Conversion price, parity, bond floor and the volatility the issuer is really selling, worked on one illustrative five-year deal.
The conversion premium is the conversion price divided by the share price at pricing, minus one. On an illustrative five-year convertible priced with the share at 40.00 and a 30.0 per cent premium, the conversion price is 52.00, each EUR 100,000 bond converts into 1,923.08 shares and parity is 76.92. With a bond floor of 89.03, investors are paying 10.97 points for the conversion option, an implied volatility of 27.3 per cent.
Worked in full in Capital Markets by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
The issuer, its share and its credit are fictional. The deal is EUR 500.0 million at par, a 2.00 per cent annual coupon, five years, bullet, no issuer call and no investor put, so the arithmetic is not clouded by features that need a tree to value. The issuer could borrow straight for five years at an illustrative 4.50 per cent. Market inputs for the option are a 2.50 per cent risk-free rate, a 2.00 per cent dividend yield and an illustrative historical volatility of 30.0 per cent.
| Term | Value |
|---|---|
| Reference share price | 40.00 |
| Conversion premium | 30.0% |
| Issue size, EUR million | 500.0 |
| Denomination, EUR | 100,000 |
| Coupon, annual | 2.00% |
| Straight debt yield, five years | 4.50% |
| Risk-free rate / dividend yield | 2.50% / 2.00% |
Conversion price = share price × (1 + premium) = 40.00 × 1.30 = 52.00
Conversion ratio = denomination / conversion price = 100,000 / 52.00 = 1,923.08 shares per bond
Parity = conversion ratio × share price / denomination = 1,923.08 × 40.00 / 100,000 = 76.92 per cent of par
Conversion premium at issue = issue price / parity − 1 = 100 / 76.92 − 1 = 30.0%
Bond floor = coupons and principal discounted at the straight yield = 89.03 per cent of par
Option value paid by investors = 100 − 89.03 = 10.97 points
Excel: =PV(4.5%,5,-2,-100) returns the bond floor of 89.03; the conversion price is =B1*(1+B2).
The premium is the easy number. What matters is what the investor is buying for the 10.97 points between the bond floor and par: a five-year call on 1.9231 shares per 100 of face, struck at 52.00. Valued with Black-Scholes on the share, that option is worth 9.55 points at 25 per cent volatility, 12.64 points at 30 per cent, and 10.97 at 27.3 per cent. So 27.3 per cent is the implied volatility of the issue, and it is the figure convertible investors and bookrunners actually argue about.
The issuer saves 2.50 percentage points of coupon a year against straight debt, EUR 12.50 million a year or EUR 62.50 million undiscounted over the five years. In present value that saving is exactly the 10.97 points of option value, EUR 54.9 million on the deal. Nothing is free: the coupon saving is the price of the option.
Whether the issuer sold the option at a good price depends on what it is worth. At the 30.0 per cent historical volatility it is worth 12.64 points, so the issuer handed over 1.66 points more than it was paid for, EUR 8.3 million, or about 0.38 points a year spread over the 4.3900 annuity factor of the straight curve. That gap, the cheapness of the option, is the convertible's new issue concession and, like a bond concession, it never appears on an invoice.
The 9.62 million shares a full conversion would issue are the dilution side of the trade. They are issued at 52.00, not 40.00, which is the point of the premium; the comparison with a straight placing is in how much a discounted placing dilutes shareholders.
| Premium | Conversion price | Parity | Shares per bond | Shares, million | Implied vol at par | Coupon for 30% vol |
|---|---|---|---|---|---|---|
| 20% | 48.00 | 83.33 | 2,083.33 | 10.42 | 23.1% | 0.94% |
| 25% | 50.00 | 80.00 | 2,000.00 | 10.00 | 25.2% | 1.30% |
| 30% | 52.00 | 76.92 | 1,923.08 | 9.62 | 27.3% | 1.62% |
| 35% | 54.00 | 74.07 | 1,851.85 | 9.26 | 29.3% | 1.90% |
| 40% | 56.00 | 71.43 | 1,785.71 | 8.93 | 31.2% | 2.15% |
Raise the premium and the option is further out of the money, so the same 10.97 points buy a less valuable call and the implied volatility climbs. At a 40 per cent premium and a 2.00 per cent coupon the issuer would be selling volatility at 31.2 per cent, above the 30.0 per cent the share has shown, and investors would not buy it. The last column is the coupon that makes each premium fair at 30.0 per cent: 1.62 per cent at a 30 per cent premium, 2.15 per cent at 40. Premium and coupon are one decision, not two, and the headline premium on its own says nothing about whether the deal was rich or cheap.
Once the bond trades, the quoted premium is the price over parity, and it moves with the share. Holding the bond floor and the 27.3 per cent volatility constant, which assumes the credit does not change:
| Share price | Parity | Bond price | Premium over parity | Premium over bond floor |
|---|---|---|---|---|
| 32.00 | 61.54 | 94.45 | 53.5% | 6.1% |
| 40.00 | 76.92 | 100.00 | 30.0% | 12.3% |
| 48.00 | 92.31 | 107.35 | 16.3% | 20.6% |
| 52.00 | 100.00 | 111.60 | 11.6% | 25.4% |
| 60.00 | 115.38 | 121.06 | 4.9% | 36.0% |
A falling share pushes the premium over parity up and the bond towards its floor: it behaves like debt. A rising share compresses the premium towards zero and the bond behaves like equity. In practice a falling share often comes with a wider credit spread, which lowers the floor too, so the downside row flatters the bond.
Treating the premium as the cost of the deal. A 30 per cent premium sounds like the issuer selling shares 30 per cent above market, and it is, but the investor has been paid for that with an option the issuer funds through the coupon. The cost is the option given away net of what investors paid for it, which here is the 1.66-point gap between 12.64 and 10.97. The second mistake is quoting premium over the bond floor as if it were the conversion premium: 12.3 per cent and 30.0 per cent describe the same bond at the same moment.
Compute the conversion price, the ratio and parity, then the bond floor, and back out the volatility the price implies. A 30 per cent premium with a 2.00 per cent coupon is a 27.3 per cent volatility sale, EUR 8.3 million cheaper than the share's own history on this deal. The fee and concession arithmetic for the book's own bond issue is in the free workbook for this case, and the option mechanics in the Black-Scholes worked example.
Premiums are usually set somewhere between the 20s and the 40s in per cent, and the choice trades off against the coupon. In the worked case, keeping a 2.00 per cent coupon, raising the premium from 20 to 40 per cent lifts the implied volatility investors pay from 23.1 to 31.2 per cent, which is why a higher premium usually comes with a higher coupon.
At pricing they are the same number: the bond is issued at 100 and parity is 100 divided by one plus the premium, 76.92 for a 30 per cent premium. Afterwards the premium over parity moves with the share. If the share rises to 48.00, parity becomes 92.31, the bond trades around 107.35 and the premium over parity falls to 16.3 per cent.
Issue size divided by the conversion price. A EUR 500.0 million deal converting at 52.00 issues 9.62 million shares if fully converted; each EUR 100,000 bond converts into 1,923.08 shares. A lower premium of 20 per cent would raise that to 10.42 million shares, which is the dilution side of the premium decision.
This article is one calculation from Capital Markets. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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