S&P 500 Relative to Trend

When considering the current valuation of the S&P 500 index, it's useful to consider the longer-term context.

The graph below shows the S&P since the beginning of 1928, with the red point representing Friday's ending value.

The time series appears as if it's growing exponentially over time, so the natural logarithm of the S&P is shown in the next graph.

This gives the impression of being subject to a long-term upward trend, so we show the series again in the next graph, this time with a trend line added.

This gives the impression that the log of the S&P 500 is reverting around a long-term upward trend. In fact, Robert Merton, in his well-known 1970 paper, Optimal Portfolio and Consumption Rules in A Continuous-Time Model, proposed just a process for the natural logarithm of stock prices. In particular, if yt is the natural logarithm of the stock price, he proposed the stochastic differential equation:

dyt = k (mxt + b - yt) dt + S dWt 

where

• xt is the index used to represent time (eg, could simply be t)
• mxt + b is the linear time trend, with constants m and b
• dt is an infinitesimal change in time, t
• dWt is the infinitesimal change in a standard Wiener process, {Wt}
• S is a constant that scales the Wiener process.

In fact, if we estimate the parameters of this process using the data shown in these graphs, it appears as if the log of the S&P 500 is reverting around a time trend that is increasing at a rate of 6.5% per year, with a half-life of four years.

The next graph shows the de-trended natural log of the S&P 500 -- ie, the difference between the blue line and the green line in the graph above.

The S&P 500 was as much as 34% above trend as recently as February 19 of this year. But as governments responded to the continuing spread of the SARS-CoV-2 virus, the S&P declined to the point that it was 11.9% below trend on March 23. In the latter half of last week, in the wake of fiscal and monetary support in a number of countries, the index regained a significant portion of its recent losses, increasing 13.6% -- essentially bringing it back to trend. (As of Friday's close, the S&P was 0.1% below trend.)

To help put this in perspective, the de-trended index is shown again in the graph below -- this time shown along with recessions, as determined by the National Bureau of Economic Analysis business cycle dating committee.

Note that most recessions involved the log of the S&P 500 index being well below trend at some point during the recession. In particular, the worst recessions saw the de-trended natural log of the S&P 500 down to -0.5. For example, during the great financial crisis, the de-trended log of the S&P 500 index reached -0.60 on March 9, 2009 -- 45% below trend.

In the event that the natural log of the S&P 500 index were to to decline to 0.6 below trend now, that would correspond to an index value of 1,395 -- ie, 45% below Friday's ending value of 2,541.47.

This exercise isn't intended to provide a precise valuation metric for US equities. Clearly, there are a number of important factors that would be expected to effect equity valuations in the present environment, including unusually low long-term risk-free interest rates, heightened volatility, and elevated risk premia.

On the the other hand, this exercise does put the current index value in to broader context. More specifically, it appears:

• the index was well above trend as recently as mid-February
• It declined to 11.9% below trend early last week
• It moved back to trend by the end of the week.

My sense is that the current recession is likely to be at least as severe as the one associated with the financial crisis of 2007-09. For that reason alone, I'd expect the S&P 500 to hit a level more than 11.9% below trend. 

But in addition, the mathematical models of virus transmission considered in previous posts all point in the same direction -- with the pandemic increasing considerably during the next few weeks. For example, between March 18 and March 29, reported deaths due to COVID-19 increased at an average daily rate of 13.4% in Italy, 24.9% in Spain, 29.4% in the UK, and 31.5% in the US.

As of March 29, deaths reported outside of China (where the virus currently appears contained) totaled 28,939. If this figure increased at a daily rate of 24.6% (the geometric mean of the death rates for Italy, Spain, the UK, and the US), we'd expect an additional 600,00 deaths to be reported between now and April 12. 

Over the same period, reported deaths globally ex China have increased at a daily rate of 15%. This is probably an underestimate of the death rate that will be experienced over the next few weeks, as the virus has only recently been reported in some regions, and death rates lag reported infection rates. But if reported deaths increased 'only' at a daily rate of 15%, this would still correspond to more than 175,000 additional deaths.

In the event that hundreds of thousands of additional deaths are reported in the next two weeks, I expect risk aversion levels among investors and traders would increase dramatically. In particular, I believe typical levels of risk aversion among investors and traders would easily exceed the levels experienced during the financial crisis of 2007-09.

Given my expectation that this recession is likely to be worse than the recession of 2007-09, and given my expectation that levels of risk aversion are likely to exceed those of 2007-09, it wouldn't be at all surprising for equity valuations to decline further below trend in this downturn than they did during the bear market of 2007-09.

None of this analysis is sufficient to provide a point forecast for this cycle's low in the S&P 500, but it seems unlikely to me that the S&P 500 is going to remain at or above trend as we go into this next, quite ugly phase of the current crisis in the next few weeks.