Determination of Forward and Futures Prices

After completing this reading, you should be able to:

  • Differentiate between investment and consumption assets.
  • Define short-selling and calculate the net profit of a short sale of a dividend-paying stock.
  • Describe the differences between forward and futures contracts and explain the relationship between forward and spot prices.
  • Calculate the forward price given the underlying asset’s spot price, and describe an arbitrage argument between spot and forward prices.
  • Explain the relationship between forward and futures prices.
  • Calculate a forward foreign exchange rate using the interest rate parity relationship.
  • Define income, storage costs, and convenience yield.
  • Calculate the futures price on commodities incorporating income/storage costs and/or convenience yields.
  • Calculate, using the cost-of-carry model, forward prices where the underlying asset either does or does not have interim cash flows.
  • Describe the various delivery options available in the futures markets and how they can influence futures prices.
  • Explain the relationship between current futures prices and expected future spot prices, including the impact of systematic and nonsystematic risk.
  • Define and interpret contango and backwardation, and explain how they relate to the cost-of-carry model.

Investment Asset vs. Consumption Asset

An investment asset is an asset held for the purposes of investing. The holder takes a position in the asset in the hope of earning an income or capital gain. Examples include stocks and bonds issued by various financial institutions.

A consumption asset is an asset primarily held for the purpose of consumption, and not for investment or resale. Examples include oil, coffee, tea, corn, e.t.c.

Short-selling

Short selling involves the sale of a security which the investor does not own. The investor borrows the security from the owner, the lender, with a promise to return it as of a specified date. But why would anyone want to short sell?

The short seller has reason to believe that the security is overpriced or there are some other factors that make it highly likely that the security will lose value in the near future. Their goal, therefore, is to sell high and buy low and get to keep the difference (profit).

frm-short-sellingWhen the short sale is closed out, the short seller must return the security to the lender. The lender may also request to have the asset even before closeout, depending on the initial agreement. There’s always the risk that the security’s price will actually rise, forcing the investor to reacquire it at a higher price and thus make a loss.

Short sales are transacted through a broker. The short seller must deposit some collateral to guarantee the eventual return of the security to the owner. In addition, the short seller is required to pay all accrued dividends to the lender. Thus, the net profit is equal to:

$$ Net\quad profit=sale\quad price–borrowing\quad price–dividend\quad paid. $$

For example, if a trader shorts a stock today at $100, a dividend of $4 is paid next month, and the trader closes the short position the following month at $90, the net profit will be $100 – $90 – $4 = $6. His return would normally have been $10/$100 = 10%, but the dividend that he had to pay to the long position decreased his return to only ($10 – $4)/100 = 6%.

Forwards vs. Futures

Forwards and futures contracts have several similarities:

  • Are priced such that they have zero value at initiation
  • Can be either deliverable or cash settlement contracts.

However, unlike forwards which are non-standardized over-the-counter instruments, futures contract are standardized, exchange-tradable obligations to buy or sell a certain amount of an underlying good at a specified price, on a specified date.

Other Key Differences:

  • Exchange-tradable: Unlike forwards which trade on OTC markets, futures contracts are traded on an organized exchange with a designated physical location.
  • Standardization: With respect to forward contracts, specific details about quality to be delivered, price, and delivery date are subjects of negotiation between the buyer and the seller. In futures contracts, however, the choice of expiry dates is limited, and trades have fixed sizes. This standardization paves the way for an active secondary market where trades can be executed. However, perhaps the most pronounced benefit is increased liquidity.
  • Marking to market: Since the clearinghouse must monitor the credit risk between the buyer and the seller, it performs daily marking to market. This is the settlement of the gains and losses on the contract on a daily basis. It avoids the accumulation of large losses over time, something that can lead to a default by one of the parties.
  • Margins: Daily settlements may not provide a buffer strong enough to avoid future losses. For this reason, each party is required to post collateral that can be seized in the event of default. The initial margin must be posted when initiating the contract. If the equity in the account falls below the maintenance margin, the relevant party receives a margin call – a requirement to provide additional funds to restore the margin account to the initial level.
  • Clearinghouse: The clearinghouse is an interposed party between the buyer and the seller which ensures the performance of the contract. In essence, futures contracts have no credit risk. Each exchange has a clearinghouse. The clearinghouse splits each trade and acts as the opposite side of each position. It’s the buyer to every seller and seller to every buyer. In other words, there is no direct contact between the short and long parties. It’s the clearinghouse that makes margin calls whenever the need arises. In OTC markets, clearinghouses play a similar role.
  • Position limits: The number of contracts that a speculator can hold is capped at a certain value by the exchange. The aim is to prevent speculators from having an undue influence in the market.

The Forward Price vs. the Spot Price

If we assume that:

  • All earnings/profits are subject to the same tax rate
  • There’s unlimited borrowing and lending at the risk-free rate
  • There are no transaction costs
  • The market is efficient (arbitrage opportunities are quickly closed out to restore equilibrium)

Then, the relationship between spot prices and forward prices can be expressed as follows:

$$ { F }_{ 0 }={ S }_{ 0 }{ e }^{ rt }……………..equation\quad 1 $$

Where:

\({ F }_{ 0 }\)=forward price today,i.e,at \(t = 0\)

\({ S }_{ 0 }\)=Underlying asset (spot) price today

\(r\)=continuously compounded riskfree annual rate

\(T\)=time to maturity of the forward contract in years

The forward price (left side of Eq. 1) must equal the right side, i.e., the cost of borrowing funds to buy the underlying asset and carrying it forward to time \(T\).

If \({ F }_{ 0 }<{ S }_{ 0 }{ e }^{ rt }\), an arbitrageur can make a risk-free profit by selling the asset, lending out the proceeds, and buying the forward.

If \({ F }_{ 0 }>{ S }_{ 0 }{ e }^{ rt }\), an arbitrageur can make a risk-free profit by selling the forward and buying the asset with borrowed funds.

Forward Price with Carrying Costs

Carrying costs are any cash flows associated with the underlying asset over the life of the forward contract. The owner of the forward contract does not receive any of these cash flows, and therefore, their present value, call it\(P\), must be deducted from the spot price when determining the forward price. Thus,

$$ { F }_{ 0 }=\left( { S }_{ 0 }-P \right) { e }^{ rT } $$

If the cash flows are in the form of dividends paid at a continuously compounded rate \(q\), then:

$$ { F }_{ 0 }={ S }_{ 0 }{ e }^{ \left( r-q \right) T } $$

Value of a Forward Contract

It’s important to note that at initiation, a forward contract has zero value. The contract can only gain value once it has already commenced. If \(K\) represents the obligated delivery price, then the value of the contract to the long is given by:

\({ value }_{ forward }={ S }_{ 0 }-K{ e }^{ -rT }\) if the underlying has no carrying costs, or

\({ value }_{ forward }={ S }_{ 0 }-P-K{ e }^{ -rT }\) if the underlying has cash flows with a present value of \(P\),or

\({ value }_{ forward }={ S }_{ 0 }{ e }^{ -qt }-K{ e }^{ -rT }\) if the underlying pays dividends at a continuously compounded rate \(q\)

Computing Foreign Exchange Rates Using the Concept of Interest Rate Parity

If we let:

\({ F }_{ 0 }\) to be the forward exchange rate

\({ S }_{ 0 }\) to be the spot exchange rate

And \(\left( r-{ r }_{ f } \right) \) to be the interest rate differential between the domestic currency and the foreign currency,

Then,

$$ { F }_{ 0 }={ S }_{ 0 }{ e }^{ \left( r-{ r }_{ f } \right) T } $$

Example

A German trader invests in a 1.5-year currency futures contract on the U.S. dollar. The risk-free interest rate in the Eurozone is 1.25%. The U.S. risk-free rate is 1.5% and the spot exchange rate is 1.098 USD per Euro (USD 1.098/EUR). What is the 1.5-year futures exchange rate?

In this case, the domestic rate is the U.S. and the foreign rate is the Euro.

$$ { F }_{ 0 }={ S }_{ 0 }{ e }^{ \left(US \quad rate-Euro \quad rate \right) T } $$

$$ { F }_{ 0 }={ 1.098 }{ e }^{ \left(1.5\%-1.25\% \right) 1.5 } $$

Since the U.S. risk-free rate is greater than the Euro risk-free rate, the futures exchange rate must be greater than the spot exchange rate.

Income, Storage Costs, and Convenience Yield

The above relationship between forward prices and spot prices are only valid for investment assets. When it comes to consumption assets, we have what we call storage costs. For example, a forward contract on several tonnes of corn must have warehouse costs.

If the storage cost is a fixed cost \(U\) that’s independent of the value of the underlying asset, then:

$$ { F }_{ 0 }=\left( { S }_{ 0 }+U \right) { e }^{ rT } $$

If the storage cost \(u\) is a percentage of the underlying asset (yield), then:

$$ { F }_{ 0 }={ S }_{ 0 }{ e }^{ \left( r+u \right) T } $$

Convenience yield is the additional value that comes with holding the asset rather than having a long forward or futures contract on the asset. A good example of a consumption asset that has convenient yield is oil. If you hold oil, you’ll have the convenience of selling it at a higher price during a shortage.

If a forward contract has a storage cost, \(u\) expressed as a percentage of the underlying, as well as a convenient yield \(y\), then:

$$ { F }_{ 0 }={ S }_{ 0 }{ e }^{ \left( r+u-y \right) T } $$

Example

The price of a 3-month crude oil futures contract (CL) is USD 62.50. The risk-free rate is 2%, the storage cost is 10%, and the convenience yield is 1%. What is the current price of crude oil?

$$ { F }_{ 0 }={ S }_{ 0 }{ e }^{ \left( r+u-y \right) T } $$

$$ { S }_{ 0 } = \frac { { F }_{ 0 } } { { e }^{ \left( r+u-y \right) T } } $$

$$ { S }_{ 0 } = \frac { 62.50 } { { e }^{ \left(2\%+10\%-1\% \right) 0.25 }  }  = 60.80 $$

Since the risk-free rate and the storage cost outweigh the convenience yield, the spot is lower than the futures price.

How Backwardation and Contango Relate to the Cost-of-carry Model

Backwardation refers to a situation where the futures price is below the spot price. It occurs when the benefits of holding the asset outweigh the opportunity cost of holding the asset as well as any additional holding costs.

Contango refers to a situation where the futures price is above the spot price. It is likely to occur when there are no benefits associated with holding the asset, i.e., zero dividends, zero coupons, or zero convenience yield.

frm-part-1-backwardation-contangoQuestions

Question 1

Consider a forward contract on a stock index such as the S&P 500. Everything else being constant, which of the following statements is least accurate?

  1. The forward price will fall if interest rates rise
  2. The forward price is directly linked to the level of the stock market index
  3. If the time to maturity is increased, the forward price will rise
  4. The forward price will fall if dividend payments on the underlying stocks increase

The correct answer is A.

Increasing the level of interest rates \(r\) makes the forward contract more appealing to investors. Thus, the forward price will increase.

Question 2

The one-year \(U.S.\quad dollar\) interest rate is \(1.5\%\), and the one-year \(GBP\) interest rate is \(2.0\%\). The current USD/GBP spot exchange rate is \(0.85\). Assuming annual compounding, calculate the one-year USD/GBP forward rate.

  1. 0.8825
  2. 0.7575
  3. 0.8520
  4. 0.8458

The correct answer is D.

If we assume annual compounding, then:

$$ { F }_{ 0 }={ S }_{ 0 }\frac { \left( 1+r \right) }{ \left( 1+R \right) } $$

Where:

\({ F }_{ 0 }\)=forward exchange rate

\({ S }_{ 0 }\)=spot exchange rate

\(r\)= domestic interest rate

\(R\) = foreign interest rate

\({ F }_{ 0 }=0.85\frac { 1.015 }{ 1.02 } =0.8458 \)

Exam tip: All prices \(\left( { S }_{ 0 },{ F }_{ 0 } \right) \) are measured in the domestic currency. Unless directed otherwise, you’re supposed to apply the indirect quotation methodology in the exam. Under the method, an \({ A }/{ B }\) quote has \(A\) as the base currency, and \(B\) as the quoted currency, and the base currency is ALWAYS the domestic currency. The base currency (in this case, the U.S. dollar) is always equal to one unit (in this case, US$1), and the quoted currency (in this case, the GBP) is what that one base unit is equivalent to in the other currency. That is,\(1\quad USD=0.85\quad GBP\).


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