pyppr.matching_ppr_extra_value#

pyppr.matching_ppr_extra_value(ua_cagr, nper, ua_tr, ppr_costr, ppr_tcr, ppr_standard_withdrawal)#

Compute the after-tax extra value, expressed as a (positive or negative) percentage, generated by a PPR comparing to an investment made directly its Underlying Assets.

Given:

  • the CAGR of the PPR’s underlying assets, ua_cagr, compounded once per period, of which there are

  • nper total,

  • the tax rate, ua_tr, the investor would pay on the capital gains in the underlying assets, had he invested in them directly,

  • the PPR’s total costs, ppr_costr, including its management commission and others, expressed as a percentage of the invested capital,

  • the tax credit rate, ppr_tcr, the investor gets in his IRS for investing in a PPR,

  • and whether the PPR will be withdrawn in standard conditions or not, ppr_standard_withdrawal.

Return:

The difference between the PPR’s and its Underlying Assets’ tax-net final value.

Parameters:
ua_cagrscalar or array_like of shape(M, )

Cumulative annual growth rate of the PPR’s underlying assets, in percentage Example: 0.05

nperscalar or array_like of shape(M, )

Number of compounding periods

ua_trscalar or array_like of shape(M, )

Tax rate the investor would pay on the capital gains in the underlying assets Example: 0.28

ppr_costrscalar or array_like of shape(M, )

PPR’s total costs, including its management commission and others, expressed as a percentage of the invested capital Example: 0.0075

ppr_tcrscalar or array_like of shape(M, )

Tax credit rate the investor gets in his IRS for investing in a PPR Example: 0 or 0.2

ppr_standard_withdrawalbool

Whether the PPR will be withdrawn in standard conditions or not.

Returns:
outarray_like

The after-tax extra value generated by the PPR, in percentage.

Warning

matching_ppr_extra_value considers the PPR’s tax credit is reinvested into the PPR, itself not generating any tax credit.

See also

matching_ppr_cost_rate, matching_underlying_assets_cagr

Notes

Returns the result of

\[V_0 + (PPR_n - V_0)\,(1 - tr_{PPR}) + TC_0 + (TC_n - TC_0)\,(1 - tr_{PPR}) = [ \, V_0 + (UA_n - V_0)\,(1 - tr_{UA}) \, ] \, (1 + ev_{PPR})\]

which, according to The Math Behind the Functions, in its longest form, can be decomposed to

\[,V_0\,(1 + tcp_{PPR})\,(1 + r)^n\,(1 - cr_{PPR})^n\,(1 - tr_{PPR}) + V_0\,(1 + tcp_{PPR})\,(tr_{PPR}) = V_0\,(1 + r)^n\,(1-tr_{ETF})\,(1 + ev_{PPR}) + V_0\,(tr_{ETF})\,(1 + ev_{PPR})\]

solved to \(ev_{PPR}\), which gives the following formula:

\[ev_{PPR} = \frac{(1 + r)^n\,(1 - cr_{PPR})^n\,(1 + tcp_{PPR})\,(1 - tr_{PPR}) + tr_{PPR}\,(1 + tcp_{PPR})} {(1 + r)^n\,(1 - tr_{ETF})+tr_{ETF}} - 1\]

You should take care to define the PPR’s tax credit rate according to the law. We recommend you to only consider inputting 0 or the current credit rate (at the time of this writing, 0.2) in the \(ppr_{tcr}\) parameter, even if only part of the investment will generate the credit. In that case, split the investment into two parts, one with \(ppr_{tcr}\) = 0 and the other with \(ppr_{tcr} = 0.2\).

Examples

>>> import pyppr

There’s a PPR that tracks the ETF I’m interested in investing in. Should I invest in the PPR or directly in the ETF? Here’s the scenario:

  • I expect the ETF to grow at a CAGR of 7%.

  • I expect to hold this investment for the next 20 years.

  • When withdrawing, I expect to pay 28% on capital gains.

  • Given the limitations and hurdles of the PPR, I require it to generate an extra 8% return, to choose it over the ETF.

  • I already invest in another ETF and I max out my tax credit when investing in it.

  • This PPR charges a management commission of 0.75% per year, and no other costs.

  • I will withdraw the PPR under standard conditions.

>>> match = pyppr.matching_ppr_extra_value(0.07, 20, 0.28, 0.0075, 0, True)
>>> if ( match < 0.08 ):
...     print('I should invest directly in the underlying assets.')
>>> else:
...     print('I should invest in the PPR.')

How much more tax-net value would investing in the PPR provide if we expect the underlying assets to grow somewhere between 6% to 8%?

>>> import numpy as np
>>> matches = pyppr.matching_ppr_extra_value(np.array([0.06, 0.08]), 20, 0.28, 0.0075, 0, True)
>>> print(matches)
[0.01120129 0.03652754]
>>> print(f'{matches[0]:.2%} - {matches[1]:.2%}')
1.12% - 3.65%