# Demystifying Pricing and Valuation of Forward Contracts | CFA Level I Derivatives

Welcome back! In this lesson, we’ll explore the mechanisms behind the pricing and valuation of forwards, focusing on forward rate agreements, which are forward contracts with interest rate as underlying. Let’s dive in!

## Understanding the Difference Between Price and Value

For derivatives, it’s crucial to distinguish between price and value. The price of a forward or futures contract is the agreed-upon payment for the underlying on the settlement date, denoted by “big F”. On the other hand, the value of a contract is the gain or loss to the contract party, denoted by “big V”. The value to the short is opposite that of the long. We use “S” to denote the spot price of the underlying.

## Determining the Price and Value of a Forward Contract

Let’s examine how the price and value of a forward contract are determined at initiation, during the life of the contract, and at expiration.

At initiation, the forward contract price must be set such that the contract has zero value to both parties, according to risk-neutral pricing. The forward price is the spot price, compounded at the risk-free rate:

Forward price = Spot price * (1 + Risk-free rate)^T

This relationship can be generalised for any time T before expiration. The actual spot price will fluctuate according to market demand on the underlying, affecting the value to the long.

Value to the long at time t = Spot price – PV of the forward price

At expiration, the value to the long is simply the spot price minus the contract price:

Value to the long at expiration = Spot priceForward price

## Considering Benefits and Costs in Holding the Underlying Asset

The short in a forward contract holds on to the underlying until the settlement date, receiving all benefits and bearing all costs. These benefits and costs should be considered when pricing and valuing forward contracts.

The present value of the forward price at initiation should account for estimated costs and benefits:

Present value of the forward price = (Spot price + Present value of benefits) – Present value of costs

At time t, the value to the long can be expressed as:

Value to the long at time t = Spot price – (Present value of forward price + Present value of benefits – Present value of costs)

At expiration, the costs and benefits of holding the asset until expiration are zero, so the value to the long is simply the spot price minus the contract price.

In summary, these are the generalised formulas to compute the value to the long at any time T. The value to the short is simply the opposite of the value to the long. For level 1, just be aware that costs and benefits need to be considered. You are unlikely to have to make calculations.

EXAMPLE

The price of a stock is currently \$100. A 2-year forward contract on the stock is to be initiated today. The stock is not expected to pay any dividend. Determine the no-arbitrage price of the forward if the risk-free rate is currently 5%.

Using the risk-neutral pricing formula, the forward price is the future value of the spot price, using the risk-free rate of 5%. There are no costs or benefits to consider as the stock does not pay a dividend. You should get a forward price of \$110.25.

F0 = S0 x (1+Rf)^T
= \$100 × (1+0.05)^2
= \$110.25

EXAMPLE

Let’s say the 2-year forward contract was initiated with a forward price of \$110.25. 1.5 years later, the stock price has risen to \$106, and the risk-free rate has dropped to 4%. What is the value of the forward to the long at this instance?

To find the value to the long at any time t, we use the formula:

Value to the long at time t = Spot pricePresent value of the forward price
= St – F0/ (1 + Rf)T-t
= \$106 – \$110.25/(1+0.04)0.5
= -\$2.11

Plug in the figures, and you should get a value of -\$2.11. Be careful to use the remaining time to expiration, which is half a year in this case.

## Understanding Forward Rate Agreements

Next, we’ll dive into forward rate agreements (FRAs) and their pricing and valuation. But first, let’s review forward rates and their relationship with spot rates.

### Forward Rates Recap

Forward rates refer to borrowing or lending rates for loans to be made at a future date. For instance, a 2Y3Y rate indicates the forward rate for a 3-year loan to be made in 2 years. These rates must be consistent with the principle of no-arbitrage when compared to spot rates.

## Forward Rate Agreements (FRAs)

Forward rate agreements (FRAs) are derivative contracts that use forward rates as their underlying. Their primary purpose is to lock in a certain forward rate between a borrower and a lender for a future loan.

Remember that while you may not be asked to perform complex calculations on the exam, it’s essential to understand how the calculations are made and how they differ from interest rate futures. Now you’re well-equipped to tackle questions related to FRAs in the CFA Level I Fixed Income exam.

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