# Comparing rates of return for different assets using $V=Pe^{rT}$

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

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