The Volatility Switch That Only Flips on Bad Days: GJR-GARCH

Finance has a founding intuition: more risk should mean more reward. If a month is expected to be wild, investors should demand a fatter expected return to sit through it. It sounds so obvious that for years researchers treated it as settled and went looking for the size of the effect rather than its existence.

In 1993, Lawrence Glosten, Ravi Jagannathan and David Runkle went looking too, with a more careful volatility model than anyone had used before. What they found was awkward. And along the way, almost as a by-product, they gave the world one of the most widely used volatility models in existence, the one everybody now calls GJR-GARCH.

The problem: the risk-reward link kept coming out wrong

Earlier work had tried to test the risk-reward relationship using a version of GARCH that lets today's expected return depend on today's forecast volatility. The results were a mess. Some studies found the relationship was positive, some found it negative, some found nothing. Each study used a slightly different setup, and the answer seemed to depend on which setup you picked.

Glosten, Jagannathan and Runkle suspected the problem was not the question but the machinery. The volatility models being used were too crude, and were missing real features of the data. If your volatility forecast is wrong, then any conclusion you draw about how volatility relates to returns is built on sand. So they set out to build a better volatility model first, and only then ask the big question.

The key idea via analogy: a switch that flips only on losses

Their headline modelling contribution is beautifully simple. Standard GARCH says: tomorrow's volatility depends on today's squared return. Squaring erases the minus sign, so a bad day and an equally big good day count the same.

Think of it as a machine with one input pipe. Whatever the size of today's move, it goes down that pipe and cranks tomorrow's volatility.

GJR bolted on a second pipe with a valve on it. The valve is closed when the market goes up and open when the market goes down. So:

• On an up day, only the first pipe feeds through, and volatility rises the normal amount.
• On a down day, both pipes feed through, and volatility rises by more.

That is the entire idea: one extra term that only switches on when today's return is negative. If the extra term comes out at zero when you fit it to data, the model quietly collapses back into ordinary GARCH and no harm is done. If it comes out positive, the data is telling you that bad news genuinely raises volatility more than good news.

When they ran it on stock market data, the switch mattered a great deal. Unexpected losses pushed volatility forecasts up. Unexpected gains actually pushed them down. The asymmetry was not a small refinement, it was a central feature of how volatility behaves.

Along with the asymmetry switch, they allowed for two other things earlier models ignored: seasonal patterns in monthly volatility, and the possibility that the level of interest rates carries information about how volatile the coming month will be.

The uncomfortable finding: risk may not pay, month to month

With their better volatility model in hand, they went back to the founding intuition. And they found evidence pointing the wrong way: months forecast to be more volatile did not come with higher expected returns, if anything the relationship leaned negative.

This is not a claim that risk is never rewarded, and the authors were careful about that. Over long horizons and across assets, riskier things do tend to pay more. But the specific, simple story that "the market's expected return next month rises and falls with next month's forecast volatility" did not survive careful testing. Other channels, hedging demand, changing investment opportunities, the possibility that people can shift consumption around, can break the tidy link.

A second surprise: once the asymmetry switch was in the model, volatility looked less persistent than earlier studies had claimed. A lot of what looked like "volatility shocks last forever" turned out to be the old models mis-handling the asymmetry.

Why it mattered

• GJR-GARCH became a workhorse. It sits alongside Nelson's EGARCH as one of the two standard ways to capture the leverage effect, and it is arguably the more popular of the two because it is so easy to explain and to fit. Almost every risk system and volatility library offers it.
• It showed that model choice changes conclusions. The paper is a case study in how a subtle misspecification, ignoring asymmetry, can flip the sign of the economic answer you get. That lesson travels well beyond volatility.
• It complicated the risk-return story. The idea that expected return moves one-for-one with forecast volatility is intuitive and, at monthly horizons, not well supported. That tension still animates research today, including work on the low-volatility anomaly.

The honest limitations

• The switch is abrupt. The model treats every negative return the same way, whether the market dipped 0.1% or fell 8%. A tiny loss flips the valve fully open. That is a crude approximation to whatever the true response looks like.
• Monthly data is thin. The core tests used monthly returns, which means a limited number of observations and correspondingly wide error bars. Conclusions about the sign of the risk-return relation are less sharp than they sound.
• "Expected return" is unobservable. Testing whether expected return depends on forecast volatility requires you to model both. If either model is wrong, the test is compromised. The authors improved the volatility side, but the expected-return side remains hard.
• Asymmetry is not universal. The leverage effect is strong in equities and weak or absent in currencies and some commodities. GJR-GARCH is not automatically the right choice everywhere.

The one-line takeaway

By adding one on-off switch that gives volatility an extra kick only after a down move, Glosten, Jagannathan and Runkle built the most-used asymmetric volatility model in finance, and found that the comforting idea that a riskier month must offer a higher expected return does not hold up as cleanly as everyone assumed.