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Valuation and analysis of zero-coupon contingent capital bonds

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Abstract

We consider the valuation and analysis of zero-coupon contingent capital bonds (CCBs) in the structural framework. Making virtually no assumptions on asset value dynamics, the terms of conversion or the conversion trigger, we express the value of the CCB in terms of the effective loss imposed on CCB investors at conversion and quantify the impact that contingent capital has on traditional debt and equity. We show how a variety of conversion terms can be incorporated into a single framework and describe how they can be calibrated to ensure that seniority is respected and/or equity investors are not rewarded for poor performance. We provide numerical evidence indicating that the terms of conversion can fundamentally alter the nature of the CCB, a phenomenon that is of clear interest to investors, issuers and regulators.

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Notes

  1. These include, but are not limited to, Lloyd’s of London (2009), Credit Suisse Group (2011) and Royal Bank of Canada (2014). See [2] for an excellent summary of global CCB issuance up to 2013.

  2. The terms of conversion refer to the specific rules governing the exchange of debt for equity upon conversion, while the conversion trigger refers to the conditions that must be satisfied in order for conversion to occur. See [2] for terms that are frequently used in practice. According to [2] conversion triggers are almost always based on first passage of the firm’s Core Tier I Equity Ratio to a predefined lower threshold, potentially subject regulatory discretion.

  3. In our examples we assume that conversion is triggered by the firm’s asset-liability ratio falling below a predefined threshold. By the “location of the conversion trigger” we simply mean that value of the asset-liability ratio that triggers conversion.

  4. We motivate this feature of the model with a simple example. Suppose that firms are liquidated when they become insolvent, i.e. when the value of their assets falls below the notional value of their liabilities. Upon conversion the liabilities of the firm that uses contingent capital are reduced, which means that its asset value has farther to fall before it becomes insolvent, which means that liquidation occurs at a later date than it would have been in the absence of contingent capital.

  5. The use of “surprise” here is not to be confused with the probabilist’s notion of unpredictability.

  6. In the academic literature [14] and [17] consider fixed dilution factors which, in the case of coupon bonds, are equivalent to fixed conversion prices. A difference arises in the present zero coupon setting due to our definition of notional value. Our own numerical evidence indicates that for realistic parameter values there is little difference between a fixed conversion price and a fixed dilution factor.

  7. In the academic literature fixed intended losses are considered, either implicitly or explicitly, by [1, 16], and [10]. Barucci and Viva[3] consider redemption at par, tantamount to a fixed loss of zero at conversion.

  8. In 2011 the Office of the Superintendent of Financial Institutions, a Canadian regulator, issued an advisory on so-called non-viability contingent capital, requiring that “. . . the conversion method should take into account the hierarchy of claims . . . ”

  9. These numbers are computed by fitting straight lines to the curves in Fig. 2 (in all cases the fits are extremely good) and looking at the magnitude of the ratio of the resulting slopes.

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Acknowledgments

The authors wish to thank the NSERC Discovery Grant program for generous financial support.

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Correspondence to A. Metzler.

Appendices

Appendix 1: Proof of Proposition 1

In preparation we note that since \(\{\tau _{\mathrm{C}}<\infty \}=\{\tau _{\mathrm{C}}<\tau _{\mathrm{L}}\}\) by assumption, it follows that \(\{\tau _{\mathrm{C}}<\infty \}\) and \(\{\tau _{\mathrm{C}}=\infty \}\) lie in \(\mathcal {F}_{\tau _{\mathrm{C}} \wedge \tau _{\mathrm{L}}}\). And since the \(\mathcal {F}_{\tau _{\mathrm{C}}}\)-measurable random variable \(w\) is constant on \(\{\tau _{\mathrm{C}}=\infty \}=\{\tau _{\mathrm{L}}<\tau _{\mathrm{C}}\}\) it follows that \(w\) is \(\mathcal {F}_{\tau _{\mathrm{C}} \wedge \tau _{\mathrm{L}}}\)-measurable.

Proof

Let \(CCB_{t}\) be the time-\(t\) value of the CCB. Since \(e^{-rt}CCB_{t}\) is a martingale the OST ensures that

$$\begin{aligned} CCB_{0} = \mathbb {E}[e^{-r(\tau _{\mathrm{C}} \wedge \tau _{\mathrm{L}})}CCB_{\tau _{\mathrm{C}} \wedge \tau _{\mathrm{L}}}] \end{aligned}$$
(3)

and

$$\begin{aligned} e^{-r(\tau _{\mathrm{C}} \wedge \tau _{\mathrm{L}})}CCB_{\tau _{\mathrm{C}} \wedge \tau _{\mathrm{L}}} = \mathbb {E}[e^{-r\tau _{\mathrm{L}}}CCB_{\tau _{\mathrm{L}}}|\mathcal {F}_{\tau _{\mathrm{C}} \wedge \tau _{\mathrm{L}}}]. \end{aligned}$$

Now

$$\begin{aligned} \mathbb {E}[e^{-r\tau _{\mathrm{L}}}J_{\tau _{\mathrm{L}}}\mathbf {1}_{\{\tau _{\mathrm{C}}=\infty \}}|\mathcal {F}_{\tau _{\mathrm{C}} \wedge \tau _{\mathrm{L}}}] = e^{-r(\tau _{\mathrm{C}} \wedge \tau _{\mathrm{L}})}J_{\tau _{\mathrm{C}} \wedge \tau _{\mathrm{L}}}\mathbf {1}_{\{\tau _{\mathrm{C}}=\infty \}} = e^{-r\tau _{\mathrm{L}}}J_{\tau _{\mathrm{L}}}\mathbf {1}_{\{\tau _{\mathrm{C}}=\infty \}}, \end{aligned}$$

since \(e^{-rt}J_{t}\) is a martingale and \(\{\tau _{\mathrm{C}}=\infty \}\) lies in \(\mathcal {F}_{\tau _{\mathrm{C}} \wedge \tau _{\mathrm{L}}}\). Moreover

$$\begin{aligned} \mathbb {E}[e^{-r\tau _{\mathrm{L}}}wR_{\tau _{\mathrm{L}}}(L_{S},\tau _{\mathrm{L}})\mathbf {1}_{\{\tau _{\mathrm{C}}<\infty \}}|\mathcal {F}_{\tau _{\mathrm{C}} \wedge \tau _{\mathrm{L}}}]&= w e^{-r(\tau _{\mathrm{C}} \wedge \tau _{\mathrm{L}})}R_{\tau _{\mathrm{C}} \wedge \tau _{\mathrm{L}}}(L_{S},\tau _{\mathrm{L}})\mathbf {1}_{\{\tau _{\mathrm{C}}<\infty \}} \\&= we^{-r\tau _{\mathrm{C}}}R_{\tau _{\mathrm{C}}}(L_{S},\tau _{\mathrm{L}}), \end{aligned}$$

since \(e^{-rt}R_{t}(L_{S},\tau _{\mathrm{L}})\) is martingale and both \(w\) and \(\mathbf {1}_{\{\tau _{\mathrm{C}}<\infty \}}\) are \(\mathcal {F}_{\tau _{\mathrm{C}} \wedge \tau _{\mathrm{L}}}\)-measurable. Thus

$$\begin{aligned} e^{-r(\tau _{\mathrm{C}} \wedge \tau _{\mathrm{L}})}CCB_{\tau _{\mathrm{C}} \wedge \tau _{\mathrm{L}}}&= e^{-r\tau _{\mathrm{L}}}J_{\tau _{\mathrm{L}}}\mathbf {1}_{\{\tau _{\mathrm{C}}=\infty \}} + w e^{-r\tau _{\mathrm{C}}}R_{\tau _{\mathrm{C}}}(L_{S},\tau _{\mathrm{L}})\mathbf {1}_{\{\tau _{\mathrm{C}}<\infty \}} \nonumber \\&= L_{J}e^{-rT}\left[ 1-\beta _{J,\mathrm {L}}-\beta _{J,\mathrm {C}}\right] . \end{aligned}$$
(4)

Inserting (4) into (3) yields the desired result. \(\square \)

Appendix 2: Computing residual value in the Black–Cox model

In this appendix we provide a closed-form expression for the function \(h(z,k,y,\sigma ,\nu )\) that is used throughout Section 7.

Proposition 2

Suppose that \(z_{t}\) is zero-drift geometric Brownian motion, i.e. a solution to \(dz_{t}=\sigma z_{t}dW_{t}\). For fixed \(\nu >0\) and \(y>0\) define

$$\begin{aligned} \tau := \min \left( \nu ,\inf \{t \in (0,\nu ): z_{t} \le y\}\right) \end{aligned}$$

and

$$\begin{aligned} h(z,k,y,\sigma ,\nu )=\mathbb {E}[(z_{\tau }-k)^{+}|z_{0}=z]. \end{aligned}$$

Then

$$\begin{aligned} h = \left\{ \begin{array}{lll} (z-k)^{+} &{}\quad \mathrm {if} &{} z \le y, \\ z-k &{}\quad \mathrm {if} &{} z>y>k, \\ z\Phi (\psi + \eta ) - k\Phi (\psi -\eta )-d\Phi (\theta +\eta )+(kz/y)\Phi (\theta -\eta ) &{}\quad \mathrm {if} &{} y<\min (k,z), \end{array} \right. \end{aligned}$$

where \(\xi =\frac{\log (z/y)}{\sigma \sqrt{T}}\), \(\psi =\frac{\log (z/k)}{\sigma \sqrt{T}}\), \(\theta =\frac{\log (y^{2}/(zk))}{\sigma \sqrt{T}}\) and \(\eta =\frac{\sigma \sqrt{T}}{2}\). Moreover if \(y<\min (k,z)\) then \(h(z,k,y,\sigma ,\nu )>z-k\).

Proof

If \(z \le y\) then \(\tau =0\) and \((z_{\tau }-k)^{+}=(z-k)^{+}\). If \(z>y>k\) then \(z_{\tau } \ge k\) almost surely and \((z_{\tau }-k)^{+}=(z_{\tau }-k)\); since \(z_{t}\) is a martingale it follows that \(\mathbb {E}[z_{\tau }|z_{0}=z]=z\). Finally, if \(y<\min (k,z)\) then \((z_{\tau }-k)^{+}=(z_{\nu }-k)^{+}\mathbf {1}_{\{\tau = \nu \}}\), which is the payoff to a down-and-out call option in the Black–Scholes model with zero interest rate. A closed-form expression for the value of such an option can be found in [11]

To confirm the last statement note that if \(y<\min (k,z)\) \(\min (z_{\tau },k)<k\) with positive probability, whence \(\mathbb {E}[\min (z_{\tau },k)]<k\). Since \((z_{\tau }-k)^{+}=z_{\tau }-\min (z_{\tau },k)\) and \(z_{t}\) is a martingale we see that \(\mathbb {E}[(z_{\tau }-k)^{+}|z_{0}=z]>z-k\). \(\square \)

For the model considered in Sect. 7 the process \(v_{t}\) is a zero-drift GBM. Using this fact, Proposition 2 and the obvious fact that \(h(\delta z,\delta k,\delta y,\sigma ,\nu )=\delta h(z,k,y,\sigma ,\nu )\) for any \(\delta >0\), we get that

$$\begin{aligned} R_{t}(kL,\tau _{\mathrm{L}}^{\mathrm{oi}}) = Le^{-r(T-t)}\left[ h(v_{t},k,d,\sigma ,T-t)\mathbf {1}_{\{\tau _{\mathrm{L}}^{\mathrm{oi}}>t\}} + (d-k)^{+}\mathbf {1}_{\{\tau _{\mathrm{L}}^{\mathrm{oi}} \le t\}}\right] \end{aligned}$$

and

$$\begin{aligned} R_{t}(kL,\tau _{\mathrm{L}}) = Le^{-r(T-t)}\left[ h(v_{t},k,(1-\alpha )d,\sigma ,T-t)\mathbf {1}_{\{\tau _{\mathrm{L}}>t\}} + ((1-\alpha )d-k)^{+}\mathbf {1}_{\{\tau _{\mathrm{L}} \le t\}}\right] \end{aligned}$$

for any \(k \ge 0\) and \(t \in [0,T]\).

Appendix 3: CCB valuation in CEV and jump diffusion models

In this section we briefly describe a valuation procedure for each of the CEV and Jump Diffusion models. For a given model we let \(CALL(v,k,\nu )\) denote the value of a European call option struck \(k\) with \(\nu \) years to maturity, written on an asset whose current spot value is \(v\), noting that we suppress the dependence of this value on all model parameters.

1.1 Appendix 3.1: CEV model

Recall that in the CEV model we have

$$\begin{aligned} dV_{t}=rV_{t}dt + \nu V_{t}^{1+a}dW_{t}, \end{aligned}$$
(5)

for constants \(a \in (-1,0)\) and \(\nu >0\).

To price the CCB we begin by using an Euler scheme with time step \(\delta >0\) (measured in years) to simulate an approximate discrete skeleton \(\{\hat{V}_{t_{i}}:0 \le i \le N\}\), where \(t_{i}=i\delta \) and \(N=T/\delta \) and we use the notation \(\hat{V}\) to reflect the fact that our simulations are not exact. If the discrete skeleton remains above the barrier over the entire time horizon, i.e. if \(\hat{V}_{t_{i}} > cLe^{-r(T-t_{i})}\) then conversion does not occur and we set \(\hat{\beta }_{J,\mathrm {C}}\) and \(\hat{\beta }_{J,\mathrm {L}}=[L_{J}-(\hat{V}_{T}-L_{S})^{+}-(\hat{V}_{T}-L)^{+}]/L_{J}\), where we note that \(\hat{V}_{T}=\hat{V}_{t_{N}}\). If the discrete skeleton does cross the barrier we set \(\hat{\beta }_{J,\mathrm {L}}=0\), \(\hat{\tau }_{\mathrm{C}}=\min (t_{i}:\hat{V}_{t_{i}} \le cLe^{-r(T-t_{i})})\) and use the formula in [18] to compute \(\hat{R}:=CALL(cLe^{-r(T-\hat{\tau }_{\mathrm{C}})},(1-\alpha )L,T-\hat{\tau }_{\mathrm{C}})\). In the case of a fixed intended loss we set

$$\begin{aligned} \hat{\beta }^{l}_{J,\mathrm {C}} = \max \left( l,1-\frac{\hat{R}}{L_{J}e^{-r(T-\hat{\tau }_{\mathrm{C}})}}\right) \end{aligned}$$

and in the case of a fixed conversion price we set

$$\begin{aligned} \hat{\beta }^{p}_{J,\mathrm {C}} = 1 - \frac{1}{1+[e^{-r\hat{\tau }_{\mathrm{C}}}pe_{0}/(\alpha \ell )]} \cdot \frac{\hat{R}}{L_{J}e^{-r(T-\hat{\tau }_{\mathrm{C}})}}, \end{aligned}$$

where \(e_{0}=CALL(V_{0},L,T)/V_{0}\) is also computed using the formula in [18].

1.2 Appendix 3.2: Jump diffusion algorithm

The jump diffusion model considered here is of the form

$$\begin{aligned} dV_{t} = (r-\lambda \kappa )\,V_{t-}\,dt + \sigma \,V_{t-}\,dW_{t}\,+V_{t-}\,dM_{t}, \end{aligned}$$
(6)

where \(M_{t}=\sum _{i=1}^{N_{t}}(Y_{i}-1)\), \(N_{t}\) is a homogeneous Poisson process (independent of \(W\)) with intensity \(\lambda \) and the \(Y_{i}\) are i.i.d. (and independent of both \(W\) and \(N\)) lognormal random variables with \(\kappa =\mathbb {E}[Y_{i}-1]\) representing the average jump size as measured by the percentage change in the value of the firm’s assets.

To price the CCB we begin by simulating the pair \((T \wedge \tau _{\mathrm{C}},V_{T \wedge \tau _{\mathrm{C}}})\) exactly using the algorithm described below. If \(\tau _{\mathrm{C}}=\infty \) we set \(\beta _{J,\mathrm {C}}=0\) and \(\beta _{J,\mathrm {L}}=[L_{J}-(V_{T}-L_{S})^{+}-(V_{T}-L)^{+}]/L_{J}\). If \(\tau _{\mathrm{C}}<\infty \) then \(\tau _{\mathrm{C}}=T \wedge \tau _{\mathrm{C}}\), we set \(\beta _{J,\mathrm {L}}=0\) and use the formula in [11] to compute \(R:=CALL(V_{\tau _{\mathrm{C}}},(1-\alpha )L,T-\tau _{\mathrm{C}})\). Note that since it is possible that the process jumped over the barrier we cannot necessarily replace \(V_{\tau _{\mathrm{C}}}\) with \(cLe^{-r(T-\tau _{\mathrm{C}})}\) when computing \(R\). The algorithm for simulating \((T \wedge \tau _{\mathrm{C}},V_{T \wedge \tau _{\mathrm{C}}})\) is as follows.

  • Simulate the number of jumps \(N_{T}\), the jump sizes \((Y_{1},\ldots ,Y_{N_{T}})\) and the jump times \((\tau _{1},\ldots ,\tau _{N_{T}})\).

  • If \(N_{T}=0\) simulate \(V_{T}=\exp ((r-\sigma ^{2}/2)T+\sigma W_{T})\) and move to the next step. Otherwise, for \(i=1,\ldots ,N_{T}\) simulate

    $$\begin{aligned} V_{\tau _{i}-}=V_{\tau _{i-1}}\exp \left( \left( r-\sigma ^{2}/2\right) \left( \tau _{i}-\tau _{i-1}\right) +\sigma \left( W_{\tau _{i}}-W_{\tau _{i-1}}\right) \right) \end{aligned}$$

    and set \(V_{\tau _{i}}=V_{\tau _{i}-}Y_{i}\), and then simulate \(V_{T}=\exp ((r-\sigma ^{2}/2)(T-\tau _{N_{T}})+\sigma (W_{T}-W_{N_{T}}))\).

  • Determine

    $$\begin{aligned} i^{*}=\min \left\{ 1 \le i \le N_{T}:\,V_{\tau _{i}-} > cLe^{-r(T-\tau _{i})},\;V_{\tau _{i}} \le cLe^{-r(T-\tau _{i})}\right\} , \end{aligned}$$

    with the convention that \(\min \emptyset =\infty \). If \(i^{*}<\infty \) set \(\tau _{\mathrm{C}}^{(1)}=\tau _{i^{*}}\) and if \(i^{*}=\infty \) set \(\tau _{\mathrm{C}}^{(1)}=\infty \). Then \(\tau _{\mathrm{C}}^{(1)}\) is the first time that asset value jumps over the conversion barrier.

  • For \(i=0,1,\ldots ,N_{T}\) simulate

    $$\begin{aligned} \tau _{\mathrm{C}}^{(2),i}=\min \left\{ t \in (\tau _{i},\tau _{i+1}):V_{t} \le bLe^{-r(T-t)}\right\} , \end{aligned}$$

    where we define \(\tau _{0}=0\) and \(\tau _{N_{T}+1}=T\). As described in more detail below this amounts to simulating the hitting time of a Brownian bridge to a fixed level, an algorithm for which is provided in Sect. 6 of [4]. Setting \(\tau _{\mathrm{C}}^{(2)}=\min (\tau _{\mathrm{C}}^{(2),0},\ldots ,\tau _{\mathrm{C}}^{(2),N_{T}})\) we obtain the first time at which asset value “diffuses” to the conversion barrier.

  • Set \(\tau _{\mathrm{C}}=\min (\tau _{\mathrm{C}}^{(1)},\tau _{\mathrm{C}}^{(2)})\). If \(\tau _{\mathrm{C}}=\infty \) then set \(V_{T \wedge \tau _{\mathrm{C}}}=V_{T}\), if \(\tau _{\mathrm{C}}=\tau _{\mathrm{C}}^{(1)}<\infty \) then set \(V_{T \wedge \tau _{\mathrm{C}}}=V_{\tau _{i^{*}}}\) and if \(\tau _{\mathrm{C}}=\tau _{\mathrm{C}}^{(2)}<\infty \) then set \(V_{T \wedge \tau _{\mathrm{C}}}=cLe^{-r(T-\tau _{\mathrm{C}}^{(2)})}\).

In order to simulate the hitting time \(\tau _{\mathrm{C}}^{(2),i)}\) it is convenient to use the fact that, conditional on \(V_{\tau _{i}}\) and \(V_{\tau _{i+1}-}\), we have the representation

$$\begin{aligned} \tau _{\mathrm{C}}^{(2),i}=\tau _{i} + \theta (\log (cLe^{-rT})/\sigma ,\log (e^{-r\tau _{i}}V_{\tau _{i}})/\sigma ,\log (e^{-r\tau _{i+1}}V_{\tau _{i+1}-})/\sigma ,\tau _{i+1}-\tau _{i}), \end{aligned}$$

where \(\theta (a,b,c,s)\) is the hitting time, to the fixed level \(a\), of a Brownian bridge of length \(s\) beginning at \(b\) and terminating at \(c\).

To see that this representation is valid begin by defining \(X_{t}=e^{-rt}V_{t}\), so that the hitting time of \(V_{t}\) to the given time-dependent barrier corresponds to the hitting time of \(X_{t}\) to the constant barrier \(cLe^{-rT}\). For \(i=0,1,\ldots ,N_{T}\) define the process \(Z^{(i)}\) via \(Z^{(i)}_{0}=X_{\tau _{i}}\) and \(Z^{(i)}_{t}=X_{\tau _{i}+t-}\) for \(t>0\); thus for \(t \in [0,\tau _{i+1}-\tau _{i}]\) we have

$$\begin{aligned} Z^{(i)}_{t}=X_{\tau _{i}}\exp \left( -\sigma ^{2}t/2+\sigma (W_{\tau _{i}+t}-W_{\tau _{i}})\right) , \end{aligned}$$
(7)

in particular \(Z^{(i)}\) is continuous on \([0,\tau _{i+1}-\tau _{i}]\), beginning at \(X_{\tau _{i}}\) and terminating at \(X_{\tau _{i+1}-}=X_{\tau _{i+1}}/Y_{i+1}\). From the representation (7) it is clear that, conditional on \(V_{\tau _{i}}\) and \(V_{\tau _{i+1}-}\), the conditional law of \(\log (Z^{(i)})/\sigma \) over \([0,\tau _{i+1}-\tau _{i}]\) is that of a standard Brownian bridge of length \(\tau _{i+1}-\tau _{i}\), beginning at \(\log (X_{\tau _{i}})/\sigma \) and terminating at \(\log (X_{\tau _{i+1}-})/\sigma \). Moreover the hitting time of interest is simply the hitting time of this new process to the level \(\log (cLe^{-rT})/\sigma \).

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Metzler, A., Reesor, R.M. Valuation and analysis of zero-coupon contingent capital bonds. Math Finan Econ 9, 85–109 (2015). https://doi.org/10.1007/s11579-014-0135-z

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