Before we move to valuing a forward contract, it’s important to understand the connection between the spot price of an asset and its forward price.

Let’s start with the simplest case — if we take an asset that does not generate any income, for example a zero-coupon bond, then its forward price is determined by the following formula:

First of all, note that the formula looks this way because the risk-free interest rate is taken with continuous compounding. If it were a discrete rate, it would look like this:

So, the forward price of an asset with no income is simply the future value of its spot price, where the risk-free rate is applied. This formula is explained by the logic of no arbitrage. That is, if the equation above turned into an inequality, arbitrage opportunities would immediately appear, meaning riskless profit could be made. In the end, such opportunities force the inequality back to equality.

Below is an example of arbitrage:

It should also be mentioned that the arbitrage argument and the corresponding equation are primarily used for investment assets (stocks, bonds, gold, silver), rather than consumption assets (copper, oil), because in the case of consumption assets, other factors have significant influence.

Forward price of an income-generating asset

The formula gets a little more complicated if the asset generates income, for example a dividend-paying stock. The income can be defined in the contract either as an absolute amount or as a percentage yield. Accordingly, the formulas differ.

  • For income defined in absolute terms, the formula is:

Here, II is the present value of dividends to be received during the life of the contract.

  • If the asset provides a proportional yield (as a percentage of the asset’s value over the period), the formula is:

Here, q is the annual percentage income on the asset, taken with continuous compounding.

Valuing a forward contract

At the moment of initiation, the value of a forward contract is zero. But as time passes, its value can become either positive or negative.

For example: if I have a forward contract to purchase an apartment under construction, I may be able to sell that contract later for more or less than the price fixed in the contract. That difference represents the intrinsic value of the forward contract.

Thus, the value of a forward contract depends on how market conditions change after initiation:

  • What happened to the asset’s current spot price?
  • What is the risk-free interest rate?

Another example: imagine you entered into a 6-month forward contract to buy gold. After 2 months, a new 4-month forward contract for gold becomes available. The maturities of the two contracts overlap, but the forward price from the first contract will not match the forward price of the second, because the spot price of gold has changed. That difference represents the value of the already existing forward contract.

  • For a non-income-generating asset, the value of the forward contract is calculated as:

Here, KK is the delivery price (fixed at contract initiation and unchanged), while FF is the forward market price of the asset at the time of valuation.

The formula becomes a little more complicated for income-generating assets:

  • For an asset with fixed income: (formula shown)
  • For an asset with percentage yield: (formula shown)

p.s.
Forward Pricing and Valuing Forwards — Excel

Source: Options, Futures & Other Derivatives, John C. Hull