Chapter 6 - Commidity futures and forwards

Question 1

most important consideration of commodities

Answer 1

they have indivuidual features that affect their price specifically. for instance storage costs.

Question 2

Recall the established forward price for a financial forwrd contract

Answer 2

F_{0,T} = S_0 e^(r-∂)T

Question 3

the book highlight some key differences between futures/forwards on financials and on commidities. These are?

Answer 3

Storage costs

Carry markets

Lease rate

convenience yield

Question 4

define a carry market

Answer 4

A carry market is one where the forward price compensates the commodity owner for storage costs.

Question 5

two terms on the forward curves

Answer 5

Contango, Backwardation

Question 6

classify commidities in two broad categories

Answer 6

extractive and renewable

Question 7

give the formula for the prepaid forward price for a commidity

Answer 7

similar as for financial forwards:

F^P_{0,T} = E[S_T] e^(-aT)

note that we discount ising alpha.

Question 8

Relate the prepaid forward price of a commidity to the forward price

Answer 8

As alwayss, the forward price is the future value of the prepaid forward price:

F_{0,T} = e^(rT) F^P_{0,T}

F_{0,T} = e^(rT)e^(-aT)E[S_T]

F_{0,T} = E[S_T] e^((r-a)T)

or alternatively:

F_{0,T} = E[S_T] e^(-(a-r)T)

So, the forward price is equal to the expected value of the actual future spot price multuplued the by net effect of the risk free return advantage we have vs the discount effect.

Question 9

elaborate on the equilibrium condition for commodities like copper, and relate it to stocks

Answer 9

The equilibrium condition for copper is primarily based on extraction costs, not storage costs. This is because the copper is stored safely in the ground. we can extract it if we need it.

Stock on the other hand, has to be stored.

so the thing to understand about commodities like copper is that no one will store them (at least for long periods of time) because this is simply a large storage cost and nothing else.

Question 10

give the “apparent” arbitrage opportunity on a commidity like copper.

Also give why this is not piossible

Answer 10

Assuming the price is equal to the forward price, we have this case:

Long the forward contract.
Short the underlying.
Invest the shit we get from the short in risk free profit.
At expiration, we would have the amount needed for the contract AND additional risk free rate.
this means that we have earned arbitrage.

This is not possible because of the short sale.
First of all, it is difficult to find copper to short, because very few people will have it available without specific intentions on using it. Therefore, your broker will struggle to find it.
Second, since there is a cost of storage, if we find some assets to short, the lender will require a premium on it, which we refer to as the lease.

Question 11

elaborate on the lease rate in regards to what is should be

Answer 11

Reference case:
Spot price copper: 3.00
Forward price according to our calculations: 3.30 (just simpel 10%)
Actual forward price: 3.00

The lender of the commodity will identify that we could make arbitrage if his lease is not appropriate. Since we’d make 10% on the price differential when we earn the risk free interest, the lender will take this.

Lease = (1+r)xSpotPrice - F_{0,T}
Lease = 3.30 - 3.00
Lease = 0.3
This payment of 0.30 is the same as the amount that we’d make in profits otherwise.

we can compute the net present value of the investment from the perspective of the lender:

The lender acquire the asset for the current spot price S_0.
Then he gives away. When we receives it back, it will have appreciated in value, and is now worth: E[S_T]. to find the present value of this, we discount using alpha:
E[S_T]e^(-aT)
However, we lend it out and require a lease rate to compensate.
Therefore, we end up with a position that is worth:
E[S_T]e^(-aT)e^(lT), and we finally subtract the spot price we used to purchase the asset:

NPV = E[S_T]e^(-aT)e^(lT) - S_0

We can find the lease rate by setting NPV to 0:

S_0 = E[S_T] e^(-aT)e^(lT)

S_0/E[S_T] = e^(l-a)T

l-a = 1/T ln(S_0/E[S_T])

Question 12

elaborate on storage costs

Answer 12

Sometimes, storage is not feasible. for instance strawberries, electricity

Question 13

if you are a commodity merchant, and you have the option of either selling a commodity now or storing it to time T and sell it then, what will you do=

Answer 13

You will store only if the present value of the forward price is at least as great as the spot price today.

Question 14

when is cash and carry arbitrage NOT profitable

Answer 14