**FORMULA AND DERIVATION**

Following a rights issue, the price of a company’s share falls below its existing level due to the extra number of shares and the discount allowed on the rights issue price. This price after a rights issue is called the TERP(Theoretical Ex-Rights Price). Investors or shareholders who do not wish to exercise their rights will have the same number of shares, but now worth less than before. Therefore, the theoretical value of these rights must be that value which compensates them for this fall in share price. So, when a shareholder doesn’t wish to exercise his right, he can opt to sell it and receive an amount equal to the amount lost due to the fall in the share price.

Therefore, the theoretical value of a nil-paid right is the difference between the TERP and the rights issue price.

For an \(n-for-m\) rights issue, i.e. \(n\) number of extra shares issued for every \(m\) number of shares held, the TERP can be calculated as follows:-

\[TERP=\frac{mP+nQ}{m+n}\]

where, \(P\) is the share price before rights issue and \(Q\) is the rights issue price.

The theoretical value of the nil-paid right is the difference between TERP and the rights issue price.

\[Value\ of\ nil\ paid\ right=\frac{mP+nQ}{m-n}-Q\]
\[\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\frac{mP+nQ}{m+n}-\frac{\left(m+n\right)Q}{m+n}\]
\[\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\frac{mP+nQ-mQ-nQ}{m+n}\]
\[\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\frac{m\left(P-Q\right)}{m+n}\]
\[\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\frac{\left(P-Q\right)}{\frac{m+n}{n}}\]
\[\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\frac{\left(P-Q\right)}{1+\frac{n}{m}}\]

**EXAMPLES**

Example 1

A company’s share is currently trading at \(\textrm{₹}100\). The company decides to make a \(1-for-4\) rights issue to raise additional capital. The rights issue price was fixed at \(\textrm{₹}50\). One investor, holding 4 shares in the company, does not wish to exercise his right to purchase these extra shares. He therefore decides to sell his rights separately and wishes to know the value of the nil-paid right that he intends to sell.

In this case,

\[P=100\]
\[Q=50\]
\[n=1\]
\[m=4\]

The value of the nil-paid right can be calculated as follows:-

\[Value\ of\ nil\ paid\ right=\frac{\left(P-Q\right)}{1+\frac{n}{m}}\]
\[\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\frac{\left(100-50\right)}{1+\frac{1}{4}}\]
\[\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\frac{50}{{5}/{4}}\]
\[\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =40\]

In order to understand how \(\textrm{₹}40\) is the theoretical value of the nil-paid right, we must consider the TERP.

\[TERP=\frac{mP+nQ}{m+n}\]
\[\ \ \ \ \ \ \ \ \ \ \ \ =\frac{(4\times 100)+(1\times 50)}{4+1}\]
\[\ \ \ \ \ \ \ \ \ \ \ \ =90\]

This means that the share price immediately after the rights issue would fall from\(\textrm{₹}100\) to \(\textrm{₹}90\). The investor in this example has four shares that were worth \(\textrm{₹}100\) each before the rights issue, but are worth only \(\textrm{₹}90\) each after the rights issue. Therefore his total worth(of four shares) would fall from \(\textrm{₹}400\) to \(\textrm{₹}360\). By selling his rights at \(\textrm{₹}40\) , his total worth will again be \(\textrm{₹}400\).