Example: Zero Curve E.g., Suppose that: 6-month zero rate is 4.0%. 12-month zero rate is 4.5%. 18-month zero rate is 4.8%. 2-year swap rate is 5.0%.
Then, the 2-year zero rate π π satisfies the following equality:
2.5ππβ 0.04 0.5 + 2.5ππβ 0.045 1.0
+ 2.5ππβ 0.048 1.5 + 102.5ππβπ π 2 = 100 Calculation gives π π = 4.953%.
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Interest Rate Swaps as Bonds An interest rate swap can be characterized as the
difference between a fixed-rate bond and a floating-rate bond. For a fixed-rate payer, the swap can be regarded as a long
position in a floating-rate bond and a short position in a fixed-rate bond.
ππswap = π΅π΅float β π΅π΅fixed For a floating-rate payer, the swap can be regarded as a
long position in a fixed-rate bond and a short position in a floating-rate bond.
ππswap = π΅π΅fixed β π΅π΅float
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Valuation of Floating-Rate Bonds
The floating-rate bond is worth the notional principal πΏπΏ immediately after an interest payment when the bond is fairly priced. Since the payments rely on LIBOR, which is the
discounting rate.
Therefore, the value of the floating-rate bond is as same as the present value of πΏπΏ and the first payment ππβ that will be made at time π‘π‘β.
π΅π΅float = πΏπΏ + ππβ ππβπππ‘π‘β
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Example: Valuation of Swaps
Suppose that: A financial institution has agreed to pay 6-month LIBOR
and receive 8% per annum on a notional principal of $100 million.
The swap has a remaining life of 1.25 years. The LIBOR rates with continuous compounding for 3-
month, 9-month, 15-month maturities are 10%, 10.5%, 11%, respectively.
The 6-month LIBOR rate at the last payment date was 10.2% with semiannual compounding.
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Example: Valuation of Swaps Then, the value of the fixed-rate bond is
π΅π΅fixed = 4ππβ0.10Γ0.25 + 4ππβ0.105Γ0.75
π΅π΅fixed = +104ππβ0.11Γ1.25
π΅π΅fixed = $98.238ππ The first payment is
ππβ =0.102
2100 = $5.1ππ
Therefore, The value of the floating-rate bond is
π΅π΅float = 100 + 5.1 ππβ 0.1 0.25 = $102.505ππ
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Example: Valuation of Swaps
Hence, the value of the swap is
ππswap = 98.238 β 102.505 = β$4.267ππ
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Valuation in Terms of FRAs
An interest rate swap can be characterized as a portfolio of forward rate agreements. Each exchange of payments in an interest rate swap is an
FRA.
Therefore, an interest rate swap can be valued on the assumption that todayβs forward rates are realized.
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Example: Valuation of Swaps
In the previous example, the floating payment in 3 months is
1000.102
2= $5.1ππ
The net cash flow in 3 months is
4 β 5.1 = β$1.1ππ
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Example: Valuation of Swaps The forward rate for the period between 3 months and 9 months
is
πππΉπΉ,3 =0.105 0.75 β 0.10 0.25
0.75 β 0.25 = 10.75%
In semi-annual compounding,
πππΉπΉ,3 = 2 exp0.1075
2 β 1 = 11.044%
Therefore, the floating payment in 9 months is
1000.11044
2 = $5.522ππ
The net cash flow in 9 months is
4 β 5.522 = β$1.522ππ
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Example: Valuation of Swaps
Similarly, the net cash flow in 15 months is
4 β 6.051 = β$2.051ππ Therefore, the value of the swap is
ππswap = β1.1ππβ0.1Γ0.25 β 1.522ππβ0.105Γ0.75
ππswap = β2.051ππβ0.11Γ1.25
ππswap = β$4.267ππ Which is exactly as same as the previous result.
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Term Structure and Swaps
The value of a FRA underlying an interest rate swap is determined by the comparison between the forward rate and the fixed rate. If forward rate is higher than the fixed rate, the fixed-rate
payer wins. If forward rate is as same as the fixed rate, both payers tie. If forward rate is lower than the fixed rate, the floating-
rate payer wins.
Since forward rates are affected by the term structure of interest rates, the value of an interest swap is affected by the term structure.
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Term Structure and Swaps
When the term structure is upward-sloping: The floating-rate payer is favored by early payments. The fixed-rate payer is favored by later payments.
When the term structure is downward-sloping, The floating-rate payer is favored by later payments. The fixed-rate payer is favored by early payments.
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Overnight Indexed Swaps
The overnight rate is the interest rate that banks use to borrow and lend from one another in the overnight market. It originates from the excess or the shortage of reserves
due to the transactions among banks in a day. Similar to the call rate by Bank of Korea.
An overnight indexed swap, or OIS, is a swap where a fixed rate for a period is exchanged for the average of the overnight rates during the period. The overnight indexed swap rate, or OIS rate, is the
fixed rate which is exchanged for the overnight rates.
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OIS Rate vs. LIBOR
Overnight indexed swap rates are generally lower than LIBOR. Since the overnight borrowing is more short-term, it is
regarded as more safer. LIBOR increases by the default risks of banks. The difference is called LIBOR-OIS spread.
OIS rate is regarded as a better proxy for the risk-free rate than LIBOR.
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