27. Subjective Expectations and Investor Decisions

27.1.1 Key Points

Greenwood-Shleifer (2014) study the nature of investors' subjective expectations of future stock-market returns, using six data sources from 1963–2011. They find:

• expectations from the six sources are highly positively correlated (surveys are meaningful and consistent despite noise);
• expectations are positively correlated with mutual fund inflows (investors mean what they say);
• expectations are positively correlated with past stock returns and the stock price level (price-dividend ratio) (extrapolation);
• but expectations are negatively correlated with model-implied expected returns (evidence against rational-expectations representative-investor models).

27.1.2 Data

Six data sources (full list in the book's Figure 27.1/27.2): (1) Gallup investor survey ("very optimistic/optimistic/neutral/pessimistic/very pessimistic", %Bullish−%Bearish; also asks minimum acceptable return 1998–2000 and estimated % return 1998–2003); (2) Graham-Harvey CFO survey (since 1998, 200+ CFOs quarterly, expectations of next-year US stock returns); (3) American Association of Individual Investors (weekly bullish/neutral/bearish); (4) Investor Intelligence (since 1963, 120+ independent newsletters); (5) Shiller investor survey (Yale, since the 1980s, individual confidence index = % expecting the market to rise next year); (6) Michigan Survey Research Center (surveying consumer beliefs since 1946; 2000–2005 asked about broader market return expectations over the next 2-3 years).

27.1.3 Empirical Analysis

• Six sources highly positively correlated (Figure 27.3 correlation matrix, most \(p\)-values ≈ 0; only Michigan and a few series are weakly correlated due to short samples).
• Positively correlated with mutual fund inflows (Figure 27.4: Gallup %Bullish−%Bearish moves with inflows into equity as a percentage of equity market capitalization).
• Positively correlated with past returns (Figure 27.5: Gallup expectations move with the CRSP value-weighted 12-month rolling nominal return; investor expectations depend more strongly on the most recent market returns).

• Negatively correlated with model-implied expected returns: e.g. the log dividend-price ratio (a measure of expected returns) has a time-series correlation of $-0.33$ with Gallup %Bullish−%Bearish.
• Higher expectations predict lower future returns at long horizons: Figure 27.6 gives the coefficient estimates of regression (27.1):

\(R^e_{t+k}\) is the \(k\)-period stock-market excess return, \(X_t\) one of the investor expectations or expected returns. Result: higher investor expectations predict lower future returns at long horizons. More IPOs also occur when investors are optimistic (Figure 27.7: Gallup %Bullish−%Bearish positively correlated with the number of IPOs).

27.1.4 / 27.1.5 Contribution & Discussion

Contribution: the first paper to use a comprehensive set of survey sources for empirical analysis; despite classical concerns about survey data, the authors show compelling evidence of mutual positive correlations among the six sources and a positive correlation with fund flows, proving survey data are meaningful (not noise) and reflect investors' true thoughts; it sets a validation paradigm for later studies using survey data. Discussion: the sources have relatively small samples, with incomparable sample periods and methodologies, and some lack consecutive reports (nothing to do about it, we have what we have); the IPO-timing argument (Figure 27.7) is not very convincing (there could be other factors simultaneously correlated with both IPOs and current price). Possible extension: use more granular individual position holdings (e.g. Terrance Odean's dataset) to study how individual investors react to past returns, form beliefs, and guide trading.

27.2 Learning from Inflation Experience: Malmendier and Nagel (2016)

27.2.1 Key Points

Malmendier-Nagel (2016) study individuals' expectations of future inflation. They propose that individuals overweight the inflation experienced in their lifetimes: the young update more strongly than the old (recent experiences carry larger weight for the young), so different age groups disagree substantially in highly volatile inflation periods (e.g. the 1970s). Using 57 years of Reuters/Michigan consumer survey inflation expectations, they find: differences in experience predict differences in expectations; these findings explain household borrowing/lending patterns and mortgage choices.

27.2.2 Framework

Two individuals, one born at \(s\), the other at \(s+j\). Inflation \(\pi_{t+1}\) for period \(t+1\) is formed in period \(t\) (\(t>s+j\)). Suppose \(\pi_{t+1}\) follows an AR(1) (27.2): \(\pi_t=\alpha+\phi\pi_{t-1}+\eta_t\), i.e. \(\pi_t=\mathbf b'\mathbf x_t+\eta_t\), \(\mathbf b=(\alpha,\phi)'\), \(\mathbf x_t=(1,\pi_{t-1})'\). Cohort \(s\)'s updated posterior estimate of \(\mathbf b\) at \(t\), \(\mathbf b_{t,s}\) (27.3):

where \(\mathbf R_{t,s}=\mathbf R_{t-1,s}+\gamma_{t,s}(\mathbf x_t\mathbf x_t'-\mathbf R_{t-1,s})\), with OLS-case gain \(\gamma_{t,s}=\frac1t\) (derivation in the collapsible proof).

The paper's innovation is to assume an age-dependent gain (27.4):

\(\theta>0\) a constant determining the shape of the weight function on past observations. Intuition: the learning gain decreases with age after the agent reaches a certain age threshold \(\theta\). The weight on time-\(t\) information is \(\frac{\gamma_{t,s}}{1+\gamma_{t,s}}\) (Figure 27.8 gives the period gain and weight for different \(\theta\)).

Learning starts at some \(t>s\) (ignoring data before the agent's birth). One can also let other information \(f_t\) affect cohort \(s\)'s \(t+1\) inflation forecast made at \(t\) (27.5): \(\tilde\pi_{s+1|t,s}=\beta\tilde\pi_{t+1|t,s}+(1-\beta)f_t\), \(\tilde\pi_{t+1|t,s}=\mathbf b_{t,s}'\mathbf x_t\) being the learning from past experience. \(f_t\) could be professional forecasters' opinions, media, etc.; \(f_t\) is observable to the decision maker (cohort \(s\)) but might not be to the econometrician (who has no idea what \(f_t\) is used).

27.2.3 Data

CPI (long history, 1872–2009 from Shiller's website; annualized quarterly log inflation rates, Figure 27.9); inflation expectations micro-data from the Reuters/Michigan Survey of Consumers (MSC, since 1953; asked about direction up/down or expected percentage change, authors use the quantitative data; Figure 27.10 gives the four-quarter moving average of one-year inflation expectations by young < 40, mid-aged 40–60, old > 60, showing disagreement across age groups during volatile periods).

27.2.4 Empirical Analysis

The first analysis is based on a modified version of (27.5): \(\tilde\pi_{t+1|t,s}=\beta\tilde\pi_{t+1|t,s}+\boldsymbol\delta'\mathbf D_t+\varepsilon_{t,s}\), \(\tilde\pi_{t+1|t,s}\) the measured inflation expectations from survey data, \(\mathbf D_t\) time dummies (absorbing the unobserved \(f_t\)), \(\varepsilon_{t,s}\) uncorrelated with \(\tilde\pi_{t+1|t,s}\) (measurement error or an idiosyncratic factor not specified by the econometrician). Use nonlinear least squares to jointly estimate \(\theta\) and \(\beta\) with aggregate cohort data (aggregation by birth year): \(\theta\) captures cross-sectional differences in learning strength, \(\beta\) captures the learning-from-experience effect (Figure 27.11 gives estimates \(\theta\approx3.044\), \(\beta\approx0.672\), with SPF-forecast and restricted comparisons). Column 1's fitted values match the survey one-year inflation expectations well (Figure 27.12 fitted vs actual by age group).

The second analysis is how beliefs affect household financial decisions: investors with higher learning-based inflation expectations \(\tilde\pi_{t+1|t,s}\) are less likely to invest and more likely to borrow. Regression (27.6):

\(y_{t,s}\) measures fixed-rate liabilities or fixed-rate assets held by cohort \(s\) at \(t\); \(\tilde\pi_{t+1|t,s}\) is the learning-from-experience inflation forecast computed from the \(\theta\) in (Figure 27.11 column 1); \(\mathbf X_{t,s}\) cohort characteristics, \(\mathbf A_{t-s}\) age dummies, \(\mathbf D_t\) time dummies. Data from the 1960–2007 Survey of Consumer Finances (SCF). Figure 27.13's coefficient estimates verify the prediction (higher inflation expectations → more fixed-rate liabilities / variable-rate mortgages).

Finally the authors note: aggregate learning (equally weighted average across cohorts) is well-approximated by constant-gain learning with gain \(\gamma_{t,s}=\gamma=0.0180\) (Figure 27.14). This makes sense because despite heterogeneous gain parameters across cohorts, the aggregate gain parameter that matters for asset pricing (when each cohort's share is constant) should be constant.

27.2.5 / 27.2.6 Contribution & Discussion

Contribution: a learning scheme matching survey data well; extension to household financial decisions, showing decisions are indeed affected by future inflation expectations (intuitive and contributes to the literature). Discussion (motivation): aggregating young (high-gain) and old (low-gain) cohorts gives constant gain, premised on investors really using Kalman filtering — but constant gain could also be micro-founded by "individual constant gain (recency bias)". The two arguments could generate opposite directions of change in a single gain function with age, hard to tell which is true. The single threshold \(\theta\) in (27.4) is a simplification of Kalman decreasing gain, but the simplification itself needs more theoretical motivation (why a single cutoff? what determines it?). Possible extension: combine the learning model with survey-data identification, applied to macro variables other than inflation (unemployment, GDP growth, stock returns, dividend growth); study how different agent types (Fed/central-bank officials, CEOs) learn and decide.

27.3 Five Facts about Beliefs and Portfolios: Giglio et al. (2019)

27.3.1 Key Points

Giglio et al. (2019) use a newly designed survey of wealthy retail investors (explicitly asking beliefs about the economy and financial markets, matched with portfolio composition) to identify five facts:

1. Beliefs are reflected in portfolio allocations: (a) the sensitivity of portfolio allocation to beliefs is small; (b) but allocations vary significantly with wealth, attention, trading frequency, and confidence.
2. Belief changes do not predict the timing of trading; but for those who trade, belief changes affect the direction and magnitude of trade.
3. Beliefs are mostly explained by large persistent individual heterogeneity; demographic characteristics have very small explaining power for optimism and pessimism.
4. Expected cash-flow growth and expected returns are positively related (both within and across investors).
5. Expected returns and the subjective probability of rare disasters are negatively related (both within and across investors).

27.3.2 Data

A newly designed online expectations survey sent to a large panel of Vanguard retail investors with substantial wealth in financial markets (average over half a million dollars at Vanguard), every two months since February 2017, to a random sample. Beliefs asked: future stock returns (expected annualized return over the coming year/ten years, and subjective probabilities of falling into each bucket); GDP growth (expected real GDP growth over the coming three years/ten years, and probabilities); bond returns (expected 1-year return of a 10-year US zero-coupon bond); at the end of each block, confidence (five-point) and difficulty (five-point). Focus on the first 15 waves, 32,198 total. Figure 27.15 survey response rate; Figure 27.16 demographics of respondents/non-respondents and differences; Figure 27.17 summary statistics of responses.

27.3.3 Empirical Analysis

• Fact 1 (beliefs reflected in allocations) regression (27.7):

\(\tilde{\mathbb E}_{i,t}[R_{1y}]\) the surveyed expected annualized 1-year stock return, \(\mathbf X_{i,t}\) investor characteristics, \(\psi_t\) time fixed effects. Figures 27.18/27.19 (two control sets) coefficients: higher expected return corresponds to higher equity share (Figure 27.20 shows the positive relation visually); larger accounts (over 100k USD) have higher sensitivity; tax-advantaged accounts, high-trading-frequency accounts, and high-attention, high-confidence accounts have higher sensitivity.

• Fact 2 (belief changes don't predict trade timing but affect direction/magnitude) regression (27.8): \(\Delta\text{Equity Share}_{i,w}=\alpha+\beta\tilde{\mathbb E}_{i,w-}[R_{1y}]+\gamma\Delta\tilde{\mathbb E}_{i,w}[R_{1y}]+\delta\text{Equity Share}_{i,w-}+\boldsymbol\phi'\mathbf X_{i,t}+\epsilon_{i,t}\) (\(w-\) the beginning-of-window value). Figure 27.21: expected returns have no significant effect on whether to trade (extensive margin, column 2); but for those who trade, trade direction (column 4) and magnitude (column 5) are explained by expected returns (intensive margin).
• Fact 3 (beliefs mostly explained by persistent individual heterogeneity) regressions (27.9)–(27.11): \(B_{i,t}=\chi_t+\varepsilon_{i,t}\) (27.9, time FE), \(B_{i,t}=\phi_i+\varepsilon_{i,t}\) (27.10, individual FE), \(B_{i,t}=\chi_t+\phi_i+\varepsilon_{i,t}\) (27.11). \(B_{i,t}\) several surveyed belief variables of individual \(i\), \(\chi_t\) time FE, \(\phi_i\) individual FE. Figure 27.22 \(R^2\): individual FE explains most of the variation. Second regression (27.12): \(\phi_i=\alpha+\boldsymbol\Gamma'\mathbf X_i+\epsilon_i\), \(\phi_i\) the individual FE estimated from (27.11), \(\mathbf X_i\) including age group/wealth quantile/region/gender/confidence/quantile of monthly log-in days. Figure 27.23 \(R^2\) shows the variation in individual FE is not much explained by characteristics.
• Fact 4 (expected cash-flow growth and expected returns positively related): regress expected stock growth (1-year and 10-year) on expected GDP growth (3-year and 10-year) (and vice versa), all coefficients significantly positive.
• Fact 5 (expected returns and subjective rare-disaster probability negatively related) regression (27.13):

Figure 27.24: a higher subjective rare-disaster probability corresponds to a lower expected 1-year stock-market return, robust across different settings of (27.13).

27.3.4 / 27.3.5 Contribution & Discussion

Contribution: a new dataset with a large sample of survey responses from wealthy investors, overcoming certain drawbacks of survey-data studies, very useful for future research; a deep dive into the heterogeneity of investor position sensitivity to belief changes, identifying several interesting facts that motivate new macro-finance models. Discussion: the five-bucket design for future stock returns and real GDP growth is asymmetric and may create framing effects (buckets tilted toward the upside, implicitly pushing answers toward optimism); the bucket design should not be based on actual historical frequency (respondents won't look that far back) but on preliminary test surveys; the controls in (27.18)/(27.19) should be combined into one comprehensive regression; the regressions are reduced-form, so it would be interesting to use this new dataset for structural estimation. Possible extension: design surveys with different wordings/choices distributed randomly to test framing effects and how they affect results; build an individual-investor structural model to explain the large individual fixed effects; although demographics seem not to explain individual beliefs, the individual fixed effects could be explained by demographics in a highly nonlinear way, explorable with machine-learning tools like neural nets.