Welcome to the World of Swaps!

Hi there! Welcome to one of the most important chapters in the CFA Level I Derivatives section. If you have ever felt a bit intimidated by Swaps, don't worry—you are not alone. Many students find this topic tricky because it involves a lot of "moving parts."

In this lesson, we are going to demystify how we determine the "price" of a swap at the beginning and how we calculate its "value" as time passes. Think of a swap as a series of Forward Contracts all bundled together. By the end of these notes, you will see that while the math looks complex, the logic is actually quite simple!

1. Pricing vs. Valuation: The "Fair Deal" Concept

Before we dive into the math, we must understand the difference between Pricing and Valuation. This is a common point of confusion for many students.

Pricing: This happens at the very beginning (Time 0). When two parties enter a swap, they want it to be a "fair deal." This means the initial value of the swap must be zero to both parties. No one pays anyone anything to start the contract. "Pricing" a swap means finding the fixed rate that makes the present value of the fixed side equal to the present value of the floating side.

Valuation: This happens after the swap has started. As interest rates or market prices change, one side of the swap will become more "valuable" than the other. If you are the one receiving more than you are paying, the swap has a positive value to you.

Quick Review Box:
• Pricing: Setting the fixed rate so Value = 0 at the start.
• Valuation: Measuring the change in value during the life of the swap.

2. Interest Rate Swaps

In a standard (plain vanilla) interest rate swap, one party pays a fixed rate and receives a floating rate (usually based on a benchmark like MRR - Market Reference Rate).

How to "Price" an Interest Rate Swap

To find the fixed rate (the swap rate), we use Discount Factors. A discount factor, \( Z_t \), is simply the present value of \$1 received at time \( t \). It is calculated as \( Z_t = \frac{1}{1 + (r \times \frac{days}{360})} \).

The formula for the annual fixed swap rate \( R_{fix} \) is:
\( R_{fix} = \frac{1 - Z_n}{\sum_{i=1}^{n} Z_i} \times \frac{360}{days} \)

Step-by-Step Breakdown:
1. Calculate the discount factors (\( Z \)) for every payment date.
2. Subtract the last discount factor (\( Z_n \)) from 1.
3. Divide that result by the sum of all the discount factors.
4. Annualize the rate (usually by multiplying by 360 divided by the number of days in a period).

Analogy: Think of the fixed swap rate as the "average" of the expected future floating rates, adjusted for the time value of money.

How to "Value" an Interest Rate Swap

After some time has passed, the fixed rate we agreed upon might be higher or lower than the new market swap rate. The value to the fixed-rate payer is:
Value = (Sum of PV of remaining floating payments) - (Sum of PV of remaining fixed payments)

Key Takeaway: If interest rates rise, the party paying fixed and receiving floating wins! The swap becomes an asset (positive value) for them because they are paying a "cheap" old rate and receiving a "high" new rate.

3. Currency Swaps

Currency swaps are a bit different because they involve two different currencies (e.g., USD and EUR) and, crucially, the principal is exchanged at the beginning and the end.

Pricing Currency Swaps

Pricing a currency swap is like pricing two separate interest rate swaps—one for each currency. We calculate a fixed rate for Currency A using its interest rate term structure, and a fixed rate for Currency B using its own term structure.

Valuing Currency Swaps

To value the swap later, you calculate the present value of all remaining payments in each currency and then convert them into a single currency using the current spot exchange rate.

Value = \( PV_{currency A} - (S_t \times PV_{currency B}) \)
(Where \( S_t \) is the current spot exchange rate).

Common Mistake to Avoid: Don't forget to include the notional principal in your PV calculations for currency swaps! Unlike interest rate swaps (where the principal is "notional" and never moves), currency swaps actually involve the exchange of principal.

4. Equity Swaps

In an equity swap, one party typically pays the return on a stock or an index (like the S&P 500) and receives either a fixed or floating interest rate.

Valuation of Equity Swaps

Equity swaps are usually "priced" at par at the start (Value = 0). As the stock price moves, the value changes.

The "Value" to the party receiving the equity return:
\( Value = (Notional \times \frac{New Index Level}{Old Index Level}) - PV(Fixed/Floating Payments) \)

Did you know? Unlike interest rate swaps where payments are usually known at the start of a period (for the fixed side), an equity swap's payment isn't known until the very end of the period because we don't know how the stock price will move!

5. Summary and Memory Aids

Memory Trick: The "Fixed Payer" perspective

If you are the Fixed Payer, you are effectively "Short" a bond (you owe fixed interest) and "Long" a floating asset.
• Rates go UP? Your value goes UP (because you are paying an old, lower rate).
• Rates go DOWN? Your value goes DOWN (because you are stuck paying a rate that is now higher than the market).

Summary Table

Swap Type: Interest Rate Swap
Key Pricing Factor: Discount factors of the yield curve.
Value Driver: Changes in market interest rates.

Swap Type: Currency Swap
Key Pricing Factor: Yield curves of both countries.
Value Driver: Interest rate changes AND exchange rate changes.

Swap Type: Equity Swap
Key Pricing Factor: Usually set to zero value at start.
Value Driver: Performance of the underlying stock/index.

Don't worry if this seems tricky at first! The most important things to remember for the exam are:
1. Swaps have zero value at initiation.
2. To price them, we use discount factors to make the PV of both "legs" equal.
3. To value them, we find the difference between the PV of what we receive and what we pay.

Keep practicing those discount factor calculations, and you'll master this in no time!