# Define a sample of Taiwan 50 ETF constituents (illustrative subset)
tickers <- c("2330.TW", # TSMC
"2317.TW", # Hon Hai (Foxconn)
"2454.TW", # MediaTek
"2308.TW", # Delta Electronics
"2412.TW", # Chunghwa Telecom
"2882.TW", # Cathay Financial
"2881.TW", # Fubon Financial
"1303.TW", # Nan Ya Plastics
"1301.TW", # Formosa Plastics
"2002.TW", # China Steel
"2891.TW", # CTBC Financial
"3711.TW", # ASE Technology
"2303.TW", # UMC
"2207.TW", # Hotai Motor
"1216.TW", # Uni-President
"2884.TW", # E.SUN Financial
"2886.TW", # Mega Financial
"2885.TW", # Yuanta Financial
"3008.TW", # LARGAN Precision
"2357.TW") # AsustekR Note 5 - Behavioral Finance and Lottery-Like Stock Strategies
Introduction
In Note 4, we explored a moving-average crossover strategy and benchmarked it against a buy-and-hold position on 0050.TW. Those strategies rest on technical analysis — the idea that past price patterns convey information about future prices. In this note, we step beyond pure technicals and explore behavioral finance-motivated strategies that exploit systematic biases in investor preferences.
The behavioral finance literature documents that many investors — particularly retail investors — exhibit a preference for stocks with lottery-like payoffs: assets offering a small probability of a large positive return, even at the cost of a low or negative expected return. This preference distorts prices in predictable ways and gives rise to cross-sectional return anomalies that academic researchers and quantitative traders have studied extensively.
Three lines of empirical work guide this note:
Bali, Cakici, and Whitelaw (2011) propose the maximum daily return over the previous month (MAX) as a proxy for lottery-like features and document that high-MAX stocks subsequently underperform by more than 1% per month in U.S. data.
Lin and Liu (2018) show that this MAX–return relation is concentrated among stocks preferred by individual investors, with the spread effectively zero among institutionally-held large-caps.
Kumar, Motahari, and Taffler (2024) broaden this evidence by combining several skewness proxies — MAXRET, lottery index (LIDX), jackpot probability (JACKPOT), and expected idiosyncratic skewness (ESKEW) — and showing that skewness sentiment amplifies a wide range of cross-sectional anomalies, primarily through the underperformance of overpriced (short-leg) stocks.
The pedagogical aim of this note is fourfold:
- Introduce a cross-sectional portfolio sorting framework using a basket of Taiwan 50 (
0050.TW) constituents. - Implement a MAX-based long-short strategy following Bali, Cakici, and Whitelaw (2011).
- Implement a lottery index (LIDX) strategy combining price, volatility, and skewness, following Kumar, Page, and Spalt (2016) and Kumar, Motahari, and Taffler (2024).
- Implement an idiosyncratic-volatility (IVOL) strategy, drawing on the seminal puzzle of Ang et al. (2006) — but using Note 3’s simpler market-model residuals to keep the implementation tractable.
Note on scope: The original studies use thousands of U.S. stocks across decades. With a small Taiwanese basket and a short window, our portfolio sorts are illustrative rather than statistically definitive. The goal is to learn the methodology, not to make causal claims about Taiwanese market efficiency.
1. Theoretical Background
1.1 Why Should Lottery-Like Stocks Underperform?
Classical asset pricing (CAPM) predicts that only systematic risk (covariance with the market) is priced. Idiosyncratic skewness and extreme tail outcomes should not matter, because rational investors hold well-diversified portfolios and diversify these features away.
Behavioral models depart from this view in two complementary ways:
Cumulative Prospect Theory (Barberis and Huang 2008): Investors overweight small probabilities of large gains, leading them to overpay for positively-skewed (lottery-like) assets. In equilibrium, these assets earn lower expected returns.
Optimal Beliefs / Skewness Preference (Brunnermeier, Gollier, and Parker 2007; Mitton and Vorkink 2007): When investors are under-diversified, idiosyncratic skewness enters their utility directly. They willingly hold positively-skewed stocks at a discount to expected return.
Both frameworks predict the empirical pattern documented by Bali, Cakici, and Whitelaw (2011): stocks with extreme positive recent returns earn lower subsequent returns.
1.2 The Role of Individual Investors
Lin and Liu (2018) argue that this anomaly should concentrate among stocks where individual investors are most active, because retail investors are the natural carriers of the behavioral biases above. Institutional investors are typically well-diversified and treat lottery-like holdings as informed bets rather than skewness preference (Kumar and Page 2014).
1.3 Skewness Sentiment as a Common Mispricing Driver
Kumar, Motahari, and Taffler (2024) extend this view by showing that skewness preference is not just a single-anomaly phenomenon — it is a common driver of cross-sectional mispricing across at least 11 distinct anomaly strategies. Their evidence works through two channels:
Multiple skewness proxies tell the same story. They compare four skewness measures — MAXRET, LIDX, JACKPOT (the out-of-sample probability of a stock generating a log return \(> 100\%\) in the next 12 months), and ESKEW (expected idiosyncratic skewness from a predictive regression according to Boyer, Mitton, and Vorkink (2010)) — and find that all four interact significantly with mispricing. A one-standard-deviation increase in skewness adds 30%–60% to the predictive power of the combined mispricing measure.
The amplification is short-leg-driven. Skewness has essentially no effect on the long leg (underpriced stocks). The interaction comes entirely from skewness-loving investors bidding up the short leg.
We will not implement JACKPOT or ESKEW directly — both require either a fitted logit panel or a cross-sectional predictive regression that is beyond the scope of a single note. But we will implement LIDX, which is conceptually closely related and computationally tractable, and we will discuss JACKPOT and ESKEW as natural extensions.
1.4 The Idiosyncratic Volatility Puzzle
Ang et al. (2006) document a separate but related puzzle: stocks with high idiosyncratic volatility, measured as the residual standard deviation from a Fama-French three-factor model fit to daily returns within the previous month, earn lower future returns. This contradicts standard intuition that volatility should be compensated, and it has spawned a large literature.
The connection to MAX/skewness is empirically tight. Bali, Cakici, and Whitelaw (2011) show that controlling for MAX reverses the IVOL puzzle, and Kumar, Motahari, and Taffler (2024) conclude that IVOL and skewness exacerbate anomalies for different reasons (IVOL deters arbitrageurs; skewness attracts speculators), but that both effects are present.
A practical simplification. The Ang et al. (2006) IVOL measure uses the daily Fama-French three-factor regression, which requires importing the FF factors and estimating a regression separately for every stock-month. To keep the implementation light, we use the simpler market-model residual standard deviation introduced in Note 3 (regressing the stock on a single market index). This is not identical to Ang et al.’s measure, but it captures the same economic intuition — the part of return volatility unrelated to systematic market movement — and is sufficient for pedagogical purposes. Students interested in the original measure should consult Ang et al. (2006) directly.
2. Building a Cross-Sectional Stock Universe
Because MAX, LIDX, and IVOL anomalies are inherently cross-sectional — they require sorting stocks into portfolios — we cannot apply them to a single asset. Instead, we construct a basket of liquid Taiwanese large-caps drawn from the Taiwan 50 constituents.
# Pull daily prices for the basket
start_date <- "2022-01-01"
end_date <- "2024-04-30"
stocks_data <- tq_get(tickers, from = start_date, to = end_date) |>
select(symbol, date, adjusted, volume) |>
arrange(symbol, date)
# Compute daily returns by stock
stocks_ret <- stocks_data |>
group_by(symbol) |>
mutate(ret = adjusted / lag(adjusted) - 1) |>
ungroup() |>
drop_na(ret)
# Pull TAIEX as the market proxy for IVOL estimation
market_data <- tq_get("^TWII", from = start_date, to = end_date) |>
select(date, mkt_close = adjusted) |>
arrange(date) |>
mutate(mkt_ret = mkt_close / lag(mkt_close) - 1) |>
drop_na(mkt_ret)
# Merge stock returns with market returns
stocks_ret <- stocks_ret |>
inner_join(market_data |> select(date, mkt_ret), by = "date")
datatable(head(stocks_ret, 12)) |>
formatRound(c("adjusted", "ret", "mkt_ret"), digits = 4)We will need monthly holding-period returns repeatedly, so we compute them once:
stocks_monthly_ret <- stocks_ret |>
mutate(year_month = floor_date(date, "month")) |>
group_by(symbol, year_month) |>
summarise(monthly_ret = prod(1 + ret) - 1, .groups = "drop")3. Strategy A: The MAX Effect (Bali, Cakici, and Whitelaw 2011)
3.1 Constructing the MAX Variable
Following Bali, Cakici, and Whitelaw (2011), we define for each stock \(i\) at the end of month \(t\):
\[ \text{MAX}_{i,t} = \max\{R_{i,d}\}, \quad d \in \text{month } t \]
That is, MAX is the single largest daily return observed during the formation month. High-MAX stocks are those that recently experienced an extreme up-day — the kind of payoff that, behaviorally, attracts lottery-seeking investors.
stocks_max <- stocks_ret |>
mutate(year_month = floor_date(date, "month")) |>
group_by(symbol, year_month) |>
summarise(MAX = max(ret, na.rm = TRUE),
n_days = n(),
.groups = "drop") |>
filter(n_days >= 15)3.2 Forming Tercile Portfolios on MAX
Each month we sort stocks into terciles (3 groups, since our universe is small — Bali et al. use deciles with thousands of stocks). The strategy goes long the low-MAX group and short the high-MAX group, rebalanced monthly.
max_signal <- stocks_max |>
group_by(symbol) |>
arrange(year_month) |>
mutate(MAX_lag = lag(MAX)) |>
ungroup() |>
drop_na(MAX_lag)
panel_max <- max_signal |>
inner_join(stocks_monthly_ret, by = c("symbol", "year_month")) |>
group_by(year_month) |>
mutate(MAX_tercile = ntile(MAX_lag, 3)) |>
ungroup()
port_max <- panel_max |>
group_by(year_month, MAX_tercile) |>
summarise(port_ret = mean(monthly_ret, na.rm = TRUE), .groups = "drop")
ls_max <- port_max |>
pivot_wider(names_from = MAX_tercile, values_from = port_ret,
names_prefix = "T") |>
arrange(year_month) |>
mutate(LS_ret = T1 - T3,
cum_low = cumprod(1 + T1),
cum_LS = cumprod(1 + LS_ret))
cat(sprintf("MAX L-S monthly mean: %.2f%% | Sharpe (ann.): %.3f\n",
100 * mean(ls_max$LS_ret, na.rm = TRUE),
mean(ls_max$LS_ret, na.rm = TRUE) /
sd(ls_max$LS_ret, na.rm = TRUE) * sqrt(12)))MAX L-S monthly mean: -1.02% | Sharpe (ann.): -0.6974. Strategy B: The Lottery Index (Kumar, Page, and Spalt 2016; Kumar, Motahari, and Taffler 2024)
4.1 Why a Composite Lottery Index?
Any single skewness measure — MAX, total skewness, or anything else — is noisy. Kumar, Page, and Spalt (2016) and Kumar, Motahari, and Taffler (2024) argue that bundling characteristics gives a more reliable identifier of “lottery stocks.” Their lottery index (LIDX) combines three features that retail investors collectively associate with gambling-style payoffs:
- Low price — lottery stocks are typically cheap in absolute terms, since retail investors are often capital-constrained and prefer to buy whole shares cheaply.
- High idiosyncratic volatility — they offer dispersion in outcomes.
- High idiosyncratic skewness — they offer the right-tail “jackpot” possibility.
A stock that scores high on all three is the canonical lottery stock. The construction is similar in spirit to the individual-investor preference index in Lin and Liu (2018) — average percentile ranks across multiple characteristics to wash out idiosyncratic noise in any single proxy.
4.2 Estimating Idiosyncratic Volatility (Note 3 Style)
We follow Note 3’s approach: regress each stock’s daily returns on the TAIEX market return over a rolling one-month window, and take the residual standard deviation as IVOL. This is not the Ang et al. (2006) measure (which uses daily Fama-French three factors), but it captures the same economic idea with much less infrastructure.
# Estimate IVOL and idiosyncratic skewness for each stock-month from a daily
# market-model regression (LIDX needs both quantities)
estimate_ivol_iskew <- function(df) {
if (nrow(df) < 15 || sd(df$ret) == 0) return(tibble(IVOL = NA_real_, ISKEW = NA_real_))
fit <- lm(ret ~ mkt_ret, data = df)
tibble(IVOL = sd(residuals(fit)), ISKEW = skewness(residuals(fit)))
}
stock_ivol_iskew <- stocks_ret |>
mutate(year_month = floor_date(date, "month")) |>
group_by(symbol, year_month) |>
group_modify(~ estimate_ivol_iskew(.x)) |>
ungroup() |>
drop_na(IVOL, ISKEW)
datatable(head(stock_ivol_iskew, 12)) |>
formatRound(c("IVOL", "ISKEW"), digits = 4)4.3 Constructing LIDX
The standard recipe is: each month, rank every stock on each lottery feature (low price, high IVOL, high ISKEW) within the cross-section, then average the percentile ranks. Stocks scoring highest on the average rank are the most lottery-like.
# Get month-end price for each stock
stock_price <- stocks_ret |>
mutate(year_month = floor_date(date, "month")) |>
group_by(symbol, year_month) |>
summarise(price = last(adjusted), .groups = "drop")
# Combine the three lottery features
lidx_raw <- stock_price |>
inner_join(stock_ivol_iskew, by = c("symbol", "year_month"))
# Cross-sectional percentile ranks within each month
# Lottery preference: LOW price (negate to flip), HIGH IVOL, HIGH ISKEW
lidx <- lidx_raw |>
group_by(year_month) |>
mutate(
rank_price = percent_rank(-price), # negate so low price gets high rank
rank_ivol = percent_rank(IVOL),
rank_iskew = percent_rank(ISKEW),
LIDX = (rank_price + rank_ivol + rank_iskew) / 3
) |>
ungroup() |>
select(symbol, year_month, price, IVOL, ISKEW, LIDX)
datatable(head(lidx, 12)) |>
formatRound(c("price","IVOL","ISKEW","LIDX"), digits = 3)4.4 LIDX Long-Short Strategy
lidx_signal <- lidx |>
group_by(symbol) |>
arrange(year_month) |>
mutate(LIDX_lag = lag(LIDX)) |>
ungroup() |>
drop_na(LIDX_lag)
panel_lidx <- lidx_signal |>
inner_join(stocks_monthly_ret, by = c("symbol", "year_month")) |>
group_by(year_month) |>
mutate(LIDX_tercile = ntile(LIDX_lag, 3)) |>
ungroup()
port_lidx <- panel_lidx |>
group_by(year_month, LIDX_tercile) |>
summarise(port_ret = mean(monthly_ret, na.rm = TRUE), .groups = "drop")
ls_lidx <- port_lidx |>
pivot_wider(names_from = LIDX_tercile, values_from = port_ret,
names_prefix = "T") |>
arrange(year_month) |>
mutate(LS_ret = T1 - T3,
cum_low_lidx = cumprod(1 + T1),
cum_LS_lidx = cumprod(1 + LS_ret))
cat(sprintf("LIDX L-S monthly mean: %.2f%% | Sharpe (ann.): %.3f\n",
100 * mean(ls_lidx$LS_ret, na.rm = TRUE),
mean(ls_lidx$LS_ret, na.rm = TRUE) /
sd(ls_lidx$LS_ret, na.rm = TRUE) * sqrt(12)))LIDX L-S monthly mean: -0.66% | Sharpe (ann.): -0.4605. Strategy C: The IVOL Anomaly (Ang et al. 2006)
Ang et al. (2006) document that high-IVOL stocks earn lower subsequent returns — a result that runs counter to standard risk-return intuition and remains an active research puzzle. We test it here using our market-model IVOL from Section 4.2.
Pedagogical caveat. Ang et al. (2006) use the residual standard deviation from a daily Fama-French three-factor regression. We use the residual standard deviation from a daily one-factor market-model regression (the same form as in Note 3). The two measures are highly correlated in practice, but a careful replication should use the original specification.
ivol_signal <- stock_ivol_iskew |>
group_by(symbol) |>
arrange(year_month) |>
mutate(IVOL_lag = lag(IVOL)) |>
ungroup() |>
drop_na(IVOL_lag)
panel_ivol <- ivol_signal |>
inner_join(stocks_monthly_ret, by = c("symbol", "year_month")) |>
group_by(year_month) |>
mutate(IVOL_tercile = ntile(IVOL_lag, 3)) |>
ungroup()
port_ivol <- panel_ivol |>
group_by(year_month, IVOL_tercile) |>
summarise(port_ret = mean(monthly_ret, na.rm = TRUE), .groups = "drop")
ls_ivol <- port_ivol |>
pivot_wider(names_from = IVOL_tercile, values_from = port_ret,
names_prefix = "T") |>
arrange(year_month) |>
mutate(LS_ret = T1 - T3,
cum_low_ivol = cumprod(1 + T1),
cum_LS_ivol = cumprod(1 + LS_ret))
cat(sprintf("IVOL L-S monthly mean: %.2f%% | Sharpe (ann.): %.3f\n",
100 * mean(ls_ivol$LS_ret, na.rm = TRUE),
mean(ls_ivol$LS_ret, na.rm = TRUE) /
sd(ls_ivol$LS_ret, na.rm = TRUE) * sqrt(12)))IVOL L-S monthly mean: -0.41% | Sharpe (ann.): -0.2716. Comparing All Strategies
We now bring all three behavioral strategies together with three passive benchmarks: an equal-weighted basket of our 20 stocks, the 0050.TW ETF (the Taiwan 50 ETF used in Note 4), and the broader TAIEX index. Including 0050 keeps continuity with Note 4 and gives students a directly investable passive alternative — unlike the TAIEX index, which is not itself tradeable. To make returns comparable, we focus on the long leg of each strategy with a NT$1,000,000 initial investment.
# Equal-weighted basket (buy-and-hold all 20 stocks)
bh_basket <- stocks_monthly_ret |>
group_by(year_month) |>
summarise(bh_ret = mean(monthly_ret, na.rm = TRUE), .groups = "drop") |>
arrange(year_month) |>
mutate(cum_bh = cumprod(1 + bh_ret))
# 0050.TW ETF buy-and-hold (Note 4's primary asset)
etf_0050 <- tq_get("0050.TW", from = start_date, to = end_date) |>
select(date, etf_close = adjusted) |>
mutate(year_month = floor_date(date, "month")) |>
group_by(year_month) |>
summarise(etf_close = last(etf_close), .groups = "drop") |>
arrange(year_month) |>
mutate(etf_ret = etf_close / lag(etf_close) - 1) |>
drop_na(etf_ret) |>
mutate(cum_etf = cumprod(1 + etf_ret))
# TAIEX benchmark
taiex <- market_data |>
mutate(year_month = floor_date(date, "month")) |>
group_by(year_month) |>
summarise(taiex_close = last(mkt_close), .groups = "drop") |>
arrange(year_month) |>
mutate(taiex_ret = taiex_close / lag(taiex_close) - 1) |>
drop_na(taiex_ret) |>
mutate(cum_taiex = cumprod(1 + taiex_ret))
# Stitch all series together
comparison <- ls_max |> select(year_month, cum_LowMAX = cum_low) |>
inner_join(ls_lidx |> select(year_month, cum_LowLIDX = cum_low_lidx), by = "year_month") |>
inner_join(ls_ivol |> select(year_month, cum_LowIVOL = cum_low_ivol), by = "year_month") |>
inner_join(bh_basket |> select(year_month, cum_bh), by = "year_month") |>
inner_join(etf_0050 |> select(year_month, cum_etf), by = "year_month") |>
inner_join(taiex |> select(year_month, cum_taiex), by = "year_month")
# Scale to NT$1,000,000 starting capital
capital <- 1000000
comparison <- comparison |> mutate(across(starts_with("cum_"), ~ . * capital))
datatable(tail(comparison, 8)) |>
formatRound(columns = c("cum_LowMAX","cum_LowLIDX","cum_LowIVOL",
"cum_bh","cum_etf","cum_taiex"),
digits = 0)highchart() |>
hc_title(text = "Strategy Comparison: Behavioral vs. Passive (NT$1,000,000 Initial)") |>
hc_xAxis(type = "datetime", title = list(text = "Month")) |>
hc_yAxis(title = list(text = "Portfolio Value (NT$)")) |>
hc_add_series(data = comparison, type = "line",
hcaes(x = year_month, y = cum_LowMAX),
name = "Low-MAX (Bali et al. 2011)") |>
hc_add_series(data = comparison, type = "line",
hcaes(x = year_month, y = cum_LowLIDX),
name = "Low-LIDX (Kumar et al. 2024)") |>
hc_add_series(data = comparison, type = "line",
hcaes(x = year_month, y = cum_LowIVOL),
name = "Low-IVOL (Ang et al. 2006)") |>
hc_add_series(data = comparison, type = "line",
hcaes(x = year_month, y = cum_bh),
name = "Equal-Weighted Buy-and-Hold") |>
hc_add_series(data = comparison, type = "line",
hcaes(x = year_month, y = cum_etf),
name = "0050.TW ETF Buy-and-Hold") |>
hc_add_series(data = comparison, type = "line",
hcaes(x = year_month, y = cum_taiex),
name = "TAIEX Benchmark") |>
hc_tooltip(valuePrefix = "NT$", valueDecimals = 0) |>
hc_legend(align = "center", verticalAlign = "bottom") |>
hc_add_theme(hc_theme_smpl())7. Summary Statistics Across Strategies
strategy_summary <- tibble(
Strategy = c("Low-MAX (BCW 2011)",
"Low-LIDX (Kumar et al. 2024)",
"Low-IVOL (Ang et al. 2006)",
"Buy-and-Hold (Equal-Weighted)",
"0050.TW ETF Buy-and-Hold",
"TAIEX Benchmark"),
Mean_Monthly_Return_pct = c(
mean(ls_max$T1, na.rm = TRUE),
mean(ls_lidx$T1, na.rm = TRUE),
mean(ls_ivol$T1, na.rm = TRUE),
mean(bh_basket$bh_ret, na.rm = TRUE),
mean(etf_0050$etf_ret, na.rm = TRUE),
mean(taiex$taiex_ret, na.rm = TRUE)
) * 100,
Volatility_pct = c(
sd(ls_max$T1, na.rm = TRUE),
sd(ls_lidx$T1, na.rm = TRUE),
sd(ls_ivol$T1, na.rm = TRUE),
sd(bh_basket$bh_ret, na.rm = TRUE),
sd(etf_0050$etf_ret, na.rm = TRUE),
sd(taiex$taiex_ret, na.rm = TRUE)
) * 100
) |>
mutate(Annualized_Sharpe = Mean_Monthly_Return_pct / Volatility_pct * sqrt(12))
datatable(strategy_summary) |>
formatRound(c("Mean_Monthly_Return_pct","Volatility_pct","Annualized_Sharpe"),
digits = 3)8. Caveats and Connections to the Literature
Several caveats are worth emphasizing to students:
Sample size and power. With ~20 stocks and ~28 months, our terciles contain about 7 stocks each. Bali, Cakici, and Whitelaw (2011) use thousands of stocks and four decades; Kumar, Motahari, and Taffler (2024) use the full CRSP universe from 1963–2015. Any “anomaly” in our small sample is best treated as a teaching artifact, not evidence about the Taiwanese market.
Conditioning matters. Lin and Liu (2018) show that even in the U.S., the MAX anomaly is concentrated in stocks dominated by individual investors. Our basket of Taiwan 50 large-caps is heavily institutionally held, so we might expect the MAX/LIDX/IVOL strategies to underperform here — itself a Lin-and-Liu-style empirical prediction.
Risk adjustment. Production-quality tests adjust returns using factor models (Fama-French 3/5, Carhart momentum). The four-factor and six-factor alphas in Bali, Cakici, and Whitelaw (2011) Table III and Kumar, Motahari, and Taffler (2024) Table 4 are the relevant magnitudes for academic claims. We omit factor adjustment here for clarity.
Transaction costs and short-selling. Our long-short strategies implicitly assume costless short-selling. In Taiwan, shorting individual stocks involves locate costs, securities-borrowing constraints, and tick-rules. Both Bali, Cakici, and Whitelaw (2011) and Kumar, Motahari, and Taffler (2024) emphasize that short-selling frictions help explain why these anomalies persist.
IVOL measurement. Our IVOL is the residual standard deviation from a single-factor market-model regression on daily returns within each month — much simpler than the daily Fama-French three-factor specification of Ang et al. (2006). The two are highly correlated, but a publishable replication would use the original specification.
9. Suggested Extensions
- Implement JACKPOT (Conrad, Kapadia, and Xing 2014): Estimate a logit/probit model predicting whether a stock generates a log return above some threshold (e.g., 50% over 6 months in a Taiwanese context) using lagged firm characteristics, and use the fitted probability as a sort variable.
- Implement ESKEW (Boyer, Mitton, and Vorkink 2010): Run a monthly cross-sectional predictive regression of realized skewness on lagged firm characteristics, then sort on the fitted (expected) skewness.
- Replicate BCW’s IVOL/MAX horse-race (Bali, Cakici, and Whitelaw 2011 Table 9): Double-sort on MAX and IVOL to test whether MAX subsumes the IVOL puzzle. This is the most influential single result in BCW, and replicating it locally is informative.
- Extend to MIN as a sanity check. Bali, Cakici, and Whitelaw (2011) report that minimum daily return generates the opposite sign — consistent with prospect-theory loss aversion. This provides a useful internal validity check.
- Combine into a composite mispricing index. Following Kumar, Motahari, and Taffler (2024), form a single MIS-style index by averaging decile ranks across MAX, LIDX, and IVOL, and sort on the composite.
10. Student Exercise
Design and implement your own behavioral-finance strategy for a Taiwanese stock universe. Specifically:
- Choose a behavioral signal beyond MAX, LIDX, or IVOL (suggestions: 52-week high proximity, total skewness over a 6/12-month window, turnover-based proxies for retail attention, short-interest ratios).
- Form portfolios by sorting on your signal each month using terciles or quintiles.
- Benchmark against the three strategies in this note plus the equal-weighted buy-and-hold and the TAIEX index.
- Discuss whether the results support or contradict the Lin and Liu (2018) prediction that lottery anomalies should be weak in institutionally-held large-caps.
A successful answer demonstrates (a) careful construction of the cross-sectional signal, (b) honest acknowledgment of small-sample limitations, and (c) thoughtful interpretation in light of the conditioning argument in Lin and Liu (2018) and Kumar, Motahari, and Taffler (2024).
References
Ang, Andrew, Robert J. Hodrick, Yuhang Xing, and Xiaoyan Zhang. 2006. “The Cross-Section of Volatility and Expected Returns.” Journal of Finance 61 (1): 259–99.
Bali, Turan G., Nusret Cakici, and Robert F. Whitelaw. 2011. “Maxing Out: Stocks as Lotteries and the Cross-Section of Expected Returns.” Journal of Financial Economics 99 (2): 427–46.
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