Electronic copy available at: http://ssrn.com/abstract=2196339

Contingent Conversion Convertible Bond:

New avenue to raise bank capital

Francesca Di Girolamoa,b∗, Francesca Campolongoa, Jan De Spiegeleerb,c,Wim Schoutensb

a European Commission, JRC, Via E. Fermi 2749, 21027 Ispra (VA), Italyb Department of Mathematics, KU Leuven, Leuven, Belgium

c Head of Risk Management, Jabre Capital Partners - Geneva

Abstract

This paper provides an in-depth analysis into the structuring and the pricingof an innovative financial market product. This instrument is called contingentconversion convertible bond or ”CoCoCo”. This hybrid bond is itself a combi-nation of two other hybrid instruments: a contingent convertible (”CoCo”) anda convertible bond. This combination introduces more complexity in the struc-ture but it now allows investors to profit from strong share price performances.This upside potential is added on top of the normal contingent convertible me-chanics whereas CoCos only expose the investors to downside risk. This setsup a new avenue for the banks to create new capital.First, we explain how the features of the contingent convertible bonds on oneside and the features of the standard convertible bonds on the other side arecombined. Thereafter, we propose a pricing approach which moves away fromthe standard Black&Scholes setting. The CoCoCos are evaluated using the He-ston process to which a Hull-White interest rate process has been added. Wedemonstrate the importance of using a stochastic interest rate when modelingthis instrument. Finally we quantify the loss absorbing capacity of this instru-ment.JEL Classification: G13, G18, G21, C15

Keywords: Convertible bonds, Contingent convertible bonds, Monte Carlo sim-ulations, Sensitivity analysis

∗Contact information: Phone: +39 0332 78 5187, Fax: +39 0332 78 5733, Email address:francesca.di-girolamo@jrc.ec.europa.eu

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Electronic copy available at: http://ssrn.com/abstract=2196339

1 Introduction

In the light of the 2008 credit crisis which has hit the whole financial market, somecapital instruments used to fund the operations of a financial institution have dis-appointed. Their quality and consistency showed fundamental flaws.

A bank funds itself by issuing equity (shares) and debt instruments. Its regula-tory capital can be decomposed into Tier 1 and Tier 2. Tier 1 consists of a core Tier1 component and a layer of additional Tier 1. The core Tier 1 layer is top qualitycapital. It consists of equity and retained earnings and is fully loss absorbing. Theother component, additional Tier 1, consists for example of preferreds, perpetualsoften combined with features such as cancellable coupons. The Tier 2 bucket ona bank’s balance sheet ranks above Tier 1 debt. Shareholders or investors in Tier1 debt would first have to loose their investment before Tier 2 instruments startsuffering. Tier 2 contains for example subordinated debts.

In the aftermath of the credit crisis one can state that the equity capital heldby banks was far too little in relation to the risks they were running (IndependentCommission on Banking (2011)). To make matters worse, the additional Tier 1did not fulfil the role it was designed for. It failed to be loss absorbing and banksneeded to be bailed out by the tax payers. This has driven the Basel Committee(Basel Committee (2010); Basel Committee (2011a)) to be stricter on the allowanceof certain hybrid instruments that used to be part of this layer of additional Tier 1.A debt instrument can have the status of regulatory capital if it is loss absorbing.Practically this means there has to be a conversion into shares or a write down ofthe face value of the bond when the bank reaches a state of non-viability.

In this context, the contingent convertible bonds (”CoCos”) tick all the boxes andthey are a kind of automatic reinforcement of the bank’s balance sheet, a mechanismwhich would reduce the overall systemic risk. They are loss absorbing and imposelosses to their investors as soon as the bank gets into a threatening situation. Theloss absorption takes place with a conversion of these bonds into equity. The balancesheet is hence automatically reinforced and the bank sees its debt load to be reduced.

LLOYDS RABO CREDIT SUISSE BANK OF CYPRUS UBS

Name Enhanced Capital Notes Senior Contingent Notes Buffer Capital Notes CoCoCo

Issue Date November 2009 March 2010 January 2011 February 2011 March 2012 May 2011 February 2012

Issue Size (m) $13,700 e 1,250 $2,000 $2,000 CHF 750 e 890 $2,000

Regulatory Ranking Tier 2 Tier 2 Tier 1 Lower Tier 2

Trigger Core Tier 1 < 5% Equity Capital Ratio < 7% Equity Capital Ratio < 8% Core Tier 1 < 7% Core Tier 1 < 5% Core Tier 1 < 5%

Write-Down - 75% with immediate Write down depends - 100%

repayment of 25% on the capital needs.

Conversion Full Conversion in Shares - Full Conversion in Shares Full Conversion in Shares -

Conversion Price 59 GBp - max(20USD,20CHF, S*) max(80% S* , 1 EUR) -

Maturity (years) 10 - 22 10 30 10 Perpetual

Callable - - After 5.5 years After 5 years After 5 years After 5 years

Fixed Coupon 6.385-16.125 6.875 8.375 7.875 7.125 6.5 7.25

Optional Conversion - - Yes at 3.3 No

Table 1: Features of some issued CoCo bonds

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The pioneer in this new asset class has been the Lloyds Banking Group whenit exchanged in November 2009 series of contingent convertible bonds (O’Doherty(2009)). Investors would receive shares when the core Tier 1 ratio of the bankfalls below a trigger level of 5%. Lloyds was soon followed by Rabobank in March2010. This CoCo issue handled the loss absorbency through a haircut of 75%. TheRabobank issue was very well accepted by the market which lead Credit Suisse toembrace the contingent convertible concept in February 2011 (Hughes and Jenkins(2011); Swiss National Bank (2010)).The possibility of using contingent convertibles to count towards the regulatorycapital fuelled the interest for this new bond (see Table 1 where the most importantissued CoCo bonds have been presented). For a while, a consensus was growingamong the market participants that contingent convertible bonds were going to allowto meet the extra capital surcharge of the so-called systemically important financialinstitutions (SIFIs). Although, the Basel Committee disallowed early July 2011the use of CoCos to handle the SIFIs capital surcharge (Basel Committee (2011b)),CoCos would definitively find their way in the bucket of addition Tier 1 or Tier 2.Their loss absorbency is going to play a relevant role in the capital requirementsimposed by national regulators: (see Figure 1).

Figure 1: Capital Requirements under Basel III

Standardisation will not be the norm in CoCo-land since different forms of con-tingent convertible bonds already have emerged. Up till the writing of this paper,there has been a large variety in trigger levels, host instruments, use of regulatorytriggers, and coupon structures.

The Bank of Cyprus for example came in April 2011 with a special convertiblebond called Convertible Enhanced Capital Securities (CECS) 1. This newcomer has

1CECSs had been exchanged via the issue of another hybrid called Mandatory Convertible Notes(”MCNs”) as part of a recapitalization plan on 20 March 2012. The MCNs had a duration of eightcalendar days and they had been redeemed with new shares on 27 March 2012 (Bank of Cyprus

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set out a twist which opens the door to a new investor base (Khasawned and Agnew(2011); Whittaker (2011)). The CECS is not a standard contingent convertible bondbut it embraces some features of traditional convertible bonds (from here the name”CoCoCo”). Although this introduces more complexity in the structure, the mostinteresting idea is to allow investors to profit from good times. The traditional CoCobond forces losses onto the investor when a trigger event is materialised. A CoCoinvestor typically becomes a shareholder in bad times or suffers a write-down. Thereis no particular upside for the CoCo investor, the best outcome is to receive all thecoupons and the face value of the bond up to the expiry of this debt instrument. Theholder of a CoCoCo has an extra feature on top of the mandatory conversion. Theinvestor also has an optional conversion possibility: in case of a good share priceperformance, the investor can convert the bond into a pre-determined amount ofshares. Summarizing we say that the conversion of this CoCoCo bond can thus takeplace in two different situations: bad times when the bank is becoming insolvent orthe opposite case when shares are performing well.

The research on convertible bonds is well developed. A number of studies hasbeen dedicated to this subject (Kind and Wilde (2005); Jan de Spiegeleer andSchoutens (2011b); Davis and Lischka (2002)) and the interest dates back to 1980when Schwartz proposed a pricing approach for such bonds.CoCos as well have started to be a topic of research unlike they are novel financialinstruments. The literature proposes some detailed descriptions on their featuresand some pricing models have already been proposed (see for example Maes andSchoutens (2012); Jan de Spiegeleer and Schoutens (2011a); Jan de Spiegeleer andSchoutens (2011c); McDonald (2010)). The contributions in Pennacchi (2011) andBarucci and Del Viva (2012) focus on investigating the structure of a general com-pany issuing CoCo bonds. Moreover, Pennacchi and Vermaelen (2011) propose toissue a special kind of CoCos called COERC, Flannery (2009) proposes to issueContingent Capital Certificates.

Concerning with the CoCoCo bonds instead, we are not aware of any otherquantitative analysis. This paper tries to provide an in-depth analysis into thevaluation and the dynamics of this innovative financial market product.

The remaining of the paper is organized as follows: Section 2 focuses on thestructure of the CoCoCo bonds and their features; Section 3 describes our pricingmethodology; Section 4 discusses the results when applying the proposed method-ology to a CoCoCo prototype; Particular attention has been given to the stochasticinterest rate by investigating how it affects the CoCoCo price; A quantification ofthe loss absorbing capacity of this instrument is provided here as well; Section 5concludes.

(2012)).

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2 CoCoCo: Contingent Conversion Convertible

Standard contingent convertible bonds (CoCos) are debt instruments on top of whicha loss absorption mechanism has been implemented. This loss absorption takesplace through a mandatory conversion into equity (shares) or a write down. Upona conversion, the bank immediately receives new equity and all future coupons arecanceled.

The combination of a CoCo and a convertible bond creates a contingent conver-sion convertible bond (CoCoCo). The mandatory conversion will take place duringa distressed situation. In such a case a trigger measuring the financial health ofthe bank forces this conversion into shares. This is the downside risk of a CoCoCowhich is the same as the risk carried by a CoCo. The optional conversion is in thehand of the investor and will be exercised when the share price is far enough abovethe conversion price.

2.1 Risk Profile

The risk profile of a CoCoCo is a combination of the profile of a CoCo and a con-vertible.

For CoCos the bond nature would predominate in ”good” states of the economycapping the maximum possible payoff to the face value of the bond. This is exactlythe reason why buying CoCos puts investors in a situation of getting a limited gainwith a probability of ending up with high losses. This carries negative convexity forthe investor, the worse the financial condition of the bank the more the investor isexposed to the share price of the bank.

Standard convertibles have the opposite convexity profile. The investor has lessdirect exposure to price movements of the underlying share in downturns. In fact,he will never convert into shares when these are trading below the conversion price.

Figure 2 shows the two risk profiles: ”unlimited downside - limited upside”for CoCos and the ”limited downside - unlimited upside” for convertibles (Jan deSpiegeleer and Schoutens (2011b); Jan de Spiegeleer and Schoutens (2011a)). Notethat the upper boundary of the first one acts at the same time as a lower boundaryfor the second one. The bond floor of the convertible is a bond ”ceiling” for theCoCo.

2.2 Framework

The value of CoCoCos depends on their anatomy. There are two faces that have tobe carefully considered. On one side there is a distressed situation which requires toset up an automatic mechanism to get a mandatory conversion. On the other side,there is a time in which investors can enjoy of an optional conversion into shares

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Bond Floor

Stock Price, S

Bon

d P

rice

Convertible BondCoCo Bond

Figure 2: CoCo vs Convertible Risk Profile

whenever it would be profitable. The structuring of a CoCoCo is a trade-off toreconcile the interests of investors, regulators, and shareholders. Investors have tobe persuaded that the coupon is attractive enough, regulators have to be convinced,and shareholders have to be reassured that an useless dilution will be avoided.

These conditions under which the conversion would take place compromise thefair value of the bond and are crucial to ensure the CoCoCos would achieve theintended objectives.

2.2.1 Mandatory Conversion

Under an hypothesis of a threatening situation, CoCoCos behave like standard CoCobonds. They give the opportunity to stabilize the precarious equilibrium of the bankby using an automatic mechanism of conversion as soon as the bank fails to meetfor example minimum regulatory capital levels. This is a the mandatory conversion.An important and ongoing discussion is the choice of the trigger event which setsoff this emergency conversion. A second topic of a similar lively debate, is relatedto the conversion price of the shares which is the implied purchase price Cp of theshares when the trigger takes place.

Trigger event

The are different trigger events possible:

- Accounting trigger:Balance sheet ratios are used to determine the financial health of the financialinstitution. A trigger event would correspond to such a ratio failing to meet aminimum level. An example are the Credit Suisse CoCos which convert into

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shares when the core Tier 1 ratio drops below the minimum level of 7% andthe CoCoCo issued by the Bank of Cyprus which would convert as soon as thecore Tier 1 falls below 5%.

- Market trigger:An observable parameter such as the level of the share price or the creditdefault swap spread of the bank could be used to trigger a conversion intoshares or a write down of the bond. The market based trigger has not beenused so far.

- Regulatory trigger:This is also called a non-viability trigger and is actually pulled by the nationalregulator whenever this authority deems the situation of the bank to be fragileto survive. The CoCoCo issued by the Bank of Cyrpus has for example anextra regulatory trigger in addition to the accounting core Tier 1.

Relevant is also the level of the trigger. A ”high” trigger means that the bondis converted long before a dangerous threshold is reached. The trigger takes placewhen the bank is still a going-concern and the CoCo or CoCoCo are as a matterof fact tools to prevent the bank to slide in a more precarious situation. A ”low”trigger on the other hand refers to an ultimate conversion just before a gone-concernor bank failure would be reached. The contingent convertibles are thus no longerprevention tools, they become an ultimate tool to reinforce a bank’s balance sheet.

The conversion price

The number of shares received upon a mandatory conversion is equal to the con-version ratio, CrMC . This is implicity related to another key element which is theconversion price, Cp, of the shares:

CrMC =F

Cp, (1)

where F is the face value of the bond. Several possibilities exist when it comes tothe determination of the conversion price. Let us denote by t∗ the time at which theCoCoCo bond is triggered (mandatory conversion):

- Cp = S∗t = S(t∗)

One natural choice is to fix the conversion price equal to the share price,S∗t , observed on the trigger moment. Under this setting, the investor does

not suffer any loss at all because the total amount of the delivered shares isequal to the face value of the bond, F . For the shareholders, this leads to asignificant potential dilution of the equity because a large number of shareshas to be distributed to the ”new owners”. The lower the share price, the more

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shares will created and the more dilution the exiting shareholder will have toundergo.

- Cp = S0 = S(t0)The other option is to provide conversion into equity based on the share price,St0 , at the issue time. This has an opposite effect. The conversion price willtypically be much higher than the real value of the stock at time t∗. The bondholder is forced to buy shares priced at Cp which is way above the value of theshares when a trigger takes place. This choice might be relatively generousto the shareholders not suffering from a substantial dilution of their equityholdings.

- Cp = max(S∗t , SF )

Another possibility is to combine the previous choices giving a compromise.The conversion price is set equal to the price at the conversion time but it isnot allowed to drop below a floor value, SF .

2.2.2 Optional Conversion

When the solvency and the liquidity profile of a bank is distant from a distressedsituation, the CoCo investor is dealing with the payoff of a corporate bond. TheCoCo holder cannot profit from any upside. The maximum payout is restricted tothe face value and the coupons. The optional convertibility of the CoCoCo, on theother hand, gives the opportunity to the holder to participate on the upside and geta profit by converting the bond into shares. This is not a mandatory conversion, itis just a call option granted to the investor to terminate the bond and receive sharesinstead. The number of shares received per bond is the conversion rate, CrOC . Thevalue of the shares received is the conversion value which is equal to CrOC · St.

We suppose that the option can be exercised over the life of the bond and theinvestor has to check daily if the optional conversion would be more profitable thankeeping the contract without an immediate exercise. When no immediate exercisetakes place the value of the contract is the expected cash flow of the future payment,called continuation value P (t). The value of the CoCoCo bond at time t beforeexpiration is thus the following:

max(P (t), CrOC · St). (2)

At maturity, if the bond has not been converted, the CoCoCo investor still has thepossibility to swap the total face value into shares leading the terminal value to be:

max(F,CrOC · ST ). (3)

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3 Valuation of a CoCoCo bond

In this section we propose a pricing method able to handle jointly the optional andmandatory conversion. Both conversions depend on the path followed by the under-lying share during the life of the bond and typically also apply to the expiration.

The starting point is the generation of several stock paths for which the CoCoCobond is evaluated. The consideration of trading in periods when the market can befollowed by turbulent times calls for the stock price volatility to be stochastic. Due tothe fact that the CoCoCo risk profile is characterized by a double convexity, we alsodo require a link between the stock price and the stochastic volatility: under changesof the stock price, the positive convexity leads the appreciation of the CoCoCoprice to be bigger than a possible depreciation; the negative convexity presents theopposite behavior and the best solution for an investor would be to have a lowvolatility stock price. The Heston model allows us to incorporate the stochasticvolatility and a non-gaussian distribution for the share price returns exhibiting fattails. We combine this stochastic volatility model with a one-factor Hull-Whitemodel for the interest rate.

Next, we proceed with the evaluation of the bond taking care of the mandatoryconversion and of the optional conversion. The mandatory conversion for exampleused by the Bank of Cyprus was depending on an accounting trigger, the core Tier 1.Here, we work with a market trigger because it is not possible to model the core Tier1 in a Monte Carlo simulation as already demonstrated in the literature (see Jan deSpiegeleer and Schoutens (2011a)): the core Tier 1 is in fact not observable nor canbe traded or hedged. This means that the risk coming from it cannot be eliminated.The best solution is thus to model the mandatory conversion with respect to shareprice which without any doubt is well linked to the financial health of the issuer.

We make the assumption that as soon as the stock price falls below a certaintrigger level corresponding to an emergency situation, the conversion in shares wouldtake place.

3.1 Heston Model with stochastic interest rate

Moving away from Black-Scholes and using the Heston model, we assume that thestock follows a process with a stochastic volatility and a stochastic interest rate(Heston (1993)):

dS(t) = r(t)S(t)dt+√

V (t)S(t)dWS(t)

dV (t) = α(β − V (t))dt+ η√

V (t)dWV (t)

E(dWS(t)dWV (t)) = ρSV dt

(4)

where S(t) and V (t) are the price and the variance processes, respectively, and WS

and WV are correlated Brownian motions. The β is the long term variance to which

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the variance will revert through its mean reverting property. The α is the rate ofmean reversion which puts a constraint on how far the variance can drift away fromthe long-term variance. The η is the volatility of the variance.

The stochastic interest rate is driven by a Hull-White process (HW):

dr(t) = (Θ(t)− ar(t))dt+ σdWr(t)

E(dWr(t)dWS(t)) = ρrSdt

E(dWr(t)dWV (t)) = ρrV dt

(5)

Here a and σ are constants, Θ(t) is deterministic and satisfies the following equation(Brigo and Mercurio (2006)):

Θ(t) =∂fM (0, t)

∂T+ afM (0, t) +

σ2

2a(1− e−2at), (6)

where fM is the market instantaneous forward rate.Let us denote the correlation matrix among the stock, variance, and interest rate

by ρ. Its positive definiteness is necessary for performing Monte Carlo simulations.

ρ =

1 ρSV ρrSρSV 1 ρrVρrS ρrV 1

3.1.1 Generation of stock paths

The valuation of a CoCoCo bond, will be implemented using Monte Carlo. Toprice a CoCoCo bond we have to use ’n’ Monte Carlo paths for the stock, for thevariance, and for the interest rate. In order to practically generate one of them letus to introduce a finite time horizon [0, T ] sliced into m steps. Each step j has asmall time increments ∆t = T

m and we set tj = j∆t. To take into account that thestock distributes some dividend payment along the life of the bond, we define asDiv the percentage dividend payment and as D̄ the times of the dividend payments.We assume for simplicity that dividend payment always coincide with our points tj ,j = 0, ...,m. At each step j the amount which has to be paid is the following:

D(j) =

{Div if tj ∈ D̄,

0 otherwise.

According to the Equation 7, we simulate the ith stock path at step j using aMilstein discretization (Glasserman (2004)).

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S(i, j + 1) = S(i, j)(1−D(j))e(r(t)−V (i,j))∆t+√

V (i,j)∆tZS,i,j+1

V (i, j + 1) = V (i, j) + α(β − V (i, j))∆t+ η√V (i, j)∆tZV,i,j+1 +

η

4∆t(ZV,i,j+1 − 1)

r(i, j + 1) = r(i, j) + (Θ(j)− ar(i, j))dt+ σ√∆tZr,i,j+1

(7)

The ZS,i,j , ZV,i,j , and Zr,i,j are standard normal variables generated according totheir correlations between the share price, the variance, and the interest rate and forexample S(i, j + 1) denotes the stock price of the ith simulated path at time tj+1.

3.1.2 Calibration of the model

The parameters, α, β, and η of the stock price process, a, σ, and Θ(t) of the interestrate process and the correlation matrix ρ have to be calibrated according to theavailable market data (Brigo and Mercurio (2006)).

We start finding out the values for the parameters in the interest rate model. Θ(t)is estimated according to the Equation 6 exactly fitting the observed term structureof the instantaneous forward rate, fM . The values for a and σ are determined via acalibration to a set of market prices for caps observed in the market: we minimize theroot mean square error between the value of these contracts according to Black’76and the value of these contracts according to the the Hull White closed formula.

After the determination of the stochastic interest rate dynamic, we proceed withthe calibration of the Heston model. The common approach is to find out thosevalues for the parameters which produce the correct market prices of vanilla options:we minimize the root mean square error between the model price (closed formulaavailable under Heston model) and the market price.

The stock - interest rate correlation ρrS will be fixed a priori and for simplicitywe assume here that the interest rate is uncorrelated with the stochastic volatility(ρrV = 0).

3.2 Steps for pricing a CoCoCo bond

We now outline the steps for pricing a CoCoCo bond after the generation of 600, 000stock paths. The value of a CoCoCo bond is equal to the expected cash flow at issuetime. Let us define CF (i, j) as the cash flow at step j for the ith path.

Let’s fix the trigger level, S∗. We start selecting those paths in which the stockfalls below S∗. For each of them we find out the first time step as soon as thishappens. As example, Figure 3 illustrates three paths of our simulation with S∗ =50. The third path and the second one are above S∗ overall the life of the bond.The Path 1 falls below S∗ at time j∗. The mandatory conversion takes place, the

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bond stops to exists and from here on we are not longer interested in the behaviourof Path 1.

0 1 2 j* 3 4 50

50

100

150

200

250

300

350

Years

Sto

ck p

aths

Path 2

Path 3

Mandatory ConversionPath 1

1

S*

Figure 3: Example of three stock paths in our simulation

Step 0

Let us now introduce the following spaces:

I⋆ = {path i: S(i, j) < S∗ at some step j}.I⋆c = {path i: S(i, j) ≥ S∗ ∀j}.

(8)

I⋆ is the set of paths for which the stock price falls below the trigger level and I⋆c

is the complementary.

Step 1

For each path i ∈ I⋆ we select the earliest step j⋆i for which the trigger sets off theconversion and we calculate the cash flow as equal to the amount of shares deliveredto the investor after the conversion:

j⋆i = {first step j for path i : S(i, j⋆i ) < S∗}. (9)

CF (i, j⋆i ) = CrMC · S(i, j⋆i ) (10)

Step 2

For each path i ∈ I⋆c, we calculate the cash flow at maturity (step m) as equal orto the bond face value increased by the final coupon payments, or to the optionalconversion value (see Equation 3):

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CF (i,m) = max(F + Final Coupon, CrOC · S(i,m)) (11)

Step 3

Starting from the time of maturity, we discount the cash flow with a backwardinduction and at each step we check if the optional conversion can be exercised. Theapproach will be dynamic because the number of paths will change as soon as themandatory conversion appear. For example, in Figure 3 we observe that at maturitywe have two cash flows: one for Path 3 and one for Path 2. As soon as we arrive attime j⋆, we need to include in the backward process the cash flow coming from themandatory conversion in Path 3: we need in fact to take into account the possibilityof an optional conversion before the mandatory in j⋆ would take place.

Let us suppose we are at step s with 0 < s < m − 1. We define the followingspaces:

J = {paths i: CrOC · S(i, s) ≤ F}Jc = {paths i: CrOC · S(i, s) > F}

(12)

A rational investor will only convert when the option is in money, CrOC ·S(i, s) > F .For paths i ∈ J it does not apply so we discount the cash flow from the previousstep according to the Equation 13:

CF (i, s) = CF (i, s+ 1)e−r(s+1)·ts+1 for i ∈ J (13)

For the other paths i ∈ Jc, the cash flow comes from max(P (i, s), CrOC · Si,s)(see Equation 2). We select the best strategy which provides the highest cash flowbetween exercising the optional conversion or continuing the contract. The evalua-tion of CrOC ·Si,s is straightforward because we exactly know the share value at steps from the stock simulation. To estimate instead the continuation value2 (P (i, s))we make use of the Longstaff Schwartz approach (Longstaff and Schwartz (2001))approximating the continuation value as a combination of basis functions:

P (i, j) = E[m∑

k=j+1

e−

∫ tktj

r(i,s)dsCF (i, k)]

==L∑

t=1

atlt(Si, Vi, ri)

(14)

2We remind that the continuation value is the value of the contract if the option will not beimmediately exercised.

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where Si is the stock price for the ith path, at are constant coefficients, and lt arethe basis functions. In our model, the randomness comes from the share S, thestochastic volatility V , and the stochastic interest rate r. We have opted for usingthese three variables and their interactions up to the fourth power as basis functions:

l(S, V, r) =[1 S′ S′2 S′3 S′4..

V ′ V ′2 V ′3 V ′4..

r′ r′2 r′3 r′4..

S′V ′ S′V ′2 S′V ′3 S′2V ′ S′3V ′ S′2V ′2..

S′r′ S′r′2 S′r′3 S′2r′ S′3r′ S′2r′2..

V ′r′ V ′r′2 V ′r′3 V ′2r′ V ′3r′ V ′2r′2..

S′V ′r′ S′V ′r′2 S′V ′2r′ S′2V ′r′],

(15)

where S′, V ′, and r′ have been normalized with respect to their initial values S0,V0, and r0.

According to the Longstaff method, the coefficients are estimated using a leastsquare regression of the discounted CF (i, j) evaluated by Equation 13 on the basisfunctions li. Inside the regression we include all the paths i ∈ Jc together. By sub-stituting these coefficient values into Equation 14 we get the estimated continuationvalue at time step j for all paths i ∈ Jc. Summarizing, the cash payment at step sis:

CF (i, s) =

{CF (i, s+ 1)e−r(s+1)·ts+1 for i ∈ J

max(P (i, s), CrOC · Si,s) for i ∈ Jc(16)

Step 4

Finally, we need to calculate the average, overall the Monte Carlo paths, of thediscounted cash flow at issue time in order to get an estimate of the CoCoCo price.

4 Results and discussion

In this section we use a simplified CoCoCo structure (see Table 2) in order to applythe methodology outlined in Section 3.

The CoCoCo bond has a 5-year maturity with a face value of 100. It distributesno coupon payments over time and no accrued interest is hence introduced. As soonas the share price falls below 50, a mandatory conversion takes place converting thebond in an amount of shares equal to 50 ·CrMC = 25. The CoCoCo bond holder hasthe right of exercising an optional conversion whenever it would be convenient forhim, using a conversion rate CrOC of 0.5.

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In order to get the price, we consider 600, 000 stock paths generated by MonteCarlo simulations 3. The initial value of the share is S0 = 100 with an initial interestrate of 0.03 and an initial variance of 0.04. We set the correlation ρrS equal to 0.50,the correlation ρrV equal to zero, and we for simplicity use a flat forward rate fM

of 0.03. All the other parameters are obtained from the calibration of the Hestonmodel and are provided in Table 3: α equal to 0.100, β equal to 0.040, η equal to0.200, and ρSV equal to −0.100 for the stochastic volatility; a equal to 0.010, and σequal to 0.012 for the stochastic interest rate.

CoCoCo bond Stock Price

Trigger Level 50 S0 100

Maturity 5 years r0 0.03

Face Value 100 V0 0.04

Coupon - ρrS 0.50

CrMC 0.5 ρrV 0.00

CrOC 0.5 fM 0.03

Table 2: Features of the CoCoCo prototype and of the stock price

Parameters of the model

α 0.100

β 0.040

η 0.200

ρSV −0.100

a 0.010

σ 0.012

Table 3: Calibrated parameters for the Heston model with stochastic interest rate

4.1 Stochastic interest rate or constant interest rate?

Under this setting the price of the CoCoCo bond results to be 85.39. Do we reallyneed to introduce a stochastic interest rate?

The consensus in convertible bond pricing is to stick to deterministic interestrate. Brennan and Schwartz for example present a study (see Brennan and Schwartz(1980)) on convertible bonds and they argue that for a reasonable range of interestrate the errors of using deterministic instead of stochastic interest rates are likely to

3To improve the Monte Carlo simulation we have also used the antithetical variable techniquefor the stock and the volatility.

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be small and therefore for practical purpose it may be preferable to use deterministicrates which are simpler to simulate.

In this section we advocate the use of uncertainty analysis. By using quasirandom numbers, we generate 256 values for the volatility of the stochastic interestrate, σ in the Equation 5. For each of them, we run our algorithm in order to pricethe CoCoCo bond. Figure 4 illustrates how the CoCoCo price changes when varyingthe volatility of the interest rate.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.160

65

70

75

80

85

90

Volatility Interest Rate ( σ )

CoC

oCo

Pric

e

Figure 4: Variability of the CoCoCo price under changing of the interest rate volatil-ity

The figure shows evident dispersion in the price whenever the volatility of theinterest rate varies from 0% to 10%. The price ranges from 85 to 60. The volatilityis here an important component when pricing CoCoCo bond.

Let’s now verify if the volatility is really of influence in pricing CoCoCo andwhether some other inputs play a relevant role. We thus investigate how muchchanges in the parameters of our quantitative model really affect the CoCoCo price inorder to detect which ones are the most influential when evaluating a CoCoCo bond.Several techniques can be applied which belong to the world of global sensitivityanalysis. Among them we have decided to use the Saltelli’s variance based in orderto quantify the contribution of each input parameter to the total uncertainty ofthe price. For details on this method we refer to Saltelli et al. (2004), Saltelli et al.(2008), Saltelli (2002), Saltelli et al. (2010), and Ratto and Pagano (2010). A typicaloutput of the sensitivity analysis is a number called total sensitivity index which isable to detect how much the output varies due to changes of one input parameterconsidering its interaction with the other inputs.

In our model the sources of uncertainty are the stochastic volatility and thestochastic interest rate whose dynamics are driven by Equation 4 and Equation 5.

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The first one is depending on four parameters: α, β, η, and ρSV . The second one4

is depending on four parameters: a, σ, and ρrS5. In Section 3 we have fixed a priori

ρrS and we have calibrated all the other parameters according to the market data(see Table 3). In this section, we let ρrS vary inside the range [−1 1]; for all theother parameters we pick percentage variations of the calibrated values provided inTable 3. These percentage perturbations are selected using quasi random number in[−1, 1] according to an uniform distribution. We use 256 percentage perturbation.This number has been demonstrated in Ratto and Pagano (2010) to produce valuableresults in a general application of the variance based method. For each of them weprice our CoCoCo bond according to the methodology presented in Section 3 andfinally we calculate the sensitivity indices.

Figure 5: The most influent parameters

Figure 5 illustrates how the volatility of the variance, η, the interest rate-stockcorrelation, ρrS , and the volatility of the stochastic interest rate, σ, have the highertotal sensitivity index values. The first one does not come a surprise because wehave already well explained how the use of the Heston model (stochastic volatility)is relevant when pricing an instrument with a double convexity. An interestingresults is the importance of ρrS and σ. This tells us that the stochastic interestrate, through its correlation with the stock and its volatility parameter, is the firstresponsible of the CoCoCo price variability. This numbers takes into account the

4We remind that in our prototype we have chosen a zero volatility interest rate correlation anda flat forward rate fM so that the parameter Θ(t) results to be deterministic according to theEquation 6.

5We remind that in our prototype we have chosen of setting the correlation ρrV equal to zero.

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individually contribution of r and the interaction effect among the others inputs.Hence, using a constant interest rate is absolutely not recommendable in the Co-

CoCo pricing, although it is a well practise in the convertible bond pricing. Althoughthis introduces more complexity into the modeling, CoCoCos require a stochasticinterest rate model.

4.2 Loss absorbing capacity of the bank

In this section we want to make clear and estimate the loss absorbing capacity of aCoCoCo when the issuer gets into a threatening situation. We thus focus on pathswhere a mandatory conversion takes place. We discount their cash flows and wemake an average of them. The difference between the face value and this averagewill provide us with an estimation of the loss absorbed as soon as a conversiontakes place. For example, suppose that after a mandatory conversion, the amountof shares delivered to the investor is 80% of the face value. This means that 20% ofthe face value of the bond has been absorbed by the bank because it must not bedelivered anymore. Table 4 highlights the relation between the price of a CoCoCobond and its loss absorbing capacity which depends on the structure.

Let us use a trigger level of 50; The upper part in Table 4 highlights how in-creasing the conversion ratio CrMC the price increases leading to decrease of the lossabsorbing capacity of the CoCoCo bond. Increasing the conversion rate means thatthe investors will receives shares at higher price with less losses. Lower losses forinvestors translates into a lower absorption of losses from the issuer side.

Now, let us fix the CrMC and vary the trigger level: in this case Table 4 showshow the both price and loss absorbing capacity of the bond increase. Increasing thetrigger level, the probability of a mandatory conversion increases so that the investortakes more risk investing in this bond. This well explains why the price decreasesunder this scenario. From the issuer side, an high trigger level is able to prevent areally dangerous situation but at the same time the capability of absorbing lossesdecreases because the conversion in shares takes place at high stock prices.

The CoCoCo bonds are thus a kind of automatic reinforcement of the bank’sbalance sheet but their success strictly depend on a trade-off between the interestsof the investor and the interest of the issuer. The investor has to be persuaded thatthe price of the bond is fair enough and the issuer has to be convinced in the successof the loss absorbency of this instrument in crisis time.

5 Conclusion

In this paper, we have discussed a particular contingent convertible bond called, ”Co-CoCo” proposed by some financial landscapes directed to save the banking system.Up to now, several conferences and debates have placed as core of the discussion theneed of issuing debt instruments with loss absorption features. CoCoCo bonds fulfill

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CrMC CoCoCo Price Loss Absorbing Capacity

0.10 84.58 95.70

0.30 84.98 87.11

0.50 85.39 78.51

0.70 85.79 69.92

0.90 86.19 61.33

Trigger Level CoCoCo Price Loss Absorbing Capacity

10 88.43 95.89

30 88.00 87.47

50 85.39 78.51

70 76.83 68.78

90 58.78 57.88

Table 4: CoCoCo Price and Loss absorbing capacity of the CoCoCo bond whenvarying CrMC at S∗ = 50 and varying S∗ for CrMC = 50.

this requirement. They are loss absorbing in crisis time by means of a mandatoryconversion of the debt into equity. At the same time they let investors to profit fromthe good performance of the shares by means of an optional conversion.

In this paper, we have outlined the mechanism and the features of this novelbond and we have proposed a pricing methodology based on an American MonteCarlo simulation model under an assumption of stochastic volatility and stochasticinterest rate.

The paper makes clear in a qualitative and quantitative way that the volatilityof the stochastic interest rate results to be an important component when pricingthis debt instrument leading to the conclusion that pricing models based on constantinterest rate are not really appropriate when dealing with this kind of instruments.

Finally, we have quantified the loss absorbing capacity of this instrument and wehave highlighted how the success of CoCoCo bonds is a really trade off between theinterest of the investor and the interest of the issuer.

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