<p>I don't think there is a unique best way to do this.</p>
<p>A flexible approach might be to compute the <em>average</em> growth rate by regression. Let <span class="math-container">$P_i, V_i$</span> and <span class="math-container">$T_i$</span> be the variables for house <span class="math-container">$i$</span>. Then we have the formula:
<span class="math-container">$$
\ln(V_i) - \ln(P_i) = r_i T_i 
$$</span>
Putting this into a regression framework gives:
<span class="math-container">$$
\ln(V_i) - \ln(P_i) = r T_i + \varepsilon_i 
$$</span>
where now <span class="math-container">$r$</span> is the &quot;average&quot; growth rate and, by definition <span class="math-container">$\varepsilon_i = (r_i - r) T_i$</span>.</p>
<p>If we assume that <span class="math-container">$\mathbb{E}(\varepsilon_i T_i) = 0$</span> (e.g. this holds if deviations of <span class="math-container">$r_i$</span> from <span class="math-container">$r$</span> are zero conditional on <span class="math-container">$T_i$</span>). Then we can multiply this equation by <span class="math-container">$T_i$</span> and take expectations to obtain:
<span class="math-container">$$
\mathbb{E}[(\ln(V_i)- \ln(P_i)) T_i] = r \mathbb{E}[T_i^2]
$$</span>
So:
<span class="math-container">$$
r = \frac{\mathbb{E}[(\ln(V_i) - \ln(P_i))T_i]}{\mathbb{E}[T_i^2]} = \frac{\mathbb{E}(r_i T_i^2)}{\mathbb{E}(T_i^2)}.
$$</span></p>
<p>An estimate can be constructed by replacing the expectation by the sample mean.
<span class="math-container">$$
\hat r = \frac{\sum_i r_i T_i^2}{\sum_i T_i^2}
$$</span>
For your data set, this gives an estimate of 0.065</p>
<p>An alternative might be to also include an intercept.
<span class="math-container">$$
\ln(V_i) - \ln(P_i) = \alpha + r T_i + \varepsilon_i.
$$</span>
Doing this gives an estimate of 0.0742</p>

CC BY-SA 4.0
2021-06-02T18:27:28.660Z
2021-06-02T18:27:28.660Z
tdm
<p>I have been playing with <span class="math-container">$V=Pe^{rT}$</span> and have been thinking about how to apply it to situations involving different assets growing at different rates.</p>
<p>Suppose I had data for the historic sale prices of a row of houses in a street, with two data points per house. My <strong>assumption</strong> here is going to be that each house's price grows according to <span class="math-container">$V=Pe^{rT}$</span>, with <span class="math-container">$r$</span> constant.</p>
<div class="s-table-container">
<table class="s-table">
<thead>
<tr>
<th>House no.</th>
<th>Year A</th>
<th>Sold for</th>
<th>Year B</th>
<th>Sold for</th>
</tr>
</thead>
<tbody>
<tr>
<td>1</td>
<td>2000</td>
<td>\$100,000</td>
<td>2015</td>
<td>\$500,000</td>
</tr>
<tr>
<td>2</td>
<td>2005</td>
<td>\$60,000</td>
<td>2019</td>
<td>\$111,000</td>
</tr>
<tr>
<td>3</td>
<td>1999</td>
<td>\$30,000</td>
<td>2018</td>
<td>$80,000</td>
</tr>
<tr>
<td>4</td>
<td>2002</td>
<td>\$125,000</td>
<td>2005</td>
<td>\$126,000</td>
</tr>
</tbody>
</table>
</div>
<p>Using <span class="math-container">$V=Pe^{rT}$</span>, I could calculate each house's instantaneous rate of interest <span class="math-container">$r=\frac{\ln{V}-\ln{P}}{T}$</span>, and its APR <span class="math-container">$\hat{r}=e^{r}-1$</span>, where <span class="math-container">$T=B-A$</span> is the number of years elapsed.</p>
<div class="s-table-container">
<table class="s-table">
<thead>
<tr>
<th>House no.</th>
<th>Year A</th>
<th>Sold for</th>
<th>Year B</th>
<th>Sold for</th>
<th><span class="math-container">$r$</span></th>
<th><span class="math-container">$\hat{r}$</span></th>
</tr>
</thead>
<tbody>
<tr>
<td>1</td>
<td>2000</td>
<td>\$100,000</td>
<td>2015</td>
<td>\$500,000</td>
<td>10.73%</td>
<td>11.33%</td>
</tr>
<tr>
<td>2</td>
<td>2005</td>
<td>\$60,000</td>
<td>2019</td>
<td>\$111,000</td>
<td>4.39%</td>
<td>4.49%</td>
</tr>
<tr>
<td>3</td>
<td>1999</td>
<td>\$30,000</td>
<td>2018</td>
<td>$80,000</td>
<td>5.16%</td>
<td>5.30%</td>
</tr>
<tr>
<td>4</td>
<td>2002</td>
<td>\$125,000</td>
<td>2005</td>
<td>\$126,000</td>
<td>0.27%</td>
<td>0.27%</td>
</tr>
</tbody>
</table>
</div>
<p>My question is, how should I meaningfully combine the growth rates of the different houses? Say I want a figure for the street's overall growth rate in order to evaluate whether a particular house has been growing more quickly or more slowly than this rate. I have various ideas:</p>
<ul>
<li>Take the average of the <span class="math-container">$r_{i}$</span>, which is 5.138%</li>
<li>Use <span class="math-container">$-1+e^{5.138\%}=5.272\%$</span></li>
<li>Take the average of the <span class="math-container">$\hat{r}_{i}$</span>, which is 5.346%</li>
<li>Work out the houses' prices in two base years (say, 2000 and 2021), using <span class="math-container">$P=Ve^{-rT}$</span>, and work out what my <span class="math-container">$r$</span> or <span class="math-container">$\hat{r}$</span> would have been if I'd bought the entire street in 2000 and kept the houses until 2021.</li>
</ul>
<div class="s-table-container">
<table class="s-table">
<thead>
<tr>
<th>House no.</th>
<th><span class="math-container">$r$</span></th>
<th>2000</th>
<th>2021</th>
</tr>
</thead>
<tbody>
<tr>
<td>1</td>
<td>10.73%</td>
<td>\$100,000</td>
<td>\$951,827</td>
</tr>
<tr>
<td>2</td>
<td>4.39%</td>
<td>\$48,165</td>
<td>\$121,197</td>
</tr>
<tr>
<td>3</td>
<td>5.16%</td>
<td>\$31,589</td>
<td>\$93,400</td>
</tr>
<tr>
<td>4</td>
<td>0.27%</td>
<td>\$124,338</td>
<td>\$131,470</td>
</tr>
<tr>
<td><strong>Total</strong></td>
<td>-</td>
<td><strong>\$304,092</strong></td>
<td><strong>\$1,297,894</strong></td>
</tr>
</tbody>
</table>
</div>
<p>Using the totals above as my new <span class="math-container">$P$</span> and <span class="math-container">$V$</span>, I get that <span class="math-container">$r=6.91\%$</span> and <span class="math-container">$\hat{r}=7.15\%$</span>.</p>
<p>Is this a legitimate method? If so, what is it called? I am concerned that it might not be legitimate, because changing the two base years gives different results:</p>
<div class="s-table-container">
<table class="s-table">
<thead>
<tr>
<th>House no.</th>
<th><span class="math-container">$r$</span></th>
<th>2005</th>
<th>2008</th>
</tr>
</thead>
<tbody>
<tr>
<td>1</td>
<td>10.73%</td>
<td>\$170,998</td>
<td>\$235,930</td>
</tr>
<tr>
<td>2</td>
<td>4.39%</td>
<td>\$60,000</td>
<td>\$68,455</td>
</tr>
<tr>
<td>3</td>
<td>5.16%</td>
<td>\$40,892</td>
<td>\$47,741</td>
</tr>
<tr>
<td>4</td>
<td>0.27%</td>
<td>\$126,000</td>
<td>\$127,008</td>
</tr>
<tr>
<td><strong>Total</strong></td>
<td>-</td>
<td><strong>\$397,890</strong></td>
<td><strong>\$479,134</strong></td>
</tr>
</tbody>
</table>
</div>
<p>Here, <span class="math-container">$r=6.19\%$</span> and <span class="math-container">$\hat{r}=6.39\%$</span>, which seems strange, since each individual house's growth rate was constant. Does this mean that the &quot;What if I'd bought the whole street in year <span class="math-container">$X$</span> and waited until year <span class="math-container">$Y$</span>&quot; method is wrong, or does it still have an interpretation?</p>
<p>What is the best way of defining the &quot;overall&quot; rate of growth for the whole street of houses?</p>
<p>Thanks</p>

<interest-rate><discount-rate>
Comparing rates of return for different assets using $V=Pe^{rT}$
s7eqx9f74nc4
# Comparing rates of return for different assets using $V=Pe^{rT}$

(Asked by: username_0 on 2021-05-29)

I have been playing with $V=Pe^{rT}$ and have been thinking about how to apply it to situations involving different assets growing at different rates.

Suppose I had data for the historic sale prices of a row of houses in a street, with two data points per house. My **assumption** here is going to be that each house's price grows according to $V=Pe^{rT}$, with $r$ constant.

| House no. | Year A | Sold for | Year B | Sold for |
| --- | --- | --- | --- | --- |
| 1 | 2000 | \$100,000 | 2015 | \$500,000 |
| 2 | 2005 | \$60,000 | 2019 | \$111,000 |
| 3 | 1999 | \$30,000 | 2018 | $80,000 |
| 4 | 2002 | \$125,000 | 2005 | \$126,000 |

Using $V=Pe^{rT}$, I could calculate each house's instantaneous rate of interest $r=\frac{\ln{V}-\ln{P}}{T}$, and its APR $\hat{r}=e^{r}-1$, where $T=B-A$ is the number of years elapsed.

| House no. | Year A | Sold for | Year B | Sold for | $r$ | $\hat{r}$ |
| --- | --- | --- | --- | --- | --- | --- |
| 1 | 2000 | \$100,000 | 2015 | \$500,000 | 10.73% | 11.33% |
| 2 | 2005 | \$60,000 | 2019 | \$111,000 | 4.39% | 4.49% |
| 3 | 1999 | \$30,000 | 2018 | $80,000 | 5.16% | 5.30% |
| 4 | 2002 | \$125,000 | 2005 | \$126,000 | 0.27% | 0.27% |

My question is, how should I meaningfully combine the growth rates of the different houses? Say I want a figure for the street's overall growth rate in order to evaluate whether a particular house has been growing more quickly or more slowly than this rate. I have various ideas:

* Take the average of the $r\_{i}$, which is 5.138%
* Use $-1+e^{5.138\%}=5.272\%$
* Take the average of the $\hat{r}\_{i}$, which is 5.346%
* Work out the houses' prices in two base years (say, 2000 and 2021), using $P=Ve^{-rT}$, and work out what my $r$ or $\hat{r}$ would have been if I'd bought the entire street in 2000 and kept the houses until 2021.

| House no. | $r$ | 2000 | 2021 |
| --- | --- | --- | --- |
| 1 | 10.73% | \$100,000 | \$951,827 |
| 2 | 4.39% | \$48,165 | \$121,197 |
| 3 | 5.16% | \$31,589 | \$93,400 |
| 4 | 0.27% | \$124,338 | \$131,470 |
| **Total** | - | **\$304,092** | **\$1,297,894** |

Using the totals above as my new $P$ and $V$, I get that $r=6.91\%$ and $\hat{r}=7.15\%$.

Is this a legitimate method? If so, what is it called? I am concerned that it might not be legitimate, because changing the two base years gives different results:

| House no. | $r$ | 2005 | 2008 |
| --- | --- | --- | --- |
| 1 | 10.73% | \$170,998 | \$235,930 |
| 2 | 4.39% | \$60,000 | \$68,455 |
| 3 | 5.16% | \$40,892 | \$47,741 |
| 4 | 0.27% | \$126,000 | \$127,008 |
| **Total** | - | **\$397,890** | **\$479,134** |

Here, $r=6.19\%$ and $\hat{r}=6.39\%$, which seems strange, since each individual house's growth rate was constant. Does this mean that the "What if I'd bought the whole street in year $X$ and waited until year $Y$" method is wrong, or does it still have an interpretation?

What is the best way of defining the "overall" rate of growth for the whole street of houses?

Thanks

**Tags:** interest-rate, discount-rate

---

## 1 answers

### Answer by username_1 on 2021-06-02. Score: 1

I don't think there is a unique best way to do this.

A flexible approach might be to compute the *average* growth rate by regression. Let $P\_i, V\_i$ and $T\_i$ be the variables for house $i$. Then we have the formula:
$$
\ln(V\_i) - \ln(P\_i) = r\_i T\_i
$$
Putting this into a regression framework gives:
$$
\ln(V\_i) - \ln(P\_i) = r T\_i + \varepsilon\_i
$$
where now $r$ is the "average" growth rate and, by definition $\varepsilon\_i = (r\_i - r) T\_i$.

If we assume that $\mathbb{E}(\varepsilon\_i T\_i) = 0$ (e.g. this holds if deviations of $r\_i$ from $r$ are zero conditional on $T\_i$). Then we can multiply this equation by $T\_i$ and take expectations to obtain:
$$
\mathbb{E}[(\ln(V\_i)- \ln(P\_i)) T\_i] = r \mathbb{E}[T\_i^2]
$$
So:
$$
r = \frac{\mathbb{E}[(\ln(V\_i) - \ln(P\_i))T\_i]}{\mathbb{E}[T\_i^2]} = \frac{\mathbb{E}(r\_i T\_i^2)}{\mathbb{E}(T\_i^2)}.
$$

An estimate can be constructed by replacing the expectation by the sample mean.
$$
\hat r = \frac{\sum\_i r\_i T\_i^2}{\sum\_i T\_i^2}
$$
For your data set, this gives an estimate of 0.065

An alternative might be to also include an intercept.
$$
\ln(V\_i) - \ln(P\_i) = \alpha + r T\_i + \varepsilon\_i.
$$
Doing this gives an estimate of 0.0742


---

