Implications of Negative Swap Spreads - PDF document

Implications of Negative Swap SpreadsTi
Implications of Negative Swap SpreadsTianyu Gu, Xingjian Yu, Yisu ZhouIgnoring market frictions, a negative swap spread gives a theoretical arbitrage opportunity. We studied the possibilities of creating an arbitrage portfolio in the setting of the long existing negative swap spread between the 10-yr US treasury issued par coupon bonds and the 10-yr LIBOR-based interest rate swaps. However, since there should be no arbitrage opportunities in the nancial market, it is worthwile to inves-tigate the factors that obstruct the construction of the arbitrage portfolio and diculties in implementing the arbitrage. Some factors that we considered include posting collateral on entering the interest rate swaps, borrrowing at LIBID instead of LIBOR, and mismatch in payment times of dierent securities. We con-clude that these are the main reasons why the negative swap spread exists in the supposedly arbitrage-free market.Arbitrage Portfolio:Denition: An arbitrage portfolio is a portfolio that satises the following 3 conditions:1. No capital is put in the portfolio2. Pr(X0) = 13. Pr(X��0) 0Remark: In the real world nancial market, theoretically there should be no possibility of building an ar-bitrage portfolio, for the reason that if there exists such opportunities, the increase in demand would drive the arbitrage away.Interest Rate Swap:Denition: A (LIBOR-based) interest rate swap is a contract in which two parties agree to exchange cash ows based on interest rate, notional face amount and maturity. Forthe two parties that enter the swap con-tract, one party takes the “receiver” position that receives a xed payment from the counterparty twice a year, and pays the counterparty a oating payment based on the current LIBOR rate 4 times a year and re-ceives the xed payment twice a year.Swap Spread:Denition: e swap spread is equal to the coupon rate of the par coupon bond minus the xed rate of the interest rate swap with same maturity and payment times.Remark-came negative. Aer that, the 10-yr swap spread became negative as well. In the scope of our problem, we say there exists a negative swap spread for the US treasury 10-yr par coupon bond and the LIBOR based 10-yr interest rate swap.Let F denote the face value of a treasury issued par coupon bond. Let Xt denote the value of our portfolio, then the portfolio is dened as the following:At time t=0:• Long(purchase) a par coupon bond paying treasury with maturity T and face value F• Enter payer swap with maturity T where the xe payment pays 2 times a year and the oat payment pays 4 times per year. Assume the xed payment pays at rate qswap and the oating payment pays at LIBOR.• Short(sell) oat note with face value F and maturity T that pays at LIBOR 4 times a year.Note that X0=0 because the time 0 value of par coupon bond is F, time 0 value of short position on oat note is -F, and entering payer swap takes 0 initial capital.At time t=T, assuming that we deposit all the dierence in payments from coupon bonds and interest rate swap in the bank, the payout of this portfolio is,where qparswap is the xed rate of interest rate swap, and rt,T is the interest rate between time t and time T. is is an arbitrage portfolio because by the setting of our problem qpar�qswap. us we have Pr(X10�0)=1.e Need For Posting CollateralDenition: collateral is a property or other asset that a borrower oers as a way for a lender to secure the loan. Certain amount of collaterals must be posted in order to enter either position of the interest rate swap. In this case, we post k*F as collateral. e rate we borrow the amount k*F is treasury rate. However, the rate that we earn interest on the amount k*F is fed rate, which is lower than treasury rate, causing the decrease in value of our portfolio. Aer considering the need for posting collaterals, the payout of our portfolio becomese LIBOR/LIBID SpreadDenition: e London Interbank Oered Rate (LIBOR) is the rate at which banks can borrow unsecured funds Denition: e London Interbank Bid Rate (LIBID) is the rate that banks are willing to pay for unsecured funds from other banks in the London interbank market. In the real world, the payer swap actually pays our oating payment at LIBID rate, which is slightly lower than LIBOR rate. However, the 10-yr oat note that we short pays at LIBOR, casuing us lose F*(LIBOR-LIBID)/4 amount of money during each period. Aer considering the LIBOR/LIBID spread, the payout of our portfolio becomese Non-Matching Float Payment timesDenition: Repo rate is the rate at which one party lends money to another in the event of any shortfall of funds. In our portfolio, the repo rate is viewed as the short-term(day count) borrrowing rate. Payment of the payer swap and the oat note may not match exactly. If we need to pay for the short position of the oat note several days before we receive oat payment from payer swap, we need to rst borrow money at repo rate to pay the short position of the oat note. Aer considering the need for posting repo, the payout of our portfolio becomes the following equation, assuming  is the dierence of the non-matching payment times.Initiating the trade(zero-cost portfolio) Portfolio at the ende Cash Flow During the ProcessAssuming there is no arbitrage, we set the terminal capital X to 0, then compute the ratio of the face val-ue that we need to pay as collateral, i.e. k. Note that we can use the magnitude of k as an indicator of the chance of making an arbitrage. A larger k means we need to post more collaterals to initiate the portfolio, which in turn means our portolio is able to make more money since we need a larger k to cancel out the prot to keep our model arbitrage-freeIf we assume the fed rate to be 0.5%, the libor/libid spread to be 0.1%, swap spread to be 0.12%, and assume the term structure stays constant and look up the predicted future interest rates from Bloomberg terminal, then we are able to get the following results:• Considering only the need for posting collaterals: k = 6.0221%• Considering the need for posting collaterals and the LIBOR/LIBID spread: k = 0.9877%• Considering the need for posting collaterals, the LIBOR/LIBID spread and the non-matching oat pay-ments: k = 0.9876%Based on our result of calculation, we are able to put forward the following discussions:1. e change of swap spread will signicantly inuence the result of k. If the swap spread is less than 0.09%, then k becomes negative in the calculation, considering all the factors we discussed above.2. If we only consider the need for posting collateral, the calculation of k is 6.0221%, which is signicantly higher than the typical collateral. is means we can make prots in this setting. If we consider the LIBOR/LIBID spread, the calculation of k becomes 0.9877%, which is lower than the typical collateral. is means the LIBOR/LIBID spread decreases our prots signicantly. Aer considering the non-matching oat pay-ment times, there is only a minor change to the result of k, which means the non-matching oat payment times is not a crucial factor in reducing the arbitrage opportunity. 3. Based on the above calculation and discussion, we think the chance of making an arbitrage portfolio based on the negative swap spread is low since the factors we considered above reduce the prot we make from the negative swap spread signicantly.4. future, we will t the current term structure to a stochastic interest rate model(eg. Ho-Lee model) and run simulations. We appreciate the support from the Department of Mathematical Sciences.Special thanks to Dr. William Hrusa, who provided signicant amount of time and invaluble suggestions for our project. AbstractFactors Inuencing the Arbitrage PortfolioArbitrage Portfolio with Negative Swap Spread CalculationDiscussionIntroductionAcknowlegment