# ARBITRAGING DEFAULT AND ASSET SWAP SPREADS

The default swap premium, floating-rate note and asset swap spreads reflect compensation required for bearing default risk.

The default swap premium, floating-rate note and asset swap spreads reflect compensation required for bearing default risk. This article explores the theoretical relationship among them using simple arbitrage arguments. The basis spread (the spread between the default swap premium and the asset swap spread) can deviate substantially from this theoretical relationship due to supply and demand, illiquidity, tax and other technical factors.

DEFAULT SWAP PREMIUMS VS. FRN SPREADS

Par FRN: The dealer writes a default swap on a floating-rate note trading at par in return for a periodic premium
*P.*
To hedge he enters into a reverse repo and sells short the underlying FRN. Under the reverse repo, money is lent at the repo rate and the FRN is received as collateral. It is assumed that the general collateral rate equals the risk free rate
*L*. If the FRN is on special then its repo rate will be
*L-M*, where
*M*
is a positive number. The FRN pays a spread
*S*
above the risk-free rate
*L*. The dealer will have to make a payment
*L+S*
to the reverse repo counterparty at every coupon payment date. At the end of the term repo, or in case of default, the dealer receives principal plus interest on the loan and returns the bond to its owner. It is assumed that the term repo terminates when the FRN defaults. Otherwise the arbitrage relationship is not exact. If the FRN defaults the dealer pays the loss
*100 -*
*
to the protection buyer. The cash flows are shown in the following table. In order to avoid arbitrage
*P=M+S.*

If the dealer is buying instead of selling protection he needs to buy the FRN to hedge himself. He funds the bond in the repo market. The above relationship still holds.

The above argument is exact if default can only occur at a coupon payment date. If default happens at an intermediate date the value of the loan on the reverse repo will depend on the interest rate movements since the last fixing. The expected value at inception of the trade will be 100 and therefore we should still get
*P=M+S.*

If default happens between coupon payment dates the protection buyer will have to pay the accrued default swap premium to the default swap seller. If that is not the case then
*P*
must be slightly greater than
*M+S*
to compensate for the reduced coupon in case of default between coupon payment dates.

Non-par FRN: Assume an FRN trading at a premium or discount
*K*
(positive or negative respectively) paying a spread S above the risk-free rate. The cash flows of the portfolio will be
*K*
at inception and
*P-M-S*
on the coupon payment dates before default.

*P*
will be the solution to the equation
*K=PV*_{0}*(S+M-P)*, where
*PV*_{0}
is the present value at inception of the trade of the future cash flows discounted at the risky rate. The cash flows will be received until default of the issuer.
*P*
will be smaller (greater) than
*S+M*
if the risky FRN is trading at a premium (discount).

DEFAULT SWAP PREMIUMS VS. ASSET SWAP SPREADS

The asset swap buyer buys the bonds and simultaneously enters into an interest rate swap. He pays the coupon
*C*
and receives LIBOR plus the asset swap spread
*S*. The swap is not canceled even if the bond defaults. In case of default the asset swap buyer, being the bond holder, receives the recovery price
*
of the bond.

If the bond is trading at a discount
*K*
the asset swap buyer lends the discount to the dealer. The asset swap spread not only compensates the asset swap buyer for bearing credit risk but also repays the loan. If the bond is trading at a premium the asset swap buyer borrows the premium from the dealer. The loan is repaid through the above-market fixed payment
*C*
on the swap. It is assumed that both parties fund at LIBOR.

Par bond: To investigate the relationship between the asset swap spread and the default swap premium we construct the following portfolio. Assume the dealer is buying protection through a default swap. To hedge he buys the asset swap package which he funds in the repo market at
*L-M*. If the bond defaults the asset swap holder does not suffer losses on the bond, but he is exposed to the interest rate swap that can have a positive or negative mark-to-market. Cash flows are shown in the following table.

*MTM*
is the mark-to-market on the swap when default occurs. The default swap premium
*P*
will be the solution to the following equation:

*PV*_{0}*(S+M-P)+P(def)x
PV*

_{0}(expected forward swap MTM/ default)=0

where
*P(def)*
is the default probability and
*PV*_{0}
is the present value of the forward mark-to-market on the swap conditional on default. To compute
*PV*_{0}, a model is needed for the evolution of the risk-free and the credit spread curves. The slope of the risk-free curve and the correlation between interest rates and a default event will determine the relationship between*S*
and
*P.*
If the forward risk-free curve is downward sloping or the correlation is negative, then S+M>=*P*
to compensate for possible exposure in case of default into an interest rate swap with negative mark-to-market.

Non-par bond: Assume the bond is trading at a premium or discount
*K*. This is the amount borrowed or lent to the asset swap buyer.

*PV*_{0}*(S+M-P)+P(def)x
PV*

_{0}(expected forward swap MTM/ default)=0

If
*K>0*
then if the bond defaults it is expected that MTM <0. Therefore
*P<S+M*.

*This week's Learning Curve was written by**Angelo Arvanitis,**head of quantitative credit, insurance & risk research at**Paribas**.*