2.2a Forward Contracts

Forward Contract

A forward contract is a derivative contract between two parties to buy or sell an asset for another asset or currency at a specified forward price on a future maturity date. By Covered Interest Parity, the per-unit forward price of an asset that prevents arbitrage $F_T$, given a time to maturity $T$ (in years), can be expressed as
$F_0=S_0{e}^{(r_f-{\rho }_f)T},$
where $S_0$ is the spot rate (price) of the underlying asset, $r_f$ is the risk-free rate over period $T$, and ${\rho }_f$ is the carry rate (e.g. a dividend yield on a stock, or the foreign interest rate in a currency forward). Note that both are continuously compounded.

Then, the value of a forward contract at time $T$ can be expressed as

(F_{0}-S_{T})e^{-r_{f}T} & \text{long position}, \
(S_{T}-F_{0})e^{-r_{f}T} & \text{short position},
\end{cases}$$

where $S_T$ is the spot rate at time $T$. More particularly, if $F_0>S_T$, a long forward contract is in-the-money. The contract can also be at-the-money or out-of-the-money.

Forward contracts trade exclusively on OTC markets via major banks, and assets only change hands at the maturity date. At maturity, assets can be paid out in two ways:

1. Physical delivery requires that all of the asset is shipped/wired on the maturity date.
2. Cash-settlement requires that the difference between the spot price and forward price is paid to the “winning” party on the maturity date.

Example

I. Hedging with Forward Contracts

Andrew has a side business importing handcrafted furniture from Japan. He just collected payment in dollars from your US-based client for a custom table that will be shipped to the US in 90 days, at which point he will owe his Japanese business partner 12M JPY.

Andrew is concerned about the volatility of the JPY-USD exchange rate and would like to lock in his gross profit by hedging with a forward contract.

A bank offers him a rate of 145 JPY/USD for 12M JPY to be delivered in 90 days, and he accepts.

A) How much must you Andrew the bank right now?

A) Solution.

Since assets in forward contracts only change hands at the maturity date, Andrew is not obligated to pay anything before the maturity date.

Thirty days go by, and the 60-day forward rate is 140 JPY/USD. A 60-day T-bill on this day yields 4.0%.

B) How much is the forward contract now worth from Andrew’s point of view?

B) Solution $F_0=12\text{M}/140>12\text{M}/145$), and so the forward contract is in-the-money, and therefore positive for Andrew. Given the 60-day forward rate, $r_f=0.04$, and therefore after 60 days Andrew saves $12\text{M}\left(\frac{1}{140}-\frac{1}{145}\right)e^{-0.04\cdot 60/365}=\$2,\mathrm{936.29.}$

Since the Yen gained value (i.e.,

C) How much is the forward contract worth to the bank?

C) Solution. \2,936.29$,theforwardcontractisworthexactly$-$2,936.29$ to the bank.

Since Andrew gained

Sixty more days go by (i.e. you have reached the 90-day maturity date) and the JPY spot rate is 150 JPY/USD.

D) How much must Andrew pay the bank now?

D) Solution. $F_0=1/145$. As such, Andrew pays $12\text{M}/145=82,\mathrm{758.62.}$

Andrew must pay the bank the value in USD agreed by the forward rate of

E) How much money did Andrew make (or lose) by entering into the forward contract?

E) Solution. $F_0=1/150<1/145=S_0$), Andrew's net gain is precisely $12\text{M}(1/150-1/145)=-\$2,\mathrm{758.62.}$

Since the forward contract is out-of-the-money (i.e.,

F) Did Andrew do the right thing by hedging?

F) Solution.

Andrew did the right thing by hedging, since JPY-USD is a volatile rate, and hedging is about eliminating risk rather than making profit. Not hedging could have done him worse.

II. Covered Interest Parity & Arbitrage

You are an arbitrageur looking for opportunities, and believe that there may be some misalignment in the fixed income and foreign exchange markets in New Zealand relative to the US. Let $r_f=0.045$ be the one-year USD risk-free rate, and let ${\rho }_f=0.035$ be the one-year NZD risk-free rate. The USD/NZD spot is currently $S_0=0.6$, and the USD/NZD one-year forward rate is ${\hat{F}}_0=0.601$.

A) Is there an arbitrage opportunity?

A) Solution. $F_0=S_0{e}^{T(r_f-{\rho }_f)}=0.6e^{0.01}\approx 0.606>0.601={\hat{F}}_0,$ and therefore there is an arbitrage opportunity. More particularly, the one-year forward for NZD is overvalued, since the theoretical (by CIP) is larger than the quoted rate. Therefore,

By CIP,

1. Borrow NZD and sell at $S_0=0.6$ USD/NZD.
2. Invest the US at $0.6e^{-0.045}=$
3. Buy NZD at the one-year forward rate ${\hat{F}}_0=0.601$ USD/NZD.
4. A year from now, you obtain $(0.606-0.601)e^{-0.035}=0.0048$ USD/NZD of profit.

B) Suppose you have the financial resources to implement your trade in part A) at the 1M USD level (1M USD “notional”). How much profit would you make?
As calculated previously, I will make a profit of \57,000$.