CHAPTER 7SWAPSDerivatives SecuritiesJunho Park

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 𝑅𝑅 satisfies the following equality:

2.5𝑒𝑒− 0.04 0.5 + 2.5𝑒𝑒− 0.045 1.0

+ 2.5𝑒𝑒− 0.048 1.5 + 102.5𝑒𝑒−𝑅𝑅 2 = 100 Calculation gives 𝑅𝑅 = 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.

𝑉𝑉swap = 𝐵𝐵float − 𝐵𝐵fixed 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.

𝑉𝑉swap = 𝐵𝐵fixed − 𝐵𝐵float

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Valuation of Floating-Rate Bonds

The floating-rate bond is worth the notional principal 𝐿𝐿 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 𝐿𝐿 and the first payment 𝑘𝑘∗ that will be made at time 𝑡𝑡∗.

𝐵𝐵float = 𝐿𝐿 + 𝑘𝑘∗ 𝑒𝑒−𝑟𝑟𝑡𝑡∗

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Valuation of Floating-Rate Bonds

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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

𝐵𝐵fixed = 4𝑒𝑒−0.10×0.25 + 4𝑒𝑒−0.105×0.75

𝐵𝐵fixed = +104𝑒𝑒−0.11×1.25

𝐵𝐵fixed = $98.238𝑀𝑀 The first payment is

𝑘𝑘∗ =0.102

2100 = $5.1𝑀𝑀

Therefore, The value of the floating-rate bond is

𝐵𝐵float = 100 + 5.1 𝑒𝑒− 0.1 0.25 = $102.505𝑀𝑀

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Example: Valuation of Swaps

Hence, the value of the swap is

𝑉𝑉swap = 98.238 − 102.505 = −$4.267𝑀𝑀

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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 today’s 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= $5.1𝑀𝑀

The net cash flow in 3 months is

4 − 5.1 = −$1.1𝑀𝑀

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Example: Valuation of Swaps The forward rate for the period between 3 months and 9 months

is

𝑟𝑟𝐹𝐹,3 =0.105 0.75 − 0.10 0.25

0.75 − 0.25 = 10.75%

In semi-annual compounding,

𝑟𝑟𝐹𝐹,3 = 2 exp0.1075

2 − 1 = 11.044%

Therefore, the floating payment in 9 months is

1000.11044

2 = $5.522𝑀𝑀

The net cash flow in 9 months is

4 − 5.522 = −$1.522𝑀𝑀

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Example: Valuation of Swaps

Similarly, the net cash flow in 15 months is

4 − 6.051 = −$2.051𝑀𝑀 Therefore, the value of the swap is

𝑉𝑉swap = −1.1𝑒𝑒−0.1×0.25 − 1.522𝑒𝑒−0.105×0.75

𝑉𝑉swap = −2.051𝑒𝑒−0.11×1.25

𝑉𝑉swap = −$4.267𝑀𝑀 Which is exactly as same as the previous result.

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Example: Valuation of Swaps

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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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