MORE investors fear the American stockmarket is overvalued than at any moment since 2000, according to Robert Shiller of Yale University. Recent research by Deutsche Bank, meanwhile, suggests that government bonds are as expensive as they have ever been. So it makes sense for investors to consider what assets they should buy to hedge against a sudden plunge in the value of equities or bonds.

New research by AQR, a fund-management group, looks at the ten worst quarters for global equities and government bonds between 1972 and 2014. On average, equities lost more than 18% during such quarters while bonds, a less volatile asset, lost 3.9%.

These two asset classes are often seen as complementary: the classic “balanced” portfolio comprises 60% equities and 40% government bonds. Shares are riskier and benefit from economic growth; bonds are safer but their value is eroded by inflation. The AQR numbers show that government bonds do act as a useful hedge for equities, earning an average return of 4.8% in the quarters when shares plummeted. That is good news: both assets may look overvalued but they are unlikely to fall in tandem.

But government bonds were not the only asset to do well; commodities and, in particular, gold, also produced positive returns. Corporate bonds were of little help: they lost value, relative to government bonds, when equities fell (see chart).

When government bonds fall, equities are less consistent as a hedge: they gained ground in only six of bonds’ ten worst quarters. Nevertheless, thanks to their phenomenal performance in the second quarter of 2009, when the post-crisis rally got going, equities delivered an average positive return of 3.5% in bad bond quarters. Commodities also gained ground. Although it does not feature in AQR’s table, cash would have delivered a small positive return during sell-offs. But the current return on cash is close to zero, making it less appealing as a hedge.

What about hedge funds? In the past, these high-charging vehicles claimed to be able to prosper in good times and bad. Proper hedge-fund indices did not begin until 1990, so do not cover the full period. Even so, their record in the eight bad equity quarters since then has been disappointing: an average loss of 5.2%. A lot of hedge funds are exposed to the stockmarket.

AQR suggests that this may not tell the full story. There are a number of asset-picking strategies that have been shown to work over time. A combination of five of those strategies (value, momentum, carry, defensive and trend-following) would have produced very good returns during past stockmarket sell-offs.

Since many hedge funds operate variants of these strategies, it might seem odd that the overall industry numbers are not as good. However, the AQR composite figures do not allow for trading costs or fees. Its numbers are also the result of back-testing, a process that can produce marvellous results but is nonetheless flawed. Test enough strategies on past data and you are bound to find a few that work brilliantly. This would be so even with a truly random series, such as winning lottery numbers. There is no guarantee that such strategies will be as successful in future.

Is AQR’s exercise worth attempting at all? Many investors take a philosophical view. First, they are unlikely to predict the exact timing of the next market plunge; if the signs were obvious, they would have sold already. Second, long-term investors should ignore short-term market falls. Crashes may come and go but in the long run, asset prices will rise. American equities overcame both the bursting of the dotcom bubble and the 2008 financial crisis to reach record peaks earlier this year.

But the challenge for investors this time may not be a sudden collapse like 1987’s Black Monday. It could be a long slow decline, like the air leaking out of a deflating tyre. The obvious case study is Japan, which was the first rich country to grapple with deflation and zero interest rates.

Japanese equities are still at just half the level of the end of 1989, despite many promising rallies in the interim; bond yields have been stuck at very low levels since the late 1990s.

At least Japanese investors had the option of buying foreign assets as a way of escaping the doldrums in their home market. But the conditions they faced may now set in across the developed world.

Emerging markets were the big hope, but even they have let investors down in recent years.

The options are narrowing.