Discounting Jeroen Kerkhof 22 September 2010 c Copyright VAR Strategies BVBA 1 / 53
Overview c Copyright VAR Strategies BVBA 2 / 53
Time Value of Money c Copyright VAR Strategies BVBA 3 / 53
Time Value of Money Time Value of Money We all know that a euro today is worth more than a euro tomorrow, but how much... This depends on Type of investment opportunities, e.g. In order to uniquely define the value of a euro in the future a benchmark investment is needed. Finance textbooks introduce the so-called risk free rate on a bank account Not that clearly defined in practice... c Copyright VAR Strategies BVBA 4 / 53
Time Value of Money Time Value of Money Not everyone has the same investment opportunities... For derivatives valuation risk-free is not clearly defined Used to assumed to be Libor Nowadays Overnight Index more relevant or maybe repo? valuation should not use an assumption for the risk-free rate It should use the terms in the bilateral contracts (CSA) c Copyright VAR Strategies BVBA 5 / 53
Interest-rate Swaps The Old Days Valuation Interest-rate Swaps Curve Construction Recipe c Copyright VAR Strategies BVBA 6 / 53
Interest-rate Swaps Interest-rate Swaps The Old Days Valuation Interest-rate Swaps Curve Construction Recipe Standard Fixed-Floating interest rate swap: 6M Libor 03/06/11 03/06/12 03/06/13 03/06/14 03/06/15 K K K K K Leg Day count basis Date rule Frequency Fixed 30/360 Mod. Following Annual Floating Act/360 Mod. Following Semi-Annual c Copyright VAR Strategies BVBA 7 / 53
The Old Days Interest-rate Swaps The Old Days Valuation Interest-rate Swaps Curve Construction Recipe Use Libor fixings (up to 1 year) Futures (1 year up to 3 years) Fixed-Floating (6M) Swaps (the long end of the curve) Derive the swap curve (bootstrapping or solver) See, for instance, your favorite edition of Hull: Options, Futures and Other The curve will provide D(t,T) for all T (up to end of curve) t = today, T = maturity date and D(t,T) represents todays value of one euro receivable at T c Copyright VAR Strategies BVBA 8 / 53
Valuation Interest-rate Swaps Interest-rate Swaps The Old Days Valuation Interest-rate Swaps Curve Construction Recipe Value other linear interest rate derivatives using this curve For instance, suppose we need to value the following swap 3M Libor 03/06/11 03/06/12 03/06/13 03/06/14 03/06/15 K K K K K Use the swap curve to find F 3M (t,t i ) = 1 δ i ( D(t,Ti ) D(t,T i+1 ) 1 ) where T i+1 = T i +3M Determine K c Copyright VAR Strategies BVBA 9 / 53
Valuation Interest-rate Swaps Interest-rate Swaps The Old Days Valuation Interest-rate Swaps Curve Construction Recipe The value of a Libor payment equals δ i D(t,T i +3M) F 3M (t,t i ) = δ i D(t,T i +3M) 1 δ i ( D(t,Ti ) D(t,T i +3M) 1 ) = D(t,T i ) D(t,T i +3M) Iteration gives the value of the floating leg equals For the 6m index we had something similar As T n = T 2k the values of the 2 floating legs are the same Hence fixed rate should be the same c Copyright VAR Strategies BVBA 10 / 53
Old days are gone... Let s look at the market for 3M-6M tenor basis swaps Interest-rate Swaps The Old Days Valuation Interest-rate Swaps Curve Construction Recipe c Copyright VAR Strategies BVBA 11 / 53
History Libor 3M - OIS spread: Euro Interest-rate Swaps The Old Days Valuation Interest-rate Swaps Curve Construction Recipe c Copyright VAR Strategies BVBA 12 / 53
History Libor 3M - OIS spread: US Interest-rate Swaps The Old Days Valuation Interest-rate Swaps Curve Construction Recipe c Copyright VAR Strategies BVBA 13 / 53
Market Data Interest-rate Swaps The Old Days Valuation Interest-rate Swaps Curve Construction Recipe These days the market trades OIS vs fixed / floating Tenor basis swaps (e.g. 3M vs 6M) Cross-currency basis swaps OIS swaps on ECB dates (Euro), MPC dates (UK) fixed-floating swaps We should use this information to build our interest rate curve... c Copyright VAR Strategies BVBA 14 / 53
Overview Interest-rate Swaps The Old Days Valuation Interest-rate Swaps Curve Construction Recipe c Copyright VAR Strategies BVBA 15 / 53
Eonia Swap data Interest-rate Swaps The Old Days Valuation Interest-rate Swaps Curve Construction Recipe c Copyright VAR Strategies BVBA 16 / 53
Eonia Swap data Interest-rate Swaps The Old Days Valuation Interest-rate Swaps Curve Construction Recipe c Copyright VAR Strategies BVBA 17 / 53
Eonia Swap data Interest-rate Swaps The Old Days Valuation Interest-rate Swaps Curve Construction Recipe c Copyright VAR Strategies BVBA 18 / 53
Fixed-Floating Swaps Interest-rate Swaps The Old Days Valuation Interest-rate Swaps Curve Construction Recipe c Copyright VAR Strategies BVBA 19 / 53
Tenor basis swaps Interest-rate Swaps The Old Days Valuation Interest-rate Swaps Curve Construction Recipe c Copyright VAR Strategies BVBA 20 / 53
Interest-rate Swaps 3M vs fixed Interest-rate Swaps The Old Days Valuation Interest-rate Swaps Curve Construction Recipe c Copyright VAR Strategies BVBA 21 / 53
Interest-rate Swap Interest-rate Swaps The Old Days Valuation Interest-rate Swaps Curve Construction Recipe Rather than having just D(t,T) for all T we will have D(t,T) for all T D 1M (t,t) for all T D 3M (t,t) for all T D 6M (t,t) for all T D 12M (t,t) for all T The value of e.g. the 12M index will be F 12m (t,t i ) = δ i ( D12m (t,t i ) D 12m (t,t i+1 ) 1 ) Its value today is given by D(t,T i +12m) F 12 (t,t i ) = D(t,T i +12m) δ i ( D12m (t,t i ) D 12m (t,t i+1 ) 1 ) What is appropriate for D(t,T) c Copyright VAR Strategies BVBA 22 / 53
Curve Construction Recipe Interest-rate Swaps The Old Days Valuation Interest-rate Swaps Curve Construction Recipe In order to construct discount curves we follow the following steps: Select relevant instruments Get the relevant market data (e.g. ICAE) Select an interpolating scheme for D(t,T),D 1M (t,t),d 3M (t,t),d 6M (t,t),d 12M (t,t) Use a solver for D(t,T),D 1M (t,t),d 3M (t,t),d 6M (t,t),d 12M (t,t) c Copyright VAR Strategies BVBA 23 / 53
Curve Construction Recipe Interest-rate Swaps The Old Days Valuation Interest-rate Swaps Curve Construction Recipe Suppose we have fixed rate OIS swaps out to end of the curve Then we can iteratively determine D(t,T) for all T on which there is a fixed cash flow. Using the interpolating scheme we have modelled D(t, T) for all T Using these discount factors we can determine value of the fixed legs of fixed-floating (6M Euribor) swaps. By iteration and the fact that swaps are 0-NPV at inception we find the value of the D 6M (t,t) for all T on which there is a cash flow. Using the interpolating scheme we have modelled D 6M (t,t) for all T c Copyright VAR Strategies BVBA 24 / 53
Curve Construction Recipe Interest-rate Swaps The Old Days Valuation Interest-rate Swaps Curve Construction Recipe Armed with D(t,T) and D 6M (t,t) we value the 6M floating index leg of 3M-6M tenor basis swaps. We can bootstrap D 3M (t,t) using an iterative procedure. Again interpolation does the rest. Continue with 1M-3M basis swaps and 6M-12M basis swaps This finally gives D(t,T),D 1M (t,t),d 3M (t,t),d 6M (t,t),d 12M (t,t) for all T Unfortunately, the market does not quote fixed OIS swaps beyond 2y, but OIS-3M Euribor swaps As such there is a circular dependency between OIS, 3M, and 6M index Therefore, a solver is to be preferred c Copyright VAR Strategies BVBA 25 / 53
Curve Construction Recipe Interest-rate Swaps The Old Days Valuation Interest-rate Swaps Curve Construction Recipe We know how to build a curves in line with market instruments How do we discount client cash-flows? Uncollateralised Collateralised (different types) c Copyright VAR Strategies BVBA 26 / 53
CVA: Credit Valuation Adjustment DVA: Debt Valuation Adjustment Funding Wrong-way risk Netting c Copyright VAR Strategies BVBA 27 / 53
CVA: Credit Valuation Adjustment CVA: Credit Valuation Adjustment DVA: Debt Valuation Adjustment Funding Wrong-way risk Netting For quite some time (some) banks realized that Not all counterparties are of Libor -credit quality CVA adjustments are needed to reflect credit quality of the counterparty Typically, CVA adjustments were unilateral (only default risk of the client was considered) CVA desks (if existent) were not very sophisticated c Copyright VAR Strategies BVBA 28 / 53
CVA: Credit Valuation Adjustment CVA: Credit Valuation Adjustment DVA: Debt Valuation Adjustment Funding Wrong-way risk Netting Nowadays, Every (self-respecting) bank has a CVA desk however, sophistication levels vary widely most banks are looking at DVA as well What is DVA? c Copyright VAR Strategies BVBA 29 / 53
DVA: Debt Valuation Adjustment CVA: Credit Valuation Adjustment DVA: Debt Valuation Adjustment Funding Wrong-way risk Netting Not only the client can default Own default probability should be a factor in pricing This leads to so-called bilateral CVA What happens if your own credit rating deteriorates? Which banks can be most aggressive in pricing? c Copyright VAR Strategies BVBA 30 / 53
DVA: How to monetize CVA: Credit Valuation Adjustment DVA: Debt Valuation Adjustment Funding Wrong-way risk Netting My credit rating deteriorates, I am making money! Really? How to lock-in this profit? Sell protection on yourself? Ideally, yes, but practically no (more later) c Copyright VAR Strategies BVBA 31 / 53
Funding CVA: Credit Valuation Adjustment DVA: Debt Valuation Adjustment Funding Wrong-way risk Netting Often ignored (for valuation purposes) Critical component of derivatives business What does it cost to run a derivatives business? How should this be reflected in derivatives prices c Copyright VAR Strategies BVBA 32 / 53
CVA/DVA/Funding: Example CVA: Credit Valuation Adjustment DVA: Debt Valuation Adjustment Funding Wrong-way risk Netting How does it work? Assume party A (that s us) extends a loan to party B payable at time T with a value X Risks party B can default. How to account for this? use party B s CDS spread Funding costs loan needs to be funded. At which rate? use party A s unsecured bond spread Note: party A s unsecured bond spread can be different from it s CDS s spread c Copyright VAR Strategies BVBA 33 / 53
CVA/DVA/Funding: Example CVA: Credit Valuation Adjustment DVA: Debt Valuation Adjustment Funding Wrong-way risk Netting D(t,T) = e (r+r cds,b r ubs,a )(T t) where r is the reference discount rate r cds,b denotes the cds spread for company B r ubs,a denotes the unsecured bond spread for company A (1) c Copyright VAR Strategies BVBA 34 / 53
What with derivatives? CVA: Credit Valuation Adjustment DVA: Debt Valuation Adjustment Funding Wrong-way risk Netting What do loans have to do with derivatives? Trade in-the-money = loan to counterparty Most trades start ATM. Need a model for future values of derivative Typical CVA formula (no wrong-way risk) T CVA = D(0, t)s(t)ep E(t)dt (2) 0 where D(0, t) denotes the relevant risk-free discount factor for t s(t) denotes the relevant credit spread at t EP E(t) denotes the expected positive exposure of the trade at time t c Copyright VAR Strategies BVBA 35 / 53
Wrong-way risk CVA: Credit Valuation Adjustment DVA: Debt Valuation Adjustment Funding Wrong-way risk Netting Can we separate derivatives valuation and credit charge? I.e. Compute exposures Multiply by survival probabilities Multiply by discount rate Unfortunately, not payoff/exposure can be (typically is) correlated to survival probability Possible big jumps in case of default liquidity issues c Copyright VAR Strategies BVBA 36 / 53
Netting CVA: Credit Valuation Adjustment DVA: Debt Valuation Adjustment Funding Wrong-way risk Netting A way to mitigate counterparty risk is netting over all trades with a particular counterparty This introduces Some non-linearities Hence, simple CVA formula no longer holds c Copyright VAR Strategies BVBA 37 / 53
DVA: How to monetize? CVA: Credit Valuation Adjustment DVA: Debt Valuation Adjustment Funding Wrong-way risk Netting Back to DVA. We can t sell protection on ourselves, so... Sell protection on (group of) competitors Obviously, less than perfect correlation Who is going to buy my protection? If my competitor goes under, I am likely in trouble as well c Copyright VAR Strategies BVBA 38 / 53
CSA Agreement Reference in Europe: Eonia R and Euribor R Funding Collateralised c Copyright VAR Strategies BVBA 39 / 53
OTC CSA Agreement Reference in Europe: Eonia R and Euribor R Funding Collateralised Derivative contracts are typically privately negotiated contracts (e.g. no exchange in between) between 2 counterparties. In order to help standardize these contracts ISDA (International Swaps and Association) has set-up standard master agreements. Before two counterparties enter into OTC transactions they typically set-up a so-called CSA (Credit Support Annex) The CSA serves to mitigate counterparty credit exposure It specifies the collateral posting procedure in case the trade moves away from the money c Copyright VAR Strategies BVBA 40 / 53
OTC in General CSA Agreement Reference in Europe: Eonia R and Euribor R Funding Collateralised Big picture: CP that is out-of-the money (OTM) posts collateral equal to the value of the swap. CP that is in-the-money (ITM) pays interest on the received collateral. (Some) details: Which collateral is posted? Cash (which currency?) Bonds (which bonds?) CDOs etc How frequent? Which interest rate? Minimum adjustment of position c Copyright VAR Strategies BVBA 41 / 53
CSA Agreement CSA Agreement Reference in Europe: Eonia R and Euribor R Funding Collateralised What is standard? Practically all banks will trade with one another via the London Clearing House (LCH, www.lchclearnet.com). The LCH CSA says the counterparties need to post cash in the local currency and will receive the overnight index (e.g. EONIA for Euro and SONIA for Sterling) Is this the CSA that Banks have with their clients as well? Typically no c Copyright VAR Strategies BVBA 42 / 53
Reference in Europe: Eonia R and Euribor R CSA Agreement Reference in Europe: Eonia R and Euribor R Funding Collateralised Reference rates within the Eurozone www.euribor.org Euribor R (Euro Interbank Offered Rate) is the rate at which euro interbank term deposits within the euro zone are offered by one prime bank to another prime bank. Eonia R (Euro OverNight Index Average) is an effective overnight rate computed as a weighted average of all overnight unsecured lending transactions in the interbank market, initiated within the euro area by the contributing panel banks. http://www.euribor.org/html/content/panelbanks.html c Copyright VAR Strategies BVBA 43 / 53
Eonia R CSA Agreement Reference in Europe: Eonia R and Euribor R Funding Collateralised Daycount convention: Act / 360 3 decimals Overnight means from one TARGET day (i.e. day on which the Trans-European Automated Real-Time Gross-Settlement Express Transfer system is open) to the next TARGET day total volume of all unsecured overnight lending transactions that day and the weighted average lending rate for these transactions Eonia be published between 6.45 p.m. and 7.00 p.m. (CET) on the same evening the Eonia will be computed as a weighted average of all (without exceptions) overnight unsecured lending transactions in the interbank market c Copyright VAR Strategies BVBA 44 / 53
Euribor R CSA Agreement Reference in Europe: Eonia R and Euribor R Funding Collateralised Euribor R (Euro Interbank Offered Rate) is the rate at which euro interbank term deposits are offered by one prime bank to another prime bank and is published at 11.00 a.m. CET for spot value (T+2). Euribor R is quoted for spot value (T+2) and on an Act/360 day-count convention. It is displayed to three decimal places. Panel Banks contribute for one, two and three weeks and for twelve maturities from one to twelve months Reuters shall, for each maturity, eliminate the highest and lowest 15% of all the quotes collected. The remaining rates will be averaged and rounded to three decimal places. c Copyright VAR Strategies BVBA 45 / 53
Multi-currency CSA CSA Agreement Reference in Europe: Eonia R and Euribor R Funding Collateralised What if the counterparty is allowed to post other currencies? Say $ on Euro swaps In order to properly value the euro swap we need to know the cost of it in dollars As such we need to translate the cost of Euro cash flows into dollars Use cross-currency basis swaps c Copyright VAR Strategies BVBA 46 / 53
Small CSA Test CSA Agreement Reference in Europe: Eonia R and Euribor R Funding Collateralised Lets do some qualitative tests on CSA conditions You have a euro swap which is massively in the money Currently you have a LCH type CSA with the bank What happens to the value of your position if we change The frequency to 1m and the index to 1M Euribor? The minimum increments from 0 to 1mln? Allow the option that either EUR or GBP cash is posted? Allow the option that subprime CLOs are posted? c Copyright VAR Strategies BVBA 47 / 53
Valuation Collateralised CSA Agreement Reference in Europe: Eonia R and Euribor R Funding Collateralised Value of collateral during the trade C(t) = [V(t) T cpty ]I(V(t) T c MT c ) [ V(t) T b ]I(V(t) T b MT b ) = (V(t) T c MT c ) + +MT c I(V(t) T c MT c ) ( V(t) T b MT b ) + +MT b I( V(t) T b MT b ),(3) where V(t) denotes the value of the derivative MT x denotes the Minimum Transfer amount for bank, counterparty T x denotes the Threshold for bank, counterparty c Copyright VAR Strategies BVBA 48 / 53
Valuation Collateralised CSA Agreement Reference in Europe: Eonia R and Euribor R Funding Collateralised The value of a collateralised derivatives therefore depends on dynamics of V(t) MT x denotes the Minimum Transfer amount for bank, counterparty T x denotes the Threshold for bank, counterparty c Copyright VAR Strategies BVBA 49 / 53
Funding Collateralised CSA Agreement Reference in Europe: Eonia R and Euribor R Funding Collateralised If your credit rating is low, collateral is expensive asset in-the-money you receive collateral asset out-of-the-money you pay collateral and receive OIS reward however, it costs (much) more to fund the collateral payment! Hence, not a symmetric trade volatile assets require higher credit charge c Copyright VAR Strategies BVBA 50 / 53
Linear Discounting CSA Agreement Reference in Europe: Eonia R and Euribor R Funding Collateralised Proper discounting requires serious modelling time of bootstrapping and rootsolving is over proper valuation requires dynamics modelling and treatment of non-linearities c Copyright VAR Strategies BVBA 51 / 53
c Copyright VAR Strategies BVBA 52 / 53
Clearly, this has an effect on Non-linear derivatives as well Nowadays, swaption premiums are quoted without discount factor e.g. V(t) D(t,T) = IET [(S(T) K) + g(s(t))] (4) where g(s(t)) denotes the IRR-settled PV01 at maturity and S(T) the swap rate at maturity. c Copyright VAR Strategies BVBA 53 / 53