> we don't do any maths on appreciation rates. Since it's a simulation, I believe the combinatorics sort themselves out!
That is never a valid inference. At the risk of sounding harsh, you cannot simply throw up your hands and say 'the simulation said so!' without understanding what is actually being simulated, or else you will get nonsense results.
> what gave you the impression that the appreciation maths was off
The 'probability cone' is clearly the wrong shape.
> so didn't want to add volatility there
Many fields the calculator, such as appreciation, have distributions attached to them. The way returns in real life get a distribution is via volatility, so it's highly misleading to both have a distribution width and say that you don't want to add volatility.
This is what your simulation is actually doing:
1. Pick a single normally distributed number
2. Pretend that is the return of your investment, linearly applied over many years. E.g. if the number I drew in step 1 is 4%, then I assume I got 4% return every single year until year N.
3. Repeat a number of times and graph the boundaries
By contrast, this is the norm in finance and financial economics:
1. Actually simulate a financial return process over a number of years. E.g. in year one, I returned between 0-6%, repeat for year 2 ... N, and keep track of the entire history
2. Repeat many times
3. Draw the distribution
Note that the 'correct' simulation procedure draws paths that go up and down, just like asset prices do in real life, while your simulation procedure only ever creates asset price paths that grow or shrink at a constant rate.
I strongly recommend either you give up on simulating returns entirely and make it a single point estimate, or do the simulation properly, otherwise your results are badly misleading. Here's a (very simplified) simulation approach:
Hey, maybe we haven't communicated this well, but we are actually doing exactly what you've described as "the norm in finance and financial economics". We run 1,000 simulations to produce the model output. In each simulation, there are 30 years. Each year, we sample a new "Annual Appreciation" rate. So each simulation does indeed have paths that go up or down.
I didn't mean to suggest that simulations are always correct. Just that I believe ours is, in this case :)
Can you help me understand your point about the "probability cone" please?
I think you're right, just did some careful testing and it looks like the 'probability cone' (basically, the shape you see if you only look at the shape drawn out by the 1 std dev / 2 std dev lines) does have the right shape. My apologies for jumping to conclusions there :)
However, I can't replicate the numbers that you create for a simple test case:
Here we've basically zeroed out rent, mortgage, and everything else, in order to purely isolate the appreciation factor, and your calculator shows a year-30 range of $1m +/- 285k at 95% confidence.
However, if you take 30 independent draws using a 0% +/- 3.2% range @ 95% confidence, that implies each draw has 1.63% std deviation (3.2% / 1.96).
The 95% CI for the sum of 30 independent draws with 1.63% stddev is given by (1.63% * sqrt(30) * 1.96) = +/- 17.5% return.
Applied to your initial amount, I'd expect to see a $1m +/- $1m * e^(17.5%) = ~$190k range for the 95% CI, not the +/- $285k range that you show.
Note the power of just using basic stats and identifying edge cases to validate whether your simulation is giving reasonable results -- here I think either an input or output might be mislabeled (is the input really 3.2% at 95% CI, or is a different confidence?).
That is never a valid inference. At the risk of sounding harsh, you cannot simply throw up your hands and say 'the simulation said so!' without understanding what is actually being simulated, or else you will get nonsense results.
> what gave you the impression that the appreciation maths was off
The 'probability cone' is clearly the wrong shape.
> so didn't want to add volatility there
Many fields the calculator, such as appreciation, have distributions attached to them. The way returns in real life get a distribution is via volatility, so it's highly misleading to both have a distribution width and say that you don't want to add volatility.
This is what your simulation is actually doing:
1. Pick a single normally distributed number
2. Pretend that is the return of your investment, linearly applied over many years. E.g. if the number I drew in step 1 is 4%, then I assume I got 4% return every single year until year N.
3. Repeat a number of times and graph the boundaries
By contrast, this is the norm in finance and financial economics:
1. Actually simulate a financial return process over a number of years. E.g. in year one, I returned between 0-6%, repeat for year 2 ... N, and keep track of the entire history
2. Repeat many times
3. Draw the distribution
Note that the 'correct' simulation procedure draws paths that go up and down, just like asset prices do in real life, while your simulation procedure only ever creates asset price paths that grow or shrink at a constant rate.
I strongly recommend either you give up on simulating returns entirely and make it a single point estimate, or do the simulation properly, otherwise your results are badly misleading. Here's a (very simplified) simulation approach:
https://www.investopedia.com/articles/07/montecarlo.asp