Futures prices are affected by various costs and benefits associated with holding the underlying asset.

Different types of costs are:

• Opportunity cost: all assets have this cost - the opportunity cost of money tied up in the asset. The effect, however, has been included in the present value calculation.
• Storage costs: costs other than the opportunity cost to holding an asset. Also called carrying costs. Examples are costs to store physical underlying assets (i.e., oil, wheat), insurance costs, etc.
  • They are generally a function of the physical characteristics of the underlying asset, the quantity of the asset to be stored, and the length of time in storage.
  • Financial assets have virtually no storage costs.

If we take storage costs into account, to avoid an arbitrage opportunity, the futures price must equal the spot price compounded over the life of the futures contract at the risk-free rate, plus the future value of the storage costs over the life of the contract.

f₀(T) = S₀(1 + r)^T + FV(SC, 0, T)

Where:

• FV(SC, 0, T): the value at time T(expiration) of the storage costs (excluding opportunity costs) associated with holding the asset over the period 0 - T. It's assumed that when storage is initiated, these costs are known.

When a trader buys an asset and sells a futures contract to create a risk-free position, the futures price must be higher by enough to cover the storage costs.

Different types of benefits are:

• Cash flows paid on the underlying asset, such as interest and dividends.
  f₀(T) = S₀(1 + r)^T - FV(CF, 0, T)
  Where:
  • FV(CF, 0, T) is the future value of all cash flows at time T(expiration).

• Non-monetary benefits of holding the underlying asset, referred to as the convenience yield (The benefit or premium associated with holding an underlying product or physical good, rather than the contract or derivative product). Sometimes, due to irregular market movements such as an inverted market, the holding of an underlying good or security may become more profitable than owning the contract or derivative instrument, due to its relative scarcity versus high demand. An example would be purchasing physical bales of wheat rather than future contracts. Should there be a sudden drought and the demand for wheat increases, the difference between the first purchase price of the wheat versus the price after the shock would be the convenience yield.

As a cost incurred from holding the asset increases the futures price, a benefit (monetary or non-monetary) generated from holding the asset should result in a lower futures price. They are just different sides of the same token.

f₀(T) = S₀(1 + r)^T + FV(CB, 0, T)

Where:

• FV(CB, 0, T) is the cost of carry: the future value of the costs of storage minus benefits (costs of storage - convenience yield). It can be either a positive or negative number.

This model is called the cost-of-carry model. It is the most general form of the futures pricing formula we shall encounter.

Practice Question 1

If an asset is priced at $45, the interest rate is 4.35%, the future value of the storage costs is $1.15, and the future expires in 9 months, what would the futures price be?

Correct Answer: F₀(T) = 45 (1.0435) ^9/12 + 1.15 = $47.61.

Practice Question 2

If an asset is priced at $45, the interest rate is 4.35%, the future value of the storage costs is $1.15, the future value of positive cash flows on the underlying asset is $0.80, and the future expires in 9 months, what would the futures price be?

Correct Answer: F₀(T) = 45 (1.0435) ^9/12 + 1.15 - 0.8 = $46.81.

Practice Question 3

Oil is currently trading at $30/barrel in the spot market and its 6-month futures price is currently trading at $31/barrel. If there are not storage costs or income from holding iol, what would be the implied repo rate?

A. 3.33%.
B. 5.24%.
C. 6.77%.

Correct Answer: C

According to the cost-of-carry model: F₀ = S₀[1 + (repo rate)]^{(contract life / 360)} + (Storage Cost) - (Income from Asset)

31 = 30 [ 1 + (repo rate)]^(180/360) + 0 => repo rate = 6.77%.

Practice Question 4

Assume that markets are perfect in the sense of being free from transaction costs and restrictions on short selling. The spot price of gold is $450/ounce. Current interest rates are 4.35%, compounded monthly. The present value of the storage costs per ounce is $11.5. According to the cost-of-carry model, what should the price of a one-ounce gold futures contract be if expiration is nine months away?

A. $476.1.
B. $481.2.
C. $476.5.

Correct Answer: C

F₀(T) = 450 (1.0435) ^9/12 + 11.5 (1.0435) ^9/12 = $476.5

Note that the present value of the storage costs is given so you need to compound it to get the future value in nine months.

Practice Question 5

If an asset is priced at $45, the interest rate is 4.35%, the future value of the storage costs is $1.15, the previous settlement price was $46, and the future expires in 9 months, what would be the value of a long futures contract an instant before marking to market at the end of today?

A. $1.61.
B. $1.65.
C. $0.46.

Correct Answer: A

The price of the futures contract should be F0(T) = 45 (1.0435) ^9/12 + 1.15 = $47.61. The value of a long futures contract equals the difference between the two prices: $47.61 - $46 = $1.61.

Practice Question 6

In the spot market, an oil dealer is quoting a bid of $30.25/barrel and an ask of $30.83/barrel. At the same time, funds may be borrowed at 9% and lend at 8%. If there are no storage costs or income involved from holding oil, what would be the no-arbitrage price range for the 6-month oil contract?

A. 30.54-31.57
B. 31.44-32.19
C. 31.94-33.60

Correct Answer: B

High end price assumes that oil is bought at the high price and the higher borrowing rate is used. F_HI = S₀ [ 1 + R_F (t/360)] + Storage Cost + Income = 30.83 [1 + (0.09)]^(180/360) + 0 = $32.19.

Low and futures price uses the lower price of oil and the low rates at which funds may be lent out. F_LOW = 30.25 [1 + (0.08)]^(180/360) + 0 = $31.44.