The DynamicPortfolio command creates new dynamically updated portfolio.

The parameter update is a procedure used for calculating the updated weights of the portfolio. This procedure will be called every time the portfolio is rebalanced. The following five arguments will be passed to the procedure: $W$, $W_0$, $X$, $X_0$ and $t$, where $W$ is the vector of new weights, $X$ is the vector of new values, $W_0$ is the vector of old weights, $X_0$ is the vector old values and $t$ is the current time.

The parameter weights is the vector of initial weights.

The parameter components is either a multi-dimensional stochastic process or a vector of one-dimensional stochastic processes.

Finally, the (optional) parameter updates controls how often the portfolio should be updated. Possible values are continuous or any positive integer. This integer value will specify the number of updates per year. If the value of this parameter is continuous, then the portfolio will be updated at every point in the discretization time grid.

$\mathrm{with}\left(\mathrm{Finance}\right)\:$

$\mathrm{X1}≔\mathrm{GeometricBrownianMotion}\left(1\,0.05\,0.3\right)\:$

$\mathrm{X2}≔\mathrm{DeterministicProcess}\left(1.0\right)\:$

$\mathrm{Value}≔t\mapsto t\left[1\right]\cdot t\left[3\right]+t\left[2\right]\cdot t\left[4\right]$

$\mathrm{Value}≔t\mapsto t_1\cdot t_3+t_2\cdot t_4$

U := proc(w, w0, x, x0, t) if x[1] < 1.05 then w[1] := 0; w[2] := 1; else w[1] := 1; w[2] := 0; end if; end proc;

$U≔\mathbf{proc}\left(w\&comma;\mathrm{w0}\&comma;x\&comma;\mathrm{x0}\&comma;t\right)\mathbf{if}x\&lsqb;1\&rsqb;<1.05\mathbf{then}w\&lsqb;1\&rsqb;≔0\&semi;w\&lsqb;2\&rsqb;≔1\mathbf{else}w\&lsqb;1\&rsqb;≔1\&semi;w\&lsqb;2\&rsqb;≔0\mathbf{end\; if}\mathbf{end\; proc}$

$W≔⟨1.0\&comma;0.⟩$

$W≔\left[\begin{array}{c}1.0\\ 0.\end{array}\right]$

$X≔⟨\mathrm{X1}\&comma;\mathrm{X2}⟩$

$X≔\left[\begin{array}{c}\mathrm{\_X0}\\ \mathrm{\_P}\end{array}\right]$

$P≔\mathrm{DynamicPortfolio}\left(U\&comma;W\&comma;X\right)\&colon;$

$\mathrm{AP}≔\mathrm{SamplePath}\left(P\left(t\right)\&comma;t=0..1\&comma;\mathrm{timesteps}=50\&comma;\mathrm{replications}=10\right)$

$\mathrm{PathPlot}\left(\mathrm{AP}\&comma;1\&comma;\mathrm{thickness}=3\&comma;\mathrm{color}=\mathrm{red}..\mathrm{blue}\&comma;\mathrm{axes}=\mathrm{BOXED}\&comma;\mathrm{gridlines}=\mathrm{true}\right)$

$\mathrm{PathPlot}\left(\mathrm{AP}\&comma;2\&comma;\mathrm{thickness}=3\&comma;\mathrm{color}=\mathrm{red}..\mathrm{blue}\&comma;\mathrm{axes}=\mathrm{BOXED}\&comma;\mathrm{gridlines}=\mathrm{true}\right)$

$\mathrm{PathPlot}\left(\mathrm{Value}\left(P\left(t\right)\right)\&comma;t=0..1\&comma;\mathrm{thickness}=3\&comma;\mathrm{markers}=\mathrm{false}\&comma;\mathrm{timesteps}=50\&comma;\mathrm{replications}=10\&comma;\mathrm{color}=\mathrm{red}..\mathrm{blue}\&comma;\mathrm{axes}=\mathrm{BOXED}\&comma;\mathrm{gridlines}=\mathrm{true}\right)$

Q := DynamicPortfolio(proc() end, W, X):

$\mathrm{AQ}≔\mathrm{SamplePath}\left(Q\left(t\right)\&comma;t=0..1\&comma;\mathrm{timesteps}=50\&comma;\mathrm{replications}=10\right)$

$\mathrm{PathPlot}\left(\mathrm{AQ}\&comma;1\&comma;\mathrm{thickness}=3\&comma;\mathrm{color}=\mathrm{red}..\mathrm{blue}\&comma;\mathrm{axes}=\mathrm{BOXED}\&comma;\mathrm{gridlines}=\mathrm{true}\right)$

$\mathrm{PathPlot}\left(\mathrm{AQ}\&comma;2\&comma;\mathrm{thickness}=3\&comma;\mathrm{color}=\mathrm{red}..\mathrm{blue}\&comma;\mathrm{axes}=\mathrm{BOXED}\&comma;\mathrm{gridlines}=\mathrm{true}\right)$

$\mathrm{PathPlot}\left(\mathrm{Value}\left(Q\left(t\right)\right)\&comma;t=0..1\&comma;\mathrm{thickness}=3\&comma;\mathrm{markers}=\mathrm{false}\&comma;\mathrm{timesteps}=50\&comma;\mathrm{replications}=10\&comma;\mathrm{color}=\mathrm{red}..\mathrm{blue}\&comma;\mathrm{axes}=\mathrm{BOXED}\&comma;\mathrm{gridlines}=\mathrm{true}\right)$

R := DynamicPortfolio(proc() end, <0.0, 1.0>, X):

$\mathrm{AR}≔\mathrm{SamplePath}\left(R\left(t\right)\&comma;t=0..1\&comma;\mathrm{timesteps}=50\&comma;\mathrm{replications}=10\right)$

$\mathrm{PathPlot}\left(\mathrm{AR}\&comma;1\&comma;\mathrm{thickness}=3\&comma;\mathrm{color}=\mathrm{red}..\mathrm{blue}\&comma;\mathrm{axes}=\mathrm{BOXED}\&comma;\mathrm{gridlines}=\mathrm{true}\right)$

$\mathrm{PathPlot}\left(\mathrm{AR}\&comma;2\&comma;\mathrm{thickness}=3\&comma;\mathrm{color}=\mathrm{red}..\mathrm{blue}\&comma;\mathrm{axes}=\mathrm{BOXED}\&comma;\mathrm{gridlines}=\mathrm{true}\right)$

$\mathrm{PathPlot}\left(\mathrm{Value}\left(R\left(t\right)\right)\&comma;t=0..1\&comma;\mathrm{thickness}=3\&comma;\mathrm{markers}=\mathrm{false}\&comma;\mathrm{timesteps}=50\&comma;\mathrm{replications}=10\&comma;\mathrm{color}=\mathrm{red}..\mathrm{blue}\&comma;\mathrm{axes}=\mathrm{BOXED}\&comma;\mathrm{gridlines}=\mathrm{true}\right)$

$\mathrm{Value}≔\left(t\&comma;u\right)\mapsto t\left[1\right]\cdot t\left[3\right]+t\left[2\right]\cdot t\left[4\right]-\max \left(u\&comma;0\right)$

$\mathrm{Value}≔\left(t\&comma;u\right)\mapsto t_1\cdot t_3+t_2\cdot t_4-\max \left(u\&comma;0\right)$

$\mathrm{ExpectedValue}\left(\mathrm{Value}\left(P\left(1\right)\&comma;\mathrm{X1}\left(1\right)-1.05\right)\&comma;\mathrm{timesteps}=50\&comma;\mathrm{replications}={10}^4\&comma;\mathrm{output}=\left[\mathrm{value}\&comma;\mathrm{standarddeviation}\right]\right)$

$\left[\mathrm{value}=1.021297741\&comma;\mathrm{standarddeviation}=0.3242635701\right]$

$\mathrm{ExpectedValue}\left(\mathrm{Value}\left(Q\left(1\right)\&comma;\mathrm{X1}\left(1\right)-1.05\right)\&comma;\mathrm{timesteps}=50\&comma;\mathrm{replications}={10}^4\&comma;\mathrm{output}=\left[\mathrm{value}\&comma;\mathrm{standarddeviation}\right]\right)$

$\left[\mathrm{value}=0.9307660907\&comma;\mathrm{standarddeviation}=0.3878610911\right]$

$\mathrm{ExpectedValue}\left(\mathrm{Value}\left(R\left(1\right)\&comma;\mathrm{X1}\left(1\right)-1.05\right)\&comma;\mathrm{timesteps}=50\&comma;\mathrm{replications}={10}^4\&comma;\mathrm{output}=\left[\mathrm{value}\&comma;\mathrm{standarddeviation}\right]\right)$

$\left[\mathrm{value}=0.8736073177\&comma;\mathrm{standarddeviation}=0.2214176439\right]$

$\mathrm{ExpectedValue}\left(\max \left(\mathrm{X1}\left(1\right)-1.05\&comma;0\right)\&comma;\mathrm{timesteps}=50\&comma;\mathrm{replications}={10}^4\&comma;\mathrm{output}=\left[\mathrm{value}\&comma;\mathrm{standarddeviation}\right]\right)$

$\left[\mathrm{value}=0.1246581605\&comma;\mathrm{standarddeviation}=0.2178360124\right]$

The Finance[DynamicPortfolio] command was introduced in Maple 15.

For more information on Maple 15 changes, see Updates in Maple 15.