Pricing and Valuation of Interest Rates and Other Swaps

Derivatives

Swaps vs. Forwards

Learning Outcome Statement:

describe how swap contracts are similar to but different from a series of forward contracts

Summary:

This LOS explores the similarities and differences between swap contracts and forward contracts. Swap contracts involve the exchange of a series of future cash flows between two parties, while forward contracts involve a single future exchange. Swaps, like interest rate swaps, have periodic settlements and a constant fixed rate over their life, contrasting with forwards which may have varying rates and a single settlement.

Key Concepts:

Swap Contracts

A swap contract is an agreement to exchange a series of future cash flows between two parties. It typically involves periodic settlements and a constant fixed rate throughout the life of the swap.

Forward Contracts

A forward contract is an agreement for a single future exchange of value at a specified date. It involves a single settlement and the rate can vary based on the term structure of interest rates.

Implied Forward Rate

Implied forward rates are derived from spot rates and represent the no-arbitrage forward rates for future periods. These rates are used in forward rate agreements (FRAs) and are crucial for setting the fixed rates in these contracts.

Par Swap Rate

The par swap rate is the fixed rate in a swap contract that equates the present value of expected future floating cash flows to the present value of fixed cash flows, ensuring no arbitrage opportunities.

Formulas:

Implied Forward Rate Calculation

(1 + z_A)^A \times (1 + IFRA,B-A)^{B-A} = (1 + z_B)^B

This formula calculates the implied forward rate between two periods based on the respective spot rates.

Variables:

$z_A$ :: Spot rate for the shorter period A
$z_B$ :: Spot rate for the longer period B
$IFRA,B-A$ :: Implied forward rate from period A to B

Units: percentage

Par Swap Rate Calculation

\sum_{i=1}^N \frac{PMT}{(1 + z_i)^i} = 100

This formula calculates the fixed payment (PMT) in a swap that would make the price of a hypothetical bond equal to par (100) using the spot rates as discount factors.

Variables:

$PMT$ :: Fixed payment in the swap
$z_i$ :: Spot rate for period i
$N$ :: Total number of periods

Units: currency

Swap Values and Prices

Learning Outcome Statement:

contrast the value and price of swaps

Summary:

This LOS explores the differences between the value and price of swaps, focusing on how these metrics are calculated and influenced by market conditions. It explains the concept of a swap rate, the role of implied forward rates, and how changes in interest rates affect the market-to-market (MTM) value of swaps over time.

Key Concepts:

Swap Rate

The swap rate is the fixed rate in a swap contract that equates the present value of expected future floating cash flows with the present value of fixed cash flows, ensuring a no-arbitrage condition.

Market Reference Rate (MRR)

MRR is the floating rate used in the calculation of periodic swap payments. It is set at the beginning of each interest period and used to determine the floating cash flows in a swap.

Implied Forward Rates (IFRs)

IFRs are derived from zero rates and represent the expected future floating rates at each period of the swap's duration. They are crucial for calculating the present value of future floating payments.

Periodic Settlement Value

This value is calculated for each period as the difference between the MRR and the swap rate, multiplied by the notional amount and the period length. It represents the net cash flow for that period.

MTM Value of Swap

The MTM value of a swap is the sum of the current periodic settlement value and the present value of all remaining future swap settlements. It reflects the current worth of the swap contract.

Formulas:

Periodic Settlement Value

\text{Periodic Settlement Value} = (MRR - s_N) \times \text{Notional Amount} \times \text{Period}

This formula calculates the net cash flow for each period in a swap contract, based on the difference between the floating rate (MRR) and the fixed rate (sN).

Variables:

$MRR$ :: Market Reference Rate
$s_N$ :: Swap rate for N periods
$Notional Amount$ :: The principal amount on which the swap payments are based
$Period$ :: The time duration for each payment period

Units: currency units

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