Proof that the Arrow-Debreu model is inappropriate for valuing pension liabilities

[Dan Moore]

Here is my current draft (a cumulative list of edits is at the bottom):

FE arbitrage (definition): A set of transactions that nets a market participant a risk-free, immediate profit that is scalable (i.e., the volume of the transactions could theoretically be increased without limit, thus increasing the profit without limit), and that is available to any market participant.

Assumption: Market participants for a company’s stock use pricing models based on all relevant information about that company which is likely to affect its future earnings. Necessarily excluded from such information is any information, which, if acted upon, would produce an FE arbitrage opportunity for a counterparty.

Theorem: The Arrow-Debreu model (a model that prices a company’s pension liability as it would a security that pays the company’s accrued pension benefits) is not an appropriate model to price pension liabilities.

Consider two companies A and B, identical in all respects, each with a pension plan. Company B executes a reversible, material change to their pension plan, while Company A does not. The change to Company B's pension plan is either:
i) B freezes accruals in their pension plan; or
ii) B changes their pension investment allocation to 100% Treasuries; or
iii) B does both (i) and (ii).
(Immediately before Company B's material pension change, the accrued benefits are identical under plans A and B, and both were invested 60% in equities, 40% in bonds.) In the argument below, pension change refers to the action B has taken - either (i), (ii), or (iii).

Immediately before the Company B pension change, Companies A and B have the same share price for their stock. At the time of the Company B pension change, all market participants for the stock of Companies A and B use their own models to predict future earnings of Companies A and B, and thus deduce whether the market price of each is overpriced or underpriced (or neither), and by how much. Market participants use their own models to price stocks. These models are internally consistent, in that the information applied to each model determines the market participant's predicted stock price. Therefore, the market price of two identical companies' shares will be equal.

The market participants’ models use all relevant information about Companies A and B which is likely to affect their future earnings (but not including any information, which, if acted upon, would create an FE arbitrage opportunity for a counterparty). Consider (1) a projection of the estimated effect of the pension benefits for both companies on top-line earnings (due to attraction and retention of employees), and (2) a projection of the estimated funding contributions and accounting expense for each company. (1) and (2), if introduced into the market participants’ models, would produce different pension-related effects on top-line earnings and expense for Companies A and B because a market participant’s model would typically use a higher probability of A continuing the plan unchanged in the future than B (because B has already changed their plan).

The question is whether (1) and (2) described above meet the stated criterion for information admissible into the market participants’ models. Because these projections involve elements of future corporate earnings, they meet the criterion unless, if acted upon, they would create an FE arbitrage opportunity for a counterparty. However, if market participants use the Arrow-Debreu model, the pension-related information admitted to all market participants’ models is identical for Companies A and B, because it can only be based on accrued pension benefits, and not on any projection of future accruals, and not on the investment allocation of assets set aside to meet the obligation. Therefore, if (1) and (2) are such that, if acted upon, they would not create an FE arbitrage opportunity for a counterparty, then the Arrow-Debreu model is not an appropriate model to price the pension liability component of a company.

We conduct a thought-experiment to see whether market participants’ actions based on (1) and (2) would create an arbitrage opportunity for a counterparty. We consider what their actions would be. As the stock market is well modeled by a Walrasian market, a market participant will buy fewer shares of B than A (if he considers them both underpriced), or sell more shares of B than A (if he considers them both overpriced), or sell B and buy A (if he considers B overpriced and A underpriced), based on the magnitude of under or overpricing, if his model predicts that the price of B should be less than A. And, the reverse, if his model predicts the price of A to be less than B. If his model predicts an equal price for A and B, there will be no change in his planned buying or selling for shares of A and B.

There are three cases for the net result of all market participants acting upon (1) and (2):

Case I: The action of the Walrasian market causes the share price of B to be less than that of A.

Following the market clearing transactions, the controlling shareholder of Company B (Mr. B) could observe the drop in their share price, and conclude that it resulted from changing the B pension plan, by comparing the B share price with A. Mr. B could then devise a plan to first buy up shares of B stock, while simultaneously selling short A stock, and then reverse the decision to change the pension plan (thus making Company B identical in all respects to Company A). As the two companies are once again identical, the share prices would become equal again, and Mr. B would have made an arbitrage profit.

However, there are two aspects to Mr. B’s arbitrage strategy that fail to meet the definition of FE arbitrage. It is a scalable strategy, but it is not available to all market participants. The only market participants who would be able to pursue this strategy are the select group of persons with insider knowledge that Company B plans to reverse the decision to change the pension plan. Also, Mr. B’s arbitrage strategy is not risk-free, because the act of conducting a trade while in possession of relevant nonpublic insider information for the purpose of making a profit or avoiding a loss is against the law in the US and other jurisdictions.

Any market participant other than one with inside knowledge of Company B lacks any basis for an FE arbitrage, because they cannot reliably know that Companies A and B will return to an identical state; their only idea of the relative prices of Companies A and B are from their own model, and from what the market dictates.

Case II: The action of the Walrasian market causes the share price of A to be less than that of B.

Following the market clearing transactions, the controlling shareholder of Company A (Mr. A) could observe the rise in the share price of B, and conclude that it resulted from changing the B pension plan, by comparing the B share price with A. Mr. A could then devise a plan to first buy up shares of A stock, while simultaneously selling short B stock, and then change Company A’s pension plan (thus making Company A identical in all respects to Company B). As the two companies are once again identical, the share prices would become equal again, and Mr. A would have made an arbitrage profit.
However, there are two aspects to Mr. A’s arbitrage strategy that fail to meet the definition of FE arbitrage. It is a scalable strategy, but it is not available to all market participants. The only market participants who would be able to pursue this strategy are the select group of persons with insider knowledge that Company A plans to change the pension plan. Also, Mr. A’s arbitrage strategy is not risk-free, because the act of conducting a trade while in possession of relevant nonpublic insider information for the purpose of making a profit or avoiding a loss is against the law in the US and other jurisdictions.

Any market participant other than one with inside knowledge of Company A lacks any basis for an FE arbitrage, because they cannot reliably know that Companies A and B will return to an identical state; their only idea of the relative prices of Companies A and B are from their own model, and from what the market dictates.

Case III: The action of the Walrasian market causes the share prices of Companies A and B to be equal.

Any market participant other than one with inside knowledge of either Company A or B lacks any basis for an FE arbitrage, because their only idea of the relative prices of Companies A and B are from their own model, and from what the market dictates. Market participants with insider knowledge of either Company A or B lack any strategy to profit by making the companies identical again, because they cannot expect any change in share price of either by doing so. They also lack any basis for FE arbitrage.

In all three cases, if (1) and (2) are acted upon, an FE arbitrage opportunity is not created for a counterparty. Therefore, the Arrow-Debreu model is not an appropriate model to price the pension liability component of a company.

Edits:
-Expanded description of the sense in which market participants are assumed to be rational.
-Initially B froze the pension plan. Generalized this to be a change either to freeze the plan, go with 100% Treasuries investment, or both.