Pricing and Valuation of Forward Commitments

CFA Level II · CFA · 6 min read

Welcome to Derivatives: Pricing and Valuation of Forward Commitments!

Hello there! Welcome to one of the most mathematically intensive but rewarding chapters in the CFA Level II curriculum. If you found Level I derivatives a bit abstract, don't worry. Here, we move from the "what" to the "how." We are going to learn how to put a price tag on contracts and figure out exactly how much they are worth as time passes. Think of this as learning the "fair market math" behind some of the biggest deals in the financial world. Let's dive in!

1. The Fundamental Distinction: Pricing vs. Valuation

Before we touch a single formula, we must clear up a common point of confusion. In the world of forward commitments (forwards, futures, and swaps), Pricing and Valuation are two very different things.

Pricing: This happens at the very beginning (Time 0). It is the act of determining the Forward Price ($F_0$) that will be written into the contract. At this moment, the contract has zero value to both parties because it is a fair deal. No money changes hands.

Valuation: This happens after the contract has started. As market prices and interest rates change, one person starts "winning" and the other starts "losing." The Value ($V_t$) is the amount of money one party would owe the other to cancel the contract today.

Analogy: Imagine you agree today to buy a house in six months for \$500,000. Pricing is deciding on that \$500,000 figure. Valuation happens three months later—if house prices have shot up to \$600,000, your contract is now very valuable to you!

Quick Summary:
• At initiation ($t=0$): Value is zero.
• During the life ($t > 0$): Value fluctuates based on market movements.
• At expiration ($t=T$): Value is the difference between the Spot price and the Forward price.

2. Pricing and Valuing Forward Contracts

The core principle here is No-Arbitrage. We assume that you shouldn't be able to make a riskless profit by buying the asset and selling a forward contract simultaneously.

A. Forwards on Stocks (No Income)

The simplest case. To price a forward on a non-dividend paying stock, we just take the Spot Price ($S_0$) and compound it at the risk-free rate ($r$).

The Formula:
$F_0 = S_0 \times (1 + r)^T$

Wait! What if we use continuous compounding?
$F_0 = S_0 \times e^{rT}$

B. Forwards on Stocks with Income (Dividends)

If a stock pays a dividend, the person holding the physical stock gets the cash, but the person holding the forward contract does not. Therefore, the forward price must be lower to reflect this "lost" income.

The Formula (Discrete Income):
$F_0 = (S_0 - PV_{dividends}) \times (1 + r)^T$

The Formula (Continuous Yield, $q$):
$F_0 = S_0 \times e^{(r-q)T}$

C. How to Calculate Value ($V_t$)

Don't worry if this seems tricky at first! The value is just the current spot price minus the "present value" of the price you agreed to pay earlier.

The Formula:
$V_t = S_t - [F_0 / (1 + r)^{T-t}]$

Common Mistake to Avoid: When calculating value, always use the remaining time ($T-t$), not the original maturity ($T$).

Key Takeaway: Forward prices are just spot prices moved forward in time by the cost of carry (interest minus income).

3. Forward Rate Agreements (FRAs)

An FRA is a contract to borrow or lend at a specific interest rate in the future. In Level II, we use Advanced Set, Settled in Arrear logic, but the payment is usually discounted to the start of the loan period.

Understanding the Notation

If you see a "2 x 5 FRA", it means:
• The "loan" starts 2 months from now.
• The "loan" ends 5 months from now.
• The duration of the loan itself is 3 months (5 minus 2).

The Payoff of an FRA

The payoff is based on the difference between the Reference Rate ($L$) and the FRA Rate ($IFR$).

The Payoff Formula:
$Payoff = \frac{(L - IFR) \times Days/Year}{1 + (L \times Days/Year)} \times Principal$

Did you know? We discount the payoff because FRA payments are typically made at the start of the loan period, but the interest rate being hedged applies to the end of the period. We have to bring that future value back to the present!

4. Pricing and Valuing Swaps

Swaps can look intimidating because they involve many payments, but here is a secret: A swap is just a series of forward contracts bundled together.

A. Interest Rate Swaps

In a standard "Plain Vanilla" swap, one party pays a Fixed Rate and receives a Floating Rate (like LIBOR or SOFR). To find the fixed rate (the Swap Rate), we use Discount Factors ($Z$).

The Formula for the Fixed Swap Rate ($C$):
$C = \frac{1 - Z_n}{\sum Z_i}$

In plain English: The swap rate is (1 minus the last discount factor) divided by the sum of all discount factors.

B. Currency Swaps

Currency swaps are unique because:
1. Principal is exchanged at the beginning.
2. Interest is paid in different currencies.
3. Principal is exchanged back at the end.

Steps to Value a Currency Swap:
1. Calculate the value of the "Fixed Leg" in its own currency.
2. Calculate the value of the "Floating Leg" (or second Fixed Leg) in its own currency.
3. Convert them to a single currency using the current spot exchange rate.
4. Subtract one from the other.

C. Equity Swaps

In an equity swap, you might swap the returns of the S&P 500 for a fixed interest rate.
• Pricing: The fixed rate is calculated similarly to an interest rate swap.
• Valuation: Value = (New Equity Price / Old Equity Price) - (Value of Fixed Leg).
Trick: If the equity price has gone up 10%, the "Equity Leg" is simply 1.10 times the notional principal.

Quick Review:
• Interest Rate Swap: Find the fixed rate using discount factors.
• Currency Swap: Two different currencies; don't forget to exchange principal!
• Equity Swap: One side is a simple percentage return.

5. Futures Pricing

You might ask: "Aren't Forwards and Futures the same?" Mathematically, they are very close, but there is one major difference: Marking to Market.

Because futures are settled daily, the Value ($V_t$) of a futures contract is reset to zero at the end of every trading day. Any gains or losses are realized immediately in your margin account.

Did you know? If interest rates are constant and positively correlated with the asset price, futures prices will be slightly higher than forward prices. However, for the CFA Level II exam, we usually treat them as identical unless the question specifies otherwise.

Summary and Final Tips

1. Follow the Timeline: Always draw a timeline. Identify where you are ($t$) and how much time is left ($T-t$).
2. The Present Value Rule: Almost every valuation formula in derivatives is just: (What I have today) minus (The Present Value of what I committed to do later).
3. Don't Panic over Notation: Whether it's $S_0 \times e^{rt}$ or $S_0 \times (1+r)^T$, the logic is the same: move the money through time using the appropriate interest rate.

Key Takeaway: Pricing is about finding the "fair" rate at the start so the contract is worth nothing. Valuation is about measuring the "winner's gain" after the market moves. Master the discount factors, and you master Swaps!

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is the fundamental difference between pricing and valuation in forward commitments?

Pricing is determining the forward price ($F_0$) at initiation so the contract has zero value; valuation is determining the market value ($V_t$) of the contract during its life.

How is the forward price ($F_0$) calculated for a stock that pays no income?

$F_0 = S_0 \times (1 + r)^T$ for discrete compounding, or $F_0 = S_0 \times e^{rT}$ for continuous compounding.

How do discrete dividends affect the pricing of a forward contract on a stock?

The forward price is reduced by the present value of the dividends: $F_0 = (S_0 - PV_{dividends}) \times (1 + r)^T$.

What is the formula for the forward price of an asset with a continuous dividend yield ($q$)?

$F_0 = S_0 \times e^{(r-q)T}$

How is the value ($V_t$) of a long forward contract calculated during its life?

$V_t = S_t - [F_0 / (1 + r)^{T-t}]$

In FRA notation, such as a "2 x 5 FRA", what do the numbers represent?

The first number (2) is the time until the loan starts (in months), and the second number (5) is the time until the loan ends; the loan duration is the difference (3 months).

Why is the payoff of an FRA discounted in the standard payoff formula?

Because the payoff is typically settled at the beginning of the loan period, whereas the interest rate it hedges applies to the end of the loan period.

How is the fixed rate (swap rate) of a plain vanilla interest rate swap determined?

$C = \frac{1 - Z_n}{\sum Z_i}$, where $Z_n$ is the final discount factor and $\sum Z_i$ is the sum of all discount factors for the payment dates.

What are the unique characteristics of a currency swap compared to an interest rate swap?

Currency swaps involve the exchange of principal at both initiation and termination, and interest payments are made in two different currencies.

How is the equity leg of an equity swap valued after initiation?

The value of the equity leg is the Notional Principal multiplied by the ratio of the current equity price to the equity price at the last reset date: $NP \times (S_t / S_{last\,reset})$.

Why is the value of a futures contract typically zero at the end of every trading day?

Due to marking to market, where daily gains and losses are settled in the margin account and the contract's value is reset to zero.

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