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### Subject 6. Futures prices and expected spot prices

If you hold a risk-free asset which incurs carrying costs:

• S0 = [ST - FV (CB, 0, T)]/(1 + r)T: the spot price is the future spot price minus the future value of the carrying cost, all discounted to the present.

• By rearranging the equation, S0 = ST/(1 + r)T - FV(CB, 0, T)/(1 + r)T: the spot price is the discounted value of the future spot price minus the present value of the carrying cost.

However, at time 0 we normally don't know what ST (the spot price of the asset at time T) will be. We must form an expectation about this price: E0(ST). As this is our expectation, we must require a risk premium (denoted as Φ0(ST)) for the uncertainty. It represents a discount off the expected value that is embedded in the current price:

S0 = [E0(ST) - FV(CB, 0, T) - Φ0(ST)]/(1 + r)T

You can see that the risk premium lowers the current spot price. This reflects the fact that investors pay less for risky assets, all other things equal: investors are risk averse so they seek compensation to warrant their taking a risky position.

As f0(T) = S0(1 + r)T + FV(CB, 0, T), we can easily obtain the following formula:

f0(T) = E0(ST) - Φ0(ST)

This equation says that the futures price equals the expected future spot price minus the risk premium. It does NOT equal the expectation of the future spot price.

For example, a wheat farmer has a long position in cash wheat because he or she grows wheat. The farmer can reduce risk by selling wheat futures. Consider a speculator who is considering whether to take a long position in wheat futures.

• A rational speculator takes a long futures position only if the expected future spot price exceeds the current futures price. Otherwise the speculator must expect not to make any profit.
• The farmer, on the other hand, must be willing to sell the futures contract at a price below the expected future spot price of wheat. Otherwise the farmer cannot induce the speculator to accept the long side of the contract. From this point of view, the farmer in effect buys insurance from the speculator.

The farmer transfers his unwanted risk to the speculator and pays an expected profit to the speculator for bearing the risk. The payment to the speculator is the difference between the futures price and the expected future spot price.

There is one situation in which the risk premium could disappear or even turn negative. Suppose the farmer can find other parties who need to purchase wheat and who would like to hedge by going long. In that case it's possible for the two parties to consummate a futures transaction with the futures price equal to the expected spot price. In fact, if the parties going long exerted greater pressure than the parties going short, it might even be possible for the futures price to exceed the expected spot price.

When futures prices are lower than expected spot prices, the situation is called normal backwardation. This should not be confused with a market that is in backwardation which means that the futures price is lower than the spot price.

When futures prices are higher than expected spot prices, the situation is called normal contango.

#### Practice Question 1

Assume that the expected future spot price of a fictional fruit is \$10. The corresponding futures price is \$10.05. This market is in

A. backwardation.
B. contango.
C. normal contango.

When the futures price (\$10.05) is higher than the expected spot price (\$10), the situation is called normal contango.

#### Practice Question 2

If the risk premium in the spot price is transferred from hedgers to future traders (speculators), the futures price will be ______ the expected spot price.

A. equal to.
B. higher than.
C. lower than.
D. one of the above three situations is possible.

When the futures price is biased low(high), it is called normal backwardation (normal contango).

#### Practice Question 3

The spot price of an asset is

I. the future spot price minus the future value of the carrying cost, all discounted to the present.
II. the expected future spot price discounted to the present.
III. the discounted value of the future spot price plus the present value of the carrying cost.
IV. the discounted value of the future spot price minus the future value of the carrying cost.

See the equations presented in the notes.

#### Practice Question 4

Assume that the expected future spot price of a fictional fruit is \$10. The corresponding futures price is \$10.05. The spot price is \$10.1. This market is in

I. backwardation.
II. normal backwardation.
III. contango.
IV. normal contango.

A. I and II
B. III and IV
C. I and IV

When the futures price (\$10.05) is higher than the expected spot price (\$10), the situation is called normal contango. When the futures price (\$10.05) is lower than the spot price (\$10.1), the situation is called backwardation.

#### Practice Question 5

The spot price of an asset is

I. the future spot price minus the future value of the carrying cost.
II. the expected future spot price discounted to the present.
III. the discounted value of the future spot price plus the present value of the carrying cost.

A. I or III
B. II only