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1 WEIDONG TIAN is a professor of finance and distinguished professor in risk management and insurance the University of North Carolina at Charlotte in Charlotte, NC. wtian1@uncc.edu Contingent Capital as an Asset Class WEIDONG TIAN This short article examines the risk return profile of contingent capital from investor s perspective. Contingent capital (CC) is a debt instrument that is automatically enforced, converted into common shares when a contractual trigger event occurs. Therefore, it is significantly different from some contingent capital concepts in insurance literature, such as in Culp [2009], in which contingent capital is one contingent option to issue standard debts on the future under certain circumstance. In the wake of the global financial crisis, contingent capital has been embraced by regulators across the world. 1 According to these regulatory capital proposals, the demand on CC could be as large as 2 trillion by an analysis conducted by Goldman Sachs [2011]. Therefore, it is essential to study whether contingent capital offers a new investment opportunity. In explaining how contingent capital works, I concentrate on one type of contingent trigger that relies on the common equity Tier 1 ratio. Specifically, when the common equity Tier 1 ratio drops below a specified level, contingent capital is converted into a number of common shares. This contingent capital indenture precludes the common equity Tier 1 ratio from falling from a specified number to meet some risk management requirements. This article describes a structural approach to analyzing the risk of contingent capital. Our approach has several key features. 1. It is an arbitrage-free approach in which all contingent claims are priced simultaneously. 2. The contingent trigger event is analyzed in an endogenous way. 3. Its fundamental variable is the bank s asset value, so it shares all important features of the structural approach of credit model. See Black and Cox [1976], Merton [1974], and Leland [1994]. 4. The model parameters are the contractual parameters. No subjective estimation is required, and model risk is reduced. The arbitrage-free approach starts with an unleveraged asset value process. Suppose the bank s asset value is represented by a dynamic binomial model, the asset value moves either 20% up or 20% down in the next time period. The initial asset value is $100. Each time period represents one year, and the constant one-year interest rate is 2%. The upward risk-neutral probability is 55%, and the downward risk-neutral probability is 45%. For any contingent claim, its price is pcu + (1 pc ) C = 1+ r d (1) THE JOURNAL OF INVESTING 1 JOI-TIAN.indd 1 2/7/14 1:10:29 PM

2 where C is the current price of one contingent claim; C u, C d are the prices of this contingent claim in the next time period in up and down scenarios, respectively; r is the short rate in this time period; and p = 55%. The remainder of this article is divided into two parts. In the first part, I illustrate the arbitrage-free pricing of common stock and contingent capital when the contingent capital is introduced. In the second part, I describe the risk return profile of common stock and contingent capital from the investing perspective. CONTINGENT CAPITAL PARAMETERS As an example, we examine a contingent capital with maturing time of three years. The coupon rate is 4%, and coupon payment occurs every year with the face value of $100. We assume that this contingent capital has been issued in the market place, so its investor receives a coupon payment in the amount of $4 today, if the contingent capital has not been converted yet. The contingent trigger covenant is characterized by the common equity Tier 1 ratio, and the specified ratio is set to be 10%. That is, whenever this equity Tier 1 capital ratio fells to or below 10%, contingent capital, investor becomes equity investor automatically. According to the asset pricing theory of contingent capital (see Tian [2012]), the bank cannot choose the number of common shares arbitrarily. In fact, it has been shown that m t K t L is the condition to ensure equilibrium price, where L is the initial investment (face) value, K t is the conversion (or trigger level) for the common stock price at time t, and m t is the conversion ratio (the number of common shares one contingent capital is changed into after the conversion). This feature is intuitively appealing to make CC a attractive asset to investors, as argued and proposed in a December 2010 statement by the Shadow Financial Regulatory Committee The Case for a Properly Structured Contingent Capital Requirement. 2 Coffee [2011] provided some convincing arguments on this feature. At the current situation, the trigger level is K t = V t /10 because S t /V t 10% is equivalent to S t K t For simplicity, we assume that the conversion trigger price K t and conversion ratio m t satisfy m t K t = L for all time t. If so, both K t and m t are random numbers. For instance, V0 = $100 at time zero, so K = $10 and m 0 = 10. One contingent capital is converted into 10 common shares at time zero if the common equity ratio falls below 10%. The higher the asset value (in times of strength period), the smaller the number of common shares; the smaller the asset value (in times of stress period), the higher the number of common shares. Exhibits 1 3 depict the unleveraged asset price, the corresponding trigger price, and the corresponding conversion ratio, respectively. For E XHIBIT 1 A Tree of Asset Value E XHIBIT 2 A Tree of Conversion Trigger Price 2 CONTINGENT CAPITAL AS AN ASSET CLASS JOI-TIAN.indd 2 2/7/14 1:10:29 PM

3 simplicity we assume that the number of share can be taken as any real number. These three trees are the input of the pricing model. Stock Price at Maturing Time Consider the situation when the asset value is $172.8 at time T = 3. As the stock s residual value after its payout to CC investor is ( ) = $72.8, its corresponding common equity ratio is 72.8/172.8 = 42%, so the contingent covenant is inactive. Therefore, this residual value is the stock price in this scenario. The situation when the asset value is $51.2 at time T = 3 is different. The stock s residual value becomes zero as this asset value is smaller than the debt s face value. Consequently, the contingent trigger covenant must be active and enforcing conversion occurs. Remember that the conversion ratio is in this case. It means that one share of contingent capital is enforced to be converted into shares, and the total number of shares after enforcing conversion is Hence, each share of common stock after enforcing conversion is $2.49 in this scenario, and one share of contingent capital is worth $ By the same idea, we are able to calculate a socalled stop price or stop value of common stock at any E XHIBIT 3 A Tree of Conversion Ratio time prior to maturity. The stop price is the price when the bank plans to do liquidation. If the residual value is high enough that the corresponding residue ratio (a ratio of residue value over the asset value) is above 10%, the stop price is its residual value. On the other hand, if the corresponding residue ratio is less than 10%, the bank s liquidation decision ensures enforcing conversion. Therefore, each share price is the ratio of the asset value to the total number of shares after conversion. In this way, the bank is recapitalized into a full equity bank, and the stop price is the ex post stock price after the bank s recapitalization. Exhibit 4 calculates all stop prices of common stock in this capital structure. Stock Price We begin to delve into the stock price at any time prior to maturity. For instance, we calculate the stock price at time t = 2 and when the asset value V = $144. For this purpose we need to consider whether the bank is willing to carry on the capital structure into the next time period, instead of enforcing liquidation. Assuming, first, the bank carries on the capital structure in this time and in this scenario, the coupon payment of $4 has to be E XHIBIT 4 A Tree of Stop Price of Common Stock THE JOURNAL OF INVESTING 3 JOI-TIAN.indd 3 2/7/14 1:10:29 PM

4 paid to the debt investor. On the other hand, the bank has a tax benefit equal to = $1.6. Therefore, the leveraged asset value loses (4 1.6) = $2.4 instantly. Moreover, carrying on into the next time period, the values become $72.8 and $15.2 (the stop price at time T = 3 in Exhibit 4), respectively. Therefore, the continue value of the common stock is (1/1 + r)(p [1 p] 15.2) 2.4 = 44.8 (2) The next step is to study whether this ongoing decision is admissible or not. Giving the continue value $44.8, the corresponding continue ratio (the ratio of continue value over the asset value) is 44.8/140 = 30.9%, which is greater than 10%. Thus, we know that this continue decision is plausible under the contingent capital covenant constraint. Although the bank can decide either to stop or to continue, the best way is surely to choose the decision that yields the highest value, which is the stock price as required. In this particular case, the stock price is $44.8. The situation when the asset value drops to V = $96 at time t = 2 illustrates the essential insight of contingent capital. Similarly, we calculate the continue value, $ If the bank carried on the original capital structure into the next time period, however, the corresponding equity ratio would be 8.425/96 = 8.8%, which is less than 10%. Therefore, the contingent indenture does not allow the continue possibility for the bank. As going on is impossible because of the contingent covenant, the bank has to enforce conversion of contingent capital into the common shares at this time. Consequently, the stock price becomes its stop price, $ There is an subtle issue in this derivation of the stock price. Remember, the contingent indenture highlights specifically that whenever the common stock ratio drops to 10%, the trigger covenant is active. Therefore, the common stock price is what we are looking for endogenously, and the endogeny of the trigger covenant has to be studied carefully. If the corresponding continue equity ratio drops to 10%, it does not imply ex ante the common stock ratio drops to 10% too, as we do not know the stock price precisely. We need to know a precise relation between the stock price with its stop value and continue value. This relation is written as follows: StockPr ice max{ ContinueValue, }, ContinueRatio > 10%; or = StopValue where ContinueRatio represents the corresponding ratio of the continue value to the asset value. Exhibit 5 presents the tree of stock price. The stock price at time zero is $14.48, as shown. Trigger Covenant In addition to the endogenous stock price, the pricing procedure characterizes the best restructuring time, which is the time when the stock price is identical to the stop price of the common stock. By comparing Exhibit 4 with Exhibit 5, we see that the best restructuring time is when the asset value drops at time t = 1, and when the asset value drops at time t = 2. Obviously, enforced conversion is desirable in these bad business times. To verify, Exhibit 6 demonstrates the corresponding common equity ratio. Not surprisingly, the common equity ratio drops below 10% when the asset value moves down at time t = 1 and when it moves up at time t = 1 but then down at time t = 2. Therefore, the trigger covenant is active in these situations. Getting the CC Price We turn next to characterize the price process of contingent capital. As the restructuring time has been E XHIBIT 5 A Tree of Common Stock Price 4 CONTINGENT CAPITAL AS AN ASSET CLASS JOI-TIAN.indd 4

5 characterized, it suffices to determine the debt value in the following situations: (t,v ) = (0,100), (1,120), (2,144). In other situations, debt value is the stop price of the debt (either liquidated value or conversion value by enforced conversion), the difference between the asset value and the stop price of the common stock. Calculation of the market debt (CC) price is similar to the calculation of common stock. We start from the maturing time in which the debt price is just the liquidating or the stop price. In the first time prior to the maturity time, when the capital structure is restructured, the debt price is again its liquidating price. When the capital structure is carried on in the next time period, we calculate its continue price by Equation (1). Then, by incorporating the coupon payment, we derive the fair price of CC. It is convenient to divide the calculation of CC price into two trees separately: the tree of pure price and the tree of coupon payment. The tree of coupon payment is displayed in Exhibit 7. For simplicity, we assume no coupon payment occurs at its maturing time. The pure price is just the price without coupon payment consideration. That is, the price of corresponding CC without coupon payment, with the same maturity time and the same conversion time. It is worth E XHIBIT 6 A Tree of Common Stock Ratio noting that the pure price is not the debt price with zero coupon rate as the conversion times for both CC debts might be different. The value of CC is a sum of the pure price and the present values of all the future coupon payments. Exhibit 8 depicts the tree of the market fair price of CC at any time before maturing time. We next move to study the question: Does contingent capital offer attractive investment opportunities? IS CC AN ASSET CLASS? After explaining the pricing methodology of CC and common stock we move to study CC as an asset class. To capture variability of time to maturity and various business situations (in times of strong business or in times of stress), we use one-period investment in this dynamic world to measure the risk return profile from three different perspectives: equity investor, debt investor, and regulator. The investor faces a one-period problem, but the investor s time horizon extends one year. This approach is often used to capture the dynamics of asset returns. See Fisher [1983], Canner, Mankiw, and Weil [1987]. The comparative analysis also highlights the distinctions between investors of varying risk preferences. Our comparative analysis is documented as follows. First of all, we analyze the return risk of oneperiod investment of the common stock when CC is introduced. Then, we examine the return risk of a one- E XHIBIT 7 A Tree of Coupon Payment THE JOURNAL OF INVESTING 5 JOI-TIAN.indd 5

6 E XHIBIT 8 A Tree of CC Price E XHIBIT 9 Comparison of Risk Return Profile in One-Period Investment on Stock period investment of the common stock in a standard capital structure. Therefore, we are able to compare the risk return profile for the common stocks in these two different capital structures. Second, we perform a similar analysis from the debt investor s perspective. At last, we compare the cost of financial distress from the regulator s perspective. We use node (t, n) to denote the investment opportunity from the node (t, n) to the next time period. To illustrate, the first component in node (1,1) represents time t = 1 and the second component represents the highest possible asset value at time t = 1. By the same reason, node (1,0) represents the investment when time t = 1 and the asset value V = $80. There are six investment opportunities in one period in total: (0,0), (1,0), (1,1), (2,0), (2,1), and (2,2). Panel A in Exhibit 9 reports the expected return and standard deviation of a one-period investment of the common stock for three different kinds of investors. The first investor is relatively risk-neutral about the business, the move-up probability is 60% (note: the move-up riskneutral probability is 55%). For the initial investment at time t = 0 as illustrated by node (0,0), the expected return (mean) is 23% and the standard deviation of holding one share from time t = 0 to time t = 1 is 67%. The corresponding Sharpe ratio is Standard deviation, mean, and Sharpe ratio for the other nodes are calculated and displayed in different columns. The second investor is relatively optimistic let s say the subjective move-up probability is 80%. The last investor we consider is relatively pessimistic the subjective move-up probability is 20%. We observe two remarkable points from Panel A in Exhibit 9. First, once the investor becomes optimistic about the business, the investment of the common stock is attractive. In fact, the Sharpe ratio in each six nodes is increasing when the investor becomes more optimistic. For instance, the Sharpe ratio on node (0,0) is for optimistic investor, comparing with the Sharpe ratio of in the same investment opportunity for a relatively risk-neutral investor. Similarly, once the investor becomes pessimistic about the business in the near future, the investment on the stock is less attractive. Second, we examine whether the asset value with high or low movement affects the investment differently. Look at Panel A in Exhibit 9. The investment in the downside business environment indeed makes the invest- 6 CONTINGENT CAPITAL AS AN ASSET CLASS JOI-TIAN.indd 6

7 ment on the stock better. For instance, when the asset value at time t = 1 drops, Sharpe ratio is , which is higher than , the Sharpe ratio of the investment at time t = 1 when the asset value moves high. We observe the same patterns for both optimistic and pessimistic investors. In other words, the Sharpe ratio in a one-period investment is negatively correlated with the asset value. That is, to invest on common stock in a time of stress is better than in times of strong business, at the presence of contingent capital. For a comparison purpose, we also calculate the stock price and the corresponding return data for the standard capital structure in which CC is replaced by a standard debt with the same maturity and same coupon rate. The default cost percentage is assumed to be 60%. Our first point is preserved in the standard capital structure. Optimistic investor finds common stock more attractive than relatively pessimistic investor does in any time. The second point, however, is not satisfied anymore. By comparing nodes (1,1) and (1,0), one represents a good time while the second denotes a bad time, the optimistic investor enjoys the investment in the good time, while the risk-neutral and pessimistic investor prefers to invest in the bad time. The situation becomes just opposite at time two, say, at nodes (2,0) and (2,1). Therefore, E XHIBIT 10 A Tree of Stock Price in a Standard Capital Structure the investment on common stock in the standard capital structure is fairly mixed and ambiguous. Next, we compare the investment of common stock in both capital structures in detail. Among three different investors and six available investment opportunities, we can draw a robust conclusion that the Sharpe ratio for the contingent capital structure is always higher than the standard capital structure in the downsize market movement environment (see nodes (1,1) and (2,1)). Moreover, the Sharpe ratio for the first capital structure is always lower in an upside market movement environment (see nodes (1,0) and (2,0)). This comparison demonstrates clearly that contingent capital indeed protects equity investors during the period of severely depressed performance. However, standard capital structure benefits equity investor in a strong business situation. What about the debt investor s investment? In general, CC performs better than or relatively close to the performance of standard debt in the market downside environment (see nodes (1,0), (2,1) and (2,2)), for all three kinds of debt investors: relatively risk-neutral, optimistic, or pessimistic. In particular, this advantage becomes more pronounce when the debt investor is more optimistic about the business. This is an interesting point: CC features can be attractive to debt investors in addition to equity investors and offer a rich risk return profile with various risk preferences. On the other hand, for E XHIBIT 11 A Tree of Debt Price in a Standard Capital Structure THE JOURNAL OF INVESTING 7 JOI-TIAN.indd 7

8 nodes (1,1), (2,2) that corresponding to upward market movement, we observe that standard debt performs better than CC in general. It means that CC investors anticipate and participate in the contingent capital market in times of stress. This discovery about CC is consistent with the motivation of regulators and supervisors. But the contingent capital structure is not attractive to both equity and debt investors in a strong time period, at least in a mean variance framework. At last, we compare the cost of financial distress in both capital structures. Obviously, there is no default at all at the presence of contingent capital. Moreover, the leveraged asset value at time zero is $ Hence, the tax benefit for the bank is $ When we look at the standard capital structure, however, the bank claims default at time t = 2 when the asset values drop to $64, and at time t = 3 when the asset value falls to $76.8. Thus the cost of financial distress is $ This cost of financial distress is so large that it outweighs the tax benefits. In fact, we calculate the leveraged asset value at $ Furthermore, we observe that both common stock and debt have higher value in the contingent capital E XHIBIT 12 Comparison of Risk Return Profile in One-Period Investment on Debt structure than in the standard capital structure. Thus, contingent capital enhances the value to investors as well as taxpayers. EXTENDING THE MODEL AND CONCLUDING COMMENTS The approach outlined here enables us to examine more complex capital structures. For instance, senior standard debt can be investigated together with CC. In this case, a bank considers when to be recapitalized and restructured into a stronger capital structure without CC, by enforced conversion, if the bank does not plan to do liquidation. This additional stronger capital structure provides an enhanced loss capability for the bank by increasing the endogenous equity buffer. Trigger covenant can be also modified, say, by common stock price and other market factors or a systemic trigger, and trigger covenant can be represented by several trigger events all together. Even in the conversion procedure, the bank does not have to enforce conversion of CC completely in one time. Instead, the bank can enforce conversion of CC into common shares 25% into four times, based on a sequence of pre-specified trigger covenants. Finally, the model enables us to value options on contingent capital or some contingent capitals with embedded options for the debt investors. By using examples written on a common Tier 1 ratio, we demonstrate that contingent capital presents a new investing regime for investors, at least in times of stress. This discovery is consistent with the motivation of the contingent capital concept. We can also use the same approach to compare various CC proposals, analyze its risk management, and build an optimal investment strategy. ENDNOTES 1 These regulators include the Federal Reserve Bank of New York, Bank of England, Office of the Superintendent of Financial Institutions Canada, Basel Committee of Banking Supervision, Swiss Commission of Experts, and International Monetary Fund, among many others. 2 R. Herring, The Case for a Properly Structured Contingent Capital Requirement. Statement #303, Shadow Financial Regulatory Committee, December Available at 8 CONTINGENT CAPITAL AS AN ASSET CLASS JOI-TIAN.indd 8

9 REFERENCES Black, F., and J.C. Cox. Valuing Corporate Securities: Some Effects of Bond Indenture Provisions. Journal of Finance, 31 (1976), pp Canner, N., N.G. Mankiw, and D.N. Weil. An Asset Allocation Puzzle. American Economic Review, Vol. 87, No. 1 (1987), pp Coffee, J.C. Systemic Risk after Dodd-Frank: Contingent Capital and the Needs for Regulatory Strategies Beyond Oversight. Columbia Law Review, 111 (2011), pp Culp, C.L. Contingent Capital vs. Contingent Reverse Convertibles for Banks and Insurance Companies. Journal of Applied Corporate Finance, Vol. 21, No. 4 (2009), pp Fisher, S. Investing for Short Term and the Long Term. In Financial Aspects of the U.S. Pension System, edited by Z. Bodie and J.B. Shoven, Chicago: University of Chicago Press, 1983, pp Goldman Sachs. Contingent Capital, Possibilities, Problems and Opportunities. Technical report, March Leland, H. Corporate Debt Value, Bond Covenants, and Optimal Capital Structure. Journal of Finance, 49 (1994), pp Merton, R.C. On the Pricing of Corporate Debt: The Risk Structure of Interest Rates. Journal of Finance, 29 (1974), pp Tian, W. Contingent Capital with Endogenous Trigger. Working paper, University of North Carolina at Charlotte, To order reprints of this article, please contact Dewey Palmieri at dpalmieri@iijournals.com or THE JOURNAL OF INVESTING 9 JOI-TIAN.indd 9 2/7/14 1:10:31 PM

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