Sampling Correlated Return and Volatility Uncertainty
Summary
The document considers Monte Carlo simulation of portfolio utility when both expected returns and covariance are uncertain. Expected returns are modeled with a Gaussian shock, while covariance is scaled by a random factor centered on one. The central modeling concern is that these components may be dependent: the author expects volatility increases to coincide with lower expected returns, and volatility contractions to coincide with higher returns. Independent draws could therefore misrepresent portfolio outcomes.
The response recommends drawing the two Gaussian variables jointly from a bivariate normal distribution, using their covariance to represent dependence. It adds that a Gaussian copula can also be used to connect marginal distributions when the variables are not Gaussian. The answer does not estimate the dependence, assess whether the assumed return-volatility relationship holds in the data, or specify how to fit and validate a model. The suggested approach is a simple starting point; its adequacy depends on the distributional assumptions and on whether covariance captures the relevant dependence.
Key ideas
- Independent sampling can miss dependence between expected returns and volatility uncertainty.
- A bivariate normal draw can represent dependence between two Gaussian variables through covariance.
- A Gaussian copula can link marginal distributions when variables are not Gaussian.
- The simulation still requires empirical estimates and validation of the dependence assumptions.
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Full text
# Monte carlo portfolio risk simulation # Monte carlo portfolio risk simulation My objective is to show the distribution of a portfolio's expected utilities via random sampling. The utility function has two random components. The first component is an expected return vector which is shocked by a random Gaussian variable ($ER$). The shock shifts the location parameter of the return distribution. The second random component is a covariance matrix which is multiplied by a random scalar $S$. The scalar is drawn from a normal random variable, centered on $1$, where the variance of $S$ corresponds to our uncertainty in whether volatility increases or decreases. My initial plan was to take separate draws of the $S$ and $EV$ and simply calculate the utility. However, clearly these random variables are not conditionally independent. In particular, research suggests that when volatility is increasing (i.e. $S > 1$), the expected returns distribution has a lower overall mean. Or, if volatility is contracting rapidly it is likely that the expected returns distribution has a higher mean. In other words, I should be sampling from the joint distribution of $EV$ and $S$ as opposed to the marginal distributions of $EV$ and $S$ separately. What's a good technique to estimate and sample these random variables from their joint distribution? The "correct" approach I can think of involves defining a set of states, estimating the transition probabilities between state pairs, and sampling $EV$ and $S$ conditional on random draws of a third state variable. Seems like overkill! A crude variation of this would be to build a transition matrix such as [ High Vol to High Vol, Low to Low, Low to High Vol, High to Low Vol ] where the location and scale parameters ($ER$ and $S$) are informed by an "expert" (i.e. casual inspection of the data). Are there other techniques that I may be missing that provide a solid "80/20 rule" solution for sampling from this joint distribution, or is state-space (markov models and the like) the only way to go? For example, perhaps there is a non-parametric technique to estimate the relationship between these two variables. ## Answer by Brian B (score 5, accepted) https://quant.stackexchange.com/a/2313 Since both $ER$ and $S$ are gaussian random, why not just assume their dependence is captured by their covariance, and make your draws from the bivariate normal distribution? It is hard to construct any other way of making two marginal gaussians cointegrated. Even if the variables were not gaussian, you would probably find yourself relating them using a gaussian copula anyway.
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