EQUILIBRIUM VALUATION AND RATIONAL EXPECTATIONS

5.1 EQUILIBRIUM VALUATION AND RATIONAL EXPECTATIONS

Fundamental notions such as rational expectations, risk-neutral pricing, complete and incomplete markets, underlie the market’s valuation of risk and its pricing of derivatives assets. Both economics and mathematical finance use these concepts for the valuation of options and other financial instruments. Rational expectations presume that current prices reflect future uncertainties and that decision makers are rational, preferring more to less. It also means that current prices are based on the unbiased, minimum variance mean estimate of future prices. This property seems at first to be simple, but it turns out to be of great importance. It provides the means to ‘value assets and securities’, although, in this approach, bubbles are not possible, as they seem to imply a persistent error or bias in forecasting. This property also will not allow investors to earn above-average returns without taking above-average risks, leading to market efficiency and no arbitrage. In such circumstances, arbitrageurs, those ‘smart investors’ who use financial theory to identify returns that have no risk and yet provide a return, will not be able to profit without assuming risks.

The concept of rational expectation is due to John Muth (1961) who formulated it as a decision-making hypothesis in which agents are informed, constructing a model of the economic environment and using all the relevant and appropriate information at the time the decision is made (see also Magill and Quinzii, 1996, p. 23):

I would like to suggest that expectations, since they are informed predictions of future events, are essentially the same as the predictions of the relevant economic theory . . . We call such expectations ‘rational’ . . . The hypothesis can be rephrased a little more precisely as follows: that expectations . . . (or more generally, the subjective probability distribution of outcomes) tend to be distributed, for the same information set, about the prediction of the theory (the objective probability of outcomes).

In other words, if investors are ‘smart’ and base their decisions on informed and calculated predictions, then, prices equal their discounted expectations. This hypothesis is essentially an equilibrium concept for the valuation of asset prices stating that under the ‘subjective probability distribution’ the asset price equals the

Risk and Financial Management: Mathematical and Computational Methods. C. Tapiero C 2004 John Wiley & Sons, Ltd ISBN: 0-470-84908-8

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expectation of the asset’s future prices. In other words, it implies that investors’ subjective beliefs are the same as those of the real world – they are neither pessimistic nor optimistic. When this is the case, and the ‘rational expectations equilibrium’ holds, we say that markets are complete or efficient. Samuelson had already pointed out this notion in 1965 as the martingale property, leading Fama (1970), Fama and Miller (1972) and Lucas (1972) to characterize markets with such properties as markets efficiency.

Lucas used a concept of rational expectations similar to Muth to confirm Milton Friedman’s 1968 hypothesis of the long-run neutrality of the monetary policy. Specifically, Lucas (1972, 1978) and Sargent (1979) have shown that eco- nomic agents alter both their expectations and their decisions to neutralize the effects of monetary policy. From a practical point of view it means that an investor must take into account human reactions when making a decision since they will react in their best interest and not necessarily the investor’s.

Martingales and the concept of market efficiency are intimately connected. If prices have the martingale property, then only the information available today is relevant to make a prediction on future prices. In other words, the present price has all relevant information embedding investors’ expectations. This means that in practice (the weak form efficiency) past prices should be of no help in predicting future prices or, equivalently, prices have no memory. In this case, arbitrage is not possible and there is always a party to take on a risk, irrespective of how high it is. Hence, risk can be perfectly diversified away and made to disappear. In such a world without risk, all assets behave as if they are risk-free and therefore prices can be discounted at a risk-free rate. This is also what we have called risk- neutral pricing (RNP). It breaks down, however, if any of the previous hypotheses (martingale, rationality, no arbitrage, and absence of transaction costs) are invalid. In such cases, prices might not be valued uniquely, as we shall see subsequently.

There is a confrontation between economists, some of whom believe that mar- kets are efficient and some of whom do not. Market efficiency is ‘under siege’ from both facts and new dogmas. Some of its critics claim that it fails to account for market anomalies such as bubbles and bursts, firms’ performance and their relationship to size etc. As a result, an alternative ‘behavioural finance’ has sought to provide an alternative dogma (based on psychology) to explain the behaviour of financial markets and traders. Whether these dogmas will converge back together as classical and Keynesian economics have, remains yet to be seen. In summary, however, some believe that the current price imbeds all future information. And some presume that past prices and behaviour can be used (through technical anal- ysis) to predict future prices. If the ‘test is to make money’, then the verdict is far from having been reached. Richard Roll, a financial economist and money manager argues:

‘I have personally tried to invest money, my clients’ and my own, in every single anomaly and predictive result that academics have dreamed up. And I have yet to make a nickel on these supposed market inefficiencies. An inefficiency ought to be an exploitable opportunity. If there is nothing that investors can exploit in a systematic way, time in and time out, then it’s very hard to say that information is not being properly incorporated into stock prices. Real money investment strategies do not produce the results that academic papers say they

113 should’ . . . but there are some exceptions including long term performers that have over the

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years systematically beaten the market. (Burton Malkiel, Wall Street Journal, 28 December 2000)

Rational expectation models in finance may be applied wrongly. There are many situations where this is the case. Information asymmetries, insider trading and ad- vantages of various sorts can provide an edge to individual investors, and thereby violate the basic tenets of market efficiency, and an opportunity for the lucky ones to make money. Further, the interaction of markets can lead to instabilities due to very rapid and positive feedback or to expectations that are becoming trader- and market-dependent. Such situations lead to a growth of volatility, instabilities and perhaps, in some special cases, to bubbles and chaos. Nonetheless, whether it is fully right or wrong, it seems to work sometimes. Thus, although rational expectations are an important hypothesis and an important equilibrium pillar of modern finance, they should be used carefully for making money. It is, however, undoubtedly in theoretical finance where it is used with simple models for the valuation of options and for valuing derivatives in general – albeit this valuation depends on a riskless interest rate, usually assumed known (i.e. mostly assumed exogenous). Thus, although the arbitrage-free hypothesis (or rational expecta- tions) assumes that decision makers are acting intelligently and rationally, it still requires the risk-free rate to be supplied. In contrast, economic equilibrium the- ory, based on the clearing of markets by equating ‘supply’ to ‘demand’ for all financial assets, provides an equilibrium where interest rates are endogenous. It assumes, however, that beliefs are homogeneous, markets are frictionless (with no transaction costs, no taxes, no restriction on short sales and divisible assets) as well as competitive markets (in other words, investors are price takers) and finally it also assumes no arbitrage. Thus, general equilibrium is more elaborate than ra- tional expectations (and arbitrage-free pricing) and provides more explicit results regarding market reactions and prices (Lucas, 1972). The problem is particularly acute when we turn to incomplete markets or markets where pricing cannot be uniquely defined under the rational expectations hypothesis. In this case, a de- cision makers’ rationality is needed to determine asset prices. This was done in Chapter 3 when we introduced the SDF (stochastic discount factor) approach used to complement the no arbitrage hypothesis by a rationality that is sensitive to decision makers’ utility of consumption. In Chapter 9, we shall return to this approach in an inter-temporal setting. For the present, we introduce the financial instruments that we will attempt to value in the next chapters.