
2021-10-17T15:19:51.363Z
dm63
<p>With geometric series, I understand that, broadly, you can sum from <span class="math-container">$1$</span> to <span class="math-container">$n$</span>, or from <span class="math-container">$0$</span> to <span class="math-container">$n-1$</span>.</p>
<p>Let <span class="math-container">$r=0.03, p=\\\$10, n = 10$</span>.</p>
<div class="s-table-container">
<table class="s-table">
<thead>
<tr>
<th><span class="math-container">$\sum_{j=s}^{j=s+n-1}{pa^j}=\frac{p\left(a^{n}-1\right)a^s}{a-1}$</span></th>
<th><span class="math-container">$a=\left(1+r\right)$</span></th>
<th><span class="math-container">$a={\left(1+r\right)}^{-1}$</span></th>
</tr>
</thead>
<tbody>
<tr>
<td><span class="math-container">$s=0$</span></td>
<td><span class="math-container">$\color{red}{\\\$114.64}$</span></td>
<td><span class="math-container">$\\\$87.86$</span></td>
</tr>
<tr>
<td><span class="math-container">$s=1$</span></td>
<td><span class="math-container">$\\\$118.08$</span></td>
<td><span class="math-container">$\color{red}{\\\$85.30}$</span></td>
</tr>
</tbody>
</table>
</div>
<p>Now, in Excel, at least, I get:</p>
<p><span class="math-container">$$\text{PV}\left(\text{rate}=r, \text{nper}=n,\text{pmt}=p,\text{fv}=0 \right)=-\\\$85.30$$</span>
<span class="math-container">$$\text{FV}\left(\text{rate}=r, \text{nper}=n,\text{pmt}=-p,\text{pv}=0 \right)=\\\$114.64$$</span></p>
<p>So it looks like FV is a geometric sum from <span class="math-container">$j=0$</span> to <span class="math-container">$j=n-1$</span>, whereas PV is a geometric sum from <span class="math-container">$j=1$</span> to <span class="math-container">$j=n$</span>.</p>
<p>The final <span class="math-container">$\text{type}$</span> parameter of the formula can be used to adjust this result - but, again, PV() and FV() work in opposite ways.</p>
<div class="s-table-container">
<table class="s-table">
<thead>
<tr>
<th><span class="math-container">$f\left(r, n, \pm p, 0, t\right)$</span></th>
<th><span class="math-container">$f=\text{PV}$</span></th>
<th><span class="math-container">$f=\text{FV}$</span></th>
</tr>
</thead>
<tbody>
<tr>
<td><span class="math-container">$t=0$</span></td>
<td><span class="math-container">$-\\\$85.30$</span></td>
<td><span class="math-container">$\\\$114.64$</span></td>
</tr>
<tr>
<td><span class="math-container">$t=1$</span></td>
<td><span class="math-container">$-\\\$87.86$</span></td>
<td><span class="math-container">$\\\$118.08$</span></td>
</tr>
</tbody>
</table>
</div>
<p>Why is this? Why do the two formulas work &quot;backwards&quot; relative to each other?</p>
<p>Thanks</p>

<finance><discount>
Why do the PV and FV formulas (in Excel, at least) use slightly different geometric series?
s7eqx9f74nc4
# Why do the PV and FV formulas (in Excel, at least) use slightly different geometric series?

(Asked by: username_0 on 2021-10-17)

With geometric series, I understand that, broadly, you can sum from $1$ to $n$, or from $0$ to $n-1$.

Let $r=0.03, p=\\\$10, n = 10$.

| $\sum\_{j=s}^{j=s+n-1}{pa^j}=\frac{p\left(a^{n}-1\right)a^s}{a-1}$ | $a=\left(1+r\right)$ | $a={\left(1+r\right)}^{-1}$ |
| --- | --- | --- |
| $s=0$ | $\color{red}{\\\$114.64}$ | $\\\$87.86$ |
| $s=1$ | $\\\$118.08$ | $\color{red}{\\\$85.30}$ |

Now, in Excel, at least, I get:

$$\text{PV}\left(\text{rate}=r, \text{nper}=n,\text{pmt}=p,\text{fv}=0 \right)=-\\\$85.30$$
$$\text{FV}\left(\text{rate}=r, \text{nper}=n,\text{pmt}=-p,\text{pv}=0 \right)=\\\$114.64$$

So it looks like FV is a geometric sum from $j=0$ to $j=n-1$, whereas PV is a geometric sum from $j=1$ to $j=n$.

The final $\text{type}$ parameter of the formula can be used to adjust this result - but, again, PV() and FV() work in opposite ways.

| $f\left(r, n, \pm p, 0, t\right)$ | $f=\text{PV}$ | $f=\text{FV}$ |
| --- | --- | --- |
| $t=0$ | $-\\\$85.30$ | $\\\$114.64$ |
| $t=1$ | $-\\\$87.86$ | $\\\$118.08$ |

Why is this? Why do the two formulas work "backwards" relative to each other?

Thanks

**Tags:** finance, discount

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## 1 answers

### Answer by username_1 on 2021-10-17. Score: 0

Usually, you want to calculate the PV as of one period prior to the first cashflow (like a bond price). Whereas, usually you want to calculate the FV coincident with the last cashflow (no economic purpose beyond that). Does this explain it?


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