Download Interest rate swap using financial intermediary

Interest rate swap using financial intermediary
10.0%
LIBOR
Financial
10% Company A
Company B +1%
intermediary
LIBOR
LIBOR
9.9%
Net gain to A = 0.2%
Net gain to B = 0.2%
Net gain to financial intermediary = 0.1%
• The financial institution has two separate contracts.
If one of the companies defaults, the financial institution
still has to honor its agreement with other company.
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Exploiting comparative advantages
• Initial motivation for the interest rate swap market
was borrower exploitation of “credit arbitrage”
opportunities because of differences between the
quality spread between lower- and higher-rated
credits in the US and Eurodollar bond markets.
Query
As with any arbitrage opportunity, the more
it is exploited, the smaller it becomes.
Explanation
The difference in quality spread persists
due to differences in regulations and tax
treatment in different countries.
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Valuation of interest rate swap
•
When a swap is entered into, it typically has zero value.
•
Valuation involves finding the fixed coupon rate K such that fixed
and floating legs have equal value at inception.
•
Consider a swap with payment dates t1, t2, …, tN set in the terms
of the swap.
(ti – ti-1) ×K × N
…
0
t1
t2
…
ti
tN
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Valuation (cont’d)
• Fixed payment at ti is (ti – ti-1) × K × N where N is the notional
principal, ti – ti-1 is the tenor period. The fixed payments are packages
of bonds with par K × N.
• To generate the floating rate payments, we invest a floating rate
bond of par value $N and use the floating rate interest earned to
honor the floating leg payments. At maturity, $N remains but all the
intermediate floating rate interests are forgone.
“Assume forward rates will be realized” rule
1. Calculate the swap’s net cash flows on the assumption that LIBOR
rates in the future equal today’s forward LIBOR rates.
2. Set the value of the swap equal to the present value of the net cash
flows using today’s LIBOR zero curve for discounting.
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Valuation (cont’d)
• Let B(0, t) be the discount bond price with maturity t.
• Sum of percent value of floating leg payments = N[1 – B(0, tN)];
sum of present value of fixed leg payments =
N
( N × K )∑ (ti − ti −1 ) B(0, ti ).
i =1
• Hence, the swap rate is given by
K=
1 − B(0, t N )
N
∑ (t − t
i =1
i
i −1
.
) B(0, ti )
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Swap rate curves
•
From traded discount bonds, we may construct the implied forward
rates; then the equilibrium swap rates are determined from these
forward rates.
•
Turning around, with the high liquidity of the swap market, and
available at so many maturities, it is the swap rates that drive the
prices of bonds. That is, the fixed leg of a par swap (having zero
value) is determined by the market.
•
For swap-based interest rate derivatives, swap rates constitute the
more natural set of state variables, rather than the forward rates.
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Numerical Example: Determining the Swap Rate
Three-year swap, notional amount $100 thousand
Fixed-rate receiver
Actual/360 day count basis, quarterly payments
Floating-rate receiver
3-month LIBOR, actual/360 day count basis, quarterly payments and
reset.
Swap rate is the rate that will produce fixed cash flows whose present
value will equal the present value of the floating cash flows.
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(1)
Period
1
2
3
4
5
6
7
8
9
10
11
12
13
Total
(2)
(3)
(4)
(5)
Futures Forward Discount Floating
Price
Rate
Factor Cash Flow
4.05 1.00000
95.85
4.15
0.990
1,012
95.45
4.55
0.980
1,049
95.28
4.72
0.969
1,150
95.10
4.90
0.957
1,193
94.97
5.03
0.945
1,279
94.85
5.15
0.933
1,271
94.75
5.25
0.921
1,287
94.60
5.40
0.909
1,327
94.50
5.50
0.897
1,365
94.35
5.65
0.885
1,390
94.25
5.76
0.872
1,459
94.10
5.90
0.859
1,456
14,053
(6)
(7)
PV of
PV of
Floating CF Fixed CF
1,002
1,027
1,113
1,141
1,209
1,186
1,186
1,206
1,224
1,229
1,272
1,251
1,234
1,235
1,221
1,206
1,230
1,176
1,148
1,146
1,130
1,115
1,123
1,083
14,053
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Column (2):
The Eurodollar CD futures price.
Column (3): Forward Rate = Futures Rate.
The forward rate for LIBOR found from the futures price of the
Eurodollar CD futures contract as follows: 100.00 – Futures price
Column (4):
The discount factor is found as follows:
Discount factor in the previous period
[1 + (forward rate in previous period × number of days in period/360)]
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Column (5): The floating cash flow is found by multiplying the
forward rate and the notional amount, adjusted for the number of
days in the payment period. That is:
Forward rate previous period × number of days in period
× notional amount
360
Column (7): This column is found by trial and error, based on a guess
of the swap rate. In determining the fixed cash flow, the cash flow must
be adjusted for the day count as follows:
Assumed swap rate × number of days in period
× notional amount
360
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Determining the value of a swap after one year
(1)
Period
1
2
3
4
5
6
7
8
9
Total
(2)
(3)
(4)
(5)
Futures Forward Discount Floating
Price
Rate
Factor Cash Flow
94.27
94.22
94.00
93.85
93.75
93.54
93.25
93.15
5.25
5.73
5.78
6.00
6.15
6.25
6.46
6.75
6.85
1.00000
0.986
0.972
0.958
0.944
0.929
0.915
0.900
0.885
PV of floating cash flow
PV of fixed cash flow
Value of swap
1,370
1,448
1,445
1,516
1,554
1,579
1,668
1,706
(6)
(7)
PV of
PV of
Floating CF Fixed CF
1,352
1,408
1,385
1,432
1,445
1,446
1,502
1,510
11,482
1,284
1,225
1,195
1,190
1,172
1,153
1,159
1,115
9,498
$11,482
$9,498
$1,984
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Credit exposure on derivatives
Credit risk fluctuates over time with the variables that determine the
value of the underlying contract.
Current exposure
Replacement cost if the counterparty defaults right now.
Potential exposure
Estimation of the future replacement cost: expected exposure and
maximum exposure. At best, we provide probabilistic assessment.
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