Convertible Bonds: How Much Equity, How Much Debt?

CFA Institute

Convertible Bonds: How Much Equity, How Much Debt?Author(s): Marcelle Arak and L. Ann MartinSource: Financial Analysts Journal, Vol. 61, No. 2 (Mar. - Apr., 2005), pp. 44-50Published by: CFA InstituteStable URL: http://www.jstor.org/stable/4480655 .

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........................... .:.-.: A TxXvXX Financial Analysts Journal

LEE -;-Volume 610 Number 2 ?2005, CFA Institute

Convertible Bonds: How Much Equity,

How Much Debt?

Marcelle Arak and L. Ann Martin

Financial analysts need accurate estimates of debt, equity, leverage, and EPS. The method proposed here, based on the probability of conversion, yields new estimates of the debt and equity in a convertible bond issue. When this method is used, the value of the equity component in a hypothetical issue is found to be substantial-larger than the value of the options and clearly larger than zero, which is assigned under current accounting rules. The estimate of the debt component is smaller than recorded under current accounting rules. Thus, the leverage of convertible bond issuers is substantially lower when this method is used.

C ompanies have issued a large number of convertible bonds in recent years, reflecting the appeal of such bonds' low coupon rates for corporations and the appeal of

their embedded stock options for investors. Despite the popularity of convertible bonds, however, relatively little progress has been made in recording the potential equity value of conversion on the financial statements of companies or incor- porating this value into the financial ratios that analysts use to evaluate a corporation's stock and bond values.

Accountants in the United States still record convertible bonds entirely as debt. They do, however, add the potential shares that would arise from conversion to the total shares outstanding for com- puting fully diluted EPS. Besides the obvious inconsistency within GAAP, these extreme treat- ments of convertibles do not reflect the "optional- ity" involved in convertible bonds: Convertibles have some probability of ending up as stock and some probability of ending up as bonds.

We discuss a new approach for determining the debt and equity portions of a convertible bond. The approach is based on the assumptions and theory associated with standard option-pricing models. The results are quite different, however, from those of previous option approaches: This method does not lead to the conclusion that the equity value of a convertible bond is equal to the value of the embedded options.

In addition, financial theory implies that the value of the debt and equity portions will change over the life of the issue.1 An analyst needs to be aware of these changes. For any option-type instru- ment, changes in the price of the convertible bond's underlying security (i.e., the stock) and shrinkage of the time to maturity affect the value of the convertible bond and the equity and debt values associated with it. Our approach allows the analyst to bifurcate outstanding convertible bonds into debt and equity, both at issue and later, to obtain a dynamic, "economic" view of the company's debt, equity, leverage, and EPS.

Background on Convertible Bonds Convertible bonds have been issued for many years. These bonds have low coupons but provide value to the investor in the form of embedded call options on the stock of the issuer: The bond can be traded in for stock at a predetermined conversion ratio. That conversion ratio implies a certain stock price that is the implicit "exercise price" on the stock (call) option.

Several previous researchers have recognized the relevance of modern option theory to convertible bonds. Vigeland (1982) was early to note that option theory could be applied to the probability and timing of conversions. King (1984) actually calculated the option values for many convertible bond issues and treated that option value as if it represented the dollar value of the equity. This "equity" measure was then used to recalculate the leverage ratio by subtracting the "equity" value from the total bond value and adding it to equity outstanding. The number of shares represented by the convertibles was calculated as this "equity value" divided by the current share price.

Marcelle Arak is professor offinance at the University of Colorado at Denver and Health Sciences Center. L. Ann Martin is associate professor of accounting at the Univer- sity of Colorado at Denver and Health Sciences Center.

44 www.cfapubs.org ?2005, CFA Institute



Convertible Bonds

Gaumnitz and Thompson (1987) ran regres- sions to see how much the price of the convertible bond moved in relation to the price of the underlying equity. They proposed that a bond whose regression coefficient is equal to its conversion ratio be treated as equity.

King and Gaumnitz and Thompson were headed toward logical bifurcation methods but stopped short of reaching them. King assumed that the equity value of the convertible bonds is identical to their option value. But options generally represent far more equity value than the simple value of the options indicates; a leverage factor is inherent in options. As a consequence of this misconception, King's "equity values" are understated-and substantially so. Also, he assumed that the debt value of the bonds is identical to the value of straight bonds. That would be true if the bonds were certain to survive as bonds. If we recognize the substantial probability of conversion, however, the expected value of the debt is much lower than the value of straight bonds.

Gaumnitz and Thompson used their econo- metric approach to reach an "all or none" conclusion: Bonds that behaved exactly as the underlying equity times the conversion factor were to be treated as equity while all other issues were to be treated as debt. But the Gaumnitz-Thompson coef- ficients could have been used to bifurcate a sea- soned convertible into equity and bond pieces rather than to simply identify cases in which the debt issues were to be treated as 100 percent equity.

In the United States, no bifurcation is incorpo- rated into the accounting guidance for convertible bonds. The accounting profession still follows the guidance in Accounting Principles Board (APB) Opinion No. 14 (see AICPA 1969) for recording convertible debt, despite several efforts in recent years to alter the approach. APB Opinion No. 14 says that convertible debt is to be recorded solely as debt.

The International Accounting Standards Board (IASB) issued International Accounting Standard (IAS) No. 32 in 1998 (revised 2003). This standard requires bifurcation of convertible bonds into debt and equity but is not specific about how to calculate the debt and equity values; it offers only suggestions. One suggestion is to value the debt component as the present value (PV) of the interest payments and the principal and then subtract this amount from the total value of the debt issue to arrive at the value of the equity portion. This approach is similar to that of APB Opinion No. 10 (see AICPA 1966), an earlier U.S. approach, and to the approach in King's analysis.

Following the passage of IAS No. 32, the Finan- cial Accounting Standards Board put the convertible bifurcation issue back on its agenda. In 2000, it

issued an exposure draft (ED 213-B), Accountingfor Financial Instruments with Characteristics of Liabili- ties, Equity, or Both, that addressed the treatment of convertible debt (see FASB 2000). In May 2003, however, with the issue of Statement of Financial Accounting Standard (SFAS) No. 150, Accounting for Certain Financial Instruments with Characteristics of Both Liabilities and Equity, the accounting treatment of convertible debt was not changed. In 2004, the FASB resumed study of liabilities and equity, and the board has apparently been leaning toward a bifurcation based on the PV of the straight bond (as debt) and the value of the embedded options (as equity), a bifurcation similar to that described in APB Opinion No. 10 and IAS No. 32.

New Approach to Bifurcation The new methodology we suggest is based on the observation that convertible bonds have some chance of remaining bonds and some chance of being converted into stock. Our approach to valu- ing the debt and equity is built on the probabilities of these two events.

Separating the value of the straight bond (the PV with the options omitted) and the value of the options-one good way to bifurcate the value inherent in a convertible bond-is not a good way to divide the bond into debt and equity. The option could produce equity, or it could leave bond obli- gations intact. The option itself is neither debt nor equity. For example, if the bond is converted, the bond principal will not need to be repaid. The only debt-type payments will be the coupons that were paid prior to conversion. Clearly, then, the debt- type payments in PV terms will be smaller in the case of conversion than they would be if conversion did not occur. The expected value of the debt in the convertible issue is a weighted average of the value of the straight bond (including the principal) and its (much smaller) value as a series of coupon payments prior to conversion. Thus, the value of the straight bond overstates the expected value of the debt part of the bond.

If the bond is converted into stock, the value of the stock will be larger than the face value of the bond principal-because it will not be converted unless the stock value of the bond is greater than its value as a bond. The expected value of the equity component needs to include this substantial conversion value, weighted by the probability that conversion will occur.

The expected values that analysts need for a recalculation of debt, equity, leverage, and EPS are the expected number of shares represented by the convertible issue, the expected value of the bond's debt, and the expected value of its equity.

March/April 2005 www.cfapubs.org 45



Financial Analysts Journal

Expected Number of Shares. Typically, analysts use the total potential shares into which the convertible can be converted as the measure to use in calculating EPS. But expected shares is a better measure of the shares associated with the convertible bond.

The expected number of shares is the potential number scaled by the probability that the bond will be converted into stock. For example, if n is the potential number of shares and the probability of conversion is p, the expected number of shares represented by the issue is n(p).2

Expected Value of the Debt. The value of the straight bond-the PV of the coupons and principal using the fair market rate for a company's "plain vanilla" debt as the discount rate-is frequently used as the measure of the "debt" part of the convertible issue. But this measure is not the expected value unless the probability that the bond principal will need to be repaid is 100 percent.

If there is a positive probability that the bond principal (and perhaps some coupons) will not need to be repaid, the expected value of the debt component is smaller than the value of the straight bond. The higher the probability that the bond will be converted into equity, the lower the value of the expected debt in the convertible bond.

For example, assume that all the coupons will be paid in any event. Then, the expected value of the debt is a probability-weighted average of the straight bond (paid if there is no conversion) and the coupons alone (paid if the bond is converted at maturity); that is, the expected value of the debt is

(1 - p) (Value of the straight bond) + p (Value of the coupons).

The reader can see that the expected value of the debt is equal to the straight bond's value only if the probability of conversion, p, is equal to zero. If p is greater than zero, the expected value of the debt will be less than the value of the straight bond.

Expected Value of the Equity. An estimate of the equity portion of the convertible bond can be obtained in two ways: 1. As a residual. The total value of the convertible

bond can be regarded as the sum of its expected debt value and its expected equity value. In this method, the expected equity is the difference between the bond's total value and the expected value of the debt.

2. From an option-pricing model. In the Black- Scholes option-pricing model, the value of the option is the difference between the expected (present) value of the equity and the expected (present) value of the exercise cost. When this

relationship is rearranged, the expected equity value is the option value plus the expected value of the exercise cost. (In the case of a convertible bond, the bond is given up to "pay" the exercise cost.) If the exercise cost is discounted by using the company's fair market rate, the expected equity obtained from this method should be identical to that calculated according to the first method.3 In these calculations of expected value of the

shares, debt, and equity, the probability of conversion is a key ingredient. If the bond has no other embedded options, such as issuer calls, and can be converted only at maturity and if the stock does not pay dividends, the Black-Scholes option-pricing model can be used to obtain the probability of conversion: N(d2) is the probability of the option being in the money at the exercise date in the Black- Scholes model. And if investors can be assumed to be rational, N(d2) is the conversion probability in the case of a convertible bond.

If the convertible bond permits exercise prior to maturity or allows the issuer to call the bonds back early, forcing conversion prior to maturity, an American style option-valuation methodology will need to be used. These alternative option-valuation techniques produce a probability of conversion on each possible exercise date. These probabilities can then be used to figure the expected value of the debt and equity for complex structures.

Illustration of Proposed Methodology We illustrate the calculations for the following example: * Company ABC issues $100 million face value

convertible bonds maturing in five years. The coupon is 2.5 percent, which is 5.5 percent below the normal five-year rate of 8 percent for straight issues by this company.

* Each $1,000 bond can be converted into 40 shares of common stock at the bond's maturity; the issuer cannot call the bonds back.

* The current price of the stock is $20. The stock pays no dividends, and the expected price volatility is approximately 35 percent.

* The interest rate on five-year risk-free assets is 3 percent.

We have purposely simplified the example by assuming that: (1) the issuer cannot call the bonds back, (2) there are no embedded options except for the call options represented by the conversion fea- ture, (3) the call is European style, and (4) the exercise date is the same as the bond's maturity date. These assumptions allow straightforward use of




Convertible Bonds

the Black-Scholes model and make the exposition easier than in the case of convertibles with other embedded options, a span of exercise dates, or a time-varying exercise price. These complications can, however, be handled by more sophisticated software that is available, and the basic method would be the same.

Table 1 details the company's convertible bond issue, its embedded options, and the company's capital structure. The corporation is able to issue five-year 2.5 percent bonds at par, even though its required yield is 8 percent, because the bonds include options. The four options for each $100 par of the issue, valued at $5.58 each, add $22.29 to the value of each $100 par. With a coupon of 2.5 percent (and no options), the bond would be worth $77.70 per $100 par as shown in Column 1. For Column 2, which provides the conversion option price information, the value of each option was calculated by using the Black-Scholes model based on the parameters shown.

Table 2 shows how the calculation of diluted shares and EPS are affected by the expected value (EV) treatment. The convertible represents 1,256,800 expected shares of stock (Column 2),

although the bond is potentially convertible into 4,000,000 shares. The expected shares are the 4 million shares multiplied by the probability of conversion of 0.3142 from Table 1. The dilution with our EV method is about 8 percent. In contrast, GAAP, using 4,000,000 shares, indicates that the dilution is 27 percent. Thus, our dilution estimate is considerably smaller.

This smaller dilution when the expected shares are used obviously affects the calculation of diluted EPS. To illustrate the impact on diluted EPS, net income is first recalculated by adding the interest associated with the convertible issue, after taxes, to net income. If we used that same earnings number for the EV method, we would get a measure of diluted EPS of $1.02-an antidilutive effect. (Accountants would not show "diluted earnings" in this case.) Column 2 of Table 2 shows an alternative net income that is more consistent with our EV approach: Only the portion of the after-tax interest associated with the probability of conversion is added back. The result is a diluted EPS of $0.95 for our method. In contrast, diluted EPS calculated under GAAP is $0.87, a consider- able understatement.

Table 1. Company ABC's Convertible Bond Issue and Previous Debt and Equity Issues

Convertible Bond Issue Conversion Option Prior Debt and Equity (1) (2) (3)

Issue size (millions) $100 Current stock price $20 Debt (millions) $300

Coupon 2.5% Strike price $25 Equity shares (millions) $ 15

Maturity (years) 5 Volatility 34.92% Equity value (millions) $300

Number of potential shares in issue (millions) 4 Risk-free rate 3%

Implicit strike price $ 25 Dividend 0

Yield on vanilla bond 8% Value of each option $ 5.58

Value of straight bond (millions) $ 77.70 Conversion probability, p 0.3142

Value of options (millions) $ 22.29

Note: Options were valued by using the Black-Scholes model and the parameters shown in Column 2; p, the probability of conversion, is N(d2) in the Black-Scholes model.

Table 2. Effect of Convertible Bond Issue on Company ABC's EPS

Ratio of Col. 2 GAAP EV Method to Col. 1

Measure Calculated (1) (2) (3)

Preexisting shares 15,000,000 15,000,000 1.00

Additional shares from convertible issue 4,000,000 1,256,800 0.31

Total shares after issue 19,000,000 16,256,800 0.86

Percentage dilution 26.67 8.38 0.31

Net income after tax (NIAT) $15,000,000 $15,000,000 1.00

Basic EPS $1.00 $1.00 1.00

NIAT for diluted EPS (current accounting guidance)a $16,500,000

NIAT for diluted EPS (EV method)b $15,511,500

Diluted EPS $0.87 $0.95 1.09

alnterest expense was added back for entire convertible bond issue, net of tax of 40 percent. bOnly the interest on the portion of the bond issue assumed to be converted was added back.





Table 3 shows how the leverage calculation is affected by our methodology. For comparison, Col- umn 3 shows the leverage calculations if the straight bond/option bifurcation method, suggest- ed in IAS No.32 and in previous FASB discussions, is used, and Column 1 shows the calculations under current GAAP.

Our methodology (Column 2) assumes that the probability of conversion is 0.3142 (see Table 1). The probability-weighted average of the straight bond and the value of the coupons produces a "debt" value of $56.47 million. The straight bond is worth $77.70 million (Line 8), which is much larger than our expected value. Adding the debt contribution from the convertible bond issue to the debt prior to that issue produces $356.47 million for total debt outstanding including the convertible debt (Line 10, Column 2).

Our methodology produces an expected equity value of $43.5 million in the convertible bond (Line 4, Column 2).4 Note that the straight bond/option bifurcation method produces an EV component of only $22.3 million (Line 4, Column 3), about half of the equity value found by our method.

The debt-to-equity ratio based on the EV method (Line 12, Colunm 2) is $356.47/$343.53 or 1.04. According to GAAP, no equity is attributed to the convertible bonds. Thus, under the official accounting guidance of APB Opinion No. 14, the debt-to- equity ratio is 1.33 (Line 12, Column 1). Clearly, following GAAP results in a substantially higher debt-to-equity ratio than found in the EV approach.

Comparison of the results for the EV method with the results implied by the straight bond/ option bifurcation method is also interesting. That method uses $377.70 million for debt-too high- and $322.30 million for equity-too low; the resulting leverage ratio of 1.17 (Line 12, Column 3) is considerably above our result of 1.04.

Changes in Debt and Equity A convertible bond's debt and equity values change with the market price of the issuing company's stock and with time, reflecting the changing value of the embedded options. Assume, for example, the market price of Company ABC's stock, which was $20 when the convertible was issued, rises 50 percent to $30 after one year. The example convertible bond now has four years to maturity. All other parameters-such as interest rates, earnings- remain the same, which allows us to isolate the effects of the stock price and time.

The new market price of $30 exceeds the implicit strike price of $25. The option is now in the money and has a much higher probability of being exercised-53 percent now (from the Black-Scholes model with the $30 stock price and four years to expiration) instead of the original 31 percent.

Table 4 shows how the new situation affects the dilution and the leverage estimates for Com- pany ABC. With a 53 percent probability of conversion, 2,130,400 expected shares are related to the convertible issue; the expected equity shares of the company now number 17,134,000, higher than the

Table 3. Leverage Measures for Company ABC (dollars in millions)

Straight Bond/ GAAP EV Method Option Method

Measure Calculated (1) (2) (3)

(1) Debt prior to new issue $300 $300 $300

(2) Size of convertible debt issue 100 100 100

(3) Value of equity prior to issue of convertible bonda $300 $300 $300

(4) Equity part of new bond issueb 0 $ 43.53 $ 22.30

(5) Total equity value (Row 3 + Row 4) $300 $343.53 $322.30

(6) PV of convertible bond face $ 67.56 $ 67.56 $ 67.56

(7) PV of bond's coupons $ 10.14 $ 10.14 $ 10.14

(8) Value of straight bond $ 77.70 $ 77.70 $ 77.70

(9) Value of "debt" in issueC $100 $ 56.47 $ 77.70

(10) Total debt after issue $400 $356.47 $377.70

(11) Debt to equity before issue 1.00 1.00 1.00

(12) Debt to equity after issue 1.33 1.04 1.17

aGAAP would use the book value of equity, whereas our approach uses the market value; we assume that book value and market value are the same just prior to the convertible bond issue. bThe equity value of the issue is the total value of the issue minus the debt value of the issue. Cln our method, debt is (1 - 0.3142)(Value of straight debt) + 0.3142(Value of coupons).




Convertible Bonds

expected shares at the time of issue. The higher stock price also raises the value of the equity portion of the convertible bond, because the likelihood of exercise is higher and the expected price, if exercised, is also higher.

Company ABC's other shares outstanding have also risen in market value. Using the market value of equity, we find that the leverage of Com- pany ABC has shrunk from the 1.04 that it was just after the issue to 0.64 (shown in Line 11 of Column 2). Under GAAP (Column 1), however, the shares continue to be recorded at book value, so the leverage ratio is the same as at issue.

There are dramatic differences between the numbers produced by the EV method and the numbers from current GAAP. Column 3 of Table 4 shows, for example, that when the EV method is used, the debt in the convertible issue (Line 8) is 58 percent lower than it would be under current GAAP and total debt (Line 9) is 14 percent lower. Total equity (Line 5) under our method and with all shares valued at market value is 78 percent higher. The lower debt and higher equity resulting from our method (and the valuation of equity at market value) produce a debt-to-equity ratio (Line 11) that is 52 percent lower than under current GAAP.

Clearly, the numbers produced under GAAP signif- icantly overestimate financial leverage. Financial analysts need to be aware of this outcome.

Conclusions We have presented a new method for bifurcating a convertible debt issue into its debt and equity com- ponents. This method relies on the assumptions and theory associated with options to value the debt and equity portions of a convertible bond at the issue date and over the bond's life.

Financial analysts need measures of leverage and EPS that reflect economic reality. Neither current accounting guidance nor the bifurcation method based on straight bond and option values proposed in ED 213-B is consistent with modern finance theory. Our method is consistent with theory and provides a more realistic assessment than GAAP does of the true capital structure of a company with convertible debt issues outstanding.

Table 4. Company ABC's Leverage Measures and Diluted Shares after One Year and Stock Price Increase of 50 Percent (dollars in millions)

GAAP: EV Method: Book Value Market Value Difference between

of Equity of Equity Col. 2 and Col. 1 Measure Calculated (1) (2) (3)

(1) Debt prior to convertible issue $300 $300

(2) Convertible debt issue $100 $100

(3) Value of equity without bond issue $300 450

(4) Equity value in bond 0 $ 85.45

(5) Total equity (Row 3 + Row 4) $300 $535.45 78.48%

(6) Value of straight bond $ 81.48 $ 81.48

(7) PV of coupons $ 8.42 $ 8.42

(8) Value of "debt" in issuea $100 $ 42.27 -57.71

(9) Total value of debt including convertible issue $400 $342.27 -14.43

(10) Debt to equity without issueb 1.00 0.67 -33.00

(11) Debt to equity including issue 1.33 0.64 -51.87

(12) Shares in issue 4,000,000 2,130,400 -46.74

(13) Diluted shares 19,000,000 17,134,000 -9.82

aThe debt value in the convertible issue given the new stock price and shorter time to maturity; in Column 2 (the EV method), the straight bond is weighted by (1 - 0.533) and the PV of the coupons is weighted by 0.533. bThe debt-to-equity ratio of 1.00 in Column 1 was calculated by using the book value of equity and GAAP. The ratio of 0.67 in Column 2 was calculated by using the market value of equity but excluding the equity and debt contributed by the convertible bond issue.

We owe thanks to Beth Cooperman, Richard Foster, Steve Henning, Stan Martin, and Robert Vigeland for useful comments on earlier drafts, to Diana Wagner and Russ Gambirasi for research assistance, and to Sally Sedgwickfor editorial assistance.





Notes 1. The accounting profession records the convertible bond at

the face value of the bond issue with no subsequent changes in that value until the bond is paid off or converted.

2. The expected value is the sum of the probability-weighted outcomes. Full exercise (of n shares) occurs with probability p, and zero shares occurs with probability (1 - p) contribut- ing 0 (1 - p) to the expected value.

3. If we used the risk-free rate, we would get a higher equity value by using Method 2 than found by using Method 1.

4. This number can be easily obtained by the residual method: $100 million minus $56.47 million. Or if we start with the option value and add the PV of the exercise price, we find the equity per share is $10.88, or for 4 million shares, $43.53 million.

References AICPA. 1966. APB Opinion No. 10, Omnibus Opinion. New York: American Institute of Certified Public Accountants.

. 1969. APB Opinion No. 14, Accounting for Convertible Debt and Debt Issued with Stock Purchase Warrants. New York: American Institute of Certified Public Accountants. FASB. 2000. ED 213-B: Accountingfor Financial Instruments with Characteristics of Liabilities, Equity, or Both. Financial Accounting Standards Board: www.fasb.org/draft/ed-fi.pdf.

.2003. Statement of Financial Accounting Standards No. 150: Accounting for Certain Financial Instruments with Characteristics of Both Liabilities and Equity. Financial Accounting Standards Board: www.fasb.org/pdf/fasl5O.pdf.

Gaumnitz, Bruce R., and J.E. Thompson. 1987. "Establishing the Common Stock Equivalence of Convertible Bonds." Accounting Review, vol. 62, no. 3 (July):601-622.

IASB. 1998 (revised 2003). IAS No. 32: Financial Instruments: Disclosure and Presentation. International Accounting Standards Board.

King, Raymond. 1984. "The Effect of Convertible Bond Equity Values on Dilution and Leverage." Accounting Review, vol. 59, no. 3 (July):419-431.

Vigeland, Robert L. 1982. "Dilution of Earnings per Share in an Option Pricing Framework." Accounting Review, vol. 58, no. 2 (April):348-359.

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Convertible Bonds: How Much Equity, How Much Debt?