Introducing arbitrage opportunities in commodity forwards

The author Robert McDonald introduced about some of the commodities, whose prices remain constant over a period of time.

When the prices remain constant, does it make any sense to store them? No, because traders or investors purchase or store commodities, expecting a price hike. If there is no price hike, it does not make sense to buy them and store them and to incur storage costs.

The above arguments are not applicable for financial assets.

The author introduced the prices of pencils, assuming that prices will remain constant.

Let us assume that 10 pencils will cost us $5.00 as of today.

Presently let us assume the risk-free rate in the market is 6% for one year maturity.(USD);

Let us calculate the forward price of these pencils after one year;

The forward price of any commodity is given by: S₀ x e^rT; S₀ = $5.00, r = 6% and T = 1 year

Therefore, the forward price of these pencils is expected to be: $5.00 x e^0.06x1 = $5.31;

The forward price formula says that after one year, the price of 10 pencils is expected to be $5.31

But do we enter into a forward agreement?

No..., because, the prices of pencils is expected to be constant. No one would buy at $5.31, when the pencils even after one year are available at $5.00;

Will there be any arbitrage, if the prices remain constant?

Yes... how?

Let us borrow 10 pencils from anyone for one year, who is holding them. Sell those 10 pencils today and get $5.00 and invest the proceeds for one year at the rate of 6%; after one year, we will get from the investment an amount of $5.31; then after one year get ten pencils from the market; return those 10 pencils to the holder of the pencils. Keep the profit of $0.31;

Or, borrow 10 pencils from the holder of pencils; Sell them in the cash market and get $5.00 and invest that $5.00 at 6%; and today only enter into a long forward agreement to buy at $5.00 after one year. After one year we will get $5.31 from our investment and buy 10 pencils at $5.00 as per long forward agreement and return those pencils to the lender. Keep the profit of $0.31;

This means, if the prices remain constant, there is an arbitrage opportunity. 

What does it mean and how to deal with such a situation? First of all prices cannot be constant. Secondly, even if the prices are constant, the lender who is lending 10 pencils for one year, is it possible that he will lend those pencils, without any benefit to him?... Let us discuss all these things in the next article.

Thanks a lot... Surya

Expression of thought

I make it clear, reading and understanding core readings prescribed by GARP is the only mantra of success. 

All my efforts are to bring these core readings in small articles in an understandable language with examples and case studies. 

The blog is intended to bring the core readings more close to the students. If we have to read and understand the core readings on our own, it would be a little difficult task, as most of the FRM students are working professionals. 

I will never say that by following my blog, you can be become master of core readings. But If you read these articles, I hope many of you would get conceptual clarity.

No one is a master of the subject. Every day in life, it is a learning day. I again request you to send me your queries to my mail id mentioned below. I will try to solve your queries within very short time as much as I can.

If these articles are useful to the student community, it would give me immense happiness and satisfaction. God has blessed me a good life and I am happy with what I have.

I am not writing just because someone cares or does not care. I will surely keep on writing, till the time it gives satisfaction to me, till the time I get satisfaction of such work, without hurting anyone, without cheating anyone, without charging anyone.

I hope the message is clear and transparent.

cfa.surya@gmail.com

Case study on storable commodities

Presently the corn prices are trading at 760 cents per bushel in the spot market. The risk free rate of interest is 2.35% (USD). The trader expects to receive a price of 785 cents per bushel after one year. Using the risk-free rate, forward price quotes are available in the market for 777.85 cents per bushel. At what discount rate the expected spot price after one year E₀(S_T) can be discounted? Can there be a situation of quoting 785 cents per bushel by the trader? Under what circumstances the trader can quote more than 777.85 cents per bushel.

Please understand the concepts for commodities that can be stored.

In this case E₀(S_T) is given. That is the expected price of the commodity, according to the trader. That is given to be 785 cents per bushel. According to McDonald, the present value of the forward price should be equal to the present value of the Future expected prices.

Mathematically: F_{(0, T)} x e^-rT =  (should be equal to) E₀(S_T) x e^-αT

The present value of forward price = 777.85 x e^{-0.0235 x 1 year} = 760 cents per bushel

The present value of  E₀(S_T) = 785 x e^{-α T}; using alpha = 3.1%, it can also be equated to 760 cents per bushel.

This means, the trader is expecting more than the risk free rate of 2.35% in the forward prices. The additional rate expected by the trader is 3.1% - 2.35% = 0.75%.

An additional yield of 0.75% is expected by the trader more than the risk-free rate. The reasons are many reasons to include this 0.75%. It may be due to demand and supply or due to cost of carry or for any other reasons.

What we need to understand here in this discussion is:

According to McDonald, the forward price formula for storable commodities is given by:

F_{(0, T)} = E₀(S_T)e^(r-α)T  (In the yesterday’s discussion, “exp” was missing in the formula, today rectified. I am sorry)

Here ‘r’ is the risk free rate and alpha is the discount rate used to convert the future expected price of the commodity to present value. In reality, risk free rate is lower than the appropriate discount rate “α”. Because, the traders include other factors like cost of carry, convenience yield or other factors in the forward price. Assuming that the forward prices quoted using the risk-free rate would lead to wrong assumptions.

Formula for Commodities that can be stored

1.         It is not possible to predict the price of a commodity at Time “T”.   2.        The future expected price of a commodity at time “T” is denoted with ST. But we do not know what will be the exact price. 3.        Let us assume that price (ST) at time T0. Let this be denoted with E0(ST). E0 refers to the expectations at Time T0. 4.        Therefore, E0(ST) can be discounted to present value, using an appropriate discount rate “α”. 5.        At time T0 :  The present value of the E0(ST), then will be : E0(ST) x e-αT 6.        Now, let F(0, T) be the forward price of the commodity at Time “T”. 7.        Even this can be converted to present value, by discounting at risk free rate. 8.        The present value of the forward price of the commodity is: F(0, T) x e-rT 9.        In the absence of arbitrage, the present value of the forward price of the commodity should be equal to the present value of E0(ST) 10.      In other words: F(0, T) x e-rT =  (should be equal to) E0(ST) x e-αT 11.       Or, when we take the e-rT to the other side, the formula for F(0,T) becomes as: F(0, T) = E0(ST)e(r-α)T According to Robert McDonald, the above formula is  used when a commodity can be stored. 12.      Examples of commodities that cannot be stored is electricity (Once produced should be consumed) 13.      Reasons for non-storability: a.        Seasonal demand and supply b.        Intra-day variation of prices

Synthetic Commodity strategies

After a long time of break, I have come back. Thanks to all the readers for their continued support.

Brief discussion on forward contracts: 

Let us take two parties A and B in a commodity market. The party A is the buyer and party B is the seller in forward market. 

This means party A has agreed to buy in future or taken a long position and party B has taken a short forward position. 

Let us suppose the price of the commodity per bushel = $961.54 cents per bushel (Spot price or S₀)

Let us suppose that they agreed into a forward contract and fixed the strike price (X) = $1000

The forward price after a time T is denoted with F_(0,T).

Today’s price is denoted as S₀ and forward price of the commodity is denoted with F_(0,T).

Please understand that the price of the commodity at time “T” can take any price. That is decided by the market demand and supply factors at time “T”. 

Suppose the price of the commodity at time T is $1050. In that case, the trader with long position will buy as per the forward contract at $1000 cents per bushel from party B and sell in the market for $1050, to gain $50 cents per bushel. The gain is generally denoted as (S_T – X), where S_T is the spot price of the commodity at time T_T and “X” is the agreed price for the long forward position holder.

Suppose the price of the commodity at time T is $950. In that case, the trader with long position will have to buy as per the forward contract at $1000 cents per bushel. He is at loss, because, the same commodity is available in the market for $950.

This means that forward contracts do have the volatility risk of the underlying assets. The agreed time to maturity of the contract will also have an impact on forward contracts.

How do we reduce such affects?

Let us purchase a zero coupon bond with future maturity par value of $1000, with remaining maturity of one year. Let us suppose the price of the bond as of now (T₀) is $961.54 and on maturity the bond will pay us par value of $1000.

Let us now enter into a long forward contract where the agreed forward price is equal to $1000. Please understand at the time of entering into a forward contract, there is no cash outflow or expenses (Initiation level).

Let us suppose the present risk-free rate in the market is 5% per annum (USD).

Let us denote the forward price of the commodity as F_(0,T). This means the forward contract is for the period T₀ to T_T and F_(0,T) is equal to $1000 ( as per the above discussion).

What is the present value of this forward contract?

It is equal to S₀ x e^-rt = $1000 x e^{-0.05 x 1}  = $961.54 

(where, 0.05 is the risk free rate and 1year is the remaining period for the future contract to mature).

This means the present value of the forward contract is equal to = $961.54

Also the present price paid for buying the zero coupon bond is also equal to = $961.54

Therefore at time T₀:

The cost of buying the zero coupon bond =The present value of forward contract = F_(0,T) x e^-rt and it is equal to $961.54

At maturity or at time T_T, the total pay off of the long forward contract and the zero coupon bond will be as below:

(S_T – X) [ pay-off or gain from the long forward contract] + $1000 from bond par value 

 “$1000 from bond par value” can be written as F_(0,T). Because the par value of the bond is equal to the agreed forward price.

Or the agreed forward price “X”  = F_(0,T) = Bond par value.

Therefore the total pay-off from the long forward contract and bond can be written as:

(S_T – F_(0,T)) + F_(0,T) = S_T

The above strategy is known as synthetic commodity creation.

I summarize this strategy as below to create a synthetic commodity as:

Long zero coupon bond + Long forward contract = Synthetic commodity price at time T.

In this the par value of the bond is equal to the agreed forward price of the contract. Such strategies are used to avoid the impact of price volatilities on the underlying assets.