# Mathematical proof of rate of return after tax

It is well known in engineering economic that rate of return after tax can be computed by multiplying tax rate and nomial rate of return which is the rate of return before tax. This relation is known to be accurate when decay is not included in our analysis. Actually it makes calculation much easier than comptung IRR after tax. I need to make sure this relation is mathematically correct but never found a mathematical proof to it. Please draw a mathematical proof for

``````RoR_AT=(1-taxrate).RoR_BT
``````
• RoR_AT: rate of return after tax

• RoR_BT: rate of return before tax

• Taxrate: Tax rate

The RoR is defined by the interest rate that make the NPV=0 computed after or before tax.

A reference to the formula RoR_AT=(1-taxrate).RoR_BT is

https://www.investopedia.com/terms/a/after-tax-real-rate-of-return.asp

It's just a simplification. If you make `r` before tax, then pay `t%` tax on `r`, then your net return after tax is

``````r - t*r = (1-t)r
``````

Note that `t` in the formula is in decimal form, meaning if the tax rate is 30%, then `t` will be `0.30`.

• did you assume RoR_AT=r - t*r ? how does it obtained? – behrad mahboobi Apr 2 at 21:34
• Say you earn a return `R`. You must pay tax on that return at some rate, `T`. The amount of tax you pay is therefore `T` times `R`. So the net return after tax is paid is `R` minus `T*R`. – D Stanley Apr 3 at 2:32
• Rete ofcreturn is not like that, it is rate that whoch npw=0. The taxable income should be first computed in cashflow and then tax be applied each year. – behrad mahboobi Apr 3 at 8:44
• @behradmahboobi That's how IRR is calculated by using after tax cash flows. The tax must be computed for each period before calculating the actual rate of return. The formula you mention can be used an approximation that is not accurate in some situations (like for a depreciating asset). See an example here – D Stanley Apr 3 at 12:09
• In other words, the formula you mention can be used to compute the after-tax return for each period, which is then used to compute IRR, but it is not always appropriate to apply to a before-tax IRR to get an after-tax IRR. – D Stanley Apr 3 at 12:11

Well, p = principal, r = dividend rate, d = dividend, t = tax rate, a = after tax dividend, and y = after tax dividend yield:

p * r = d

d - (d * t) = a

a / p = y

Then working with

r - (r * t) = y

show that

r - (r * t) = a / p

as

(p * r) - (p * r * t) = a

and

(p * r) - (p * r * t) = d - (d * t)

and proven with

p * r = d .

• Rate of return is the interest rate at which npw=0. The above proof is not true. – behrad mahboobi Mar 27 at 8:44
• r - (r * t) = a / p was proven to be true. Here 'p' represents a principal value and 'a' represents an after-tax dividend. Then 'r' represents a dividend rate while 't' represents a tax rate. – S Spring Mar 27 at 9:50
• The first formula p.r=d is incorrect for ROR – behrad mahboobi Mar 27 at 11:06
• One should solve npw(r)=0 to obtain ROR – behrad mahboobi Mar 27 at 11:08
• What was posted could be represented in an example as [.04 = (1 - .20) * .05] and that is a simplified version of [.04 = .05 - (.05 * .20)] . – S Spring Mar 27 at 14:58