Propagating Estimation Error into Two-Asset Portfolio Variance
Summary
The document develops a proposed way to approximate how estimation errors in two assets’ variances and their correlation affect the estimated variance of a weighted portfolio. It starts from the standard two-asset portfolio variance formula, expresses covariance using volatilities and correlation, and applies first-order error propagation to those inputs. It then uses bounds on weights and variance errors to propose an upper estimate in quadratic-form notation.
This is a derivation attempt rather than a validated result: the document asks whether the approximation is conservative and gives no response or empirical check. Error propagation generally depends on the covariance structure among estimated inputs, and correlation near zero makes a relative-error expression divided by correlation unstable. The stated bounds on weights and variance errors do not alone establish that the proposed expression is conservative. Readers should treat the equations as a starting point for checking assumptions and deriving an appropriate uncertainty estimate, not as a confirmed formula.
Key ideas
- Two-asset portfolio variance includes both weighted asset variances and a covariance contribution.
- Covariance can be expressed through asset volatilities and their correlation.
- First-order propagation can approximate how input estimation errors affect portfolio variance.
- A conservative bound requires assumptions about dependencies among estimation errors.
- The proposed relative-error expression for correlation is unstable when correlation is near zero.
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# How to calculate the estimation error of portfolio variance using propagation results?
# How to calculate the estimation error of portfolio variance using propagation results?
I am trying to find a conservative approximation for the propagated estimation error of a investment portfolio's variance (comprising two assets), given we know the estimation error for the variance of the two underlying assets and the correlation coefficient between these assets.
I have attempted to work through this using standard results for the propagation of estimation error but I am unsure that the end result is correct. Can you confirm whether you think this is a reasonable conservative estimation.
I use the assumptions that the asset weights will always be between 0 and 1, and furthermore that the error of the variance of the two assets will also always be between 0 and 1.
[\begin{array}{l} \sigma _p^2 = w_1^2\sigma _1^2 + w_2^2\sigma _2^2 + 2{w_1}{w_2}{\sigma _{1,2}}\\ {\sigma _{1,2}} = {\sigma _1}{\sigma _2}{\rho _{1,2}}\\ {\varepsilon _{{\sigma _1}}} = \frac{{{\varepsilon _{\sigma _1^2}}}}{{2{\sigma _1}}}\\ {\varepsilon _{{\sigma _2}}} = \frac{{{\varepsilon _{\sigma _2^2}}}}{{2{\sigma _2}}}\\ {\varepsilon _{{\sigma _{1,2}}}} \approx {\sigma _{1,2}}\left( {\frac{{{\varepsilon _{{\sigma _1}}}}}{{{\sigma _1}}} + \frac{{{\varepsilon _{{\sigma _2}}}}}{{{\sigma _2}}} + \frac{{{\varepsilon _{{\rho _{1,2}}}}}}{{{\rho _{1,2}}}}} \right) = {\sigma _{1,2}}\left( {\frac{{{\varepsilon _{\sigma _1^2}}}}{{2\sigma _1^2}} + \frac{{{\varepsilon _{\sigma _2^2}}}}{{2\sigma _2^2}} + \frac{{{\varepsilon _{{\rho _{1,2}}}}}}{{{\rho _{1,2}}}}} \right)\\ {\sigma _{{\varepsilon _{\sigma _1^2}},{\varepsilon _{{\sigma _{1,2}}}}}} = \frac{{{\sigma _{1,2}}\varepsilon _{\sigma _1^2}^2}}{{2\sigma _1^2}}\\ {\varepsilon _{\sigma _p^2}} \approx w_1^2\varepsilon _{\sigma _1^2}^{} + w_2^2\varepsilon _{\sigma _2^2}^{} + 2{w_1}{w_2}{\varepsilon _{{\sigma _{1,2}}}} + 2w_1^3{w_2}\left( {\frac{{{\sigma _{1,2}}\varepsilon _{\sigma _1^2}^2}}{{2\sigma _1^2}}} \right) + 2w_2^3{w_1}\left( {\frac{{{\sigma _{1,2}}\varepsilon _{\sigma _2^2}^2}}{{2\sigma _2^2}}} \right)\\ 0 \le {w_i},{\varepsilon _{{\sigma _j}}} \le 1 \Rightarrow w_i^3 \le {w_i},\varepsilon _{{\sigma _j}}^2 \le {\varepsilon _{{\sigma _j}}}\\ {\varepsilon _{\sigma _p^2}} \le w_1^2\varepsilon _{\sigma _1^2}^{} + w_2^2\varepsilon _{\sigma _2^2}^{} + 2{w_1}{w_2}{\sigma _{1,2}}\left( {\frac{{{\varepsilon _{\sigma _1^2}}}}{{\sigma _1^2}} + \frac{{{\varepsilon _{\sigma _2^2}}}}{{\sigma _2^2}} + \frac{{{\varepsilon _{{\rho _{1,2}}}}}}{{{\rho _{1,2}}}}} \right) = {w^T}\Sigma w\\ w = \left( {\begin{array}{*{20}{c}} {{w_1}}\\ {{w_2}} \end{array}} \right),\Sigma = \left( {\begin{array}{*{20}{c}} {{\varepsilon _{\sigma _1^2}}}&{{\sigma _{1,2}}\left( {\frac{{{\varepsilon _{\sigma _1^2}}}}{{\sigma _1^2}} + \frac{{{\varepsilon _{\sigma _2^2}}}}{{\sigma _2^2}} + \frac{{{\varepsilon _{{\rho _{1,2}}}}}}{{{\rho _{1,2}}}}} \right)}\\ {{\sigma _{1,2}}\left( {\frac{{{\varepsilon _{\sigma _1^2}}}}{{\sigma _1^2}} + \frac{{{\varepsilon _{\sigma _2^2}}}}{{\sigma _2^2}} + \frac{{{\varepsilon _{{\rho _{1,2}}}}}}{{{\rho _{1,2}}}}} \right)}&{{\varepsilon _{\sigma _2^2}}} \end{array}} \right) \end{array}]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.