The mechanics, in the order a shareholder meets them. The company announces the offering and sets a record date; anyone who owns shares on that date receives one right per share, or a stated number. It sets a ratio, say four rights plus a cash payment to buy one new share, and a subscription price, almost always below the current market price so that the rights have value and holders have a reason to participate. It sets an expiration date, typically a few weeks out. If the rights are transferable, they typically trade on the exchange during that window under their own ticker symbol, so a holder who does not want more shares can sell the rights to someone who does. If they are non-transferable, the only choices are to exercise or to let them lapse. Many offerings add an oversubscription privilege, letting holders who exercise in full ask for additional shares that other holders declined, and some are backstopped by an underwriter or a large shareholder who agrees to buy whatever is left.
Why the rights have value, and what ignoring them costs. When new shares are sold below the market price, the average value of all shares afterward is lower than the price before, and the right to buy at the discount is worth the difference. A holder who exercises pays the subscription price for shares worth more than that, and the gain on the new shares offsets the decline on the old ones; the holder's wealth and percentage ownership are both unchanged by the offering itself. A holder who sells the rights collects that value in cash and accepts a smaller percentage of the company. A holder who does nothing gives the value to whoever ends up with the shares and is diluted without compensation. That is why the practical rule for a shareholder is that "do nothing" is the one choice that is almost always worse than the other two, whatever the holder thinks of the company.
The tax lives in the rights, not in the offering. Publication 550 states the starting point: "Generally, stock dividends and stock rights are not taxable to you, and you do not report them on your return," with exceptions where a holder could have chosen cash instead or where preferred holders are involved. What matters is basis. Under IRC section 307(b), if the fair market value of the rights at distribution is "less than 15 percent of the fair market value of the old stock at such time," the basis of the rights is zero, unless the taxpayer elects, on a timely return for that year, to allocate part of the old stock's basis to them. If the rights are worth 15 percent or more, the allocation is mandatory, in proportion to the two fair market values. Publication 550 then follows each path. Rights that expire "have no basis," so there is no loss to deduct. Rights that are sold produce a gain or loss measured against their allocated basis, which for most small rights is zero, and the holding period of nontaxable rights includes the holding period of the underlying stock. Rights that are exercised disappear into the new shares: "the basis of the new stock is its cost plus the basis of the stock rights exercised," and the holding period of the new shares starts on the exercise date.
Why closed-end funds do this so often. Section 23(b) of the Investment Company Act of 1940 prohibits a registered closed-end fund from selling its common stock "at a price below the current net asset value of such stock," which for a fund trading at a discount would otherwise make raising new capital impossible. The first of the section's exceptions is a sale "in connection with an offering to the holders of one or more classes of its capital stock." A rights offering is therefore the standard way a closed-end fund grows, and because the subscription price is often set below net asset value, a shareholder who does not participate suffers dilution of net asset value per share as well as of percentage ownership. The closed-end fund page covers the discount itself; the point here is that for these funds a rights offering is not an unusual event but the built-in growth mechanism, and the arithmetic above applies with extra force.
Reading the announcement. The documents disclose the ratio, the subscription price, the record and expiration dates, whether the rights are transferable, whether there is an oversubscription privilege, who is backstopping the deal and what the company intends to do with the money. For a company registered with the SEC, the offering is made under a registration statement and the prospectus is on EDGAR. The one figure that is not in the documents is the market's judgment of the shares afterward, which is what decides whether buying more at the subscription price was a good idea; the offering's structure only makes not participating the costlier choice.