Contents

Compared to the portfolio of aggregate risks, uninformed agents invest more in assets about which they are more optimistic than the informed agents. To cope with this winner’s curse problem, the uninformed agents optimally extract information from prices. Thus they hold the optimal price-contingent portfolio, that is, the portfolio that is mean–variance efficient conditional upon the information revealed by prices.

The relationship between our price-contingent strategy and size and value investing is less clear, because there is no role for firm size or book value of assets in our theoretical framework. To the extent that size and value are secondary effects, enhancement of indexing by skewing weights toward small firms or high book-to-market value ratios should not lead to outperformance relative to our price-contingent strategy. Because optimal weights for our price-contingent axitrader fees strategy are based on estimated expected returns, variances, and covariances, we inevitably introduce estimation error. When we base ex ante volatility estimates on the covariance matrix of the prediction errors from GLS projections of returns onto prices, we fail to properly account for estimation error. The ex post volatilities can readily be estimated as mean squared differences between returns actually recorded over the target months and ex ante expected returns .

## Asset Pricing and Portfolio Choice Theory

6 compares the performance of the price-contingent, size, value, and momentum portfolios. We plot the evolution of the wealth of an investor starting on June 30, 1932, with $1 and investing according to one of five possible strategies. The five strategies were constructed to have the same ex post monthly return volatility over the period of July 1932 to December 2000. Hence, the ordering in mean–variance space can be readily inferred from the relative wealth levels that the strategies generate. 30 Analogously, the value strategy exploits the difference between the returns of firms with high book value of equity relative to market value and firms with low book-to-market ratio.

This shows that asymmetric information can give rise to momentum in a dynamic noisy rational expectations equilibrium. Otherwise stated, outperformance of the index by the momentum portfolio should not be viewed as evidence of deviation from rational expectations. It should be noted, however, that in the other equilibria we simulated, the momentum portfolio did not outperform the index. Thus the excess performance of the momentum portfolio is not a robust feature of our equilibria, in contrast with the excess performance of the price-contingent portfolio. The corollary states that equilibrium returns will be positively correlated too.

Higher, investors do not discount the future as much, and hence wish to save to finance date–1 consumption. The risk-free return must fall to o↵set this inclination to save. ✓¯0 ⌃ ✓¯ is the variance of aggregate date–1 consumption. When it is larger, there is more risk, and investors expected date–1 utilities are smaller. They wish to transfer wealth from date 0 to date 1 in this circumstance, and the risk-free return must fall to o↵set that desire. For some constants A and E.7 The paraITleter B is called the cautiousness parameter.

Jimenez Garcia offers an interesting empirical analysis of asset pricing under private information, relying on industry groupings. We assume the informed agents observe a signal on the next cash flow, which is then publicly observed. Investors can perform the projection because they know the parameters of prime xtb the cash flows, endowment processes, as well as the pricing equation. Another difference is that finite horizon models are nonstationary, while we analyze stationary price equilibrium functions. Cochrane , but we combine the SMB, HML, and momentum strategy with the index, instead of the risk-free asset.

## The Financial Crisis of Our Time

We find that the optimal price-contingent portfolio outperforms the index, both economically and statistically. Some investors have private information about the future cash flows, while others are uninformed. Revelation is only partial because the demand of informed investors reflects their random endowment shocks, along with their signals. This pricing relation cannot be directly relied upon in the econometrics since the beliefs of the representative agent are not observable by the econometrician. Hence, to test our model, we instead focus on portfolio choice. We show that portfolio separation does not obtain, as investors hold different portfolios, reflecting their different information sets.

In principle, one can construct those portfolios by combining individual stocks. This requires, however, that one handles thousands of different stocks, correlating their returns to their prices, a computationally challenging exercise. A more parsimonious approach is to use groups of stocks as building blocks for our portfolios.

## Part Four Beliefs, Information, and Preferences

To document this point, we estimated the partial correlation between a portfolio’s return and its own price. Below is a list of the average slope coefficients in the GLS projections of the returns of the six FF benchmark portfolios onto prices. 33 In contrast, it was not necessary to include the risk-free asset in the price-contingent strategy to match its volatility with that of the index. Hence, simply adding these zero-investment portfolios to the market would have increased the volatility of the portfolio above that of the index.

Each chapter concludes with a notes and references section that supplies references to additional developments in the field. They study portfolio implications of particular types of information heterogeneity. Their approach is static, python distributed computing library however, as in Admati’s original model. In that overlapping generations model, prices change because of supply shocks and public information, but there are no private signals. All the random variables are assumed to be jointly normal.

## 1 Testable restriction implied by theory

Is a constant and ” is a local martingale uncorrelated with B. Is the positive root of the quadratic equation, and to derive A. Therefore, the intertemporal budget constraint (13.34) holds. By concavity, the first-order condition is sufficient for optimality. ¯ it suffices to sum this over h, noting that aggregate initial wealth is p0 ✓. This exercise is a ver’ simple version of a model of the bid-ask spread presented by Stoll .

To offset this increase, 5% of the wealth is invested in Treasury bills. 23 In particular, we determine the right combination of our price-contingent strategy with investment in the market portfolio that generates the same ex post volatility as the index. The left panel shows average momentum returns in excess of average index returns. The right panel displays the average excess returns earned by the price-contingent strategy. One should bear in mind that these restrictions are made only for the sake of simplicity. We checked that in a more complicated model, where uninformed agents could receive endowment shocks, qualitatively similar results obtain.

## Performance and Analytics

As agents seek to trade away from their undiversified endowments, to hold more balanced portfolios. For example, consider an agent working for Exxon, whose income and wealth are exposed to the risk of this firm and, more generally, to the oil industry. This agent will form his optimal portfolio taking into account his exposure to this firm and industry.

Show that if investors also disagree about the variance of w ˜m , then the sharing rule (18.4) is quadratic in w ˜m . Show that the risk premium of a discount bond can depend on r and Y . This is ccrtainly suggested by the characterization of risk premia in Section 1.4; however, the result in Section 1.4 is only an approximate result for small gambles. To go from the “local” result of Section 1.4 to global results, one has to integrate the risk aversion coefficient, as in Exercise 1.9. Exercises designed to provide practice with the concepts and also to introduce additional results.

This enables uninformed agents without endowment shocks to outperform the index. The agents with endowment shocks are willing to pay a premium to hedge their risk. Outperformance reflects the reward to providing insurance against this risk while optimally extracting information from prices. In Asset Pricing and Portfolio Choice Theory, Kerry E. Back at last offers what is at once a welcoming introduction to and a comprehensive overview of asset pricing. The book includes numerous exercises designed to provide practice with the concepts and to introduce additional results.

We test the key implication from our theory that this portfolio outperforms the index. We use monthly U.S. stock data over the period 1927–2000. We extract the information contained in prices by projecting returns onto prices. We use the corresponding expected returns and variance–covariance matrix to construct the conditional mean–variance optimal portfolio of the uninformed agent. We then compare the performance of this portfolio, as measured by its Sharpe ratio, to that of the value-weighted CRSP index.

In contrast, we combine our price-contingent strategy with investment in the market portfolio. DeMarzo and Skiadas under the assumption that the aggregate supply of risky assets is common knowledge. Note that these are zero-investment strategies, not portfolios. Therefore, returns are not defined for these strategies , and it is impossible to position them directly in a mean–return/ variance space.

Portfolio 2 also selects large companies, but with a medium book to market value. Portfolios 4 to 6 are analogous to Portfolios 1 to 3, but for small firms only. We focus on monthly returns on U.S. common stock listed on the NYSE, AMEX, and NASDAQ, as recorded by CRSP for the period from July 1927 until December 2000.

## Empirical Asset Pricing

Estimating the volatility of the price-contingent portfolio is more complicated, as discussed below. Depending on parameter values, one effect or the other can dominate. Hence, momentum or reversals can arise in our dynamic rational expectations equilibrium. These empirical findings have motivated theoretical analyses based on the assumptions that some investors are irrational. Our framework offers an opportunity to check whether momentum and predictability are consistent with equilibrium in a dynamic CAPM where all agents are rational. Calculate the risk tolerance of each of the five special utility functions in Section 1.7 to verify the formulas given in the text.

Each chapter includes a Notes and References section providing additional pathways to the literature. Each chapter includes a ”Notes and References” section providing additional pathways to the literature. Each chapter includes a “Notes and References” section providing additional pathways to the literature. This book is intended as a textbook for asset pricing theory courses at the Ph.D. or Masters in Quantitative Finance level and as a reference for financial researchers. The first two parts of the book explain portfolio choice and asset pricing theory in single‐period, discrete‐time, and continuous‐time models.

36 Our estimation of the correlation between returns and prices is based on simple linear GLS. We did not investigate more sophisticated specifications or estimation strategies, such as nonlinear least squares or conditional heteroscedasticity. No attempt was made to estimate the optimal window size on which to estimate the correlation between prices and returns. Refining the statistical analysis along those and other lines may yield more powerful information extraction and consequently superior performance. Our portfolio allocation strategy will be based on projections of a month’s returns onto the vector of relative prices at the beginning of the month.

The main differences between our analysis and his are that we analyze the multi-asset case and we consider an overlapping generations model. Also, our model is designed to set the stage for our econometric analysis, while his analysis is purely theoretical. Several papers have analyzed empirical applications of the noisy rational expectations framework. Consider an investor with log utility and an infinite horizon. Assume the capital market line is constant, so we can write J instead of J for the stationary value function defined in Section 14.10.