Using Joint Distributions to Propagate Factor Shocks in Portfolio Stress Tests
Summary
This discussion concerns equity portfolio stress testing when macroeconomic factors move together. The author has estimated asset sensitivities to several factors and a covariance matrix, and wants to shock one factor, infer changes in the others, and estimate the resulting portfolio P&L. An example is a decline in a long-term interest rate. The author also explains that a multiple-factor model and factor-shock approach are being considered, while vector autoregression is not preferred for this application.
The response emphasizes estimating the factors’ joint distribution from historical data, then deriving a distribution conditional on the shocked factor. Correlations alone generally do not determine that joint distribution, so a covariance matrix by itself may not capture all relevant dependence. The conditional calculation depends on the chosen distributional model and the details of the stress scenario. The exchange offers a general framework rather than a specific model, implementation, or empirical validation, and it does not supply an R procedure or a formula for the requested P&L estimate.
Key ideas
- A factor shock scenario needs a way to model how other factors respond.
- Historical data can be used to estimate the joint distribution of the risk factors.
- Correlations alone are generally insufficient to specify a joint distribution.
- Conditional factor distributions can support scenario impact calculations, depending on the model and requirements.
- The discussion does not select a particular model or provide implementation details.
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Full text
# How to see the impact of one variable on a set of other variables? # How to see the impact of one variable on a set of other variables? Editing my question: I have decided to use multiple factor model to model my stress test. I am using factor shock method to implement the propagation of shocks. I am doing this according to a book "Multi-Asset Investing: A Practical Guide to Modern Portfolio Management" by Yoram Lustig. According to this: "The investor shocks any risk factor in the portfolio by a chosen amount. The adjusted returns of other risk factors are modelled through a covariance matrix based on their correlation with the shocked risk factor. Finally, the hypothetical impact on the portfolio is calculated." I have run multiple factor regression on 5 factors against the assets in the portfolio and got the respective loadings (betas) for these 5 factors. I also have a covariance matrix of the factor returns. Now, I do not know how can I put a shock in this matrix. My question is how I design now that I shock one variable in the matrix and I get the returns of others. And then I can calculate the predicted/hypothetical return in that scenario. PS: I cannot use Vector Auto Regression as it is not a recommended method for modelling stock returns. The question may be naive, may be I am missing some really basic point here. Thanks ## Manurag Thanks for your reply!! So I will explain a bit more of my problem. I want to perform a stress test on my portfolio of equities. So basically, I want to see the impact of shock in macroeconomic factors (e.g. 10 year interest rate) on the portfolio P&L. Theoretically, my approach is - Obtain a set of relevant macroeconomic factors. - Shock any one of these factors. - Propagate this shock to other factors as the macroeconomic factors are generally correlated. Obtain the new values of factors. - On the basis of risk model, calculate the portfolio P&L based on new set of risk factors. So I want to know about some model or some similar thing that I can program, preferably in R, so that it becomes possible for me to do, say: Change interest rate by -10% and see how it effects the portfolio P&L. Thanks again Manurag ## Answer by g g (score 1) https://quant.stackexchange.com/a/14003 Your question is formulated in a very general way, this is why any answer will need to be general as well. In a nutshell and in full generality you need to estimate the joint distribution from your historical data since in most cases correlations alone are not sufficient to define the joint distribution. In a second step you can calculate the distribution contingent on one of the variables being shocked. Whether this is easy or difficult depends on the joint distribution and your specific requirements. Once you have the contingent distributions you can derive all convenient indicators, i.e. the "impact", you desire.
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