Pricing and Valuation of Forward Contracts and for an Underlying with Varying Maturities

Welcome to the World of Forwards!

Hello there! Today, we are diving into one of the most fundamental parts of the CFA Level I Derivatives curriculum: Pricing and Valuation of Forward Contracts. While "Pricing" and "Valuation" might sound like the same thing, in the world of derivatives, they are very different concepts. Don't worry if this seems a bit abstract at first—we’re going to break it down using simple math and real-world analogies. By the end of this, you’ll be able to tell the difference between the two and calculate them like a pro!

1. Pricing vs. Valuation: The "House" Analogy

Before we look at formulas, let’s clear up the biggest source of confusion for students: the difference between Forward Price and Value.

Imagine you agree today to buy a house from a friend in exactly one year for \$500,000.

Pricing: This is the act of deciding on that \$500,000 number today. At the moment you sign the contract, neither person owes the other any money. Therefore, the Value of the contract at the very start is zero.

Valuation: Now, imagine six months pass. Interest rates have changed, and the housing market has boomed. That same house is now worth much more. Even though you are still paying \$500,000 later, your "contract" is now a valuable asset because you're locked into a lower price. This change in "worth" is the Value.

Quick Review:
\n1. At initiation (time \(t=0\)), the Value of a forward contract is Zero.
\n2. The Forward Price is the fixed price written in the contract that makes the initial value zero.

2. The No-Arbitrage Principle

How do we decide what the Forward Price should be? We use the No-Arbitrage Principle. This assumes that market participants are rational and that there are no "free lunches."

If you want to buy an asset in the future, you have two choices:
\n1. Buy it now and store it (which costs money or ties up capital).
\n2. Enter a forward contract to buy it later.

In a fair market, these two paths should cost the same amount of money in "present value" terms. If they didn't, traders would exploit the difference until the prices aligned.

3. Pricing and Valuing a Basic Forward Contract

Let's look at the math for a basic underlying asset (like a non-dividend paying stock).

Calculating the Forward Price

To find the forward price \(F_0\), we take the current spot price \(S_0\) and "grow" it at the risk-free rate \(r\) for the time \(T\):

\( F_0(T) = S_0 \times (1 + r)^T \)

Example: If a stock is currently \$100 and the annual risk-free rate is 5%, the 1-year forward price is:
\( 100 \times (1 + 0.05)^1 = \$105 \)

Valuing the Contract During its Life

To find the value \(V_t\) at some time \(t\) before the contract ends, we compare the current spot price to the present value of the price we locked in earlier:

\( V_t(T) = S_t - \frac{F_0(T)}{(1 + r)^{T-t}} \)

Common Mistake to Avoid: When valuing the contract mid-way through, students often forget to discount the forward price back to the present. Remember: you are comparing today's price (\(S_t\)) with a future payment, so you must bring that future payment back to today's dollars!

Key Takeaway:

Pricing is about the future (compounding forward), while Valuation is about the present (comparing current spot to discounted forward price).

4. Dealing with Benefits and Costs (The "Carry" Model)

In the real world, holding an asset isn't always "free." Some assets give you money (like dividends or interest), while others cost you money (like storing gold or grain).

Benefits (Dividends or Interest)

If an asset pays a dividend, holding the forward contract is less attractive than holding the actual stock because forward holders don't get dividends. Therefore, benefits reduce the forward price.

Forward Price with Benefit \(I\):
\( F_0(T) = (S_0 - PV(I)) \times (1 + r)^T \)

Costs (Storage and Insurance)

If an asset is expensive to store, the forward contract becomes more attractive because the seller has to pay those costs while holding the asset for you. Therefore, costs increase the forward price.

Forward Price with Cost \(C\):
\( F_0(T) = (S_0 + PV(C)) \times (1 + r)^T \)

Memory Aid: "Costs climb, Benefits bend."
Costs make the forward price climb higher. Benefits make the forward price bend lower.

5. Forward Rate Agreements (FRAs)

An FRA is a forward contract on an interest rate. These can be a bit tricky because of the "m x n" notation.

Did you know? An FRA is named by when it starts and when it ends. A "3 x 9" FRA means the contract starts in 3 months and the interest rate period lasts for 6 months (ending in month 9).

The Logic of FRA Pricing

The price of an FRA is based on the yield curve. If you know the 3-month interest rate and the 9-month interest rate, you can mathematically determine what the "forward" rate for that 6-month gap should be. This is called the Implied Forward Rate.

Key Points for FRAs:
- Long Position: Benefits if interest rates rise (you locked in a low rate).
- Short Position: Benefits if interest rates fall (you locked in a high lending rate).
- Cash Settlement: Unlike physical assets, FRAs are settled in cash based on the difference between the contract rate and the actual market rate (usually LIBOR or its replacement like SOFR) at the time the contract starts.

6. The Term Structure and Varying Maturities

Forward prices aren't the same for every maturity. If you look at forwards for 1 month, 6 months, and 1 year, the prices will differ based on the Term Structure of Interest Rates.

Key Concepts:

1. Contango: When the forward price is higher than the spot price. This is common when interest rates or storage costs are high.
2. Backwardation: When the forward price is lower than the spot price. This happens when the "benefits" of holding the asset (like a high dividend or a "convenience yield") outweigh the costs of carry.

Step-by-Step: Determining the Term Structure Impact
1. Identify the risk-free rate for that specific maturity.
2. Identify any benefits (yields) or costs associated with that specific time frame.
3. Apply the formula: \( F = S \times e^{(r + cost - benefit)T} \) (using continuous compounding, often used in this section).

Quick Review Box:

- If Benefits > Costs + Interest: The forward price will be lower than the spot (Backwardation).
- If Costs + Interest > Benefits: The forward price will be higher than the spot (Contango).

Summary and Final Tips

We've covered a lot of ground! Here are the most important things to remember for your exam:

1. Value at start is always zero. If a question asks for the value at initiation, don't even do math—the answer is zero!
2. Forward Price is about "No Arbitrage." It's just the spot price adjusted for the cost of carry.
3. Discounting is key. When valuing a contract after it has started, always discount the forward price back to the evaluation date.
4. FRAs are "Start x End." The actual interest period is the difference between the two numbers.

Don't worry if the FRA math feels heavy! Focus on the relationship: if rates go up, the person who locked in the "pay fixed" rate wins. Practice a few problems on the "cost of carry" model, and you'll have this chapter mastered in no time!