Forward Contracts

Forward Contract Definition

Definition: A forward contract is an agreement to exchange an asset at a future date at a prespecified price.

  • The contract settlement date is called the expiration date.
  • The asset that is exchanged is called the underlying asset.
  • The buyer holds the long position.
  • The seller holds the short position.
  • There is no initial payment or premium.

Delivery and Settlement

There are two types of forward contract settlements.

  • Delivery: The long position pays the prespecified price to the short position, who delivers the asset.
  • Cash settlement: The long and short positions pay the net cash value to the other.

Forward Example

Two parties contract to exchange a \(\smash{\$100}\) bond for \(\smash{\$98}\) at a future date.

  • If the bond is worth \(\smash{\$98.25}\) at expiry, the short position pays \(\smash{\$0.25}\) to the long position at expiry.
  • If the bond is worth \(\smash{\$97.50}\) at expiry, the long position pays \(\smash{\$0.50}\) to the short position at expiry.
  • Cash-settled forwards are often called NDFs, or nondeliverable forwards.
  • Usually, cash settlement is used for underlying assets that are difficult to exchange (think of a stock index).

Market Prices

_images/wsjFutures.png

For current data, visit the WSJ or CME Group homepages.

Early Termination

Suppose one party in a forward contract wishes to terminate early.

  • She could engage in another forward contract on the opposite side.
  • Depending on market conditions, the new contract may be written at a new price.

Early Termination Example

Suppose a trader enters a long forward contract position to exchange a barrel of crude oil on 13 Feb 2015 and decides to terminate the contract on 16 Feb 2015.

  • On 13 Feb, the forward price is $52.78 per barrel.
  • On 16 Feb, the forward price is $52.73 per barrel.
  • She can write a forward contract for $52.73 on 16 Feb.
  • Note that she takes a $0.05 loss and is still exposed to risk of default on two different contracts.
  • Alternatively, she can ask her original counterparty to accept the present value of $0.05 to terminate.

Notation

We will use the following notation:

  • \(\smash{S_0}\): Spot price of the underlying asset today.
  • \(\smash{F_0}\): Forward price of the underlying asset today.
  • \(\smash{T}\): Time until delivery.
  • \(\smash{r}\): Risk-free rate of interest for maturity \(\smash{T}\).

Note that any units (minutes, hours, days, weeks, months, years) may be used for \(\smash{T}\), but that the interest rate, \(\smash{r}\), must be adjusted accordingly.

Forward Valuation

The price of a forward contract with maturity \(\smash{T}\) for an asset with price \(\smash{S_0}\) is:

\[\begin{split}\begin{align*} F_0 & = S_0 e^{rT}. \end{align*}\end{split}\]
  • \(\smash{r}\) is the risk-free interest rate over period \(\smash{T}\).
  • If \(\smash{r}\) is constant, \(\smash{F_0}\) is a deterministic function of the spot sprice, and has nothing to do with the unknown, future price of the asset.
  • \(\smash{e^{rT}}\) is known as the basis.
  • Intuition: the foward holder must pay the holder of the spot contract for interest that would have been earned.

Forward Valuation Example

Suppose you would like to purchase a 3-month forward contract on Coca-Cola (KO) stock on 1 Mar 2016. What is the value of the forward (assuming the stock never pays dividends)?

  • Set \(\smash{T = 0.25}\) (i.e. time units of 1 year).
  • Use Quandl to determine the (annualized) yield on the 3-month U.S. Treasury Bill: \(\smash{r = 0.0033}\).

Thus,

\[\begin{split}\begin{align*} F_0 & = S_0 e^{rT} = \$43.35 e^{0.0033 \times 0.25} = \$43.39. \end{align*}\end{split}\]

Forward Valuation with Income

Suppose the underlying asset provides income with present value \(\smash{I}\).

  • This may be a single payment or a stream of payments, all appropriately discounted:
\[\begin{split}\begin{align*} I & = \frac{d}{1+\frac{r}{m}} + \frac{d}{\left(1+\frac{r}{m}\right)^2} + \cdots + \frac{d}{\left(1+\frac{r}{m}\right)^{mT}}. \end{align*}\end{split}\]
  • This assumes \(\smash{m}\) equally spaced payments of equal size during interval \(\smash{T}\).

The value of a forward contract is now:

\[\begin{split}\begin{align*} F_0 & = (S_0 - I) e^{rT}. \end{align*}\end{split}\]

Forward Valuation with Yield

Suppose the underlying asset provides income yield (continuously compounded) \(\smash{q}\). Then:

\[\begin{split}\begin{align*} F_0 & = S_0 e^{(r-q)T}. \end{align*}\end{split}\]
  • Intuition: the holder of the spot contract now pays interest (implicitly), but earns income. The foward holder must compensate the spot holder for interest, net of income earned over period \(\smash{T}\).

Forward Valuation with Yield Example

Reconsider the previous example for Coca-Cola stock.

  • Now assume that KO has an annualized dividend yield of 3%.

The forward price is

\[\begin{split}\begin{align*} F_0 & = S_0 e^{(r-q)T} \\ & = \$43.35 e^{(0.0033 - 0.03) \times 0.25} \\ & = \$43.06. \end{align*}\end{split}\]

Forward Valuation for Currency

Suppose the underlying asset is a currency, and that the risk-free interest rate in the foreign market is \(\smash{r_f}\). Then:

\[\begin{split}\begin{align*} F_0 & = S_0 e^{(r-r_f)T}. \end{align*}\end{split}\]
  • The foreign interest is income and the rate is the income yield.

Curreny Forward Example

What is the value of a 6-month forward contract for Canadian dollars (CAD) on 1 Mar 2016?

  • Set \(\smash{T = 0.5}\) (i.e. time units of 1 year).
  • Use Quandl to determine the spot exchange rate for USD/CAD: \(\smash{S_0 = \$1.34}\).
  • Use Quandl to determine the (annualized) yield on the 3-month Canadian Treasury Bill: \(\smash{r_f = 0.0047}\). We already determined that \(\smash{r = 0.0033}\).

Thus,

\[\begin{split}\begin{align*} F_0 & = S_0 e^{(r-r_f)T} = \$1.34 e^{(0.0033 - 0.0047) \times 0.5} = \$1.339. \end{align*}\end{split}\]

Forward Valuation for Commodities

Suppose that the underlying is a physical asset that must be stored. Then:

\[\begin{split}\begin{align*} F_0 & = (S_0 + U) e^{rT}. \end{align*}\end{split}\]

or

\[\begin{split}\begin{align*} F_0 & = S_0 e^{(r+u)T}. \end{align*}\end{split}\]
  • \(\smash{U}\) is the present value of storage costs.
  • \(\smash{u}\) is the annual storage cost expressed as a fraction of commodity value.
  • Note that storage costs are like negative income.

Cost of Carry

The foregoing compounding rates are referred to as the cost of carry, \(\smash{c}\).

\[\begin{split}\begin{align*} F_0 & = S_0 e^{cT}. \end{align*}\end{split}\]
  • The cost of carry includes interest rate and storage costs, minus income.
  • For a stock index that pays a dividend yield, \(\smash{c = r-q}\).
  • For a foreign currency, \(\smash{c = r - r_f}\).
  • For a commodity that provides income, \(\smash{c = r - q + u}\).