Interrelations amongst Liquidity, Market and Credit Risks -

Transcription

1 Interrelations amongst Liquidity, Market and Credit Risks - some proposals for integrated approaches Antonio Castagna 28th February 2012

2 Index Balance Sheet Items Requiring Statistic-Financial Models 1 Balance Sheet Items Requiring Statistic-Financial Models 2

3 Index Balance Sheet Items Requiring Statistic-Financial Models 1 Balance Sheet Items Requiring Statistic-Financial Models 2

4 Index Balance Sheet Items Requiring Statistic-Financial Models 1 Balance Sheet Items Requiring Statistic-Financial Models 2

5 The Relevance of in the Banking Activity 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.

6 Overview of the Proposed Model 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.

7 Market Rates 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 where x(t) has a CIR dynamics: r(t) = x(t) + φ(t) dx t = κ[θ x t]dt + σ x tdz t and φ(t) is a deterministic function of time. An example of curves generated with: x 0 = 2%, κ = 0.5, θ = 4.5%, σ = 7.90%, φ(t) = 0 for any t Zero-rate curves and implied 6 month forward rates are shown in the picture. 5.00% 4.50% 4.00% 3.50% 3.00% 2.50% 2.00% 1.50% 1.00% 0.50% Term Structure Zero 6M 0.00%

8 Deposit Rates We assume that deposit rates are a function of the (short term) market interest rates and volumes: d n(t) = d(r(t), V(t)) where d n 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: a proportion α of market rates: d n(t) = max[r(t) α, 0] d n(t) = αr(t) a function as the two above dependent on the amount deposited m d n(t) = d j (r(t))1 {vj,v j+1 }(V(t)) j=1 where v j and v j+1 are ranges of the volume V producing different levels of the deposit s rate.

9 Volumes 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.

10 0 Balance Sheet Items Requiring Statistic-Financial Models Volumes 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. If we, for example, set α = 1.5 and β = 0.05 we have a distribution labeled as 1 in the figure. If α = 30 and β = we have a distribution 2. It is possible to model the aggregated customers behaviour, making it more or less concentrated around specific levels Disitribution 1 Distribution 2

11 Evolution of Volumes: Second Example We present the market and deposits interests rates, the evolution of the volumes of the transaction account, of the total volume of the savings accounts and the split between the savings 1 and savings 2 accounts, in a different economic cycle. At the beginning market rates are stable, then they sharply increase. The transaction volume experiences a decline due to the reallocation of the total savings in other investments, since they offer higher returns. Also the composition of the total savings account volume shows that the savings 2 account (yielding higher rates) is preferred to the savings 1 account % 9.00% 8.00% 7.00% 6.00% 5.00% 4.00% 3.00% 2.00% 1.00% 0.00% Ref. rate Trans. Depo Savings2 Savings2 Trans. Depo Savings Tot Savings1 Savings2

12 Liquidity Risk 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: M j (t) = min 0 s t V(s) In each scenario the stochastic process M j (t) specifies the minimal volume between [0, t]. This is the amount available for investment over the entire period in the given scenario. Define L ts(t, q) as the p-quantile of M j (t). The probability that the volume V drops below level L ts(t, q) in the time interval [0, t] is q. L ts(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.

13 Liquidity Risk: a Practical Example The table indicates the Term Structure of Liquidity up to 10 years for transaction deposits, given the assumptions stated above, with a probability of 99%. Time Amnt The Term Structure of Liquidity up to 10 years for the transaction deposits is also shown in the figure Trans. Depo

14 Liquidity Risk: a Practical Example We now show in the table the Term Structure of Liquidity for the two types of savings deposits, given the assumptions stated above, with a probability of 99%. Time Amnt Sav 1 Amnt Sav Also for savings deposits, a visual representation of the Term Structure of Liquidity is also produced Savings1 Savings

15 Valuation of 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 V(0, T) = N E Q [r(t) d j (t)v j (t)p(0, t)]dt where: j=1 N is the number of average customers; n is the number of types of deposits; 0 V j (t) and d j (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 d j (t) and receiving the floating rate r(t), on the stochastic principal V j (t), for the period between 0 and t.

16 Valuation and Interest Rate Risk: a Practical Example The table shows the value V(0, T) at time 0 of the transaction deposits. We assume we have 10 swap contract, expiring from 1 year to 10 years, as hedging insturments. We identify 10 scenarios where each swap rate is tilted up 10 bps. The table also shows the values corresponding to 10 scenarios and the sensitivities to each swap rate. Since we limited our time horizon to 10 years, the bulk of the exposure is to the 10-year swap, all other sensitivity being negligible. Value Sens +10bps

17 Adding the Risk of Bank Run 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.

18 Risk of Bank Run: a Practical Example The figure plots the evolution of the instantaneous zero-spread, assuming a CIR dynamics for the default intensity and a recovery rate of 40% of the face value on default. The trigger rate has been placed in a narrow range around 800 bps of the CDS spread; we assume customers withdraw 85% of the deposited volumes in case they loose confidence in the bank. The transaction deposits experience a sudden drop in volumes as the spread climbs towards the trigger level. We make a similar assumption of withdrawal for the savings accounts. Also their volumes quickly decreases in a bank run scenario. 8.00% 7.00% 6.00% 5.00% 4.00% 3.00% 2.00% 1.00% 0.00% CDS spread Trans. Depo Saving Tot Savings1 Savings2

19 Index Balance Sheet Items Requiring Statistic-Financial Models 1 Balance Sheet Items Requiring Statistic-Financial Models 2

20 Facts about Prepayments 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.

21 The Probability of Prepayment 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: dλ t = κ[θ λ t]dt + ν t λtdz t 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.

22 The Probability of Prepayment 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 (r t < 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: [ ] PP(t, T) = 1 e ρ(t t) E e T t λ sds if r t < r (1) PP(t, T) = 1 e ρ(t t) if r t > r (2)

23 Prepayment Probability Curves: An Example The figure plots the prepayment probabilities for different times up to the (fixed rate) mortgage s expiry, assumed to be in 10 years. The three curves refer to: the exogenous prepayment, given by a constant intensity ρ = 3.5%; the rational (interest driven) prepayment, produced assuming λ 0 = 10.0%, κ = 27%, θ = 50.0% ν = 10.0%. the total prepayment when it is rational to prepay the mortgage (r t < r ) % 80.00% 70.00% 60.00% 50.00% 40.00% 30.00% 20.00% 10.00% 0.00% Exogenous Rational Total

24 Modelling Losses upon Prepayment It is possible to compute the Expected Loss on Prepayment, defined as the Expected Loss at a time t k given that the decision to prepay (for whatever reason) is taken between t k 1 and t k : [ ELoP(t k ) = E PP(t k 1, t k ) max [ P(t k, t j )τ j A j 1 (c c ); 0 ]]. 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 λ t; j

25 Modelling Losses upon Prepayment We are now able to measure which is the expected loss occurring to the bank upon prepayment, at each possible prepayment date t k ; We define the current value, at time t 0, of all the expected losses is the Total Prepayment Cost (TPC) related to a mortgage: TPC(t 0) = k P(t 0, t k )ELoP(t k ) 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 λ t and constant exogenous intensity ρ; the TPC can be also included in the mortgage pricing when calculating the fair rate c.

26 Hedging Prepayment Exposures 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.

27 A Practical Example Assume 1Y Libor forward rates and their volatilities are those in the table besides. Assume also that the exogenous prepayment intensity is 3% p.a. and the rational prepayment intensity has the same dynamics parameters as presented above. Yrs Fwd Libor Vol % 18.03% % 18.28% % 18.53% % 18.78% % 18.43% % 18.08% % 17.73% % 17.38% % 17.03% % 16.78% We consider a 10Y mortgage, with a fixed rate paid annually of 3.95%. The fair rate has been computed wihout taking into account any prepayment effect (also credit risk is not considered, although it can be included within the framework). The amortization schedule is in the table besides. Yrs Notional

28 A Practical Example: The EL and the ELoP. Given the market and contract data above, we can derive the EL at each possible prepayment date, which we assume occurs annually. It is plotted in the figure. The closed form approximation has been employed to compute the EL. In a similar way it is possible to calculate the ELoP. We use also in this case an analytical approximation that allows for a correlation between interest rates and the rational prepayment intensity. In the figure the ELoP is plotted for the 0 correlation case and for a negative correlation set at 0.8. This value implies that when interest rates decline, the default intensity increases. Since the loss for the bank is bigger when the rates are low, the ELoP in this case is higher than the uncorrelated case Expected loss ELoP Zero Corr ELoP Negative Corr

29 A Practical Example: Hedging the Prepayment Risk The hedging of the prepayment risk (or, of the TPC) is possible with respect to the interest rates and to the Libor forward volatilities. The TPC is 48 bps. The first table shows the sensitivity of the TPC to a tilt of 10 bps for each forward rate. Those sensitivities are then translated in an equivalent quantity of swaps, with an expiry form 1 year to 10 years, needed to hedge them. The second table shows the Vega of the TPC with respect to the volatilities of each Libor forward rate. Those exposures can hedged with caps&floors, or swaptions in the Libor Market Model setting we are working in (by calibrating the Libor correlation matrix to the swaptions volatility surface). Yrs Sensitivity Hedge Qty Yrs Vega

30 Liquidity Management We have shown how the framework can be used to measure and hedge the prepayment risk in its form of replacement cost. The model can be also use to project expected cash flows due to the prepayment activity. Since the rational prepayment is stochastic and correlated to the level of the interest rate, a VaR-like approach can be adopted also in this case to calculate the maximum and minimum amount of cash flows; Expected Cash Expected Amort. Cash Flows Flows Amort Amort. Exp. Amort

31 Index Balance Sheet Items Requiring Statistic-Financial Models 1 Balance Sheet Items Requiring Statistic-Financial Models 2

32 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.

33 The model The bank has a portfolio of m different credit lines, each one with a given expiry T i (i = 1, 2,..., m). Each credit line can be drawn within the limit L i 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 L i 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 λ i (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).

34 Stochastic Withdrawal Intensity 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 L i 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.

35 Stochastic Intensity The stochastic withdrawal intensity λ i (t), for the i-th debtor, is a combination of three terms: λ i (t) = α i (t)( i(t) + λ D i (t)) A multiplicative factor α i (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. A multiplicative factor α i (t) of the stochastic default intensity λ D i (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: λ D i (t) = λ I i (t) + p i λ C (t)

36 0 Balance Sheet Items Requiring Statistic-Financial Models An Example: Usage Distribution of a Single Credit Line The default intensity parameters are chosen so that the debtor s probability of default is about 2% in 1 year, and it declines on average in the future toward a long term average of 1.5%. The deterministic intensity is constant and equal to 2%. The multiplying factor is also constant and equal to α = Given this, we have an average withdrawal intensity at time 0 equal to 1000 (2% + 2%) = 40 or 40% of the total amount of the line (since each jump corresponds to 1% of usage). The credit line nominal is equal to 5 mln, the expiry is in 1 year , , 0 0 0, , 5 0 0, , 0 0 0, , 5 0 0, , 0 0 0, , 5 0 0, , 0 0 0, , 5 0 0, , 0 0 0, The figure shows the usage distribution of the credit line over a period of 1 year. The total amount of the credit line is Euro 5 mln. The average usage is 2,058,433. We added the 99%-percentile of this distribution which has the following value: 4,050,000.

37 Joint Usage Distribution of a Several Credit Line In the present framework, the correlation amongst different debtors play a great role in determining the joint usage distribution. The p i 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.: p i s), the larger th expected usage and the 99%-percentile unexpected usage. Even if for different scenarios (created by different p i s), the expected usage is equal for the single credit lines, the expected joint usage and the percentile changes with the choice of p i.

38 0 Balance Sheet Items Requiring Statistic-Financial Models An Example: Usage Distribution of Several We choose 3 credit lines, 5 mln, each and two opposite scenarios: First Scenario: we choose p i = and calibrate and the parameters of the default intensity so as to have an expected usage of 50%: Second Scenario: we choose p i = and calibrate again α and the parameters of the default intensity so as to have an expected usage of 50%: The default intensity starts at 2% for each debtor in both scenarios, but the correlation amongst the probability of defaults and of default events is very different: almost perfect in the first case and almost nil in the second , 0 0 0, , 0 0 0, , 0 0 0, , 0 0 0, , 0 0 0, , 0 0 0, , 0 0 0, , 0 0 0, The figure shows the joint usage distribution, and the 99% precentile, for first scenario (in red) and for the second scenario (in blue). In both scenarios the expected usage is almost 7.5 mln. The 99%-percentile usage is 14, 750, 000 in the first scenario and 9, 150, 000 in the second scenario.

39 Index Balance Sheet Items Requiring Statistic-Financial Models 1 Balance Sheet Items Requiring Statistic-Financial Models 2

40 Index Balance Sheet Items Requiring Statistic-Financial Models 1 Balance Sheet Items Requiring Statistic-Financial Models 2

41 Credit Mitigation Agreements 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).

42 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.

43 Index Balance Sheet Items Requiring Statistic-Financial Models 1 Balance Sheet Items Requiring Statistic-Financial Models 2

44 Pricing OTC Derivatives with CSA In a very general fashion, the price at time 0 of a derivatives contract which is not subject to counterparty risk is: V 0 = E Q [ e T 0 r sds V T ] where V T is the terminal pay-off of the contract; r t 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): V CCP 0 = E Q [ e T 0 r sds V T ] CVA + DVA The terminal value of the contract is still discounted ad the risk-free rate r t, but then the price is adjusted with the net effect due to the losses upon default of the two counterparties involved in the trade.

45 Pricing OTC Derivatives with CSA 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 c t 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 c t replacing the risk-free rate r t.: V CSA 0 = E Q [ e T 0 c sds V T ] (3)

46 Pricing OTC Derivatives with CSA 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.

47 Pricing OTC Derivatives with CSA Assume we have a contract whose value during the life of the contract at any time 0 < t < T, V t can be positive or negative. We also assume that the bank can invest cash at a risk-free rate equal to the collateral rate r t = c t, but it has a funding spread f t when borrowing money over a short period, so that the total funding cost is r t + f t. 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: V CSA 0 = E Q [ e T 0 c u f u1 {Vu<0} du V T ] = E Q [ e T 0 c udu V T ] + FVA (4) where FVA = E Q [ T 0 ] e s 0 c udu min(v s(0), 0)f sds (5)

48 A Practical Example We show an example, assuming the following market data for interest rates: Time Eonia Fwd Spread Fwd Libor % 0.65% 1.40% % 0.64% 1.39% % 0.64% 2.39% % 0.63% 2.63% % 0.63% 2.88% % 0.62% 2.99% % 0.61% 3.11% % 0.61% 3.26% % 0.60% 3.35% % 0.60% 3.47% % 0.59% 3.59% % 0.59% 3.69% % 0.58% 3.78% % 0.58% 3.88% % 0.57% 3.97% % 0.57% 4.07% % 0.56% 4.16% % 0.56% 4.23% % 0.55% 4.30% % 0.55% 4.37% % 0.54% 4.44% 5.00% 4.50% 4.00% 3.50% 3.00% 2.50% 2.00% 1.50% 1.00% 0.50% 0.00% Euribor Fw d Eonia Fw d

49 A Practical Example Market data for caps&floors and swaptions volatilities are: Caps&Floors Expiry Volatility % % % % % % % % % % % % % % % % % % % % Swaptions Expiry Tenor Volatility % % % % % % % % % % % % % % % % % % % 10 0

50 A Practical Example: Collateralized Swap We price under a CSA agreement a receiver swap whereby we we pay the Libor fixing semi-annually (set at the previous payment date) and we receive the fixed rate annually. With market data shown above, the fair rate can be easily calculated (we are using the new market standard approach to employ the EONIA/OIS curve for discounting and the 6M Libor curve to project forward rates). We assume also that we have to pay a funding spread of 15bps over the EONIA/OIS curve. This is applied to the ENE plotted beside. FVA % Fair Swap rate % Swap Rate + Coll. Fund % Difference % - (1.0000) (2.0000) (3.0000) (4.0000) (5.0000) (6.0000) (7.0000) ENE

51 A Practical Example: Collateralized Swap We may be interested in calculating the impact of the liquidity of a collateralized swap with respect to a more conservative measure than the ENE, similarly to what happens in the counterparty risk management. We choose the Potential Future Exposure, which is the expected negative NPV of the swap at a given level of confidence, set in this example at the 99% and computed with market volatilities. The funding spread is still 15bps over the EONIA/OIS curve. The Potential Future Exposure (blue line), and the ENE (purple line, same as before) for comparison, are plotted beside. FVA % Fair Swap rate % Swap Rate + Coll. Fund % Difference % - (5.0000) ( ) ( ) ( ) ( ) ( ) PFE ENE

52 Index Balance Sheet Items Requiring Statistic-Financial Models 1 Balance Sheet Items Requiring Statistic-Financial Models 2

53 Bonds as Collateral 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.

54 A Framework to Set Haircuts 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 T b with price B(t, T b ) 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 T a For the collateral pledged, there exists a haircut h. Let us divide the interval [t, T a] in m periods of equal extension. Assume now that our counterparty goes defaulted in t m, 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(t m) = E α[b(t m, T b )1 τb >t m + R B 1 tm 1 τ B t m ] where τ B is the default time of the collateral bond and R B is its recovery on default. We also assume that the default of the counterparty is independent from the default of the bond s issuer.

55 A Framework to Set Haircuts 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 = κ r(θ r r)dt + σ r rdzt and the PD is produced by a stochastic intensity still with a CIR dynamics: dλ = κ λ (θ λ λ)dt + σ λ λdwt At the end of each period t m 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 t m. In case the collateral bond is in default, the recovery value must be considered. The collateral value in t m is then the expected value in the two states of the world.

56 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, T a], we will weight the possible values of the collateral bonds at the several times t i by the probability of default between [t m 1, t m] over the total default probability over the entire contracts period: w m = PD C(t m 1, t m) PD C (t, T a) The expected maximum loss over [t, T a] is m ELGD(t m) = E α[b(t i, T b )(1 PD B (t, t i )) i=1 + R B (PD B (t, t i 1 ) PD B (t, t i ))]w m

57 A Practical Example Assume we have an exposer E originated by a contract expiring in T a = 6M and the bank requires as collateral a bond expiring in T b = 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 intensity processes with the following parameters: r % κ r 0.75 θ r 4.50% σ r 25.00% λ % κ λ 0.75 θ λ 1.00% σ λ 25.00% Years 1Y Libor PD % 0.65% % 1.47% % 2.38% % 3.33% % 4.31% % 5.30% % 6.30% % 7.31% % 8.34% % 9.37%

58 A Practical Example The collateral bond has the characteristics shown in the table below so that its market price at time t 0 is We divide the contact period of 6 months in 6 monthly intervals and compute the minimum value of the bond at the end of each period, considering also the occurrence of default and the recovery. We weight these values as described above so that we derive the quantity α capable to match the exposure, and then we can set the haircut on the market value of the bond. The table beside shows all calculations. Face Value 100 Expiry 10 Coupon 4.50% Frequency 1 Recovery 40% PD Bond Month Cpt Price Recov % % % % % % % Min Coll W ed Coll Month Value Weight Value % % % % % % Av. Coll. Value α 1.05 Fair Haircut 5.66%

59 About Iason Iason is a company created by market practitioners, financial quants and programmers with valuable experience achieved in dealing rooms of financial institutions. Iason offers a unique blend of skills and expertise in the understanding of financial markets, in the pricing of complex financial instruments and in the measuring and the management of banking risks. The company s structure is very flexible and grants a fully bespoke service to our Clients. Iason believes that the ability to develop new quantitative finance approaches through research as well as to apply those approaches in practice, is critical to innovation in risk management and derivatives pricing. It brings into all the areas of the risk management a new and fresh approach based on the balance between rigour and efficiency Iason s people aimed at when working in the dealing rooms. Besides tailor made services, Iason offers software applications to calculate and monitor credit VaR and conterparty VaR, fund transfer pricing and loan pricing, liquidity-at-risk. c Iason This is a Iason s creation. The ideas and the model frameworks described in this presentation are the fruit of the intellectual efforts and of the skills of the people working in Iason. You may not reproduce or transmit any part of this document in any form or by any means, electronic or mechanical, including photocopying and recording, for any purpose without the express written permission of Iason ltd.