A Structure for Spanning Heterogeneous Realizations of Systemic Discount Rates

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I apply a 'no arbitrage' constraint towards the development of the martingale that spans the discount rates. A no arbitrage condition is the most rigorous condition that governs the determination of asset prices. I illustrate. The reason investors do not solely purchase the assets that promise the highest expected returns is, because in presence of the emergence of unanticipated risk those are the assets that, simultaneously are most likely to succumb to either negative returns or returns that are lower than the promised expected returns. I show discount rates - which exist solely because cash flows on risky assets are not guaranteed - are supposed to induce investors to be indifferent, in the following sense, to purchases of the assets that promise either of relatively low or relatively high expected returns; namely that agents choose between assets on the basis of their personal, i.e. subjective estimates of the possibility and severity of the unanticipated risks that feasibly emerge in future periods. In stated respect, if the discount rates that are applied to the valuation of the assets that promise high expected returns are not sufficiently higher than the discount rates that are applied to the valuation of the assets that promise relatively lower expected returns, the demand for both sets of assets becomes sub-optimal and markets are perverted. First, with the prices for the assets that promise high expected returns then too high, savvy investors choose not to buy. Second, in presence of the emergence and severity of unanticipated risks, feasibly the assets that promise lower expected returns still underperform the assets that promise high expected returns. As such, the demand for those assets from savvy investors also dry up. For concreteness, whereas an agent i purchases the lower return asset in response to a subjective prior that unanticipated risks are both likely to emerge and likely to be severe, some other agent j purchases the higher return asset in response to the alternate subjective prior that unanticipated risk is unlikely to be severe in future periods. If, ex post, agent i turns out right, the asset with the lower expected return realizes returns which outperform the returns that actually are realized by the asset with the higher expected return. If, ex post, agent j turns out right, the ordering of the returns that are realized conform with the ordering of the expected returns. If, however, the discount rates are not well formulated, a priori, such that the two events have a strictly positive probability, markets become perverted. The following finding, namely that across business cycles - the combination of market upturns and downturns - value stocks (typically, lower expected return stocks) tend to outperform growth stocks (typically, higher expected return stocks) - see for example, Fama and French (2007) - refers. Conversely, the evidence that an asset that has had high realized returns in the last 12 months tends to continue to have high realized returns in the following 12 months - see for example, Jegadeesh and Titman (1993, 2001) - refers.

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