Hello,
welcome back to Papers with Backtest podcast.
Today,
we dive into another Algo trading research paper.
Yes,
today we're looking at lottery-related anomalies,
the role of reference-dependent preferences.
It's by Ahn,
Wang,
and Yu,
originally penned back in
November 2015.
And this one tackles a really interesting question,
doesn't it,
about why these sort of lottery ticket stocks tend to do poorly.
Exactly.
Stocks with
With that small chance of a massive payoff,
you know,
high skewness,
why do they generally underperform?
And maybe more importantly,
does that change based on whether investors have,
say,
recently made money or lost money?
Right.
The whole reference point idea.
So these lottery features.
Yeah.
How did they actually pin down what makes a stock lottery like?
Well,
they used a few different measures.
Things like the maximum return a stock had on any single day in the past month.
That's one clear sign.
Okay.
Or they looked at the...
predicted probability of a huge future jump.
They call it the predicted jackpot probability.
Also expected idiosyncratic skewness,
which is a bit more technical.
And even things like failure probability or,
you know,
the chance of outright bankruptcy.
Basically different ways to capture that long shot potential.
Gotcha.
And the big takeaway,
the headline finding.
Yeah.
The key thing is this,
that underperformance,
the lottery anomaly,
it's really strong when investors have experienced losses.
But.
What?
But if investors have been winning.
If they've seen gains,
that underperformance gets much weaker or it can actually flip entirely.
The lottery stocks might even outperform.
Wow.
OK,
so it's state dependent.
It really matters what happened before.
Precisely.
It's not a constant effect.
So our mission today for this deep dive is to really unpack the trading rules and the back tests behind this state dependent anomaly and understand how this ties into reference dependent preferences.
Right.
How are recent gains or losses?
color our choices.
Exactly.
Let's start with how they actually measured those prior gains or losses.
They used a concept called capital gains overhang or CGO.
CGO.
And they didn't just use one method,
they used two distinct approaches to try and capture this feeling of being ahead or behind.
All right,
what were they?
So the first one is ECOGH.
That's based on work by Grimblatt and Hahn.
The idea is to compare the current price to a sort of weighted average of past prices.
Wait and buy.
Bird over.
how much the stock is traded,
it's meant to proxy the average price investors actually bought in at.
They look back up to five years,
provided they have enough data,
like 150 weeks minimum.
And that gives you a sense of whether the current price is above or below where people likely bought in.
Exactly.
High CGO means likely sitting on gains.
Low CGO,
likely losses.
And their data for this ran from 65 to 2014,
so a long period.
OK,
that's CGOGH.
What was the second measure?
The second one was CGOFR from Fred Zini's research.
This takes a different angle.
It looks at actual mutual fund holdings.
Ah,
using fund managers as a proxy for the average investor.
Kind of,
yeah.
The assumption is that the aggregate buying behavior of mutual funds gives you a clue about the reference prices for many investors.
This data was a bit more recent,
April 80 to October 2014.
Interesting.
Two different ways to gauge the market's gain-loss state.
So how do they connect this CGO measure to the performance of those lottery stocks?
Right.
This is the core of the back test.
They used independent double sorts.
It sounds complicated,
but the idea is straightforward.
Okay,
break it down for us.
At the end of every month,
they'd take all the stocks.
First,
sort them into five buckets,
quintiles,
based on their CGO,
from low CGO losses to high CGO gains.
Step one,
sort by CGO.
Step two,
independently sort all those same stocks into five other buckets based on one of the lottery measures,
say...
MaxRet,
the maximum daily return from the last month.
So you end up with,
what,
25 portfolios?
Yeah,
exactly.
A five by five grid.
Each cell represents stocks with a certain level of CGO and a certain level of lotteriness.
Then they track the returns of these portfolios for the next month.
And this lets you see how lottery stocks perform specifically within the loss group versus the gain group?
Precisely.
And the results,
especially using MaxRet and the CGOGH measure,
were quite striking.
All right,
let's hear them.
What happened in the low CGO group?
The stocks where investors were likely down.
Okay,
so within that bottom CGO quintile,
the stocks that were most lottery-like,
the top quintile for MaxRet,
they seriously underperformed the least lottery-like stocks,
the bottom quintile for MaxRet.
How much underperformance?
We're talking 1.38%
per month in excess returns.
It's a big difference,
and it was statistically very significant,
T-stat of negative 5.35.
Wow.
So chasing those high flyers after you've already lost money.
seemed to backfire badly according to this.
That's what the data suggests.
Now,
contrast that with the high CGO group stocks where investors were likely sitting on gains.
Okay,
what happened there?
Did the lottery stocks still underperform?
No,
this is where it flips.
In the top CGO quintile,
the high max-ret stocks actually outperformed the low max-ret stocks.
Outperformed by how much?
By 0.54%
per month,
also statistically significant,
with a T-stat of 2.30.
That's fascinating.
So the same type of stock
behaves completely differently depending on whether the average holder is likely up or down.
Exactly.
It really highlights the state dependent nature.
What if you just ignored CGO altogether?
Just looked at high max rate versus low max rate across all stocks?
Well,
the effect largely disappears.
The unconditional spread,
just P5 minus P1 for max rate,
was only 0.24%
per month.
and that wasn't statistically significant.
So you need that CGO context to see the strong anomaly.
That seems crucial,
yes.
And they quantified this further with a difference-in-differences calculation.
Right,
comparing the lottery spread,
P5-P1,
in the high CGO group versus the low CGO group.
Yes.
And that difference was a hefty 1.92%
per month using MaxRet and CGOGH.
Very statistically significant.
T-stat of 7.50.
It really drives home that the lottery effect itself changes dramatically based on prior gains or losses.
That's pretty compelling evidence for the state dependency.
Did this pattern hold up when they used the other lottery measures,
like skewness or bankruptcy risk?
Generally,
yes.
They found similar state-dependent patterns across the board,
predicted jackpot probability,
skewness,
failure probability,
bankruptcy risk.
The magnitudes varied a bit,
you know.
But the core story was consistent.
The lottery anomaly was much stronger following losses and weaker or even reversed following gains.
Okay,
but could this just be risk?
Maybe these lottery stocks are just way riskier,
especially after losses,
and the returns reflect that.
Did they check against standard risk models?
Good question.
They did.
They looked at Fama French three-factor alphas.
These alphas adjust for market,
size,
and value risk.
Did the anomaly survive?
It did.
The patterns in the alphas were very similar to the raw excess returns.
For example,
with MaxRed and CGOGH,
the alpha spread in the low CGO group was Mattis 1.76%.
percent
per month,
still very significant.
And in the high CGO group?
It was positive,
plus 0.35%
per month,
also significant.
So standard risk factors don't seem to explain away this state-dependent effect.
Sounds like they were pretty thorough.
Did they run other checks,
other ways to slice the data?
Oh yeah,
they did a lot of robustness checks.
They used Fama-Macbeth regressions to control for other stock characteristics simultaneously,
things like size,
book-to-market,
turnover,
volatility.
The state-dependent effect held.
It persisted.
They also confirmed the results using that other CGO measure,
CGOFR,
based on mutual funds.
Got similar results,
a difference in differences for max rent,
around 1.88%.
They even calculated a residual CGO,
basically,
the part of CGO not explained by other stock features,
and still found the pattern.
Used equal weighted returns instead of valuated,
results held.
Used conditional sorting instead of independent sorts,
still held.
Wow.
And finally,
those fama-
Beth regressions directly tested the interaction term between lottery proxies and CGO and found it was positive and significant,
directly supporting the state dependent hypothesis.
So across different measures,
different methods,
controlling for various factors,
the conclusion seems pretty solid.
It really does seem robust.
The underperformance of lottery stocks seems heavily tied to whether investors are in a loss domain relative to their reference point.
Which brings us back to the author's interpretation.
How do they explain why this happened?
Their explanation leans heavily on behavioral finance concepts,
reference-dependent preferences,
and mental accounting.
Meaning,
when you've lost money,
your reference point has shifted down.
You might become more risk-seeking,
specifically for gambles that offer a chance,
even a small one,
to get back to even.
Ah,
the break-even effect.
So investors might pile into lottery stocks after losses,
hoping for that quick win.
That's the idea.
This extra demand could push prices up temporarily.
above fundamental value,
leading to lower subsequent returns.
And conversely,
if you're sitting on gains.
If you're in the gain domain,
you might be more risk-averse,
especially toward volatile,
skewed stocks.
You want to protect your winnings,
perhaps,
so you might avoid lottery stocks or demand a higher return for holding them.
Which could explain why they don't underperform or even outperform in the high CGO state.
Exactly.
It fits that framework where our decisions aren't
purely rational in the traditional sense,
but influenced by whether we feel like we're winning or losing.
It certainly provides a behavioral lens on why this particular anomaly seems to behave the way it does.
Really fascinating link between market outcomes and investor psychology.
Absolutely.
It suggests these anomalies aren't static things,
but can be quite dynamic depending on the market's recent history and investor sentiment.
Thank you for tuning in to Papers with Backtest podcast.
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Happy trading.