Capital gains test

[PaperBeetle]

Q&A Bank 3, question 3.22 asks us to price a bond on which the investor will be paying income tax and perhaps capital gains tax. But there is a lag in the payment of the taxes. This ought to affect the formulation of the capital gains test? The q3.22 solution uses \[i^{(2)}>g(1-t_{1}{)}\] but working from first principles, I get the following (where w is the lag between the timing of the bond and the tax year, and where t2 may or may not be 0):
\[\require{enclose}
P=Ca_{\enclose{actuarial}{n}}^{(p)}-t_{1}v^{w}Ca_{\enclose{actuarial}{n}}+Rv^{n}-(R-P)t_{2}v^{n+w}
\]
which rearranges to
\[\require{enclose}
P(1-v^{n+w}t_{2})-Rv^{n}(1-v^{w}t_{2})=C(a_{\enclose{actuarial}{n}}^{(p)}-t_{1}v^{w}a_{\enclose{actuarial}{n}})
\]
But for capital gain, we have
\[R>P
{\implies}R(1-v^{n+w}t_{2})>P(1-v^{n+w}t_{2})\]
\[\require{enclose}
{\implies}R(1-v^{n+w}t_{2})-Rv^{n}(1-v^{w}t_{2})>P(1-v^{n+w}t_{2})-Rv^{n}(1-v^{w}t_{2}=C(a_{\enclose{actuarial}{n}}^{(p)}-t_{1}v^{w}a_{\enclose{actuarial}{n}})
\]
\[
{\implies}R(1-v^{n+w}t_{2})-Rv^{n}(1-v^{w}t_{2})>C(1-v^{n})(\frac{1}{i^{(p)}}-\frac{t_{1}v^{w}}{i})
\]
\[
{\implies}R-Rv^{n}-Rv^{n+w}t_{2}+Rv^{n+w}t_{2}>C(1-v^{n})(\frac{1}{i^{(p)}}-\frac{t_{1}v^{w}}{i})
\]
\[
{\implies}R>C(\frac{1}{i^{(p)}}-\frac{t_{1}v^{w}}{i})
\]
Sooooo... is the original version of the capital gains test just an approximation in this question? I guess there's only a small margin of capital gain that would fail one test but pass the other.
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Sorry about the length, I just really like LATEX.

 

[Mark Mitchell]

The capital gains test implicitly assumes that the income tax is payable immediately that income is received. You're right that when the payment of income tax is deferred, the capital gains test is only approximate.

The safest way to proceed in this type of question is to carry out the standard capital gains test, do the calculations, then check that the final answer agrees with the outcome of the test ie does the price calculated give the capital gain the test predicted. If not, rework the calculations.

Here (as in the majority of such cases) the capital gains test gives the correct answer.

 

[PaperBeetle]

Good plan, thanks. I originally derived this one because I reached that quetsion but couldn't remember the CG test, so I started working it out from first principles. By about 10 miuntes in, I was thinking "hmm, I don't remember it being this complicated in the notes "