Nil-paid rights arise when a firm sells new shares for cash to existing shareholders via a rights issue. So, for example, a firm might offer one new share priced at, say, £1 for every four currently held. That’s called a ‘one for four’ issue. Let’s say the current share price is £2.50. So after the new share has been issued you would expect the firm’s shares to trade at around £2.20 (4 x £2.50 = £10. And (£10 + £1)/5 is £2.20). That’s called the ex-rights price.
Any shareholder can choose not to take up their rights, in which case they can often be sold. The ‘nil-paid’ price is the difference between the issue price and the expected ex-rights price. Here that’s £2.20 – £1 = £1.20. Another investor who was not invited to participate in the original rights issue might be interested in paying for nil-paid rights instead.