A convertible holder indirectly owns a specific number of shares (for which the convertible bond will be exchanged eventually), but receives coupons instead of dividends until maturity or conversion, whichever occurs first. This is equivalent to owning a dividend swap that gives the equity investor the coupons (fixed payments) in exchange for dividends. Assuming there is an income advantage for the convertible holder, the investor will pay a premium to buy this embedded dividend swap.
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