Live models
The same maths as the Financial Modelling Dashboard and the regression estimator, rewritten in plain JavaScript so it runs in the page rather than needing Python installed. Nothing here is a screenshot. Every number is computed as you move the sliders.
Discounted cash flow
A cash flow grows each year, and each year is worth less the further away it is. Move the assumptions and watch what discounting does: the solid bars are what each year is worth today, the faint bars behind them are the nominal amounts.
And the same model, with the certainty taken out
The valuation above assumes growth lands on exactly that rate every single year. It never does. This draws each year's growth from a normal distribution around the same rate, runs the whole thing repeatedly, and discounts every resulting path back to today. One number becomes a range.
- 10th percentile
- Median
- 90th percentile
Set volatility to zero and all three percentiles collapse onto the deterministic figure above, which is the simulation agreeing with the closed-form answer. At any volatility above that, notice the median sits slightly below it: compounding a symmetric growth rate does not give you a symmetric outcome.
Beta is a regression slope
An asset's beta is not a number you look up. It is the slope of a regression of the asset's returns on the market's, which means it is estimated from a sample and carries all the uncertainty that implies. This generates returns from a beta you choose, then estimates that beta back out of the data with ordinary least squares. The dashed line is the truth. The solid line is what the regression recovers.
- True beta
- Estimated beta
- 95% interval
- R²
The residuals are the vertical gaps between each point and the solid line. Least squares is the line that makes the sum of those gaps, squared, as small as possible. Drag the observations down to twenty or push the noise up and the estimate starts missing the truth by a wide margin, while the 95% interval widens to admit it. That interval is the honest part: beta is an estimate, and a beta quoted without a sample size behind it is not telling you much.