My Finance

0. First Remarks. We need to use the Yeo-Johnson transform, not the Box-Cox transform to make data normal, since we need to ensure data can be both positive and negative. When we run modified random walk and autoregression simulations, we can always get outside of set bounds where the inverse Box-Cox transform is defined. We succeeded in this: The transformed log volatility has autoregression residuals which are IID Gaussian. And we can run simulations of such transformed volatility, and transform it back using the inverse Yeo-Johnson transform, no problem.

First, we must transform log volatility, then fit the autoregression. We should not first fit the autoregression for log volatility, then transform the residuals. This would give us IID Gaussian data, but will not allow us to make objective Bayes inference for linear regression. Transforming the original data first allows us to simulate regression coefficients and standard errors.

1. Simulation Description. We fit a one-factor model for stock returns (nominal, total, geometric) for three asset classes:

• US stocks (domestic, measured by S&P 500)
• International developed markets (measured by 88% MSCI EAFE + 12% MSCI Canada)
• International emerging markets (measured by MSCI EM)

Of course, for the latter two, we have less data: 1927-2025 for US stocks, 1970-2025 for developed, and 1988-2025 for emerging. Therefore, we expect Bayesian inference to make a difference. For each of 3 classes, we have two choices:

• Keep volatility original, as is
• Apply the Yeo-Johnson transform

We apply three modes of statistical inference:

• Classic (frequentist, with point estimates for the covariance matrix of innovations and regression coefficients)
• Bayesian lite (fixed covariance matrix for innovations, objective Bayes posterior for regression coefficients)
• Bayesian full (fixed correlation matrix for innovations, objective Bayes posterior for standard errors and regression coefficients)

For each of the 18 experiments, we run 10000 Monte Carlo simulations for 20 years. We return 10 results:

• For average returns over 20 years, mean, stdev, and percentiles 10, 30, 50 (=median), 70, and 90.
• For 3%, 4%, 5% withdrawal rules with 4% annual arithmetic increase to adjust for inflation, the survival probability.

2. Results. We do not see much difference in ruin probability.