Calibrating the Gaussian model
1. Calibrating the Gaussian model
What have you learned so far? On the one hand, you have the empirical distribution describing the frequency of past returns for stock ABC. On the other hand, you have the Gaussian model widely used in finance. How can you now use the model to describe the returns? This is done by calibrating the model to the returns.2. Calibration
To find the Gaussian model that matches your past returns, you can simply set the location and dispersion of the Gaussian model to the sample-based estimates obtained from the past returns. Hence, you set "m" to the average return and "s" to the volatility of the returns. This is called ad-hoc calibration. Then, you can visually inspect if the model fits well to the past returns. To do so, just display the histogram of probabilities of the calibrated Gaussian model together with the histogram of the relative frequencies of the empirical distribution.3. Comparing two histograms
Adding a Gaussian histogram to an existing one in spreadsheets is simple. First, double-click on the graph to open the "Chart editor". Next, in the section "Setup", under "Series", click on "Add series".4. Comparing two histograms
Then select the range of cells that include the Gaussian probabilities. Once the appropriate cells are selected, you should already see the two histograms on the same graph.5. Comparing two histograms
If this is not the case, go to "Chart type", and select "Column chart".6. Use of the Gaussian model
The Gaussian model allows you to do more than simply comparing histograms. From the model, you can extrapolate the full distribution of return thus getting the full picture of possible values returns can take. For instance, you can compute the probability of observing a positive return for the next day or you can compute risk measures such as the 5% value-at-risk for instance.7. Gaussian value-at-risk
The value-at-risk is the risk measure we saw in Chapter 2. In the case of the empirical distribution, this is simply the fifth percentile of the sample. But this can also be measured by the Gaussian model. In the case of the model, we must find the value of the return, on the horizontal axis, for which we have 5% of the total surface under the Gaussian curve on the left.8. Gaussian value-at-risk
To compute the 5% value-at-risk with the Gaussian model in spreadsheets, use the function NORMINV(). You simply input the risk level as the first argument, and plug the average return and volatility as the second and third arguments. The function outputs the value-at-risk. In this case, the model indicates that there is a 5% chance to observe returns below -7.68%.9. Gaussian value-at-risk
Graphically, the 5% value-at-risk is a point located on the horizontal axis at -7.68%. It's a key metric which indicates that 5% of the overall surface below the density curve is located below -7.68% while 95% of the surface is located above this point.10. Let's practice!
Your turn now to calibrate the Gaussian model to stock ABC returns. Time to practice!Create Your Free Account
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