Finance Theory Group

Finance Theory Insights

Asset Management Contracts and Equilibrium Prices

Andrea Buffa, Dimitri Vayanos, Paul Woolley,

Based on: Journal of Political Economy, 2022, 130(12), 3146–3201 DOI:

Constraining asset managers to choose portfolios that lie close to benchmark indices induces them to trade overvalued assets procyclically, inverts the risk-return relation, and causes the aggregate market to become overvalued.


Delegated portfolio management has grown dramatically over the past 60 years, and so has the practice of benchmarking portfolios to market indices. Many asset managers are subject to “tracking constraints,” which limit how far their portfolio can diverge from their benchmark index. Tracking constraints can take the form of a bound on tracking error, defined as the standard deviation of the difference between the return on a manager’s portfolio and the return on the benchmark index. Bounds on tracking error typically range from ±1% to ±6%, and are higher for funds invested in riskier assets, for example, in stocks rather than bonds, or in emerging market stocks rather than U.S. stocks. Tracking constraints can alternatively take the form of a bound on the difference between the portfolio weights that a manager gives to asset classes, such as stocks or bonds, or to segments within a class, such as industry-sector portfolios, and the corresponding weights in the benchmark index.

We show that the seemingly prudent practice of constraining asset managers to remain close to their benchmark indices results in significant price distortions and can shed light on well-known asset pricing puzzles.

Procyclical trading

Tracking constraints induce asset managers to trade overvalued assets procyclically. This amplifies overvaluation and price bubbles. It also causes overvalued assets to have high volatility.

To illustrate how tracking constraints generate procyclical trading, suppose that there are 10 industry-sector portfolios, out of which 5 are overvalued with 15% weight in a benchmark index, and 5 are undervalued with 5% weight. Suppose that a tracking constraint requires that managers’ portfolio weight of each sector does not deviate from the sector’s benchmark weight by more than 10%. Managers give an overvalued sector 5% weight, which is the maximum allowed negative divergence. If the overvalued sector appreciates and reaches 30% weight in the index, then its weight in managers' portfolios reaches (approximately) 10% but must rise further to 20% so that the tracking constraint is met. Managers must thus buy an overvalued sector when it appreciates, which means that they trade procyclically.

A similar argument implies that asset managers trade undervalued assets countercyclically. Managers give an undervalued sector 15% weight, which is the maximum allowed positive divergence. If that sector appreciates and reaches 10% weight in the index, then its weight in managers' portfolios reaches (approximately) 30%, but must drop to 20% so that the tracking constraint is met. Managers must thus sell an undervalued sector when it appreciates.

In our paper, all managers are identical, and over- or undervaluation is caused by noise traders. Our results would hold even without noise traders, provided that investors relax tracking constraints on managers who perform well, as is the case in practice. Returning to the previous example, suppose that some managers overweight and some underweight an overvalued sector. Since the overweighting managers perform well when the sector appreciates, their tracking constraints are relaxed and they face weak pressure to sell. Therefore, when aggregating across all managers, tracking constraints generate buying pressure for overvalued assets that appreciate. Suppose next that some managers overweight and some underweight an undervalued sector. Since the underweighting managers give the sector zero weight when the sector’s index weight is 5%, their tracking constraint does not force them to buy when the sector appreciates and reaches 10% index weight. Therefore, when aggregating across all managers, tracking constraints generate selling pressure for undervalued assets that appreciate. Intuitively, undervalued assets constitute a small fraction of benchmark indices, so constraints are less tight for them.

Our results would also hold even in the absence of formal tracking constraints. This is because asset managers are keen to avoid shortfalls relative to their benchmark index because these convey the impression of incompetence and trigger outflows of capital.

Our theoretical predictions are borne out by the data. Active mutual fund managers investing in U.S. stocks buy stocks that they underweight and do so procyclically. They sell stocks that they overweight, and such selling is slightly countercyclical. The procyclical buying of underweighted stocks is economically significant: funds with the tightest tracking constraints eliminate 40% of their underweight in overperforming stocks in the two semesters during and following the overperformance. By comparison, they eliminate only 20% of their underweight in underperforming stocks.

Risk-Return Inversion

Procyclical trading of overvalued assets implies that those assets have high volatility. Conversely, countercyclical trading of undervalued assets implies low volatility. Tracking constraints thus generate an inverted relation between risk and expected return: overvalued assets have high volatility and low expected return, and undervalued assets have low volatility and high expected return. The inverted risk-return relation is at odds with the CAPM and other standard models.

A weak or inverted relation between risk and expected return has been found to hold in equity markets and in some major bond markets. It holds when risk is measured by CAPM beta, in which case it is known as the beta anomaly, and when risk is measured by return volatility, in which case it is known as the volatility anomaly. One explanation for these anomalies is based on leverage: when investors seek to leverage their exposure to markets, they find borrowing difficult or impracticable, so they choose to do the next best thing, which is buying high-risk assets. Leverage, however, can only explain the flattening of the risk-return relation and not its inversion. Our theory can also explain the inversion.

Our theory also implies that the profitability of the volatility and beta anomalies derives primarily from the overvalued assets, and that constrained investors are more likely to be holding such assets. These predictions are consistent with a number of recent empirical findings, for example, that the profitability of anomalies comes primarily from the stocks that are sold short, and that mutual fund managers who manage pension fund assets, and hence are evaluated more tightly relative to benchmarks, hold a larger fraction of their portfolios in high-beta stocks and achieve lower CAPM alphas.

Aggregate Overvaluation

Since tracking constraints prevent managers from taking large positions in mispriced assets, they amplify the mispricing. This works in both directions: overvalued assets become more overvalued, and undervalued assets become more undervalued. We show that the effect is asymmetric and stronger for overvalued assets. Tracking constraints thus cause the aggregate market to become overvalued. Intuitively, overvaluation is harder to correct than undervaluation because overvalued assets make up a larger fraction of the market, so trading against them entails more risk and tighter constraints. Short-selling costs, often considered as a key determinant of asset overvaluation, play no role in our analysis.


The rise of delegated portfolio management and of benchmarking to market indices calls into question the conventional distinction between passive funds constrained to hold specific portfolios, and active funds investing without constraints. Viewing asset management as a continuum between active and passive, depending on the tightness of asset managers’ constraints, is a better description of reality. We provide new empirical evidence supporting the continuum view, and explore theoretically the implications of that view for equilibrium asset prices and market efficiency. We show that constrained asset managers buy overvalued assets procyclically, and this generates an association between high volatility and low expected returns. We also show that overvaluation is harder to correct than undervaluation, even in the absence of short-selling costs.

The mechanisms that we derive theoretically have attracted policy attention because of their links to asset bubbles. Our research suggests that the contracts between asset owners and delegated asset managers, and the incentives that these induce, lie at the root of the problem.

Andrea Buffa

Assistant Professor of Finance

Leeds School of Business, University of Colorado Boulder

Dimitri Vayanos

Professor of Finance

London School of Economics, CEPR and NBER

Paul Woolley

Senior Fellow

London School of Economics