Interrelations Amongst Liquidity, Market and Credit Risks - Some Proposals for Integrated Approaches

Balance Sheet Items Requiring Statistic-Financial Models Liquidity and Counterparty Risks

Interrelations amongst Liquidity, Market and Credit Risks - some proposals for integrated approaches

Antonio Castagna

www.iasonltd.com

28th February 2012

Risk - Managing Liquidity Risk under Basel III - London Antonio Castagna - Iason 2012 - All rights reserved Balance Sheet Items Requiring Statistic-Financial Models Liquidity and Counterparty Risks Index

1 Balance Sheet Items Requiring Statistic-Financial Models Deposits Prepayment of Fixed Rate Mortgages Credit Lines

2 Liquidity and Counterparty Risks Introduction Liquidity Risk Pricing in OTC Derivatives Haircut Setting

Risk - Managing Liquidity Risk under Basel III - London Antonio Castagna - Iason 2012 - All rights reserved Deposits Balance Sheet Items Requiring Statistic-Financial Models Prepayment of Fixed Rate Mortgages Liquidity and Counterparty Risks Credit Lines Index

1 Balance Sheet Items Requiring Statistic-Financial Models Deposits Prepayment of Fixed Rate Mortgages Credit Lines

2 Liquidity and Counterparty Risks Introduction Liquidity Risk Pricing in OTC Derivatives Haircut Setting

1 Balance Sheet Items Requiring Statistic-Financial Models Deposits Prepayment of Fixed Rate Mortgages Credit Lines

2 Liquidity and Counterparty Risks Introduction Liquidity Risk Pricing in OTC Derivatives Haircut Setting

Non-maturing deposits represent a significant source of funding for a financial institution. Yet, there is still no commonly accepted framework for valuation and for interest rate and liquidity risk management. Such a framework is needed even more in the current environment: non-maturing deposits are a low-cost source of funding, compared with other sources, so that in a funding-mix they contribute to abate the total cost of funding (although a consistent model is needed to manage them, such as the one proposed by Iason); the interest rate and the liquidity risks have to be properly identified, measured and hedged, in order to make possible the insertion of the deposits volumes in the funding management; liquidity crisis have to be modelled, allowing for stress-testing activity involving both idiosyncratic and market specific extreme scenarios.

We assume that the bank offers different deposits, i.e.: transaction account and different savings accounts. A transaction account is mainly used by the customer to fulfill his short term liquidity needs while he uses the savings accounts as investment opportunities with very small risk. The framework models three factors: market rates, deposits’ rates deposits’ volumes. These factors determine the customer behaviour, modelled as reasonable rules or strategies to set the total level of deposits, and then the allocation amongst the different types of deposits. Once this is done, we can calculate the value of the deposits, their sensitivities to market rates and the amounts available, thus allowing for the design of hedging and liquidity management strategies.

We can use any available model for market rates. We choose an extended Cox, Ingersoll and Ross model (CIR), with a time dependent deterministic shift parameter (to perfectly match the starting term structure of market rates). The instantaneous rate r(t) is deﬁned as

r(t) = x(t) + φ(t)

where x(t) has a CIR dynamics:

dxt = κ[θ xt ]dt + σ√xt dZt − and φ(t) is a deterministic function of time.

Term Structure

5.00%

An example of curves generated with: 4.50% x0 = 2%, κ = 0.5, θ = 4.5%, σ = 4.00% 3.50% 7.90%, φ(t) = 0 for any t 3.00% Zero 2.50% 6M Zero-rate curves and implied 6 month 2.00%

1.50%

forward rates are shown in the picture. 1.00%

0.50%

0.00% 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

We assume that deposit rates are a function of the (short term) market interest rates and volumes: dn(t) = d(r(t), V (t))

where dn is the rate for the deposit of type n, r(t) is the instantaneous rate and V (t) is the amount deposited. The function is given by the bank’s pricing policy. Examples may be: constant spread α below market rates:

dn(t) = max[r(t) α, 0] − a proportion α of market rates:

dn(t) = αr(t) a function as the two above dependent on the amount deposited m

dn(t) = dj (r(t))1{vj ,vj+1}(V (t)) Xj=1

where vj and vj+1 are ranges of the volume V producing diﬀerent levels of the deposit’s rate.

We make the following assumptions to model the customer’s behaviour determining the deposits’ volumes: A customer modifies his balance on the transaction account targeting a given fraction of its average monthly income. This level can be interpreted as the amount he needs to cover his short time liquidity needs. There is an interest rate’s strike level, specific for the customer, such that, when the market rate is above it, then the customer reconsiders the target level and redirects a higher amounts to other investments. A customer has a savings’ account policy targeting to save a fraction of his income. Again, there is a strike level, specific for the customer, such that when the market rate is above it then the customer increases the fraction saved. A customer may reconsider his savings policy and may decide to close one or more of his savings accounts. If he takes such actions he reallocates all of his money to one of the other savings accounts offered by the bank.

On an aggregate basis, we consider the “average customer” to model the total deposits’ volume. To take into account the degree of heterogeneity among the customers, concerning the level of the market rate at which the customer changes his policy for the transaction account and for the total savings volume, we adopt a Gamma distribution: (x/β)β−1 exp( x/β) − βΓ(α)

The Gamma function is very ﬂexible. α . 40 If we, for example, set = 1 5 and 35 β = 0.05 we have a distribution labeled 30 α 25 as “1” in the ﬁgure. If = 30 and Disitribution 1 20 β = 0.002 we have a distribution “2”. Distribution 2 15

It is possible to model the aggregated 10 customers’ behaviour, making it more or 5 0 0

less concentrated around speciﬁc levels. 0.1 0.2 0.02 0.04 0.06 0.08 0.12 0.14 0.16 0.18 0.22 0.24 0.25 0.27 0.29 0.31 0.33 0.35 0.37

10.00% 9.00% We present the market and deposits’ in- 8.00% 7.00% Ref. rate 6.00% Trans. Depo terests rates, the evolution of the vol- 5.00% 4.00% Savings2 umes of the transaction account, of the 3.00% Savings2 2.00% 1.00% total volume of the savings accounts and 0.00%

- 3 7 3 0 3 7 0 7 0 .6 .17 .8 .33 0.8 1 2.50 3.3 4 5.0 5 6.6 7.5 8 9.1 0.0 the split between the “savings 1” and 1 “savings 2” accounts, in a diﬀerent eco- 7000.0 nomic cycle. 6800.0 6600.0 At the beginning market rates are stable, 6400.0 6200.0 Trans. Depo 6000.0 then they sharply increase. The transac- 5800.0 5600.0 tion volume experiences a decline due to 5400.0

- 7 3 0 3 0 7 0 .83 .50 .17 .8 .33 0 1.6 2 3.3 4 5.0 5 6.67 7.5 8 9.1 0.0 the reallocation of the total savings in 1 other investments, since they oﬀer higher 140000.0 returns. 120000.0 100000.0 Savings Tot Also the composition of the total savings 80000.0 Savings1 60000.0 Savings2 account volume shows that the “savings 40000.0 20000.0 2” account (yielding higher rates) is pre- 0.0

- 7 0 7 3 7 3 .83 .5 .00 .50 .17 ferred to the “savings 1” account. 0 1.6 2 3.33 4.1 5 5.8 6.6 7 8.3 9 10.00

The behavioural modelling of the amounts of the diﬀerent deposits allows us to manage the liquidity risk, which is based on the concept of the term structure of liquidity. Generate a number S of scenarios. For each 0 t T deﬁne the process of minima by: ≤ ≤

Mj (t) = min V (s) 0≤s≤t

In each scenario the stochastic process Mj (t) speciﬁes the minimal volume between [0, t]. This is the amount available for investment over the entire period in the given scenario.

Deﬁne Lts (t, q) as the p-quantile of Mj (t). The probability that the volume V drops below level Lts (t, q) in the time interval [0, t] is q. Lts (t, q) is the amount available for investment with probability q. We name it Term Structure of Liquidity. The Term Structure of Liquidity can therefore be used to implement liquidity policies.

Time Amnt 1 5905 2 5808 The table indicates the Term Structure 3 5796 of Liquidity up to 10 years for transaction 4 5798 deposits, given the assumptions stated 5 5856 above, with a probability of 99%. 6 5919 7 6053 8 6151 9 6325 10 6430

6500.00 6400.00 6300.00 The Term Structure of Liquidity up to 10 6200.00 6100.00 6000.00 Trans. Depo years for the transaction deposits is also 5900.00 5800.00 shown in the ﬁgure. 5700.00 5600.00 5500.00 5400.00 1 2 3 4 5 6 7 8 910

Time AmntSav1 AmntSav2 1 47657 50882 We now show in the table the Term 2 47087 52014 Structure of Liquidity for the two types 3 47301 53214 4 47305 54372 of savings deposits, given the assump- 5 47810 56232 tions stated above, with a probability of 6 47990 57455 99%. 7 48743 59620 8 48979 61043 9 48496 62477 10 48572 63865

70000.0 60000.0 Also for savings deposits, a visual repre- 50000.0 sentation of the Term Structure of Liq- 40000.0 Savings1 30000.0 Savings2 uidity is also produced. 20000.0 10000.0 0.0 1 2 3 4 5 6 7 8 910

The value of the deposits is diﬀerent from their amount. It is possible to show that the value to the bank, assuming that volumes are deposited up to time T , is: n T Q (0, T ) = N E [r(t) dj (t)Vj (t)P(0, t)]dt V Z − Xj=1 0 where: N is the number of average customers; n is the number of types of deposits;

Vj (t) and dj (t) are, respectively, the amount and the rate for deposit j at time t; P(0, t) is the price, at time 0, of a pure discount bond expiring in T . The value of the deposits can be considered as the value of an exotic swap, paying the ﬂoating rate dj (t) and receiving the ﬂoating rate r(t), on the stochastic principal Vj (t), for the period between 0 and t.

The table shows the value V(0, T ) at time 0 of the transaction deposits. We Value Sens +10bps 0 1994.057 assume we have 10 swap contract, ex- 1 1994.058 0.00 piring from 1 year to 10 years, as hedg- 2 1994.052 -0.01 ing insturments. We identify 10 scenarios 3 1994.044 -0.01 where each swap rate is tilted up 10 bps. 4 1994.033 -0.02 The table also shows the values corre- 5 1994.023 -0.03 sponding to 10 scenarios and the sensi- 6 1994.013 -0.04 7 1994.002 -0.06 tivities to each swap rate. Since we lim- 8 1993.977 -0.08 ited our time horizon to 10 years, the 9 1993.911 -0.15 bulk of the exposure is to the 10-year 10 1998.343 4.29 swap, all other sensitivity being negligi- ble.

The framework can be extended so as to include a possible bank run causing a sudden drop in the deposit’s volumes. This is useful to compute a Liquidity VaR and to operate stress-testing. To that end: we assume that customers may loose confidence in the financial institution, so that they withdraw in a very short time a large percentage of the deposited volumes; we use as an indicator of the confidence the CDS spread (or a similar spread extracted from bonds issued by the bank), which is modelled by a CIR dynamics, similarly to the instantaneous interest rate; the bank run is triggered by a high level of the spread, indicating a high perceived default probability of the bank; the heterogeneity in the customers’ behaviour is modelled by a Gamma distribution, as for the behaviour determining the allocation amongst deposits and other investments.

The ﬁgure plots the evolution of the in- 8.00% 7.00% 6.00% stantaneous zero-spread, assuming a CIR 5.00% 4.00% CDS spread dynamics for the default intensity and a 3.00% 2.00% recovery rate of 40% of the face value 1.00% 0.00%

on default. - 7 3 0 7 3 .83 .50 .17 .83 .50 .17 0 1.6 2 3.3 4 5.0 5 6.6 7 8.3 9 The trigger rate has been placed in a nar- 10.00

row range around 800 bps of the CDS 7000.0 6000.0 spread; we assume customers withdraw 5000.0 4000.0 Trans. Depo 85% of the deposited volumes in case 3000.0 2000.0 they loose conﬁdence in the bank. The 1000.0 0.0

transaction deposits experience a sudden - 7 3 0 3 0 7 0 .83 .50 .17 .8 .33 0 1.6 2 3.3 4 5.0 5 6.67 7.5 8 9.1 0.0 drop in volumes as the spread climbs to- 1

wards the trigger level. 120000.0 We make a similar assumption of with- 100000.0 80000.0 Saving Tot drawal for the savings accounts. Also 60000.0 Savings1 40000.0 Savings2 their volumes quickly decreases in a 20000.0 0.0

“bank run” scenario. - 7 3 7 3 0 3 .83 .50 .00 .67 .17 .00 0 1.6 2 3.3 4.1 5 5.8 6 7.5 8.3 9 0 1

1 Balance Sheet Items Requiring Statistic-Financial Models Deposits Prepayment of Fixed Rate Mortgages Credit Lines

2 Liquidity and Counterparty Risks Introduction Liquidity Risk Pricing in OTC Derivatives Haircut Setting

Empirical features commonly attributed to mortgage prepayment include: some mortgages are prepaid even when their coupon rate is below current mortgage rates; some mortgages are not prepaid even when their coupon rate is above current mortgage rates. prepayment appears to be dependent on a burnout factor. The model we propose takes into account these features: mortgagees decide whether to prepay their mortgage at random discrete intervals. The probability of a prepayment decision taken on interest rate reasons, is commanded by a hazard function λ: the probability that the decision is made in a time interval of length dt is approximately λdt; besides reﬁnancing for interest rate reasons, the mortgagees may also prepay for exogenous reasons (e.g.: job relocation, or sale of the house). The probability of exogenous prepayment is described by a hazard function ρ: this represents a baseline prepayment level, the expected prepayment level when no optimal (interest-driven) prepayment should occur.

We model the interest rate based prepayment within a reduced form approach. This allows us to include consistently the prepayments into the pricing, the interest rate risk management (ALM) and the liquidity management. We adopt a stochastic intensity of prepayment λ, assumed to follow a CIR dynamics: dλt = κ[θ λt ]dt + νt √λt dZt − the intensity is common to all obligors and provides the probability that the mortgage rationally terminates over time; the intensity is correlated to the interest rates, so that when rates move to lower levels, more rational prepayments occur; the framework is stochastic and it allows for a rich speciﬁcation of the prepayment behaviour; The exogenous prepayment is also modelled in a reduced form fashion, by a constant intensity ρ.

Consider a mortgage with coupon rate c expiring at time T : each period, given the current interest rates, the optimal prepayment strategy determines whether the mortgage holder should refinance; for a given coupon rate c, considering also transaction costs, there is a ∗ ∗ critical interest rates’ level r such that if rates are lower (rt < r ) then the mortgagor will optimally decide to prepay; if it is not optimal to refinance, any prepayment is for exogenous reasons; if it is optimal to refinance, the mortgagor may prepay either for interest rate related or for exogenous reasons. These considerations lead to the following prepayment probability:

T −ρ(T −t) − λs ds ∗ PP(t, T ) = 1 e E e t if rt < r (1) − R −ρ(T −t) ∗ PP(t, T ) = 1 e if rt > r (2) −

The figure plots the prepayment probabilities for different times up to the (fixed rate) mortgage’s expiry, assumed to be in 90.00% 10 years. The three curves refer to: 80.00% the exogenous prepayment, given 70.00% by a constant intensity ρ = 3.5%; 60.00%

50.00% Exogenous the rational (interest driven) Rational Total prepayment, produced assuming 40.00% 30.00% λ0 = 10.0%, κ = 27%, θ = 50.0% ν = 10.0%. 20.00%

10.00% the total prepayment when it is 0.00% rational to prepay the mortgage 1 2 3 4 5 6 7 8 9 10 ∗ (rt < r ).

It is possible to compute the Expected Loss on Prepayment, deﬁned as the Expected Loss at a time tk given that the decision to prepay (for whatever reason) is taken between tk−1 and tk :

∗ ELoP(tk ) = E PP(tk−1, tk ) max P(tk , tj )τj Aj−1(c c ); 0 . − Xj the computation of the ELoP is unfortunately not possible in a closed form formula: this can be a problem since the ELoP has to be computed for all the possible exercise dates for a mortgage, and this for a likely large number of mortgages; considering the fact that we are interested also in the computation of the sensitivities of the ELoP, for hedging purposes, the computation via numerical techniques, such as Montecarlo, can be very time-consuming and unfeasible; for this reason we developed an analytical approximation to model: the future fair mortgage rate c, the correlation between the fair rate c∗ and the rational prepayment intensity λt ;

We are now able to measure which is the expected loss occurring to the bank upon prepayment, at each possible prepayment date tk ;

We deﬁne the current value, at time t0, of all the expected losses is the Total Prepayment Cost (TPC) related to a mortgage:

TPC(t0) = P(t0, tk )ELoP(tk ) Xk

the TPC is the quantity to be hedged. It is a function of: the Libor forward rates, the volatilities of the Libor forward rates, the stochastic rational prepayment intensity λt and constant exogenous intensity ρ; the TPC can be also included in the mortgage pricing when calculating the fair rate c.

The model allows also to compute sensitivities to the main underlying risk factors: sensitivity to Libor rates can be computed by tilting each forward a given amount (e.g.: 10 bps), and then recomputing the TPC; Libor exposures can be translated to swap rates exposures, since these are the most liquid and easily tradable hedging tools; by the same token, we can compute also the sensitivities to Libor volatilities: these exposures can be hedged by trading caps&ﬂoors; the sensitivity to the prepayment, both rational and exogenous, can be derived, but no market instrument exists to hedge this exposure. In this case, a VAR-like approach can be adopted and the unexpected cost included into the fair rate, or economic capital posted to cover this risk.

Yrs Fwd Libor Vol Assume 1Y Libor forward rates and their 1 3.50% 18.03% volatilities are those in the table besides. 2 3.75% 18.28% Assume also that the exogenous prepay- 3 4.00% 18.53% ment intensity is 3% p.a. and the rational 4 4.25% 18.78% prepayment intensity has the same dy- 5 4.50% 18.43% 6 4.75% 18.08% namics’ parameters as presented above. 7 5.00% 17.73% 8 5.25% 17.38% 9 5.50% 17.03% 10 5.75% 16.78%

We consider a 10Y mortgage, with a Yrs Notional ﬁxed rate paid annually of 3.95%. The 1 100.00 2 90.00 fair rate has been computed wihout tak- 3 80.00 ing into account any prepayment eﬀect 4 70.00 (also credit risk is not considered, al- 5 60.00 though it can be included within the 6 50.00 framework). The amortization schedule 7 40.00 8 30.00 is in the table besides. 9 20.00 10 10.00

Given the market and contract data 0.90000 above, we can derive the EL at each pos- 0.80000 sible prepayment date, which we assume 0.70000 0.60000

occurs annually. It is plotted in the ﬁg- 0.50000 Expected loss ure. The closed form approximation has 0.40000 been employed to compute the EL. 0.30000 In a similar way it is possible to calculate 0.20000 0.10000

the ELoP. We use also in this case an - analytical approximation that allows for 1 2 3 4 5 6 7 8 9 a correlation between interest rates and the rational prepayment intensity.

In the ﬁgure the ELoP is plotted for the 0.0900 0 correlation case and for a negative cor- 0.0800 relation set at −0.8. This value implies 0.0700 that when interest rates decline, the de- 0.0600 0.0500 ELoP Zero Corr ELoP Negative Corr fault intensity increases. 0.0400

Since the loss for the bank is bigger when 0.0300 the rates are low, the ELoP in this case 0.0200 is higher than the uncorrelated case. 0.0100 - 1 2 3 4 5 6 7 8 9

The hedging of the prepayment risk (or, of the TPC) is possible with respect to Yrs Sensitivity Hedge Qty the interest rates and to the Libor for- 1 0.02 16.82 2 0.02 9.07 ward volatilities. The TPC is 48 bps. 3 0.02 5.58 The ﬁrst table shows the sensitivity of 4 0.01 3.45 the TPC to a tilt of 10 bps for each 5 0.01 2.19 forward rate. Those sensitivities are then 6 0.01 1.38 translated in an equivalent quantity of 7 0.01 0.92 swaps, with an expiry form 1 year to 10 8 0.00 0.64 9 0.00 0.40 years, needed to hedge them.

Yrs Vega The second table shows the Vega of the 1 0.08 TPC with respect to the volatilities of 2 0.20 each Libor forward rate. Those exposures 3 0.33 can hedged with caps&ﬂoors, or swap- 4 0.45 tions in the Libor Market Model setting 5 0.53 6 0.54 we are working in (by calibrating the Li- 7 0.48 bor correlation matrix to the swaptions’ 8 0.33 volatility surface). 9 0.12

Expected Cash Expected Amort. Cash Flows Flows Amort. 22.34 13.95 81.61 90.00 20.96 13.56 64.08 80.00 We have shown how the framework can 17.59 13.16 49.44 70.00 be used to measure and hedge the pre- 14.15 12.77 37.78 60.00 11.28 12.37 28.55 50.00 payment risk in its form of replacement 9.13 11.98 21.08 40.00 7.56 11.58 14.81 30.00 cost. 6.39 11.19 9.35 20.00 The model can be also use to project ex- 5.48 10.79 4.46 10.00 pected cash ﬂows due to the prepayment activity.

Since the rational prepayment is stochas- 100.00 tic and correlated to the level of the in- 90.00 80.00

terest rate, a VaR-like approach can be 70.00

adopted also in this case to calculate the 60.00

50.00 Amort. maximum and minimum amount of cash Exp. Amort. 40.00

ﬂows; 30.00

20.00

10.00

0.00 1 2 3 4 5 6 7 8 9

1 Balance Sheet Items Requiring Statistic-Financial Models Deposits Prepayment of Fixed Rate Mortgages Credit Lines

2 Liquidity and Counterparty Risks Introduction Liquidity Risk Pricing in OTC Derivatives Haircut Setting

Loan commitments or credit lines are the most popular form of bank lending representing a high percent of all commercial and industrial loans by domestic banks. Loan commitments allow ﬁrms to borrow in the future at terms speciﬁed at the contract’s inception. The model we propose simple, analytically tractable approach that incorporates the critical features of loan commitments observed in practice: random interest rates; multiple withdrawals by the debtor; impacts on the cost of liquidity to back-up the withdrawals; interaction between the probability of default and level of usage of the line.

The bank has a portfolio of m diﬀerent credit lines, each one with a given expiry Ti (i = 1, 2, ..., m).

Each credit line can be drawn within the limit Li at any time t between today (time 0) and its expiry. We assume the there is a withdrawal intensity indicating which is the used percentage of the total amount of the line Li at a given time t: from each credit line i, the borrower can withdraw an integer percentage of its nominal: 1%, 2%,..., 100%; each withdrawal is modelled as a jump from a Poisson distribution (one speciﬁc to each credit line). This distribution can not have more than 100 jumps. As an example, a 3% withdrawal is equivalent for the Poisson process to jump 3 times; the withdrawal intensity λi (t) determines the probability of the jumps during the life of the credit line. This intensity is stochastic (so we have a doubly stochastic Poisson process).

The stochastic intensity allows to model: the documented correlation between the worsening of the creditworthiness of the debtor and the higher usage of the amount Li of the credit line; the correlation between the probabilities of default of the debtors of the m credit lines. Both eﬀects have a heavy impact on the single and joint distributions of the usage of the credit lines. The liquidity management at a portfolio level is enhanced, since the bank can properly take into account the joint distribution The correlation between debtors determines in a more precise fashion the value of credit lines and the credit VaR, relying on a framework more robust and consistent than simple credit conversion factors.

The stochastic withdrawal intensity λi (t), for the i-th debtor, is a combination of three terms:

D λi (t) = αi (t)(̺i (t) + λi (t))

A multiplicative factor αi (t) of a deterministic function of time ̺i (t), which can be used to model the withdrawals of the credit line independent from the default probability of the debtor. D A multiplicative factor αi (t) of the stochastic default intensity λi (t) (default is modelled as a rare event occurring with this intensity). We model the correlation between debtor by assuming that the default intensity is the sum of two separate components:

D I C λi (t) = λi (t) + pi λ (t)

The default intensity parameters are chosen so that the debtor’s probability of default is about 2% 0.05 0.045 in 1 year, and it declines on 0.04 0.035 average in the future toward a 0.03 0.025 long term average of 1.5%. 0.02 0.015 0.01 The deterministic intensity ̺ is 0.005 0

constant and equal to 2%. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , 0 , 0 0 0 , 0 , 0 , 0 0 0 , 0 , 0 , 0 0 0 , 0 0 , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 5 , 5 0 , 0 The multiplying factor is also 1 , 1 , 5 0 0 2 , 0 0 2 , 3 , 0 0 3 4 , 4 , 5 0 5 constant and equal to α = 1000. Given this, we have an average withdrawal intensity at time 0 The ﬁgure shows the usage distribution of equal to 1000 × (2% + 2%) = 40 the credit line over a period of 1 year. The or 40% of the total amount of the total amount of the credit line is Euro 5 line (since each jump corresponds mln. The average usage is 2,058,433. We to 1% of usage). added the 99%-percentile of this distribution which has the following value: The credit line nominal is equal to 4,050,000. 5 mln, the expiry is in 1 year.

In the present framework, the correlation amongst diﬀerent debtors play a great role in determining the joint usage distribution.

The pi ’s coeﬃcient weights the dependence of the default intensity on the common factor, and then it is an indication of the correlation amongst debtors.

The higher the correlation (i.e.: pi ’s), the larger th expected usage and the 99%-percentile unexpected usage.

Even if for diﬀerent scenarios (created by diﬀerent pi s), the expected usage is equal for the single credit lines, the expected joint usage and the percentile changes with the choice of pi .

We choose 3 credit lines, 5 mln, each and two opposite scenarios: 0.04 First Scenario: we choose 0.035 0.03 pi = 0.999 and calibrate and the 0.025 parameters of the default intensity 0.02 0.015 so as to have an expected usage 0.01 of 50%: 0.005 0

0 0 0 0 0 0 0 0 Second Scenario: we choose , 0 0 0 0 , 0 0 0 0 , 0 , 0 2 , 0 0 0 , 0 0 0 4 6 , 0 0 0 , 0 0 0 8 , 0 0 2 , 0 0 0 , 0 0 0 pi = 0.001 and calibrate again α 1 0 , 0 0 0 , 1 1 4 , 0 0 1 6 , 0 0 0 , 0 0 0 and the parameters of the default intensity so as to have an expected usage of 50%: The figure shows the joint usage The default intensity starts at 2% distribution, and the 99% precentile, for for each debtor in both scenarios, first scenario (in red) and for the second but the correlation amongst the scenario (in blue). In both scenarios the probability of defaults and of expected usage is almost 7.5 mln. The default events is very different: 99%-percentile usage is 14, 750, 000 in the almost perfect in the first case first scenario and 9, 150, 000 in the second and almost nil in the second. scenario.

Risk - Managing Liquidity Risk under Basel III - London Antonio Castagna - Iason 2012 - All rights reserved Introduction Balance Sheet Items Requiring Statistic-Financial Models Liquidity Risk Pricing in OTC Derivatives Liquidity and Counterparty Risks Haircut Setting Index

1 Balance Sheet Items Requiring Statistic-Financial Models Deposits Prepayment of Fixed Rate Mortgages Credit Lines

2 Liquidity and Counterparty Risks Introduction Liquidity Risk Pricing in OTC Derivatives Haircut Setting

1 Balance Sheet Items Requiring Statistic-Financial Models Deposits Prepayment of Fixed Rate Mortgages Credit Lines

2 Liquidity and Counterparty Risks Introduction Liquidity Risk Pricing in OTC Derivatives Haircut Setting

Agreements between counterparties aim at limiting and reducing the counterparty risk 1 Netting (e.g.: ISDA Master Agreement) a contract that allows aggregation of transactions; in the event of default of one of the counterparties, the entire portfolio included in a netting agreement is considered as a single trade. 2 Collateral Agreements (e.g.: ISDA’s Credit Support Annex (CSA)) collateral is required if unsecured exposure is above a given threshold; threshold and frequency depend on counterparty’s credit quality. 3 Early termination clauses: Termination clause: trade-level agreement that allows one (or both) counterparties to terminate the trade at fair market value at a predefined set of dates; Downgrade provision: portfolio-level agreement that forces the termination of the entire portfolio at fair market value the first time the credit rating of one (or either) of the counterparties falls below a predefined level. 4 Certain contracts, like Contingent CDS (CCDS). Risk - Managing Liquidity Risk under Basel III - London Antonio Castagna - Iason 2012 - All rights reserved Introduction Balance Sheet Items Requiring Statistic-Financial Models Liquidity Risk Pricing in OTC Derivatives Liquidity and Counterparty Risks Haircut Setting Collateral and Margin Agreements

Collateral agreement is a contract between two counterparties that requires one or both counterparties to post collateral (typically cash or high quality bonds) under certain conditions. Margin agreement is a legally binding collateral agreement with speciﬁc rules for posting collateral, which include:

1 Minimum transfer amount: defines the minimum amount of collateral that can be exchanged. If the exposure entails a collateral posting below the minimum, amount, no collateral is provided; 2 A threshold, defined for one (unilateral agreement) or both (bilateral agreement) counterparties. If the difference between the net portfolio value and already posted collateral exceeds the threshold, the counterparty must provide collateral sufficient to cover this excess (subject to minimum transfer amount); 3 Frequency: defines the periodicity of the exposure calculation and of the determination of the collateral to post. The terms of the rules depend mainly on the credit qualities of the counterparties involved.

1 Balance Sheet Items Requiring Statistic-Financial Models Deposits Prepayment of Fixed Rate Mortgages Credit Lines

2 Liquidity and Counterparty Risks Introduction Liquidity Risk Pricing in OTC Derivatives Haircut Setting

In a very general fashion, the price at time 0 of a derivatives contract which is not subject to counterparty risk is:

T Q − rs ds V0 = E e 0 VT R

where

VT is the terminal pay-off of the contract; rt is the (possibly time dependent) risk-free interest rate. When counterpaty risk is considered, then we have to include the so called CVA (the expected losses we suffer when on default of the counterparty) the and DVA (the expected losses the counterparty suffers on our default):

T CCP Q − rs ds V = E e 0 VT CVA + DVA 0 R −

The terminal value of the contract is still discounted ad the risk-free rate rt , but then the price is adjusted with the net eﬀect due to the losses upon default of the two counterparties involved in the trade.

Assume now we have a CSA agreement operating between the two counterparties. The CSA provides for a daily margining mechanism of the full variation of the NPV (nowadays a very common form of the CSA). The party that owns a positive balance on the collateral account (corresponding to a positive NPV of the contract) pays the rate ct to the other party. The pricing of the contract can be now be operated by excluding the default risk (there is still a very small residual risk between two daily margining). It can be shown that the pricing formula is very similar to the standard case we have seen above, but with the collateral rate ct replacing the risk-free rate rt .:

T CSA Q − cs ds V = E e 0 VT (3) 0 R

This result is very convenient, since we have a well deﬁned rate that has to be paid on the collateral balance (set within the contract), whereas the risk-free rate is very diﬃcult to determine in the current market environment (it used to be the Libor in the interbank market). Usually the daily margined CSA agreements set the remuneration of the collateral at the EONIA for contracts in euro (or some equivalent OIS rate for other currencies). EONIA (OIS) rates can be considered the best approximation of a risk-free rate. Nevertheless there is still one assumption that is made when deriving the pricing formula with the CSA: The rate at which the bank can lend money is the same of the one it can borrow money. This assumption can be easily accepted when we price contracts whose NPV can be replicated by a dynamic strategy. When the NPV of the contract cannot be replicated (e.g.: forward and swap contracts) then relaxing the assumption is trickier.

Assume we have a contract whose value during the life of the contract at any time 0 < t < T , Vt can be positive or negative. We also assume that the bank can invest cash at a risk-free rate equal to the collateral rate rt = ct , but it has a funding spread ft when borrowing money over a short period, so that the total funding cost is rt + ft . When considering the funding spread in the pricing of a collateralized derivatives contract, it can be shown that the valuation equation can be written as:

T T CSA Q − cu −fu 1{V <0}du Q − cu du V = E e 0 u VT = E e 0 VT + FVA (4) 0 R R

where

T Q − s c du FVA E e 0 u V f ds = R min( s (0), 0) s (5) Z0

We show an example, assuming the following market data for interest rates:

Time Eonia Fwd Spread Fwd Libor 0 0.75% 0.65% 1.40% 0.5 0.75% 0.64% 1.39% 1 1.75% 0.64% 2.39% 1.5 2.00% 0.63% 2.63% 2 2.25% 0.63% 2.88% 5.00% 2.5 2.37% 0.62% 2.99% 4.50% 3 2.50% 0.61% 3.11% 4.00% 3.5 2.65% 0.61% 3.26% 3.50% 4 2.75% 0.60% 3.35% 3.00% Euribor Fw d 4.5 2.87% 0.60% 3.47% 2.50% Eonia Fw d 5 3.00% 0.59% 3.59% 2.00% 5.5 3.10% 0.59% 3.69% 1.50% 6 3.20% 0.58% 3.78% 1.00% 6.5 3.30% 0.58% 3.88% 0.50% 7 3.40% 0.57% 3.97% 0.00% 7.5 3.50% 0.57% 4.07% 0 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 8 3.60% 0.56% 4.16% 8.5 3.67% 0.56% 4.23% 9 3.75% 0.55% 4.30% 9.5 3.82% 0.55% 4.37% 10 3.90% 0.54% 4.44%

Market data for caps&ﬂoors and swaptions volatilities are:

Caps&Floors Swaptions Expiry Volatility Expiry Tenor Volatility 0.5 30.00% 0.5 9.5 27.95% 1 40.00% 1 9 28.00% 1.5 45.00% 1.5 8.5 27.69% 2 40.00% 2 8 27.09% 2.5 35.00% 2.5 7.5 26.61% 3 32.00% 3 7 26.32% 3.5 31.00% 3.5 6.5 26.16% 4 30.00% 4 6 26.02% 4.5 29.50% 4.5 5.5 25.90% 5 29.00% 5 5 25.79% 5.5 28.50% 5.5 4.5 25.68% 6 28.00% 6 4 25.57% 6.5 27.50% 6.5 3.5 25.46% 7 27.00% 7 3 25.37% 7.5 26.50% 7.5 2.5 25.28% 8 26.00% 8 2 25.22% 8.5 25.50% 8.5 1.5 25.21% 9 25.50% 9 1 25.34% 9.5 25.50% 9.5 0.5 25.50% 10 25.50% 10 0

We price under a CSA agreement a receiver swap whereby we we pay the FVA -0.0512% Libor fixing semi-annually (set at the Fair Swap rate 3.3020% previous payment date) and we re- Swap Rate + Coll. Fund 3.3079% ceive the fixed rate annually. With Difference 0.0059% market data shown above, the fair rate can be easily calculated (we are ENE

- using the new market standard ap- 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 proach to employ the EONIA/OIS (1.0000) (2.0000) curve for discounting and the 6M Li- (3.0000)

bor curve to project forward rates). (4.0000)

We assume also that we have to pay (5.0000) a funding spread of 15bps over the (6.0000) EONIA/OIS curve. This is applied to (7.0000) the ENE plotted beside.

We may be interested in calculating the impact of the liquidity of a collateralized swap with respect to a FVA -0.2156% more conservative measure than the Fair Swap rate 3.3020% ENE, similarly to what happens in Swap Rate + Coll. Fund 3.3264% the counterparty risk management. Diﬀerence 0.0244% We choose the Potential Future Ex-

posure, which is the expected nega- - 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5

tive NPV of the swap at a given level (5.0000)

of conﬁdence, set in this example at (10.0000)

the 99% and computed with market PFE (15.0000) volatilities. ENE (20.0000) The funding spread is still 15bps over the EONIA/OIS curve. The Potential (25.0000) Future Exposure (blue line), and the (30.0000) ENE (purple line, same as before) for comparison, are plotted beside.

1 Balance Sheet Items Requiring Statistic-Financial Models Deposits Prepayment of Fixed Rate Mortgages Credit Lines

2 Liquidity and Counterparty Risks Introduction Liquidity Risk Pricing in OTC Derivatives Haircut Setting

Collateral can be also made of bonds, usually of good credit quality. In this collateralization transforms counterparty credit risk into market risk and issuer’s credit risk, which should be relatively small for the collateral to be eﬀective. Even when the collateral itself can be defaultable (e.g.: in corporate bonds or emerging currencies sovereign bonds) the counterparty credit risk is strongly mitigated. Common practices have appeared to manage the intrinsic risk of collateral: marking to market and haircuts. The problem is now to determine the fair level of the haircut on a bond, given a chosen mark to market period. We outline a simpliﬁed framework, with no mark-to market periods a where the haircut can be set only once. The approach can be extended to a portfolio of bonds posted as collateral, but we will not pursue the analysis that further in the current discussion.

Assume that at time t a bank has a ﬁxed exposure E to a given counterparty.

The bank asks the counterparty for α units of a bond expiring in Tb with price B(t, Tb) as collateral. We assume that the time to maturity of the collateral is greater than the expiration date of the contract, originating the exposure (e.g.: loan), between the bank and the counterparty. The contract expiry is Ta For the collateral pledged, there exists a haircut h.

Let us divide the interval [t, Ta] in m periods of equal extension.

Assume now that our counterparty goes defaulted in tm, the end of one of the m periods, in which case the bank has to recover the loss on the exposure by selling the collateral bond in the market. Since the collateral bond can go defaulted also, the loss the bank still suﬀers after the selling is: LGD(tm) = E α[B(tm, Tb)1τ >t + RB 1t ≤τ ≤t ] − B m m−1 B m where τB is the default time of the collateral bond and RB is its recovery on default. We also assume that the default of the counterparty is independent from the default of the bond’s issuer.

The expected loss can be computed by considering the PD of both the counterparty and the bond’s issuer. We want to compute the maximum loss given a conﬁdence level, say 99%, if the counterparty goes defaulted. We assume that the interest rates are modelled by a CIR model of the kind:

dr = κr (θr r)dt + σr √rdZt − and the PD is produced by a stochastic intensity still with a CIR dynamics:

dλ = κλ(θλ λ)dt + σλ√λdWt −

At the end of each period tm the minimum value of the bond is determined after deriving the maximum level of the interest rate and of the default intensity at the 99% c.l., given it is still alive in tm. In case the collateral bond is in default, the recovery value must be considered. The collateral value in tm is then the expected value in the two states of the world. Risk - Managing Liquidity Risk under Basel III - London Antonio Castagna - Iason 2012 - All rights reserved Introduction Balance Sheet Items Requiring Statistic-Financial Models Liquidity Risk Pricing in OTC Derivatives Liquidity and Counterparty Risks Haircut Setting A Framework to Set Haircuts

In the case we consider we have to set the level of the haircut once for all at the beginning of the contract, so that we do not revise the quantity of collateral bond after its market value changes. Since counterparty’s default can happen anytime during the period between [t, Ta], we will weight the possible values of the collateral bonds at the several times ti by the probability of default between [tm−1, tm] over the total default probability over the entire contracts’ period:

PDC (tm−1, tm) wm = PDC (t, Ta)

The expected maximum loss over [t, Ta] is

ELGD(tm) = E α[B(ti , Tb)(1 PDB (t, ti )) − − Xi=1

+ RB (PDB (t, ti−1) PDB (t, ti ))]wm −

Assume we have an exposer E originated by a contract expiring in Ta = 6M and the bank requires as collateral a bond expiring in Tb = 10Y . The term structure of the interest rates and the PD’s of the issuers are generated by a CIR short rate and a CIR default in- Years 1Y Libor PD tensity processes with the following pa- 1 2.05% 0.65% rameters: 2 3.34% 1.47% 3 3.91% 2.38% 4 4.17% 3.33% r0 1.00% κr 0.75 5 4.28% 4.31% θr 4.50% 6 4.33% 5.30% σr 25.00% 7 4.35% 6.30% 8 4.36% 7.31% 9 4.36% 8.34% 10 4.37% 9.37%

λ0 0.50% κλ 0.75 θλ 1.00% σλ 25.00%

The collateral bond has the characteris- tics shown in the table below so that its market price at time t0 is 100.4675. We divide the contact period of 6 PD Bond Month Cpt Price Recov. months in 6 monthly intervals and com- 0 0.00% 100.47 0.00 pute the minimum value of the bond at 1 0.25% 97.94 0.04 2 0.52% 96.56 0.13 the end of each period, considering also 3 0.79% 95.38 0.15 the occurrence of default and the recov- 4 1.07% 94.20 0.13 5 1.35% 93.42 0.08 ery. We weight these values as described 6 1.64% 92.93 0.02 above so that we derive the quantity α capable to match the exposure, and Min Coll W’ed Coll Month Value Weight Value then we can set the haircut on the mar- 0 100.47 ket value of the bond. The table beside 1 97.99 15.49% 15.18 2 96.69 16.03% 15.50 shows all calculations. 3 95.53 16.51% 15.77 4 94.33 16.95% 15.99 5 93.49 17.34% 16.21 Face Value 100 6 92.95 17.69% 16.44 Expiry 10 Av. Coll. Value 95.08 α 1.05 Coupon 4.50% Fair Haircut 5.66% Frequency 1 Recovery 40%

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