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Dear colleagues,
I will try to be laconic. I am conducting a quantitative research now and two questions arose with respect to calculation of portfolio volatility and VaR measurement. Both of the questions have mathematical nature and concern the fact that logarithmic returns are usually used as initial data in such surveys according to the assumption that prices are log-normally distributed.
1. The following formulas are commonly used to measure portfolio volatility:
\[ \sqrt {\displaystyle\sum_{i=1}^{N} a_i^2\sigma_i^2+2\displaystyle\sum_{i=1}^{N}\displaystyle\sum_{j>i}^{N}a_ia_j\sigma_{ij}}=\sqrt {\displaystyle\sum_{i=1}^{N}\displaystyle\sum_{j=1}^{N}a_ia_j\sigma_{ij}}=\sqrt {A^TVA}, \]
where \( \sigma \) – portfolio volatility;
\( a_i \) – share of i asset;
\( a_j \) – share of j asset;
\( \sigma_i^2 \) – variance of i asset;
\( \sigma_{ij} \) – covariance of i asset with j asset;
\( A^T \) – transposed vector of structure;
\( V \) – covariance matrix;
\( A \) – vector of structure.
These formulas work well when random variable (portfolio return) is linear combination:
\[ \displaystyle\sum_{i=1}^{N} a_ir_i, \]
where \( r_i \) – mean return of i asset.
In case of logarithmic returns the same formulas are usually used for portfolio volatility measurement despite the fact that it is incorrect from the mathematical point of view. For practical use the difference is insignificant but I am interested in the theory now. In case of logarithmic returns of individual assets total portfolio return (also logarithmic) is not a linear combination but:
\[ r=ln(\displaystyle\sum_{i=1}^{N} a_ie^{r_i}). \]
So strictly speaking one cannot use the formulas above for variance (volatility) measurement of this variable. Does anybody have an idea how to measure the variance of this variable through its components? How can one break it down into the linear combination by analogy with non-logarithmic returns?
2. Volatility measured by the correct formula from the first question (or volatility of just individual asset calculated on historical data using logarithmic returns) is also expressed through logarithmic return. Again, according to the practice nobody converts it back to effective return. I want to know an opinion of professionals if it is correct. For example if we measure parametric VaR in dollars and received volatility value expressed in logarithmic returns then we cannot just multiply this value by sum invested USD and by quantile of normal distribution. First we need to receive volatility expressed in effective return (since the log. return is a nominal rate): \( e^\sigma-1 \). Are my considerations correct? Again from the practical point of view all this is in margins of error but I want to know if it is correct mathematically.
I will try to be laconic. I am conducting a quantitative research now and two questions arose with respect to calculation of portfolio volatility and VaR measurement. Both of the questions have mathematical nature and concern the fact that logarithmic returns are usually used as initial data in such surveys according to the assumption that prices are log-normally distributed.
1. The following formulas are commonly used to measure portfolio volatility:
\[ \sqrt {\displaystyle\sum_{i=1}^{N} a_i^2\sigma_i^2+2\displaystyle\sum_{i=1}^{N}\displaystyle\sum_{j>i}^{N}a_ia_j\sigma_{ij}}=\sqrt {\displaystyle\sum_{i=1}^{N}\displaystyle\sum_{j=1}^{N}a_ia_j\sigma_{ij}}=\sqrt {A^TVA}, \]
where \( \sigma \) – portfolio volatility;
\( a_i \) – share of i asset;
\( a_j \) – share of j asset;
\( \sigma_i^2 \) – variance of i asset;
\( \sigma_{ij} \) – covariance of i asset with j asset;
\( A^T \) – transposed vector of structure;
\( V \) – covariance matrix;
\( A \) – vector of structure.
These formulas work well when random variable (portfolio return) is linear combination:
\[ \displaystyle\sum_{i=1}^{N} a_ir_i, \]
where \( r_i \) – mean return of i asset.
In case of logarithmic returns the same formulas are usually used for portfolio volatility measurement despite the fact that it is incorrect from the mathematical point of view. For practical use the difference is insignificant but I am interested in the theory now. In case of logarithmic returns of individual assets total portfolio return (also logarithmic) is not a linear combination but:
\[ r=ln(\displaystyle\sum_{i=1}^{N} a_ie^{r_i}). \]
So strictly speaking one cannot use the formulas above for variance (volatility) measurement of this variable. Does anybody have an idea how to measure the variance of this variable through its components? How can one break it down into the linear combination by analogy with non-logarithmic returns?
2. Volatility measured by the correct formula from the first question (or volatility of just individual asset calculated on historical data using logarithmic returns) is also expressed through logarithmic return. Again, according to the practice nobody converts it back to effective return. I want to know an opinion of professionals if it is correct. For example if we measure parametric VaR in dollars and received volatility value expressed in logarithmic returns then we cannot just multiply this value by sum invested USD and by quantile of normal distribution. First we need to receive volatility expressed in effective return (since the log. return is a nominal rate): \( e^\sigma-1 \). Are my considerations correct? Again from the practical point of view all this is in margins of error but I want to know if it is correct mathematically.
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