DEFINITION

During bond valuation (pricing), the investor would have to consider capital gains tax (in case the final redemption amount would be higher than the price paid by him).

However, determining whether there would be a capital gain would depend on the price paid, and on the other hand, the price to be paid would depend on whether there will be a capital gain (and therefore some capital gains tax).

This necessitates a test that would determine if there could be a capital gain, even before we calculate the price.

This test is called the capital gains test.

**FORMULA AND DERIVATION**

Consider a fixed interest security that has a redemption of amount \(R\) and pays coupons p times a year, the annual coupon amount being \(C\), i.e. a coupon of amount \(C\) is payable in p installments. If \(t_I\) is the rate of income tax and \(i\) is the yield required, then there will be a capital gain if the redemption amount is greater than the price at which the investor would purchase the security, i.e. if \(R>P\) .

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Substituting the formula for P, we get

\[R>\ \left(1-t_I\right).C.a^{\left(p\right)}_{\overline{n|}}+Rv^n\]
\[R-Rv^n>\ \left(1-t_I\right).C.\frac{\left(1-v^n\right)}{i^{\left(p\right)}}\]
\[R\left(1-v^n\right)>\ \left(1-t_I\right).C.\frac{\left(1-v^n\right)}{i^{\left(p\right)}}\]
\[R>\left(1-t_I\right)\frac{C}{i^{\left(p\right)}}\]
\[i^{\left(p\right)}>\ \left(1-t_I\right)\frac{C}{R}\]

Where,

\[a^{\left(p\right)}_{\overline{n|}}=\ \frac{\left(1-v^n\right)}{i^{\left(p\right)}}\]
\[i^{\left(p\right)}=p\left[{\left(1+i\right)}^{{1}/{p}}-1\right]\]

So, when \(i^{\left(p\right)}\ >\ \left(1-t_I\right)\frac{C}{R}\) , there will be a capital gain.

The results of a capital gains test are summarised below:

If \(i^{\left(p\right)}\ >\ \left(1-t_I\right)\frac{C}{R}\) there is a capital gain.

If \(i^{\left(p\right)}=\ \left(1-t_I\right)\frac{C}{R}\ \) there is no capital gain nor a capital loss.

If \(i^{\left(p\right)}\ <\ \left(1-t_I\right)\frac{C}{R}\ \) there is a capital loss.

** EXAMPLES**

Example 1

Consider a bond of \(\textrm{₹}\ 1000\) nominal, paying coupon of \(10\ \%\) p.a. and redeemable at \(\textrm{₹}\ 1100\) . Income tax will be levied at the rate of \(30\ \%\) on a coupon payments. The investor requires a yield of \(10\ \%\)p.a. from the bond.

Here,

\[i^{\left(p\right)}=i=10\ \%\]
\[R=1100\]
\[C=100\left(10\ \%\ of\ 1000\right)\]
\[t_I=30\ \%\]
\[\left(1-t_I\right)\frac{C}{R}=0.7\ \times \ \frac{100}{1100}=0.0636\]
\[0.1>0.0636\]

Since \(i^{\left(p\right)}\ >\ \left(1-t_I\right)\frac{C}{R}\) , there is a capital gain.

Example 2

Consider a bond of \(\textrm{₹}\ 1000\) nominal, paying coupon of \(4\ \%\) p.a and redeemable at \(80\ \%\) . Rate of income tax on coupon payments is \(30\ \%\) . The investor requires a yield of \(3.5\ \%\) p.a.

Here,

\[i^{\left(p\right)}=i=3.5\ \%\]
\[R=800\]
\[c=40\left(4\%\ of\ 1000\right)\]
\[t_I=30\ \%\]
\[\left(1-t_I\right)\frac{C}{R}=0.7\times \ \frac{40}{800}=0.035\]

Since \(i^{\left(p\right)}=\ \left(1-t_I\right)\frac{C}{R}\ \) , there is neither a capital gain nor a capital loss.

Example 3

Consider a bond of \(\textrm{₹}\ 1000\) nominal, paying coupons of \(4\ \%\) annually and redeemable at \(80\ \%\) . The rate of income tax on coupon payments is \(30\ \%\) . The investor requires a yield of \(3\ \%\) .

Here,

\[i^{\left(p\right)}=i=3\ \%\]
\[R=800\]
\[c=40\left(4\%\ of\ 1000\right)\]
\[t_I=30\ \%\]
\[\left(1-t_I\right)\frac{C}{R}=0.7\times \ \frac{40}{800}=0.035\]
\[0.03<0.035\]

Since \(i^{\left(p\right)}\ <\ \left(1-t_I\right)\frac{C}{R}\) , there is a capital loss.