Island Economy

Quarks and leptons are considered the fundamental particles of which matters in the physical universe are composed. So one might ask: what are the elemental components of securities or other financial instruments that make up our financial universe?

Kenneth Arrow and Gerard Debreu developed an approach, called the Time-State Paradigm, which characterizes promised future payments in terms of both the times at which payments are to be made and the states of the world that must occur for payments to be made.

  An island economy powered by the almighty coconuts. (Image Credit:    Haemimont Games   ).

An island economy powered by the almighty coconuts. (Image Credit: Haemimont Games).

Let's start with the simplest example involving both time and uncertainty. For simplicity, we consider an island economy whose only commodity is the coconut. There is no concept of money on the island, and so the coconut has become its native unit of currency. Naturally, the only productive investment on the island is growing coconut trees. The yield of the coconut trees in the next season is primarily influenced by island weather over the next year. If the weather is good (i.e., sunny), the coconut trees produce a bounty of 100 coconuts. If the weather is bad (i.e., rainy), the coconut trees instead produce only 70 coconuts.

For purpose of forecasting the island economy, we consider two time periods: (i) today, and (ii) a year from today. We also consider two possible future states of the island: (a) good weather; and (b) bad weather. Keep in mind that the states of the island are mutually exclusive (only one can occur), and exhaustive (one of them must occur).

According to Arrow-Debreu, there are a total of three elemental time-state claims in the island economy:

  • One coconut today
  • One coconut a year from today if the weather is good
  • One coconut a year from today if the weather is bad

These are called atomic time-state claims. Any investment vehicle can be considered to be composed of such atomic claims. For ease of exposition, these claims can be alternately described as follows:

  • One "Current Coconut" (CC)
  • One "Sunny Coconut" (SC)
  • One "Rainy Coconut" (RC)

A Friday market exists on the island that allows such time-state claims to be traded efficiently at zero cost. Dealers in the Friday market stand ready to trade such atomic claims and do so without cost as a service to the island.

For example, a "fair-weather dealer" specializing in "Sunny Coconuts" would be willing to trade (or swap): 0.4 coconuts today for 1.0 sunny coconut a year from now, or 1.0 sunny coconut a year from now for 0.4 coconuts today.

In standard financial terms, we can understand this obligation from the certificate issued by a coconut tree grower that reads as follows:

 A  "Sunny Coconut" security  issued by a coconut grower and traded on the Friday market by a "fair-weather dealer".

A "Sunny Coconut" security issued by a coconut grower and traded on the Friday market by a "fair-weather dealer".

This piece of paper would be called a security. Should the Island Credit-Rating Agency care to examine the property of the coconut growers (i.e., coconut trees) and is able to determine that no more than 100 pieces of such securities have been issued, and that there are no other claims on the growers' assets in the event of good weather, this security would be rated AAA (or "Triple-A") and can be considered "default-free".

The security in question, under these conditions, represents a property right in an atomic time-state claim. It is thus an "atomic security", or an Arrow-Debreu security. The price for this "Sunny Coconut" security is 0.4 current coconuts, because the dealer stands ready to trade this number of current coconuts for the security. The ability to make a trade is thus central to the definition of a price.

Outside of an island economy, dealers in the real world would charge more to sell a security than they pay to buy it. The difference between the ask and bid price is called the spread, and provides compensation for market-making. In practice, the mid-point between the bid and ask price is often used as a surrogate for "the price".

To complete our picture of the island economy, there is another dealer on the island who specializes in "Rainy Coconuts". This "foul-weather dealer" is willing to trade (or swap): 0.5 coconuts today for 1.0 rainy coconut a year from now, or 1.0 rainy coconut a year from now for 0.5 coconuts today.

The trading environment on the island now comprises of dealer-operated markets for trading: (i) CC and SC, and (ii) CC and RC. Note that each such trade has the characteristics of an investment today's coconuts are traded for the prospect of coconuts in the future. For example, one who purchases a SC security can be said to have invested 0.4 coconuts today to obtain 1.0 coconut in the future if the weather is good. Similarly, one who purchases a RC security can be said to have invested 0.5 coconuts today to obtain 1.0 coconut in the future if the weather is bad.

But what of other possible types of trades on the island? For example, swapping sunny coconuts for rainy coconuts (i.e., SC for RC)? How might this be accomplished? And what are the terms of the trade?

Consider someone who wishes to trade 1.0 sunny coconut for some number of rainy coconuts. One way to do this is to trade 1.0 sunny coconut for 0.4 current coconuts, while at the same time trading 0.4 current coconuts for 0.8 rainy coconuts. The net result is to swap 1.0 sunny coconut for 0.8 rainy coconuts. If a dealer were to offer any other terms of trade (i.e., other than 1.0 sunny coconut for 0.8 rainy coconuts), an astute trader would spot an opportunity to exploit this dealer by engaging in a combination of trades that could provide:

  1. net receipt of coconuts in at least one time and state; and
  2. no net payout of coconuts at any other time and state.

Such an opportunity, termed an arbitrage, is rare and fleeting in a well-functioning capital market. An arbitrage is thus a "coconut machine", and every trader's dream. When an opportunity of this type arises, traders will rush to exploit it, in the process causing dealers to adjust their terms of trade until arbitrage disappears. A set of swap terms that does not permit arbitrage is considered "arbitrage-free".

In a broad sense, every security transaction can be considered a swap. When one side of the swap involves current coconuts (e.g., as in the purchase of an atomic security), it is called an investment. Thus one invests current coconuts in the hope of obtaining more coconuts in the future. But the swap of sunny coconuts for rainy coconuts involves no net current coconuts and is not considered an investment; thus these types of swaps are referred to as "zero-investment strategies". Arbitrage would be a zero-investment strategy; it does not tie up capital for longer than is necessary to execute the combination of trades.

What is interesting here is that although markets are being made in only future atomic time-state claims, it is possible to synthesize trades involving any current and future claims. If one can trade each possible future atomic time-state claim for current units of a numeraire (e.g., coconuts), then any desired trade can be accomplished. Think of it as traversing a connected graph “from point to point” using only its “hub-and-spoke". Thus a set of atomic security prices is sufficient for accomplishing any desired trade.

In the real world, most traded securities represent patterns of payments over many states. After all, people rarely make explicit agreements for payments contingent on just a single state of the world. Therefore, one can think of these securities as “packages of atomic time-state claims”. For example, a correctly-priced forward contract for coconuts (rain or shine) would promise to pay 0.4444 coconuts next year no matter what, in exchange for a promise from the counterparty to deliver 1.0 coconut if the weather has been good and nothing otherwise. Furthermore, by combining existing securities, one can synthesize a security that does not exist. The result is often termed a derivative security, as it is derived from existing securities.

  Nobody knows coconuts like I do!

Nobody knows coconuts like I do!

It is difficult to imagine everyone agreeing on the probabilities associated with various future states of the world. Not even the wisest coconut tree growers on the island can accurately forecast the weather over the next year. Therefore, it is believed that security prices reflect a set of “consensus probabilities,” representing weighted averages of investors’ participation in the market. The differences among the investors arise from individual preferences, circumstances, and predictions. Nevertheless, they can all agree on one thing: an additional coconut is likely to be worth more in a state of scarcity than in a state of plenty. In other words, sunny coconuts are cheap and rainy coconuts are expensive. This is how William Sharpe describes the concept of risk premium: “it is not risk per se that that is likely to be rewarded in a well-functioning capital market – only the risk of doing badly in bad times.”

References:

  1. Sharpe, William F. (1993). Nuclear Financial Economics. Research Paper 1275, Stanford University. Retrieved from: http://web.stanford.edu/~wfsharpe/art/RP1275.pdf