Libor Rates and the Credit Crunch

Transcription

1 Libor Rates and the Credit Crunch

2 Libor Rates in the pre Credit Crunch world Quick Recap Libor: an interest rate at which banks are willing to lend/borrow money without receiving/posting any kind of collateral Hypothetical Market: Libor 6m: 4% - Libor 1y: 5% Bank blue does the following: 4% 5% - Borrow money from green bank for 6 months at a 4% rate - Lend money to purple bank for 1 year at a 5%

3 Libor Rates in the pre Credit Crunch world Quick Recap (2) Resulting Cash Flows for blue bank t = 0 No cash flows t = 6m Outgoing cash flow (negative sign) - N(1 + 4%* 0.5) = - N(1 + 2%) t = 1y Incoming cash flow + N(1 + 5%)

4 Libor Rates in the pre Credit Crunch world Quick Recap (3) FinalCapital Initial Capital t Linear yield of the operation: 1/(T2-T1)* (Final Capital / Initial Capital 1) N(1 + 5%) / N(1 + 2%) = 5.882% This is the Forward Libor Rate: a rate implied by Libor Rates which is known today (t=0) A deposit between two panel banks that will start in the future, but it's traded today, will have a fixed interest rate equal to the foward libor rate This is also the fair strike of a FRA: a contract in which the future unkown libor rate will be exchanged for a fixed rate Fair strike of a fra/swap: the strike for which no premium is exchanged

5 Discounting in the pre Credit Crunch world Unsecured intebank deposits are perceived as safe and liquid, therefore they are taken as a reference for the construction of a discountig curve Given a discount curve (and an interpolation method) I can obviously determine all of the market rates: libor rates, forward libor rates, swap rates.

6 Discounting in the pre Credit Crunch world (2) To determine this discounting curve we boostrap from the most liquid instruments Short term deposits Futures (equivalent to FRA on the 3M Libor) Swaps on Libor 6m

7 Basis Swaps - Pre Credit Crunch A swaps in which the two parties exchange floating rates (libor rates) with a different tenor. Eventually the party paying the rate with the shorter tenor pays an extra margin Libor 3m + m Green bank: - pays quarterly the Libor 3m plus a margin - receives semi-annually the Libor 6M Libor 6m Which is the fair margin m? It is the margin for which the basis swap there is no premium exchange at trade inception The two legs must have the same value

8 Basis Swaps - Pre Credit Crunch (2) Value of the receiver semi annual leg Value of the payer leg (in absolute valute) The fair marging must be zero if we want that present value of the two legs match

9 Basis Swaps - Credit Crunch Credit Crunch and the financial crisis

10 Credit Crunch and Libor Rates What happend to the relationship between Libor rates and Forward Rates? 10/02/2009 Libor 3m = 1.989% Libor 6m = 2.069% Model Forward = 2.138% FRA3x6 Strike = 1.675% The forward rate implied from the formula was no longer the fair strike of a FRA. All in a sudden banking systems were not able to price the trades accordingly to the market Who is right? If the math is still correct, are the opportunities to make money? Unfortunately no

11 Credit Crunch and Libor Rates (2) In the derivation of the first formula, some assumptions were implicitely made 1. the credit risk for the panel banks is almost negligible 2. infinite liquidity: for panel banks there s no problem to borrow/lend money in any size 3. The panel banks are stable: a small probability that a panel bank credit will deteriorate and no longer be part of the panel group

12 Practitioner solution Problem Forward libor with different tenors cannot be longer calculated from one single discounting curve Solution What if I build several discounting curves, one for each tenor? Forward Libor Rate 3m Forward Libor Rate 6m

13 Practitioner solution (2) Before credit crunch: After credit crunch: Discounting Curve Discounting Curve 1m Discounting Curve 3m Discounting Curve 6m Discounting Curve 1y Forward Libor Curve 1m Forward Libor Curve 3m Forward Libor Curve 6m Forward Libor Curve 1y Forward Libor Curve 1m Forward Libor Curve 3m Forward Libor Curve 9m Forward Libor Curve 1y

14 Boostrapping a forecast curve These new curves are commonly called Forecast Curves because their purpose is to determine the forward libor rate The problem of discounting a future cash flow is postponed. Each of them is build with the aid of a boostrapping algorithm based only on a set of homogeneus derivatives. Homogeneous with respect to the underlying libor rate Libor 6M - Deposit 6M - FRA 1X7, 2X8, etc.. - Swaps vs 6m (2y,, 30y) Libor 3M - Deposit 3M - FRA 1X4, 2X5, etc.. - Futures - Swaps vs 3m (2y,, 30y)

15 Boostrapping a forecast curve (2) These curves work well when pricing "new" contracts, i.e. fra or swaps (even if non "standard") whose value at trade inception is negligible Why? Because these contracts have a little sensitivity to the discouting. Consider the NPV of a FRA traded at par If the forward rate implied by my curves is correct, in the sense that is close to the strike, that difference is zero and the discounting doesn t play any role. But for all of the other products? Which of these curves is the "right" discounting curve?

16 Discounting and Collateral

17 Deposits vs Collateralized Derivatives To understand which is the right discounting curve we have to better understand which are the characteristics of the interest rate instruments quoted in the market. Deposits FRA Swaps Futures These are unsecured instruments: They are correctly discounted on the forecast curve, which embeds the credit risk of the panel banks for a given maturity These instruments are collateralized: in the discounting of their flows it should be taken into account that the credit risk is mitigated by the collateral

18 Credit Support Annex Amongst market partecipants, nowadays it is common to trade derivatives with collateral agreements whose characteristics are collected in a document called Credit Support Annex (optional document in an ISDA Master Agreement) It states: - which collateral can be posted (currency, securities) - independent amount, minimum transfer amount - margin call frequency - interest rates payable on the collateral

19 Credit Support Annex (2) For a better understanding on how the collateral impact on the discounting of a derivative we'll assume an hypothetical CSA holds: - Deliverable collateral: cash (EUR) - No independent amount, minimum transfer equal to zero - Margining frequency: daily - payable interest rate: EONIA This is an overnight interest rate, somehow similar to the Euribor, but it s not an offered rate: it is the average rate of real transactions

20 Collateral Mechanism A time t = 0 a collateralized derivative has been traded between two banks at par: No Cashflows Derivative for G.B. = 0 Derivative for P.B. = 0 No Collateral

21 Collateral Mechanism (1) During the life of the derivative "market moves": suppose the derivative has a positive value for Green Bank Purple Bank post Eur into the collateral account Green bank, the day after, pays interest on collateral Derivative for G.B. > 0 Derivative for P.B. < 0 Collateral Interest on collateral Collateral

22 Collateral Mechanism (2) When the derivative expires: Purple Bank gets collateral back and pays the amount due to Green Bank Derivative for G.B. > 0 Derivative for P.B. < 0 Collateral

23 Collateral Mechanism (3) If Purple Bank defaults before the derivative expires: Green Bank gets the collateral Derivative for G.B. > 0 Derivative for P.B. < 0 Collateral

24 Collateral Account and Derivative Market Value Assume that some collateral has been posted at time t, and between t and t + 1 the market doesnt move Deal collateralizzato t Value (t) Cash = Value(t ) Deal collateralizzato t+1 Value (t+1)??? Cash = V(t) * (1 + c/360)

25 Collateral Account and Derivative Market Value (2) Deal collateralizzato Market value (t +1) = MV(t) * (1 + c/360) Cash = MV(t) * (1 + c/360) If the market doesn t move, the value of the derivative must still match the collateral account! MV(t) = MV(t+ 1) / (1 + c/360) In the valaution of a collateralized derivative, the discounting curve to use is the collateral yield curve

26 OIS In the Euro area, the usual rate to be paid is the EONIA. As for the Libor rates, in the market there are traded collateralized swaps whose underlying is the Eonia rate: The Overnight Index Swap (OIS). At maturity Annually

27 Value of an OIS In the Eonia/OIS world, the "old" single curve derivations still appy: The market quotes the fair strike of Overnight Index Swaps for maturities up to 30y. It s easy to verify that for these swaps the par swap rate is still given by:

28 Bootstapping OIS curve The boostrapping procedures is still the same Eonia rate Tom/Next (1d FRA) OIS

29 Multi Curve Boostrap Having identified the correct discounting curve, we have to tackle again the issue of bootstrapping the forecast curves. This is because we finally know the correct formulas to derive the fair strikes (i.e. the market quotes). Example: Libor 6M So, the idea is: - First bootstrap the discounting curve P(t,T) - Then bootstrap the forecast curve with the new formulas (where the discounting is fixed), on a set of instruments homogeneus with respect to the tenor of the underlying libor rate

30 Multi Curve Bootstrap (2) Are forecasting curve still necessary? No. I can directly boostrap the forward curves. Easier to understand with an example. Market Instruments: Libor 6M, FRA 3X9, Swap 1y maturity Derivative maturity Market Quote Libor Rate 6M 9M 1Y Deposit FRA K

31 References Interesting papers on these topics M. Bianchetti, Two Curves, One Price: Pricing & Hedging Interest Rate Derivatives Decoupling Forwarding and Discounting Yield Curves M. Morini, Solving the Puzzle in the Interest Rate Market