Asset prices and portfolio holdings encode information about investor beliefs about the future and about investors’ concerns for risk. A natural empirical exercise is to use the financial market data to extract such information. In this post, I briefly outline a few approaches that have emerged (relatively) recently.

IO Approach: Egan, MacKay, and Yang (2020)

Authors employ a revealed-preference approach to estimate investor expectations of stock market returns. The underlying idea is that demand for assets embodies the investor’s risk aversion and expectation of the asset’s financial value, so with (minor) structural assumptions we can extract the parameters of interest.

For identification, they make use of the fact that investors choose investment options from a menu of several ETFs with different risk/return profiles and fee structures.

One important assumption in the paper is that investors only care about the mean and volatility of the ETFs. This assumption seems empirically at odds with the key insight in Koijen and Yogo (2019): the size of the latent demand is quite large, which points to the existence of “investor taste” not captured by traditional financial metrics.

Reduced-Form Approach: Pflueger, Siriwardane, and Sunderam (2020)

Authors formalize the following intuition: stock prices of the riskiest, most volatile firms should be particularly sensitive to investor perceptions of risk. Therefore, they measure perceived risk indirectly through the variable \(PVS_t\), defined to be the average book-to-market ratio of low-volatility stocks minus the average book-to-market ratio of high-volatility stocks.

Given the simplicity of the construction, the measure inherits the granularity of the stock returns and can be used to construct high-frequency measure of risk appetite.

Authors note that \(PVS_t\) “captures a broad notion of perceived risk that operates simultaneously in many asset classes.” An interesting question therefore is to decompose \(PVS_t\), especially in terms of other metrics such as volatility expectations and uncertainty.

State Prices: Ross (2015) & Borovička, Hansen, and Scheinkman (2016)

The holy grail for expectation extraction is to obtain the probability distribution of future returns, commonly referred to as “natural” probabilities, as opposed to risk-neutral probabilities which also encodes information about risk aversion.

Ross (2015) claimed that it is possible to separate the market’s forecast of returns and risk aversion from state prices alone. Borovička, Hansen, and Scheinkman (2016) argued that this requires an additional assumption.

The key point of departure for the two papers is whether they view the martingale contributions to the stochastic discount factor as degenerate or not.

Life-cycle Portfolio Choice: Calvet el al. (2019)

In this paper, the authors calibrate a life-cycle model to recover the distribution of risk aversion. The household has Epstein-Zin preferences, which implies the need to estimate (at least) three structural parameters: the rate of time preference, the coefficient of relative risk aversion, and the elasticity of intertemporal substitution (EIS).

Importantly, they assume that investors hold common expectations of returns. Their model also assumes that expected returns on safe and risky assets are constant over time.

Machine-Learning Approach: Chaudhry and Oh (2020)

In my paper with Aditya Chaudhry, we rely on reinforcement learning, a branch of machine learning, to extract aggregate expectations of growth from asset prices.

Our key idea is as follows. Prices reflect many variables besides growth expectations such as discount rates or expectations of other variables related to cash flow growth. Therefore, with a single asset, we cannot extract the component of asset returns driven solely by changes in expectations of macroeconomic growth. But with multiple assets, a suitable linear combination of them can cancel the extraneous sources of return variation and deliver a estimate of the change in growth expectations.

To learn the optimal combination of assets, we propose a reinforcement learning approach and find that it outperforms the Kalman Filter or the MIDAS regression in extracting a daily series of growth expectations from asset prices.