Sample from aggregate portfolio distribution versus individual asset distributions












1















Suppose I have three assets $x_1,x_2,x_3$ in a portfolio with weights $W=begin{bmatrix} w_1 \ w_2 \ w_3 end{bmatrix} $, expected returns $R=begin{bmatrix} mu_1 \ mu_2 \ mu_3 end{bmatrix}$, and a covariance matrix $V$.



The expected return of my portfolio is $mu_p=W^TR$ and the variance of my portfolio is $sigma^2_p=W^TVW$.



I would like to run Monte Carlo simulations on my portfolio using a normal distribution.



I can do this either by:




  1. Sampling from the distribution of portfolio returns $N(mu_p,sigma^2_p)$.

  2. Sample from the three individual asset returns and use those three returns to compute my overall portfolio return.


First, how would I accomplish the second approach (would I be sampling from a multivariate normal distribution)?



Second, are these two approaches equivalent as long as I assume that the weights $W$ of my portfolio remain the same?






monte-carlo statistics modern-portfolio-theory







share|improve this question



























    1















    Suppose I have three assets $x_1,x_2,x_3$ in a portfolio with weights $W=begin{bmatrix} w_1 \ w_2 \ w_3 end{bmatrix} $, expected returns $R=begin{bmatrix} mu_1 \ mu_2 \ mu_3 end{bmatrix}$, and a covariance matrix $V$.



    The expected return of my portfolio is $mu_p=W^TR$ and the variance of my portfolio is $sigma^2_p=W^TVW$.



    I would like to run Monte Carlo simulations on my portfolio using a normal distribution.



    I can do this either by:




    1. Sampling from the distribution of portfolio returns $N(mu_p,sigma^2_p)$.

    2. Sample from the three individual asset returns and use those three returns to compute my overall portfolio return.


    First, how would I accomplish the second approach (would I be sampling from a multivariate normal distribution)?



    Second, are these two approaches equivalent as long as I assume that the weights $W$ of my portfolio remain the same?






    monte-carlo statistics modern-portfolio-theory







    share|improve this question

























      1












      1








      1








      Suppose I have three assets $x_1,x_2,x_3$ in a portfolio with weights $W=begin{bmatrix} w_1 \ w_2 \ w_3 end{bmatrix} $, expected returns $R=begin{bmatrix} mu_1 \ mu_2 \ mu_3 end{bmatrix}$, and a covariance matrix $V$.



      The expected return of my portfolio is $mu_p=W^TR$ and the variance of my portfolio is $sigma^2_p=W^TVW$.



      I would like to run Monte Carlo simulations on my portfolio using a normal distribution.



      I can do this either by:




      1. Sampling from the distribution of portfolio returns $N(mu_p,sigma^2_p)$.

      2. Sample from the three individual asset returns and use those three returns to compute my overall portfolio return.


      First, how would I accomplish the second approach (would I be sampling from a multivariate normal distribution)?



      Second, are these two approaches equivalent as long as I assume that the weights $W$ of my portfolio remain the same?






      monte-carlo statistics modern-portfolio-theory







      share|improve this question














      Suppose I have three assets $x_1,x_2,x_3$ in a portfolio with weights $W=begin{bmatrix} w_1 \ w_2 \ w_3 end{bmatrix} $, expected returns $R=begin{bmatrix} mu_1 \ mu_2 \ mu_3 end{bmatrix}$, and a covariance matrix $V$.



      The expected return of my portfolio is $mu_p=W^TR$ and the variance of my portfolio is $sigma^2_p=W^TVW$.



      I would like to run Monte Carlo simulations on my portfolio using a normal distribution.



      I can do this either by:




      1. Sampling from the distribution of portfolio returns $N(mu_p,sigma^2_p)$.

      2. Sample from the three individual asset returns and use those three returns to compute my overall portfolio return.


      First, how would I accomplish the second approach (would I be sampling from a multivariate normal distribution)?



      Second, are these two approaches equivalent as long as I assume that the weights $W$ of my portfolio remain the same?






      monte-carlo statistics modern-portfolio-theory




      monte-carlo statistics modern-portfolio-theory






      share|improve this question













      share|improve this question











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      asked Nov 23 '18 at 14:31









      cpagecpage

      8610




      8610






















          2 Answers
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          For the first case, you would directly sample $n$ random normals $x$ and compute:
          $$R^p_i = mu_p + sigma_p x_i, i in [1,n]$$



          For the second case, you can sample $n$ x $3$ independent normals, compute the Cholesky decomposition matrix $C$ of $V$, which is the matrix $C$ such that $V=C^t C$, and get $n$ samples of vectors $X$ of size 3.



          The return $R_i$ for random draw $i$ is given by:
          $$R_i = mu_p + C . X_i, i in [1,n]$$
          You can check for high values of $n$ the convergence towards the limit values:
          $$E(R_i) = R$$
          $$Cov(R_i) = V$$
          The portfolio return is then computed as:
          $$R^p_i = W.T R_i$$
          and you can check it converges towards the same mean and variance $mu_p$, $sigma_p^2$ for a large enough $n$.



          The two approaches are mathematically equivalent as a linear combination of independent normals is normally distributed. This works so long as the random normal variables generated are iid gaussian normals.



          With numpy, iid normals can be generated with np.random.normal. As pointed out below, np.random.multivariate_normal can be used to generate the multivariate gaussian vector.






          share|improve this answer





















          • 1





            How does this compare to using docs.scipy.org/doc/numpy-1.15.1/reference/generated/…?

            – cpage
            Nov 23 '18 at 16:16













          • good question from @cpage on a 5x5 historical covariance matrix, I get 0.27s for 1e6 samples from np.random.multivariate_normal and 5.3s using np.random.normal with a loop in python. The samples have similar statistics but different values.

            – Sebapi
            Nov 24 '18 at 9:52





















          0














          Basically what @sebapi said.
          "The two approaches are equivalent so long as the random normal variables generated are iid gaussian normals."



          Q: How does this compare to using docs.scipy.org/doc/numpy-1.15.1/reference/generated/…?



          A: You might use scipy.stats.multivariate_normal (rv = multivariate_normal(mean=None, cov=1, allow_singular=False))






          share|improve this answer























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            2 Answers
            2






            active

            oldest

            votes








            2 Answers
            2






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            2














            For the first case, you would directly sample $n$ random normals $x$ and compute:
            $$R^p_i = mu_p + sigma_p x_i, i in [1,n]$$



            For the second case, you can sample $n$ x $3$ independent normals, compute the Cholesky decomposition matrix $C$ of $V$, which is the matrix $C$ such that $V=C^t C$, and get $n$ samples of vectors $X$ of size 3.



            The return $R_i$ for random draw $i$ is given by:
            $$R_i = mu_p + C . X_i, i in [1,n]$$
            You can check for high values of $n$ the convergence towards the limit values:
            $$E(R_i) = R$$
            $$Cov(R_i) = V$$
            The portfolio return is then computed as:
            $$R^p_i = W.T R_i$$
            and you can check it converges towards the same mean and variance $mu_p$, $sigma_p^2$ for a large enough $n$.



            The two approaches are mathematically equivalent as a linear combination of independent normals is normally distributed. This works so long as the random normal variables generated are iid gaussian normals.



            With numpy, iid normals can be generated with np.random.normal. As pointed out below, np.random.multivariate_normal can be used to generate the multivariate gaussian vector.






            share|improve this answer





















            • 1





              How does this compare to using docs.scipy.org/doc/numpy-1.15.1/reference/generated/…?

              – cpage
              Nov 23 '18 at 16:16













            • good question from @cpage on a 5x5 historical covariance matrix, I get 0.27s for 1e6 samples from np.random.multivariate_normal and 5.3s using np.random.normal with a loop in python. The samples have similar statistics but different values.

              – Sebapi
              Nov 24 '18 at 9:52


















            2














            For the first case, you would directly sample $n$ random normals $x$ and compute:
            $$R^p_i = mu_p + sigma_p x_i, i in [1,n]$$



            For the second case, you can sample $n$ x $3$ independent normals, compute the Cholesky decomposition matrix $C$ of $V$, which is the matrix $C$ such that $V=C^t C$, and get $n$ samples of vectors $X$ of size 3.



            The return $R_i$ for random draw $i$ is given by:
            $$R_i = mu_p + C . X_i, i in [1,n]$$
            You can check for high values of $n$ the convergence towards the limit values:
            $$E(R_i) = R$$
            $$Cov(R_i) = V$$
            The portfolio return is then computed as:
            $$R^p_i = W.T R_i$$
            and you can check it converges towards the same mean and variance $mu_p$, $sigma_p^2$ for a large enough $n$.



            The two approaches are mathematically equivalent as a linear combination of independent normals is normally distributed. This works so long as the random normal variables generated are iid gaussian normals.



            With numpy, iid normals can be generated with np.random.normal. As pointed out below, np.random.multivariate_normal can be used to generate the multivariate gaussian vector.






            share|improve this answer





















            • 1





              How does this compare to using docs.scipy.org/doc/numpy-1.15.1/reference/generated/…?

              – cpage
              Nov 23 '18 at 16:16













            • good question from @cpage on a 5x5 historical covariance matrix, I get 0.27s for 1e6 samples from np.random.multivariate_normal and 5.3s using np.random.normal with a loop in python. The samples have similar statistics but different values.

              – Sebapi
              Nov 24 '18 at 9:52
















            2












            2








            2







            For the first case, you would directly sample $n$ random normals $x$ and compute:
            $$R^p_i = mu_p + sigma_p x_i, i in [1,n]$$



            For the second case, you can sample $n$ x $3$ independent normals, compute the Cholesky decomposition matrix $C$ of $V$, which is the matrix $C$ such that $V=C^t C$, and get $n$ samples of vectors $X$ of size 3.



            The return $R_i$ for random draw $i$ is given by:
            $$R_i = mu_p + C . X_i, i in [1,n]$$
            You can check for high values of $n$ the convergence towards the limit values:
            $$E(R_i) = R$$
            $$Cov(R_i) = V$$
            The portfolio return is then computed as:
            $$R^p_i = W.T R_i$$
            and you can check it converges towards the same mean and variance $mu_p$, $sigma_p^2$ for a large enough $n$.



            The two approaches are mathematically equivalent as a linear combination of independent normals is normally distributed. This works so long as the random normal variables generated are iid gaussian normals.



            With numpy, iid normals can be generated with np.random.normal. As pointed out below, np.random.multivariate_normal can be used to generate the multivariate gaussian vector.






            share|improve this answer















            For the first case, you would directly sample $n$ random normals $x$ and compute:
            $$R^p_i = mu_p + sigma_p x_i, i in [1,n]$$



            For the second case, you can sample $n$ x $3$ independent normals, compute the Cholesky decomposition matrix $C$ of $V$, which is the matrix $C$ such that $V=C^t C$, and get $n$ samples of vectors $X$ of size 3.



            The return $R_i$ for random draw $i$ is given by:
            $$R_i = mu_p + C . X_i, i in [1,n]$$
            You can check for high values of $n$ the convergence towards the limit values:
            $$E(R_i) = R$$
            $$Cov(R_i) = V$$
            The portfolio return is then computed as:
            $$R^p_i = W.T R_i$$
            and you can check it converges towards the same mean and variance $mu_p$, $sigma_p^2$ for a large enough $n$.



            The two approaches are mathematically equivalent as a linear combination of independent normals is normally distributed. This works so long as the random normal variables generated are iid gaussian normals.



            With numpy, iid normals can be generated with np.random.normal. As pointed out below, np.random.multivariate_normal can be used to generate the multivariate gaussian vector.







            share|improve this answer














            share|improve this answer



            share|improve this answer








            edited Nov 24 '18 at 9:46

























            answered Nov 23 '18 at 15:09









            SebapiSebapi

            1065




            1065








            • 1





              How does this compare to using docs.scipy.org/doc/numpy-1.15.1/reference/generated/…?

              – cpage
              Nov 23 '18 at 16:16













            • good question from @cpage on a 5x5 historical covariance matrix, I get 0.27s for 1e6 samples from np.random.multivariate_normal and 5.3s using np.random.normal with a loop in python. The samples have similar statistics but different values.

              – Sebapi
              Nov 24 '18 at 9:52
















            • 1





              How does this compare to using docs.scipy.org/doc/numpy-1.15.1/reference/generated/…?

              – cpage
              Nov 23 '18 at 16:16













            • good question from @cpage on a 5x5 historical covariance matrix, I get 0.27s for 1e6 samples from np.random.multivariate_normal and 5.3s using np.random.normal with a loop in python. The samples have similar statistics but different values.

              – Sebapi
              Nov 24 '18 at 9:52










            1




            1





            How does this compare to using docs.scipy.org/doc/numpy-1.15.1/reference/generated/…?

            – cpage
            Nov 23 '18 at 16:16







            How does this compare to using docs.scipy.org/doc/numpy-1.15.1/reference/generated/…?

            – cpage
            Nov 23 '18 at 16:16















            good question from @cpage on a 5x5 historical covariance matrix, I get 0.27s for 1e6 samples from np.random.multivariate_normal and 5.3s using np.random.normal with a loop in python. The samples have similar statistics but different values.

            – Sebapi
            Nov 24 '18 at 9:52







            good question from @cpage on a 5x5 historical covariance matrix, I get 0.27s for 1e6 samples from np.random.multivariate_normal and 5.3s using np.random.normal with a loop in python. The samples have similar statistics but different values.

            – Sebapi
            Nov 24 '18 at 9:52













            0














            Basically what @sebapi said.
            "The two approaches are equivalent so long as the random normal variables generated are iid gaussian normals."



            Q: How does this compare to using docs.scipy.org/doc/numpy-1.15.1/reference/generated/…?



            A: You might use scipy.stats.multivariate_normal (rv = multivariate_normal(mean=None, cov=1, allow_singular=False))






            share|improve this answer




























              0














              Basically what @sebapi said.
              "The two approaches are equivalent so long as the random normal variables generated are iid gaussian normals."



              Q: How does this compare to using docs.scipy.org/doc/numpy-1.15.1/reference/generated/…?



              A: You might use scipy.stats.multivariate_normal (rv = multivariate_normal(mean=None, cov=1, allow_singular=False))






              share|improve this answer


























                0












                0








                0







                Basically what @sebapi said.
                "The two approaches are equivalent so long as the random normal variables generated are iid gaussian normals."



                Q: How does this compare to using docs.scipy.org/doc/numpy-1.15.1/reference/generated/…?



                A: You might use scipy.stats.multivariate_normal (rv = multivariate_normal(mean=None, cov=1, allow_singular=False))






                share|improve this answer













                Basically what @sebapi said.
                "The two approaches are equivalent so long as the random normal variables generated are iid gaussian normals."



                Q: How does this compare to using docs.scipy.org/doc/numpy-1.15.1/reference/generated/…?



                A: You might use scipy.stats.multivariate_normal (rv = multivariate_normal(mean=None, cov=1, allow_singular=False))







                share|improve this answer












                share|improve this answer



                share|improve this answer










                answered Nov 23 '18 at 18:32









                TomDecimusTomDecimus

                998




                998






























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