What do we agree on when we disagree? Forward contracts with private forecasts

1. What do we agree on when we disagree? Forward contracts with private forecasts Eddie Anderson, University of Sydney (joint work with Andy Philpott, Auckland) Rothschild Lecture, Cambridge 29 April 2019

2. Talk outline • Introduction: Contracts between players with different forecasts • The setting: Bilateral forward contracts in wholesale electricity markets • The model: Risk-averse firms with different forecasts for the spot price • Different methods to negotiate a contract: broker or direct negotiation. • An equilibrium in supply functions. • Deductions from the other firm's behaviour. • Conclusions

3. A betting game w = -1 Player A • There are two players (A and B) and two possible outcomes (which could be future prices): w = -1 or w = +1 . • The players disagree on the probabilities of the different outcomes. • Player A believes that the probability of the outcome w = +1 isrA= 0.6. Player B believes that the probability of the outcome w = +1 isrB= 0.4. (so player A thinks the high price is more likely than player B). ? 60% Player B ? 40% w =+1

4. Contracts – and the role of risk aversion w = -1 Player A • The players consider a possible contract under which player B pays to player A an amount of $ w (positive so player A gains if the price is high, but negative and so player A loses if the price is low). This is equivalent to a contract for differences with a strike price f= 0. • Both players expect to end up making profits (on average) with this contract. • If neither player is concerned about risk, then they would agree this contract with a payment of $ Qw for as large a quantity Q as possible. ? $1 $1 Player B w =+1

5. Forward contracts in an electricity market • Participants in a wholesale electricity market also take part in a derivatives market. The most important forward contracts traded are simple contracts for differences (CfD). These are bilateral contracts with a strike price f under which one party (the seller) agrees to pay an amount w -f to the other party (the buyer) where w is the average electricity price over an agreed period. • Forward contracts are financial instruments with payments depending on the price of electricity. The ‘right’ price depends on the expected price in the future. • Most contracts are "over-the-counter" (OTC) involving a bilateral agreement (possibly arranged through a broker).

6. The hedging context • Firms buy and sell contracts with the aim both of maximizing profits and hedging risks. A generator's contract position will often cover the majority of its output. • Forward markets are driven by hedging behaviour. Retailers sell at a fixed price but buy at the spot price: forward contracts that fix the price for the contract quantity protect them from price spikes. Generators have an opposite set of incentives. Generators and retailers are natural counterparties for forward trades. • In practice bids in the physical market will depend on contract positions, with higher contract levels tending to depress spot prices. But we focus on risk trading and assume the spot market price is unaffected by the contracts in place.

7. What do we aim to do? • We investigate contract negotiation when individual firms have some private information on future prices. Typically this information comes from in-house simulation or from consultants. • Firms can profit from their private information, at the same time as trading in order to hedge their risks. Sanda, Olsen and Fleten, Energy Economics, 2013, discuss the way that Norwegian hydro power companies use derivative trading to hedge risk, but at the same time make substantial amounts of money from this activity. • Since we are looking at bilateral trades it is natural to ask how the trading stance of a firm may indicate its private information. Can one side of the negotiation infer the other's private information about future prices and gain from this?

8. A simple model • Two players (A = retailer, B = generator). We use for the future price scenario and let and be the operating profits for the two players. • Suppose there is a contract (CfD) trading at price f and player A buys from player B an amount Q. Then, if price occurs, player A receives from player B an amount in addition to normal operating profits. • We use a utility function U to capture risk aversion. Where prices vary over a range we let and be the two player’s beliefs about the probability of different price outcomes. We get the expected utilities for the two players:

9. Nash bargaining Econometrica Vol. 18, No. 2 (Apr., 1950), pp. 155-162 (8 pages)

10. Bargaining framework • Players A and B are given a set of feasible joint actions (the feasibility set). For any option s in the feasibility set there is a utility to player A of u(s) and a utility to player B of v(s) . • One of the feasible options, d, is that there is no agreement and we suppose that this gives a utility of (u(d), v(d)) for the two players. • We can think of the feasibility set as being fully represented by the set S={(u(s), v(s))} in utility space. The players can achieve any point on the line between two options (either with a money transfer or tossing coins to see which outcome is chosen) so we can take S as convex. We also suppose it is compact. • For convenience we rescale utilities so (u(d), v(d)) is at the origin. • Consider a method of reaching agreement between the two players, assuming that they both know all the utilities involved.

11. Bargaining axioms • The method produces the same answer for any equivalent utility representations (linear scaling of utility) • (Pareto optimality) The method cannot end up at a point which is dominated for both players by another option. • (Independence of irrelevant alternatives) If the method produces a choice x and some options are removed from the feasibility set S but x is still feasible, then x will still be chosen. • (Symmetry) If the problem is symmetric ( so that for any option delivering utility (u, v) there is another option delivering utility (v, u) ) then the method chooses a solution where both players have the same utility. • Given these axioms the method will choose x to maximize the product (u(x) u(d)) (v(x) v(d))

12. Alternating offers and Nash bargaining • Suppose the players negotiate directly with each other – players alternate, each making an offer, before the other either accepts or makes a counter-offer. • After each offer there is a chance that the negotiations breakdown, and we get the disagreement outcome d. Given a specific probability of breakdown we can find the best strategy for each player. • In the limit as the probability of breakdown at any stage goes to zero we reach the Nash bargaining solution. • For our problem we assume that repeated offers of (price, quantity) pairs are made and the disagreement option is that the contract quantity is zero: Q = 0. Then the Nash bargaining solution is at

13. A simple broker mechanism • An alternative: each player submits to a broker its preferred contract quantity as a function of strike price f. The broker then determines a price at which the market for contracts clears. • The preferred contract quantities as a function of price are obtained from (∂/(∂Q)) ΠA(Q,f) = 0 and (∂/(∂Q)) ΠB(Q,f) = 0 • When there are just two outcomes and, andutility is CARA, so , and is the retailer’s probability of , then the retailer’s best contract quantity is and a similar expression for the generator. • The final outcome depends on the and values.

14. An example • Take , and . At the high price outcome the retailer has net operating profit and the generator has net profit . At the low price these numbers are reversed. Generator’s offer Retailer’s offer

15. An example • Take , and . At the high price outcome the retailer has net operating profit and the generator has net profit . At the low price these numbers are reversed. Generator’s offer Retailer’s offer

16. An example • Take , and . At the high price outcome the retailer has net operating profit and the generator has net profit . At the low price these numbers are reversed.

17. Essential equivalence to Nash bargaining solution • The simple broker solution has exactly the same Q values as the Nash bargaining solution, and the values are close. For this example we cannot easily distinguish the two: they match exactly on the dashed mid line. Prices differ by less than 0.2% at any point.

18. The Nash equilibrium • A Nash equilibrium is a set of actions such that no player wants to change their action, knowing the actions of the others. • Introduced by John Nash in Proceedings of the National Academy of Sciences of the United States of America, Vol. 36, No. 1 (Jan. 15, 1950), pp. 48-49 • Then an expanded treatment appeared in Annals of Mathematics, Vol. 54, No. 2 (Sep., 1951), pp. 286-295

19. The broker mechanism is not a Nash equilibrium • If the retailer knows the generator’s offer it is best to select a different point on the generator’s offer curve. Knowing the generator will sell less contracts if they are less expensive, changes the retailer’s best offer. Generator’s offer Retailer’s offer Best point for retailer

20. Strategic Broker offers: Best response • Suppose we know the set of offers (supply functions) used by the generator for different probability estimates • Given the retailer’s estimate there are a set of points that are preferred for each curve offered by the generator, i.e. maximizing its own profit . Best for retailer with

21. Supply function equilibria • A supply function equilibrium occurs with a set of bid curves for each player which are mutually best response • This will be a complete solution which describes the joint behaviour of retailer and generator. • Best response means that with an estimate , the retailer’s profit is maximized at . • There are a continuum of supply function equilibria possible. It turns out that there are two degrees of freedom in the choice of supply function equilibrium. • We can search for the supply function equilibrium that achieves the highest expected utility for both players.

22. How do we calculate the expected utility? • Both players are aware of Nature's distribution of possible values. This is the prior. In this example we take the prior as uniform on (0.4,0.6). • Nature selects the true value of according to the prior. • Both players take a sample of size N (N =10 here). On the basis of the number of and values in the sample they make a maximum likelihood estimation of and submit their offers. • The expected utility is obtained by taking expectations over Nature's choice from the prior. There is correlation between and (when Nature chooses a high both samples are likely to have a large number of values). • The simple broker mechanism achieves an expected utility of 0.388683 on the example.

23. Supply function equilibrium for the example • Expected utility 0.385757

24. Supply function equilibrium for the example • Expected utility 0.385782

25. Supply function equilibrium for the example • Expected utility 0.385601

26. Comparison with broker solution • The best achievable expected utility is lower than the broker solution • The volume of contracts is less, and the range of contract prices is less

27. Deduction from the other player's information • If the retailer faces a generator supply function with high contract quantities, then he can deduce that the generator thinks the price will be low. This information can be combined with the retailer's own information. • Suppose that the two players weight their own information as equivalent to the other player's information. So that given for the retailer and observing a supply function implying for the generator, then the retailer updates his expectation to . • This leads to (even) smaller contract quantities than without deduction (large contract quantities arise from very different beliefs about future prices).

28. Supply function equilibrium with deduction • Expected utility 0.384884

29. Supply function equilibrium with deduction • Expected utility 0.385339

30. Supply function equilibrium with deduction • Expected utility 0.384867

31. Supply function equilibrium with deduction • Expected utility 0.384034

32. Supply function equilibrium with deduction • Expected utility 0.384776

33. A comparison across different cases • All calculations based on a range for of (0.4,0.6)

34. Conclusions • This is the first model to look at derivative trading in a context where players have private information on future prices and trade partly to hedge risks and partly to make profits from their private information. • Direct bargaining, as modelled by the Nash bargaining solution, has results very similar to a simple broker model. • When players interact strategically there is a supply function equilibria. These often produce a worse overall outcome in comparison with simpler non-strategic models. • When both players combine their private information with deductions made from the other player's offers, there is a reduction in the contract quantities and the results do not improve. EVERYONE GETTING MORE CLEVER DOES NOT HELP