Estimate of CAGR after tax and ongoing charges

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Given an idea of how much one’s strategy returns per year on average, it can be useful to estimate what this corresponds to after tax and ongoing charges (ÅOP). This estimate differs between mark-to-market taxation and taxation on realisation.

Derivation for taxation on realisation

The capital of a portfolio taxed on realisation is given by:

\[k'_{\mathrm{slut}}=k_{\mathrm{start}}\cdot\prod_{i=1}^{y}\left(1+a_{i}\right)-s\left(k_{\mathrm{start}}\cdot\left(\prod_{i=1}^{y}\left(1+a_{i}\right)-1\right)\right)\]

Here, \(a_{i}\) is the percentage return in the \(i\)‘th year, \(y\) is the number of years, and \(s\left(x\right)\) is a function that computes the tax on a return of \(x\). It can be seen that:

\[k_{\mathrm{start}}\cdot\left(\prod_{i=1}^{y}\left(1+a_{i}\right)-1\right)=k_{\mathrm{start}}\cdot\prod_{i=1}^{y}\left(1+a_{i}\right)-k_{\mathrm{start}}\]

This is thus the capital exceeding the starting capital. A simplification can now be made with the following definition:

\[\begin{eqnarray} \bar{a}&=&\left(\prod_{i=1}^{y}\left(1+a_{i}\right)\right)^{1/y}-1 \\ \left(1+\bar{a}\right)^{y}&=&\prod_{i=1}^{y}\left(1+a_{i}\right) \end{eqnarray}\]

It can be seen that this is the definition of CAGR. The final capital can now be written as:

\[k'_{\mathrm{slut}}=k_{\mathrm{start}}\cdot\left(1+\bar{a}\right)^{y}-s\left(k_{\mathrm{start}}\cdot\left(\left(1+\bar{a}\right)^{y}-1\right)\right)\]

The ongoing charge can be accounted for by subtracting it from the CAGR:

\[k_{\mathrm{slut}}=k_{\mathrm{start}}\cdot\left(1+\bar{a}-\mathrm{åop}\right)^{y}-s\left(k_{\mathrm{start}}\cdot\left(\left(1+\bar{a}-\mathrm{åop}\right)^{y}-1\right)\right)\]

Instead of having the tax as a function of the profit, the tax can also be computed as an effective tax rate multiplied by the profit. The effective tax rate can be computed as:

\[q_{\mathrm{effektiv}}=\frac{s\left(x\right)}{x}\]

Therefore:

\[q_{\mathrm{effektiv}}\cdot k_{\mathrm{start}}\cdot\left(\left(1+\bar{a}-\mathrm{åop}\right)^{y}-1\right)=s\left(k_{\mathrm{start}}\cdot\left(\left(1+\bar{a}-\mathrm{åop}\right)^{y}-1\right)\right)\]

Thereby:

\[k_{\mathrm{slut}}=k_{\mathrm{start}}\cdot\left(1+\bar{a}-\mathrm{åop}\right)^{y}-q_{\mathrm{effektiv}}\cdot k_{\mathrm{start}}\cdot\left(\left(1+\bar{a}-\mathrm{åop}\right)^{y}-1\right)\]

Recall that CAGR can also be defined as:

\[\mathrm{CAGR}=\left(\frac{k_{\mathrm{slut}}}{k_{\mathrm{start}}}\right)^{1/y}-1\]

The equation for the final capital can therefore be rewritten to this form:

\[\begin{eqnarray} \frac{k_{\mathrm{slut}}}{k_{\mathrm{start}}}&=&\left(1+\bar{a}-\mathrm{åop}\right)^{y}-q_{\mathrm{effektiv}}\cdot\left(\left(1+\bar{a}-\mathrm{åop}\right)^{y}-1\right) \\ \left(\frac{k_{\mathrm{slut}}}{k_{\mathrm{start}}}\right)^{1/y}&=&\left(\left(1+\bar{a}-\mathrm{åop}\right)^{y}-q_{\mathrm{effektiv}}\cdot\left(\left(1+\bar{a}-\mathrm{åop}\right)^{y}-1\right)\right)^{1/y} \\ \mathrm{CAGR_{effektiv}}&=&\left(\left(1+\bar{a}-\mathrm{åop}\right)^{y}-q_{\mathrm{effektiv}}\cdot\left(\left(1+\bar{a}-\mathrm{åop}\right)^{y}-1\right)\right)^{1/y}-1 \end{eqnarray}\]

The final expression is now:

\[\mathrm{CAGR_{effektiv}}=\left(\left(1-q_{\mathrm{effektiv}}\right)\cdot\left(1+\bar{a}-\mathrm{åop}\right)^{y}+q_{\mathrm{effektiv}}\right)^{1/y}-1 \label{eq3}\tag{1}\]

Derivation for mark-to-market taxation

The capital of a mark-to-market taxed portfolio after one year is given by:

\[k'_{1}=k_{0}\cdot\left(1+a_{1}\right)-s\left(k_{0}\cdot a_{1}\right)\]

Again using the effective tax rate:

\[\begin{eqnarray} k'_{1}&=&k_{0}\cdot\left(1+a_{1}\right)-q_{\mathrm{effektiv},1}\cdot k_{0}\cdot a_{1} \\ &=&k_{0}\cdot\left(1+a_{1}-q_{\mathrm{effektiv},1}\cdot a_{1}\right) \end{eqnarray}\]

The capital after two years is given by:

\[k'_{2}=k'_{1}\cdot\left(1+a_{2}-q_{\mathrm{effektiv},2}\cdot a_{2}\right)\]

Inserting the equation for the capital after one year gives:

\[k'_{2}=k_{0}\cdot\left(1+a_{1}-q_{\mathrm{effektiv},1}\cdot a_{1}\right)\cdot\left(1+a_{2}-q_{\mathrm{effektiv},2}\cdot a_{2}\right)\]

It can therefore be seen that the general expression is:

\[\begin{eqnarray} k'_{\mathrm{slut}}&=&k_{\mathrm{start}}\cdot\prod_{i=1}^{y}\left(1+a_{i}-q_{\mathrm{effektiv},i}\cdot a_{i}\right) \\ &=&k_{\mathrm{start}}\cdot\prod_{i=1}^{y}\left(1+a_{i}\cdot\left(1-q_{\mathrm{effektiv},i}\right)\right) \end{eqnarray}\]

Again, the ongoing charge can be subtracted from the percentage return:

\[k_{\mathrm{slut}}=k_{\mathrm{start}}\cdot\prod_{i=1}^{y}\left(1+\left(a_{i}-\mathrm{åop}\right)\cdot\left(1-q_{\mathrm{effektiv},i}\right)\right)\]

Now the effective CAGR can be found:

\[\begin{eqnarray} \frac{k_{\mathrm{slut}}}{k_{\mathrm{start}}}&=&\prod_{i=1}^{y}\left(1+\left(a_{i}-\mathrm{åop}\right)\cdot\left(1-q_{\mathrm{effektiv},i}\right)\right) \\ \left(\frac{k_{\mathrm{slut}}}{k_{\mathrm{start}}}\right)^{1/y}&=&\left(\prod_{i=1}^{y}\left(1+\left(a_{i}-\mathrm{åop}\right)\cdot\left(1-q_{\mathrm{effektiv},i}\right)\right)\right)^{1/y} \\ \mathrm{CAGR_{effektiv}}&=&\left(\prod_{i=1}^{y}\left(1+\left(a_{i}-\mathrm{åop}\right)\cdot\left(1-q_{\mathrm{effektiv},i}\right)\right)\right)^{1/y}-1 \end{eqnarray}\]

To get a usable formula, we must now make an approximation. It can be approximated that:

\[\prod_{i=1}^{y}\left(1+\left(a_{i}-\mathrm{åop}\right)\cdot\left(1-q_{\mathrm{effektiv},i}\right)\right)\approx\left(1+\left(\bar{a}-\mathrm{åop}\right)\cdot\left(1-\bar{q}_{\mathrm{effektiv}}\right)\right)^{y}\]

This is an approximation, and is only exact if all \(a_{i}\) have the same value and all \(q_{\mathrm{effektiv},i}\) have the same value. Thereby:

\[\widetilde{\mathrm{CAGR}}_{\mathrm{effektiv}}=\left(\left(1+\left(\bar{a}-\mathrm{åop}\right)\cdot\left(1-\bar{q}_{\mathrm{effektiv}}\right)\right)^{y}\right)^{1/y}-1\]

The tilde indicates that this is an approximate equation. The equation therefore finally becomes:

\[\widetilde{\mathrm{CAGR}}_{\mathrm{effektiv}}=\left(\bar{a}-\mathrm{åop}\right)\cdot\left(1-\bar{q}_{\mathrm{effektiv}}\right) \label{eq2}\tag{2}\]

Discussion of the equations

In equations (\ref{eq3}) and (\ref{eq2}), \(\bar{a}\) is the expected average yearly return before tax and ongoing charges, and \(q_{\mathrm{effektiv}}\) is the effective tax rate, which will be between 27% and 42% for a normal share portfolio.

It can be seen that the equation for the effective CAGR after tax for a portfolio taxed on realisation depends on the number of years. Let us see how the effective CAGR depends on the number of years for a given return before tax, over tax rates between 0% and 100%.

The graph above shows equations (\ref{eq3}) and (\ref{eq2}). The dashed lines are 27% tax and 42% tax. It can be seen that the effective CAGR for mark-to-market taxation is always lower than for taxation on realisation, which is as expected. For a portfolio taxed on realisation, the effective CAGR approaches the CAGR before tax the more years pass. This is due to the compound interest effect one gets with taxation on realisation.

A similar plot can be made, but now with a CAGR before tax of 15%.

It can be noted here that the trend is the same as for a CAGR before tax of 7%, but that the effective CAGR approaches the CAGR before tax faster as the number of years increases.

For a mark-to-market taxed portfolio, the effective CAGR will be (CAGR before tax minus ongoing charges) times (1 minus the tax rate), cf. equation (\ref{eq2}). Whereas for a portfolio taxed on realisation, the effective CAGR will depend on the number of years that pass, and over time get closer to the CAGR before tax. This is accelerated by the size of the CAGR before tax.

The code used to make the graphs can be found here: estimat_effektiv_cagr.py

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