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 &quot;What if I'd bought the whole street in year $X$ and waited until year $Y$&quot; method is wrong, or does it still have an interpretation?
What is the best way of defining the &quot;overall&quot; rate of growth for the whole street of houses?
Thanks
First of all stop using discrete interest in your intermediate steps to keep your work simple. (As a final step you can convert continuous to discrete if you want to.)
The reason you get different results when you change the start and end dates is because the longer the time period the more the result (growth rate of the totals) will approach the biggest growth rate. By implication the dates do not matter; only the length of the period, so 1901 to 1911 gives you the same result as 2001 to 2011. I didn't test this but it seems like it would be true.
You said you want the overall growth rate.
You can take the average (the mean).
You can take the trimmed mean, which is the average after discarding the highest and lowest rate because may be they are anomalies. Since you don't have a lot of data, you might not want to do the trimmed mean.
You can take the price weighted mean of all data or the trimmed data. I would use initial prices, to be specific. Since you might want to buy the whole street (big spender) the price weighted mean might be more relevant to you. If you're only interested in buying one randomly selected house, the mean or trimmed mean is better.
You can take the median if you think higher and lower rates are less useful for your purpose but you would probably want more than 4 houses in your set if you take a median. If you're only interested in buying one house and you have a median-sized budget then the median rate is more interesting for you.