What is it about?

All risky assets have value, i.e. a price, because they promise a risky package of future cash flows to investors. Given the cash flows are risky, feasibly they do not arrive exactly as promised. Discount rates are the parameters that are formulated to account for the risk that cash flows do not arrive exactly as promised, because feasibly they, the cash flows either are delayed or turn out smaller than promised. For illustration, suppose two different assets, ABC and XYZ each promise investors $1.00 to arrive in the very next period. Suppose, however, that the $1 promised by ABC is riskier than that promised by XYZ. Feasibly, whereas investors are willing to pay $.80 in exchange for the promise of $1.00 from ABC, simultaneously and necessarily they pay more than $0.80, e.g. $0.85 for the $1.00 that is promised by XYZ. Those prices translate into discount rates of 20% or 15% for the $1.00 that are promised by ABC or XYZ, respectively. But then the question arises, namely how exactly do investors figure out that the same promised amount embeds different realizations of risk, resulting in different realizations for the discount rates that are applied to the determination of asset valuations? The answer is, of course the differences in the information that are available to investors in respect of the circumstances that surround the generation of the promised $1.00 (see for example, Cochrane 2011). Since then it is not the promised amount that is the source of risk, but rather the circumstances (information) that surround the generation of the promised amount, ideally discount rates are inferred independent of the amounts that are promised to investors. The Problem? Prior to this study, there does not exist any structure for inferring discount rates in which the effort is independent of the incorporation of the amounts that are promised to investors. In presence of the weakness inherent in such structures, in Cochrane (2011), the discount rates that are generated by those structures are not solely premised on information. In Damodaran (2019), all such structures are non-substantive, that is, deviate from the ideal structure for the estimation of discount rates.

Featured Image

Why is it important?

If discount rates are to be robustly estimated, they are benchmarked to the discount rate for the market portfolio (the market discount rate, see Merton 1973); are responsive to the risks that are specific to the investment opportunity; yet are not premised on the promised future cash flows. There is not any prior study that, simultaneously incorporates all of the enumerated desirable qualities in the same structure for the determination of discount rates. This study is first to develop a structure - a martingale - which embeds the three enumerated desirable qualities, resulting in heterogeneous realizations of discount rates, all of which have characterization as systemic discount rates. Importantly, the structure does not only estimate discount rates, rather spans discount rates.


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.

Dr Oghenovo A Obrimah
Fisk University

Read the Original

This page is a summary of: A parsimonious analytically specified general equilibrium structure that spans discount rates, Finance Research Letters, March 2024, Elsevier,
DOI: 10.1016/j.frl.2024.105252.
You can read the full text:




The following have contributed to this page