Why Individual VaRs Do Not Determine Portfolio VaR
Summary
The document explains why a portfolio’s historical simulation value at risk cannot generally be recovered from the individual VaRs of its holdings. The portfolio loss distribution depends on the joint behavior of asset returns, including their dependence, while separate VaR figures describe only marginal risk. The example lists positions in four stocks and their individual VaRs, but those inputs do not reveal how the stocks move together.
It suggests two ways to proceed: calculate historical portfolio returns directly and take the relevant quantile, or impose strong assumptions such as normal returns and independence to combine risk estimates. The first route requires aligned historical price or return observations and portfolio weights. The answer gives a conceptual probability argument, not implementation steps, and its assumed alternatives can misstate risk when dependence or distributional behavior differs from the chosen model.
Key ideas
- Individual asset VaRs do not contain enough information to determine portfolio VaR.
- Portfolio risk depends on the joint distribution and dependence among asset returns.
- Historical simulation can estimate VaR from the portfolio’s combined historical returns.
- Combining individual risk estimates requires additional assumptions, such as normality and independence.
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Full text
# Calculate the historical simulation VaR of the portfolio using Python
# Calculate the historical simulation VaR of the portfolio using Python
Assume that we have 200 stocks in WeiBo (WB), 300 stocks in Netflix (NFLX), 250 stocks in Ford Motor Company (F) and 150 in Royal Dutch Shell (RDS-A) as of 31 August 2019 in the portfolio.
I have computed the individual VaR of each stock which amounts to:
1) WB: 736.6841286587987
2) NFLX: 6481.021865242725
3) F: 114.6225730582521
4) RDS-A: 114.6225730582521
However, how do I use python to find the VaR of the portfolio given the individual VaRs?
## Answer by JeanGuillaume (score 0, accepted)
https://quant.stackexchange.com/a/47545
Let us start from the definition of the $VAR_{\alpha}$ at level $\alpha$. We denote by $R_P$ the random variable representing the absolute return of the portfolio (difference in value of $P$ between two dates). We have by definition:
$P(R_p \leq Var_{\alpha}) = \alpha$
As $ R_p = P_1 - P_0 = 200(P_1^W - P_0^W) + 300(P_1^N - P_0^N) + 250(P_1^F - P_0^F) + 150 (P_1^R - P_0^R)$, we get:
$P(200(P_1^W - P_0^W) + ... + 150 (P_1^R - P_0^R) \leq Var_{\alpha}) = \alpha$
And you can see that you cannot extrapolate from the VAR of each stock the VAR of your portfolio. If you have some notions in probability, it is similar to say that, in general, you cannot from marginal distributions (the distribution of each stock) deduct the global distribution (the distribution of the market).
If you want to overcome this problem, you can either take the returns of your portfolio and calculate directly the historical VAR without using the VAR of each stock or you can make very strong assumptions (such as normality and independence between the stocks).
Hope I have helped. Have a good day!Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.