Two Claims at Maturity
A convertible bond gives its holder a choice, subject to the contract: receive the bond’s promised repayment or exchange the bond for a specified number of the issuer’s shares. The key comparison is between two dollar amounts at the decision date. One is the cash redemption amount; the other is the market value of the shares the holder would receive.
This comparison is useful when analyzing the bond’s payoff at maturity. It does not, by itself, determine the bond’s market value before maturity. Before maturity, remaining time, interest rates, credit risk, and other contract features can matter. The simplified calculations here assume the issuer can pay the promised amount, conversion is voluntary, and the holder may choose one alternative at maturity.
The conversion ratio \(m\) is a number of shares per bond, not a dollar amount. Multiplying it by the share price gives a dollar value. If the bond’s redemption amount is \(R\), the share price at which the two choices have equal value is \(S^*=R/m\). Below that price, repayment is worth more; above it, conversion is worth more. At exactly that price, the two alternatives have equal value.
Why the Maximum Gives the Payoff
The maximum formula follows directly from the holder’s choice. If the holder does not convert, the holder receives \(R\). If the holder converts, the holder receives shares worth \(mS_T\) at maturity. Since the holder can select the more valuable available alternative, the payoff is the larger of those two amounts.
To identify the relevant case, compare the quantities in the same units. The inequality \(mS_T>R\) can be divided by \(m\), which is positive, to give \(S_T>R/m\). Thus conversion is the higher-value choice when the share price exceeds the break-even price \(S^*\); repayment is the higher-value choice when it is below \(S^*\).
This result concerns the value of the alternatives at the maturity decision. It assumes the promised repayment is payable in full. If the issuer defaults, the actual debt recovery may be less than the promised amount, and this simplified comparison no longer describes the complete payoff. Contract terms can also change the available choices; for example, a bond may have a conversion restriction. Apply the maximum only when both alternatives are available and the holder can choose freely.
Worked Example: Comparing Three Maturity Share Prices
A hypothetical convertible bond has a $1,000 redemption amount and a conversion ratio of 25 shares per bond. No coupon is due at the decision date. Find the break-even share price and the maturity claim when the share price is (a) $36, (b) $40, and (c) $47.
Method. Calculate \(S^*=R/m\), then calculate \(mS_T\) at each share price. Compare the conversion value with the $1,000 redemption amount.
Break-even price. \(S^*=\$1{,}000/25=\$40\) per share. At $40, the two choices have equal value.
(a) Share price $36. Conversion value \(=25\times \$36=\$900\). Since $900 is less than the $1,000 repayment, the holder chooses repayment. The maturity claim is $1,000.
(b) Share price $40. Conversion value \(=25\times \$40=\$1{,}000\). The alternatives are equal, so either choice gives a claim worth $1,000.
(c) Share price $47. Conversion value \(=25\times \$47=\$1{,}175\). Since $1,175 exceeds $1,000, the holder converts. The maturity claim is $1,175.
Check. The selected claim is $1,000 below the $40 threshold, $1,000 at the threshold, and $1,175 above it. The choices change at the calculated break-even price. The common mistake avoided is comparing the share price itself with the $1,000 repayment rather than comparing the value of all 25 shares with that repayment.
From One Bond to a Holding
For multiple identical bonds, calculate the alternatives consistently. If an investor holds \(n\) bonds, the total cash repayment is \(nR\), and the total number of conversion shares is \(nm\). At a common share price \(S_T\), the total conversion value is \(nmS_T\). The break-even price per share remains \(R/m\), because multiplying both alternatives by the same positive number \(n\) does not change which is larger.
Keep the number of bonds separate from the conversion ratio. The conversion ratio tells you how many shares come from one bond; the number of bonds scales that amount to the investor’s total holding. If coupons are separately payable regardless of conversion, add them to the investor’s total receipts after comparing the two principal alternatives. Do not use a coupon to change the break-even comparison when it is identical under both choices.
Worked Example: Comparing Choices for an Issue Holding
An investor holds 12 identical hypothetical convertible bonds. Each bond has a $5,150 redemption amount and can be converted into 125 shares. At maturity, the share price is $42.80. Determine which choice has the higher value and calculate the difference for the holding.
Method. First compare the alternatives per bond, then multiply each by 12. This avoids mixing a per-bond amount with the total shares from the entire holding.
Break-even price. \(S^*=\$5{,}150/125=\$41.20\) per share. Since $42.80 exceeds $41.20, conversion should have the higher value.
Conversion value per bond. \(125\times \$42.80=\$5{,}350\). The difference per bond is \(\$5{,}350-\$5{,}150=\$200\).
Total repayment. \(12\times \$5{,}150=\$61{,}800\).
Total shares and conversion value. The investor would receive \(12\times125=1{,}500\) shares. Those shares are worth \(1{,}500\times \$42.80=\$64{,}200\).
Difference and check. Conversion is higher by \(\$64{,}200-\$61{,}800=\$2{,}400\), which agrees with \(12\times \$200=\$2{,}400\). The common mistake avoided is multiplying the share price by 125 but then forgetting to scale the conversion value to all 12 bonds—or scaling the shares twice.
Redemption Amounts, Coupons, and Units
The amount used in the comparison is the amount actually promised at maturity, not automatically the bond’s original face amount. If a bond with $1,250 face amount is redeemable at 103% of face, its redemption amount is \(1.03\times \$1{,}250=\$1{,}287.50\). The conversion value must be compared with $1,287.50.
Coupons require attention to the contract. If the final coupon is paid in cash under either choice, it is common to both alternatives: add it to either outcome when calculating total receipts, but it does not affect which principal alternative is more valuable. If converting changes whether a coupon is received, or if the contract includes accrued interest in a conversion payment, include those terms explicitly in the alternatives before comparing them.
The basic calculation also assumes the conversion ratio is already known and applies at the decision date. If the contract specifies an adjustment to that ratio, use the adjusted ratio. A price per share and a redemption amount must be stated in the same currency, and the share price must use the same currency unit as the redemption amount. These checks prevent unit errors before arithmetic begins.
Worked Example: Premium Redemption and a Separate Coupon
A hypothetical bond has a $1,250 face amount, is redeemable at 103% of face, and converts into 35 shares. At maturity the share price is $36.90. A $22.50 coupon is paid in cash whether or not the holder converts. For one bond, compare total receipts under the two choices and determine the higher-value alternative.
Method. Calculate the promised redemption amount from the stated percentage, compare that principal amount with conversion value, and then add the common coupon to both outcomes.
Redemption amount. \(1.03\times \$1{,}250=\$1{,}287.50\).
Conversion value. \(35\times \$36.90=\$1{,}291.50\). Conversion exceeds repayment by \(\$1{,}291.50-\$1{,}287.50=\$4.00\).
Break-even price. \(S^*=\$1{,}287.50/35=\$36.785714\ldots\), or $36.79 per share to the nearest cent. The actual share price of $36.90 is above the unrounded break-even price, consistent with conversion being higher.
Total receipts. If the holder takes repayment, total receipts are \(\$1{,}287.50+\$22.50=\$1{,}310.00\). If the holder converts, the shares are worth $1,291.50 and the coupon adds $22.50, for total receipts of \(\$1{,}291.50+\$22.50=\$1{,}314.00\). Conversion is higher by $4.00.
Check. Both alternatives include the same $22.50 coupon, so the difference in total receipts remains $4.00, exactly the difference between conversion value and redemption. The common mistake avoided is treating face amount as the repayment when the bond specifies a premium redemption amount.
Common Traps and Appropriate Use
A frequent error is to compare a share price directly with a bond’s dollar repayment. The units do not match: a per-share price is dollars per share, while repayment is dollars per bond. Multiply the price by the number of shares received per bond first. Another error is to use face amount when the contract promises a different redemption amount. A third is to treat conversion value as the bond’s market value at any date. The maximum calculation here describes the holder’s available maturity claims under the stated assumptions; it is not a general pre-maturity pricing formula.
A quick check is to calculate the break-even share price \(R/m\) and verify the comparison. If the share price is above that amount, \(mS_T\) must exceed \(R\); if below, it must be less. This check can catch an arithmetic or unit error before choosing an alternative.
Check Your Understanding
For each problem, assume the stated redemption is payable in full and conversion is voluntary at maturity. Ignore any coupons unless the problem specifies one.
- A bond is redeemable for $800 and converts into 20 shares. At a maturity share price of $39.50, calculate the conversion value, identify the better choice, and give the break-even share price. (Answer: Conversion value $790; take the $800 repayment; break-even price $40.00 per share.)
- A bond with $1,500 face amount is redeemable at 102% of face and converts into 30 shares. If the maturity share price is $52, calculate its redemption amount, conversion value, and the advantage of the better choice. (Answer: Redemption $1,530; conversion value $1,560; conversion is higher by $30.)
- An investor holds 8 bonds, each redeemable for $2,400 and convertible into 48 shares. The maturity share price is $47.50. Compare total repayment with total conversion value and state the difference. (Answer: Repayment $19,200; conversion value $18,240; repayment is higher by $960.)
- A bond is redeemable for $990 and converts into 22 shares. The maturity share price is $45.00, and a $18 coupon is paid under either choice. Compare total receipts under repayment and conversion. (Answer: Conversion value and redemption are both $990; either choice plus the coupon gives total receipts of $1,008.)
- An investor holds 25 bonds, each redeemable for $1,040 and convertible into 26 shares. At maturity the share price is $41.50. Calculate total repayment, total conversion shares, total conversion value, and which choice is higher. (Answer: Repayment $26,000; 650 shares; conversion value $26,975; conversion is higher by $975.)