Charitable Gratuitous Transfers

Hi all, I need a little help as I’m stuck with the following formula:

Re – pretax earnings of the asset if held by the giver
Tie – income tax for the asset if held by the giver
Rg – pretax earnings of the asset if held by receiver
Te – estate tax rate
Toi – current income tax benefit associated with a charitable transfer

First part of the numerator:
(1 + Rg) ^ N
– Since receiving the asset, it will grow at a tax free return of Rg. Not an issue.

Toi [ 1 + Re (1 – Tie) ] ^ N (1 – Te)
– This is what confuses me, since this is comparing giving the asset to a charity now vs transferring to a beneficiary N years later, shouldn’t the formula just add back the Toi as part of the growth to the (1 + Rg) ^ N?

In the CFAI text (Vol 2 p300 or the screen shot above), it stated that “The tax advantages of charitable giving allow the donor to either increase the charitable benefit associated with a given transfer of excess capital from the estate, or …”, so wouldn’t it be right to say that the additional funds from the tax deduction or savings that the charity could claim from the government (referring to Gift Aid Scheme in the UK) would also be part of the initial donated funds. Hence, should grow (FV) at the same rate as the (1 + Rg) ^ N instead of increasing it at the giver’s after-tax return rate [ 1 + Re (1 – Tie) ] ^ N.

Also, my second issue is with the tax deduction amount (from donating to the charity) which was increased at the giver’s after-tax return rate and was taxed by the estate tax rate at N (meaning a bequest instead of a charity). I thought this whole formula is to find out the relative value of giving the asset as a charitable gift against a bequest?

Thanks guys.