Pricing and Valuation of Forward Contracts and for an Underlying - Derivatives | CFA® Level 1 Guide

Pricing and Valuation of Forward Contracts

Learning Outcome Statement:

explain how the value and price of a forward contract are determined at initiation, during the life of the contract, and at expiration

Summary:

The learning outcome statement focuses on explaining the determination of the value and price of forward contracts at different stages: initiation, during the life of the contract, and at expiration. It covers the initial zero value of forward contracts, the fixed forward price set at inception, and the changes in value over time due to market conditions. It also discusses the mark-to-market (MTM) valuation, the no-arbitrage condition, and the effects of interest rates and other factors on the forward price and value.

Key Concepts:

Forward Contract Initial Value

At initiation, forward contracts typically have a value of zero, assuming no transaction costs and ignoring counterparty credit risk. The forward price set at this stage is based on the no-arbitrage condition and includes factors like the underlying asset's spot price and the risk-free rate.

Mark-to-Market (MTM) Valuation

As the market conditions change, the value of a forward contract changes. This MTM value reflects potential gains or losses if the contract were settled immediately, based on the current spot price and other relevant factors.

No-Arbitrage Condition

The forward price at initiation is set such that there is no arbitrage opportunity available, meaning the forward price equals the expected future spot price discounted at the risk-free rate, adjusted for any benefits or costs from holding the underlying asset.

Interest Rate Effects

Interest rates play a crucial role in determining the forward price. The forward price is generally the future value of the current spot price compounded at the risk-free rate over the term of the contract.

Formulas:

Forward Price at Initiation

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

This formula calculates the forward price at initiation, ensuring no-arbitrage conditions are met, based on the spot price, the risk-free rate, and the time to maturity.

Variables:

$F_0(T)$ :: Forward price at time T
$S_0$ :: Spot price at time 0
$r$ :: Risk-free rate
$T$ :: Time to maturity

Units: Currency

Forward Contract Value at Maturity

V_T(T) = S_T - F_0(T)

This formula calculates the value of the forward contract at maturity, which is the difference between the spot price at maturity and the initially agreed forward price.

Variables:

$V_T(T)$ :: Value of the forward contract at maturity
$S_T$ :: Spot price at maturity
$F_0(T)$ :: Forward price agreed at initiation

Units: Currency

Mark-to-Market Value during Life of Contract

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

This formula calculates the MTM value of the forward contract at any time t before maturity, based on the current spot price and the present value of the initially agreed forward price.

Variables:

$V_t(T)$ :: Mark-to-Market value of the forward contract at time t
$S_t$ :: Spot price at time t
$F_0(T)$ :: Forward price agreed at initiation
$r$ :: Risk-free rate
$T$ :: Time to maturity
$t$ :: Current time

Units: Currency

Pricing and Valuation of Interest Rate Forward Contracts

Learning Outcome Statement:

Explain how forward rates are determined for interest rate forward contracts and describe the uses of these forward rates.

Summary:

The content discusses the determination and application of forward rates in interest rate forward contracts, emphasizing the relationship between spot and forward interest rates, the process of bootstrapping to derive zero or spot rates, and the calculation of implied forward rates using no-arbitrage conditions. It also covers the valuation and pricing of forward rate agreements (FRAs) and the mechanics of settling these contracts.

Key Concepts:

Interest Rate Forward Contracts

Contracts that specify the interest rate applicable to a principal amount over a future period, allowing financial institutions to manage interest rate exposure.

Spot and Forward Interest Rates

Spot rates are current interest rates available for investing or borrowing today, while forward rates are agreed upon today for a loan or investment that will start in the future.

Bootstrapping

A method used to derive zero or spot rates from the prices of coupon-bearing bonds by sequentially solving for zero-coupon yields that equate the present value of cash flows to the bond's price.

Implied Forward Rate

The rate that equates the investment returns of two different investment strategies over different time frames under no-arbitrage conditions. It is used to link short-dated and long-dated zero-coupon bonds.

Forward Rate Agreements (FRAs)

Over-the-counter derivatives where counterparties agree on an interest rate to be applied to a notional amount for a future period. FRAs are used to hedge against interest rate changes.

Formulas:

Discount Factor

DF_i = \frac{1}{(1 + z_i)^i}

Calculates the present value of one unit of currency receivable or payable in period i.

Variables:

$DF_i$ :: Discount factor for period i
$z_i$ :: Zero rate for period i
$i$ :: Time period

Units: dimensionless

Implied Forward Rate

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

Used to calculate the implied forward rate between two periods based on zero rates, ensuring no-arbitrage conditions.

Variables:

$z_A$ :: Zero rate for period A
$z_B$ :: Zero rate for period B
$IFR_{A,B-A}$ :: Implied forward rate from period A to B
$A$ :: Start period
$B$ :: End period

Units: dimensionless