Assume a fixed expected dollar asset rate of return (ROR %($),e], uniform across all assets, domestic and foreign, and equal to the fixed US interest rate. Hence the expected ROR on US equity is fixed as well, as are, by a further assumption, dividends, both present and future, on such equity. Recall that the current yield (here the ratio of the current dividend over the current equity price) plus the expected percent capital gain (= the expected percent equity price change between the current and the next period) together constitute the expected ROR on equity. Apart from the fixed dividend, make these two additional assumptions: the expected percent capital ohm equity is positive (i.e. not zero) and it correctly predicts the actual percent capital gain that will obtain from the current to the next period (or, in continuous time, from the current moment to the next moment) -- that is, there is by myopic perfect foresight (MPFS) from period to period. In light of the preceding assumptions, answer the following three questions about US equity price dynamics: How will the equity price and its rate of change evolve over time? Use a diagram with the equity price on the horizontal axis and its (actual = expected) percent rate of change on the vertical axis to illustrate your answer. (Remember that, by assumption, both dividends and the expected ROR on US equity are to remain constant throughout.) Are the equity price dynamic thus derived stable? That Is, will the growth in equity price ultimately die down and tend towards zero as time progresses? On a separate diagram, show how the percent rate of price change (= the actual and expected percent capital gain) would have to evolve over time for the resulting equity price dynamic to be stable; discuss why this stable evolution of price is inconsistent with the assumed fixity of dividends and of the expected ROR on equity. Remark: Myopic perfect foresight means that the actual equity price moves from period to period (or moment to moment) in exactly the direction and by exactly the amount that was predicted in the antecedent period (or moment); in the above problem we start from a given current-period equilibrium price and an associated non-zero rate of expected change in that price, and then let price and expected percent capital gain evolve according to the MPFS postulate (that is, we assume the expected and the actually realized percent capital gain to be equal as we move from period to period). MPFS -- or period-to-period (hence 'myopic') perfect foresight -- differs from unqualified perfect foresight or 'rational expectations' as encountered in Jiahui Shen's class note: in the latter, unlike under MPFS, agents concern themselves with and correctly predicted not Just what happens in the next period, but in all periods beyond the current period, thereby putting themselves in a position to detect-and, by appropriate discontinuous price movements in the current period, to arbitrage away-profitable asset trades a future price jump or drop ('crash') invariably offers, thus extinguishing all anticipated future price discontinuities. By contrast, myopic perfect foresight does not ensure absence of future price jumps or of crashes; on the contrary, and as questions 1. - 3. above illustrate, it renders them unavoidable. Assume a fixed expected dollar asset rate of return (ROR %($),e], uniform across all assets, domestic and foreign, and equal to the fixed US interest rate. Hence the expected ROR on US equity is fixed as well, as are, by a further assumption, dividends, both present and future, on such equity. Recall that the current yield (here the ratio of the current dividend over the current equity price) plus the expected percent capital gain (= the expected percent equity price change between the current and the next period) together constitute the expected ROR on equity. Apart from the fixed dividend, make these two additional assumptions: the expected percent capital ohm equity is positive (i.e. not zero) and it correctly predicts the actual percent capital gain that will obtain from the current to the next period (or, in continuous time, from the current moment to the next moment) -- that is, there is by myopic perfect foresight (MPFS) from period to period. In light of the preceding assumptions, answer the following three questions about US equity price dynamics: How will the equity price and its rate of change evolve over time? Use a diagram with the equity price on the horizontal axis and its (actual = expected) percent rate of change on the vertical axis to illustrate your answer. (Remember that, by assumption, both dividends and the expected ROR on US equity are to remain constant throughout.) Are the equity price dynamic thus derived stable? That Is, will the growth in equity price ultimately die down and tend towards zero as time progresses? On a separate diagram, show how the percent rate of price change (= the actual and expected percent capital gain) would have to evolve over time for the resulting equity price dynamic to be stable; discuss why this stable evolution of price is inconsistent with the assumed fixity of dividends and of the expected ROR on equity. Remark: Myopic perfect foresight means that the actual equity price moves from period to period (or moment to moment) in exactly the direction and by exactly the amount that was predicted in the antecedent period (or moment); in the above problem we start from a given current-period equilibrium price and an associated non-zero rate of expected change in that price, and then let price and expected percent capital gain evolve according to the MPFS postulate (that is, we assume the expected and the actually realized percent capital gain to be equal as we move from period to period). MPFS -- or period-to-period (hence 'myopic') perfect foresight -- differs from unqualified perfect foresight or 'rational expectations' as encountered in Jiahui Shen's class note: in the latter, unlike under MPFS, agents concern themselves with and correctly predicted not Just what happens in the next period, but in all periods beyond the current period, thereby putting themselves in a position to detect-and, by appropriate discontinuous price movements in the current period, to arbitrage away-profitable asset trades a future price jump or drop ('crash') invariably offers, thus extinguishing all anticipated future price discontinuities. By contrast, myopic perfect foresight does not ensure absence of future price jumps or of crashes; on the contrary, and as questions 1. - 3. above illustrate, it renders them unavoidable