the relative performance of the sectors comprising the index. For exam- ple, if the credit sector is expected to outperform the other sectors, an asset manager may wish to overweight that sector. The asset manager can monetize this view by entering into a total return swap as the total return receiver. Again, as noted earlier, this is an efficient way to repli- cate the performance of the index.
Hedge funds manager can use total return swaps to create leverage in the same way described earlier when we showed how a synthetic repo can be created for a credit-risky bond. Moreover, suppose instead that a hedge fund manager believes that the credit sector will have a negative return. The manager can monetize this view by selling a total return swap. The advantage of the total return swap is that the credit sector can be shorted, a task that is extremely difficult and costly to do for individual bond issues in the credit sector.
Risk Control
Total return swaps can be sued as effective risk control instruments.
Interest rate swaps can be used to control the duration of the portfolio.
Total return swaps can be used to control the spread duration of a port- folio and, more specifically, the credit spread duration of a portfolio, that is the sensitivity of a portfolio to changes in credit spreads. Hedg- ing a position with respect to credit spread risk means creating a cash and total return swap position whereby the credit spread duration is zero. An asset manager would want to hedge a portfolio that has expo- sure to credit spread risk if the credit spread duration of the portfolio differs from that of the target index. Total return swaps can be used to bring the portfolio’s credit spread risk duration in line with the credit spread risk of the target index.
must be made by the swap buyer is Lt+s; the cash flow inflow at time t per $1 of notional amount is Rt.
As a result, the pricing of total return swap is to decide the right spread,s,to pay on the fund (i.e., LIBOR) leg. Formally,
wherer is the risk-free discount rate.
In words, the spread should be set so that the expected payoffs of the total return swap the next cash flow is equal to zero.3To make the matter simple (we shall discuss more rigorous cases later), we view r,R, andLas three separate random variables. We then rearrange the above equation as
Exchange expectation and summation of the right-hand side gives
3We employ the standard risk-neutral pricing and discounting at the risk-free rate.
EXHIBIT 5.4 Cash Flows for the Total Return Receiver
Eˆ0 exp r t( )dt
0 Tj
∫
–
Rj–(Lj+s)
[ ]
j=1
∑
n
0
=
Eˆ0 exp r t( )dt
0 Tj
∫
–
Rj–Lj
( )
j=1
∑
n
Eˆ0 exp r t( )dt
0 Tj
∫
–
j=1
∑
n
s
=
as the sum of risk-free pure discount bond prices. This implies
The next step is to use the forward measure to simplify the left-hand side of the above equation:
Later we show that the forward measure expectation of an asset gives the forward price of the asset. Hence, the left-hand side of the above equation gives two forward curves, one on the asset return, Rand the other on the LIBOR, L:
where is the forward rate of i (i = R or L) for j periods ahead. There- fore, the spread can be solved easily as
Eˆ0 exp r t( )dt
0 Tj
∫
–
j=1
∑
n
Eˆ0 exp r t( )dt
0 Tj
∫
–
j=1
∑
n=
P(0,Tj)
j=1
∑
n=
Eˆ0 exp r t( )dt
0 Tj
∫
–
Rj–Lj
j=1
∑
n P(0,Tj)s j=1∑
n=
P(0,Tj)E0F j( )[Rj–Lj]
j=1
∑
n P(0,Tj)s j=1∑
n=
P(0,Tj)[fjR–fjL]
j=1
∑
n P(0,Tj)s j=1∑
n=
fji
s
P(0,Tj)[fjR–fjL]
j=1
∑
nP(0,Tj)
j=1
∑
n---
=
The result intuitive: the spread is a weighted average of the expected difference between two floating-rate indexes. The weight is
Note that all the weights should sum to one.
Using the Duffie-Singleton Model
The difference in two floating-rates is mainly due to their credit risk, otherwise they should both offer identical rates and give identical for- ward curves. As a consequence, to be rigorous about getting the correct result, we need to incorporate the credit risk in one of the indices.
Among various choices, the reduced form Duffie-Singleton model suits the best for this situation.4The Duffie-Singleton model that will be discussed in Chapters 9 and 10 defines the present value of any risky cash flow as
where N is the notional, L is the LIBOR rate, S is the index level. As noted earlier, since both cash flows are random, it is a floating-floating swap. Also since the index is always higher than LIBOR because of credit risk, this swap requires a premium. As a result, the premium is computed as the sum of all future values, discounted and expected:
whereq is the “spread” in the Duffie-Singleton model that incorporates the recovery rate and default probability.
The Forward Measure
In this section, we show how the forward measure works and why a for- ward-adjusted expectation gives the forward value. We first state the
4Darrell Duffie and Kenneth Singleton, “Modeling the Term Structure of Default- able Bonds,” working paper, Stanford University (1997).
P(0,Tj) P(0,Tj)
j=1
∑
n---
ct St+1–St St
---–Lt+1 N
=
V E [r u( )+q u( )]du
t Tj
∫
exp cT
j j=1
∑
n=
separation principle that leads to the forward measure. Based on the no- arbitrage principle, the current value of any asset is the risk-neutral expected value of the discounted future payoff:
Separation principle states that if we adopt the forward measure, then the above equation can be written as
where is the forward measure. The derivation of this result can be found in a number of places.5Note that the first term is nothing but the zero coupon bond price:
and hence
While we do not prove this result, we should note an intuition behind it. Let C be a zero coupon bond expiring at time u. Then the above result can be applied directly and gives
or equivalently
5See: Farshid Jamshidian, “Pricing of Contingent Claims in the One Factor Term Structure Model,” Merrill Lynch Capital Market, 1987; and Ren-Raw Chen, Under- standing and Managing Interest Rate Risks(Singapore: World Scientific Publishing Company, 1996).
C t( ) Eˆt r u( )du
t T
∫
–
C T( ) exp
=
C t( ) Eˆt r u( )du
t T
∫
–
C T( )
exp EtF T( )[C T( )]
=
EtF T( )[ ]·
P t T( , ) Eˆ
t r u( )du
t T
∫
–
exp
=
C t( ) = P t T( , )EtF T( )[C t( )]
P t s( , ) = P t T( , )EtF T( )[P T u( , )]
EtF T( )[P T s( , )] P t s( , ) P t T( , ) ---
=
This is an indirect proof that the forward-adjusted expectation gives a forward value. The instantaneous forward rate can be shown to be the forward-adjusted expectation of the future instantaneous spot rate:
The discrete forward rates, fD(t,w,T) for all wandT can be shown also as the forward-adjusted expectations of future discrete spot rates:
wheret < w < T.
f t T( , ) d P t Tln ( , ) ---dT –
= 1 P t T( , ) ---Eˆ
t d
dT--- r u( )du
t T
∫
–
exp –
=
1 P t T( , ) ---Eˆ
t r u( )du
t T
∫
–
r T( ) exp
=
EtF T( )[r T( )]
=
fD(t w T, , ) 1 Ψ(t w T, , ) ---–1
≡ P t w( , ) P t T( , ) ---–1
= 1 P t T( , ) ---Eˆ
t r u( )du
t T
∫
–
1
P w T( , ) ---
exp –1
=
EtF T( ) 1 P w T( , ) ---–1
=
CHAPTER 6
119
Credit-Linked Notes
redit derivatives are grouped into funded andunfunded variants. In an unfunded credit derivative, typified by a credit default swap, the protection seller does not make an upfront payment to the protection buyer. In a funded credit derivative, typified by a credit-linked note (CLN), the investor in the note is the credit-protection seller and is mak- ing an upfront payment to the protection buyer when buying the note.
Thus, the protection buyer is the issuer of the note. If no credit event occurs during the life of the note, the redemption value of the note is paid to the investor on maturity. If a credit event does occur, then on maturity a value less than par will be paid out to the investor. This value will be reduced by the nominal value of the reference asset that the CLN is linked to. In this chapter we discuss CLNs.