Price of bond when tax is not considered:-

As stated above the price of a bond is the sum of present values of all future payments from the bond. The future payments from a bond include coupon payments received at regular time intervals and the final redemption payment received when the bond matures.

Let us assume that \(P\) is the price that the investor wishes to pay, \(C\) is the annual coupon amount, \(R\) is the redemption amount and \(i\) is the yield required by the investor.

Then, for an \(n\)-year bond, the price would be:

\[P=C{\left(1+i\right)}^{-1}+C{\left(1+i\right)}^{-2}\dots \dots \dots C{\left(1+i\right)}^{-n}+R{\left(1+i\right)}^{-n}\]
\[\ \ \ ={Cv}^1+\ {Cv}^2\dots \dots \dots \ {Cv}^n+{Rv}^n\]
\[\ \ \ =C\sum^n_{t=1}{v^t}+\ {Rv}^n\]
\[\ \ \ =C.a_{\overline{n|}}+\ {Rv}^n\]

where,

\(a_{\overline{n|}}=\ \frac{1-v^n}{i}\)
is the present value of an annuity immediate

\[v=\ {\left(1+i\right)}^{-1}\]

If the coupon payment is paid more than once per year, say p times, then the formula must be adjusted for this as follows:

\[P=C.a^{\left(p\right)}_{\overline{n|}}+\ {Rv}^n\]

where,

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

When income tax is considered on coupon payments:-

Investors would wish to consider the coupon payments net of tax, i.e. after income tax has been deducted from them. If the rate of income tax is t

_{I} then the price would be

\[P=\ \left(1-\ t_I\right)C.a^{\left(p\right)}_{\overline{n|}}+\ {Rv}^n\]

The yield that the investor would consider for this would be the net yield, i.e. the yield that takes into consideration the amount of income tax paid on the coupons.

When capital gains tax is considered:-

When the redemption amount is more than the price being paid, the investor would make a capital gain.
If \(\left(R-P\right)\) is positive it means there is a capital gain, and when there is a capital gain, a capital gains tax would be levied on the amount of gain.

While determining the price, it wouldn’t be known if there would be a capital gain or not, since the price would still be unknown. Therefore a capital gains test would have to be performed to determine whether there would be a capital gain or not. The capital gains test is performed as explained below:

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

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

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

When there is no capital gain, the formula given above (in the previous section) can be used. But when there is a capital gain, then capital gains tax must be considered.

If the rate of capital gains tax is \(t_C\) then the equation for the price can be written as follows:

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

It must be noted that the capital gains tax should not be calculated on the total redemption amount, but only the capital gain \(\left(R-P\right)\) .

The above equation can be simplified to the following form:-

\[P=\ \frac{\left(1-t_I\right)C.a^{\left(p\right)}_{\overline{n|}}+\left(1-t_c\right){Rv}^n}{1-t_cv^n}\]

**EXAMPLES**

Example 1

An investor wishes to purchase \(\textrm{₹}\ 1000\) nominal of a \(10\) year bond which is redeemable at par and pays coupons of \(6\ \%\) p.a. in arrears. The investor expects a yield of \(10\ \%\) .

Here,

\(R=1000\) (the bond is redeemable at par)

\(C=60\ \left(6\ \%\ of\ 1000\right)\)

\(i=10\ \%\)

\[P=C.a_{\overline{n|}}+{Rv}^n\]
\[\ \ \ =60.\frac{\left(1-v^{10}\right)}{i}+1000v^{10}\]
\[\ \ \ =60.\frac{\left(1-1.1^{-10}\right)}{0.1}+1000\times 1.1^{-10}\]
\[\ \ \ =754.22\]

Example 2

Consider the same case as example 1 where the investor wishes to purchase \(\textrm{₹}\ 1000\) nominal of a bond redeemable at par and paying annual coupon of \(6\ \%\) . But in this case the coupon payments are made twice a year. The investor expects the same yield of \(10\ \%\).

Here,

\(R=1000\)

\(C=60\)

\(i=10\ \%\)

\(p=2\) (since the coupon payment of \(\textrm{₹}\ 60\) is made in two installments)

\[i^{\left(p\right)}=\ i^{\left(2\right)}=2\left[{1.1}^{{1}/{2}}-
1\right]=9.7617\ \%\]

\[P=C.a^{\left(p\right)}_{\overline{n|}}+\ {Rv}^n\]
\[\ \ \ =60.\frac{\left(1-\ v^{10}\right)}{i^{\left(12\right)}}+1000v^{10}\ \]
\[\ \ \ =60.\ \frac{\left(1-{1.1}^{-10}\right)}{0.097617}+1000\times 1.1^{-10}\]
\[\ \ \ =763.21\]

The price in example 2 is higher than that in example1, the reason being that coupon payments were made twice a year in example 2 as against once in example 1, i.e. \(\textrm{₹}\ 60\) was paid once at the end of every year in example 1, whereas in example 2, \(\textrm{₹}\ 30\) was paid at the end of every six months. Since the present value of an amount that is received earlier is higher than an amount received on a later date, the price is higher in example 2.

Example 3

Consider the same case as example 2. But now the investor will be levied income tax on the coupon payments he receives, at \(\ 30\ \%\) .

Here,

\(R=1000\)

\(C=60\)

\(i=10\ \%\)

\[i^{\left(p\right)}=i^{\left(2\right)}=9.7617\ \%\]
\[{t}_I=30\ \%\]

\[P=\ \left(1-t_I\right)C.a^{\left(p\right)}_{\overline{n|}}+{Rv}^n\ \]
\[\ \ \ =\ \left(1-0.3\right)60\frac{\left(1-1.1^{-10}\right)}{0.097617}+1000\times 1.1^{-10}\]
\[\ \ \ =649.91\ \]

Example 4

Consider the same case as example 3. But now the investor is also subject to capital gains tax at the rate of \(20\ \%\) .

Here,

\(R=1000\)

\(C=60\)

\(i=10\ \%\)

\[i^{\left(p\right)}=\ i^{\left(2\right)}=9.7617\ \%\]
\[t_I=30\ \%\]
\[t_C=20\ \%\]

We first need to determine whether there would be a capital gain or not. The capital gains test says that there would be a capital gain if \({\ i}^{\left(p\right)}>\ \left(1-t_I\right)\frac{C}{R}\)

\[i^{\left(2\right)}=9.7617\ \%\]

\[\left(1-t_I\right)\frac{C}{R}=0.7\times \ \frac{60}{1000}=0.042\]

Since \(i^{\left(2\right)}>\left(1-t_I\right)\frac{C}{R}\) there is a capital gain. So, the price is given by the following formula:

\[P=\ \frac{\left(1-t_I\right)C.a^{\left(p\right)}_{\overline{n|}}+\left(1-t_C\right)Rv^n}{1-t_Cv^n}\]
\[\ \ \ =\ \frac{\left(1-0.3\right)\times 60\times {{\frac{\left(1-1.1^{-10}\right)}{0.097617}}}\ +\left(1-0.2\right)\times 1000\times 1.1^{-10}}{1-0.2\times 1.1^{-10}}\]
\[\ \ \ =\ 620.66\]