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.

Oxford University's COVID-19 Analysis

Researchers at Oxford University just released a draft working paper titled, Fundamental principles of epidemic spread highlight the immediate need for large-scale serological surveys to assess the stage of the SARS-CoV-2 epidemic. While the authors make claims with significant implications for the evolution of the epidemic in the UK, it's also been roundly criticized by other academics. With that in mind, I thought it might be useful to offer a few thoughts.

The ostensible purpose of the paper is to estimate the percentage of the UK population that already has been exposed to SARS-CoV-2 and that therefore reasonably might be expected to have developed some degree of immunity to the virus. If this proportion is sufficiently large, it may imply that the UK population already has developed a degree of  herd immunity, in which case public policy measures designed to slow the spread of the virus -- including those that are hobbling the economy -- may be relaxed.

The authors use the same sort of SIR model that I've discussed in previous posts. More specifically, they assume a range of plausible values for most of the parameters appearing in this model, and then they calibrate the model -- in particular, the day on which the virus first appeared in the population -- to a small subset of initial data. In particular, they use reported deaths for the first fifteen days after the first reported death in the population. They perform separate analyses for the UK and for Italy.

With this approach, the authors suggest that the virus was introduced into each population roughly one month before the first reported death in each population. In that case, the implication is that the virus would have spread widely among individuals in both populations by now. For the UK, they suggest that between 36% and 68% of the population would have been exposed to the virus by now. For Italy, the figure is estimated to be between 60% and 80%.

If true, this would mean that the policy of isolation is not necessary and that policy measures that have been crippling the UK and Italian economies can be relaxed. But does their analysis really support their conclusions?

One insurmountable problem with this analysis is the size of the data set: 15 days in each country. The authors choose this small subset in order to avoid using data from periods during which public policy interventions may have changed the course of the epidemic in each population.

Fifteen data points is far too small to reasonably calibrate the various parameters of their model, so the authors assume values that they believe are reasonable for many of these parameters, including

• the basic reproduction number, R0
• the duration of an infection
• the time between infection and death
• the probability of dying with severe disease
• the proportion of the population at risk of developing a severe infection.

But even if the authors had only one parameter to estimate -- the time of introduction of the virus into each population -- fifteen data points is almost surely far too few to estimate this time with any reasonable level of confidence. In fact, the authors don't report standard errors or confidence intervals for these estimates. Even with their many assumptions, I suspect these confidence intervals would be quite large. And if they were required to estimate all the parameters of their model using their fifteen data points, the results almost surely would be nonsensical, as there are simply too few data points per parameter.

A reasoned critique of the Oxford paper appeared today in a BMJ article titled, Covid-19: experts question analysis suggesting half UK population has been infected. It's useful to quote an excerpt from the BMJ critique. [Emphasis added.]

Neil Ferguson, director of the MRC Centre for Global Infectious Disease Analysis at Imperial College London, was asked about the study when he appeared before a parliamentary select committee hearing on 25 March. It was his analysis that showed that without physical distancing there would be 260 000 deaths in the UK from covid-19 and that led to change in government policy.

Ferguson said, “We’ve been analysing data from a number of Italian villages at the epicentre for the last few weeks where they did a viral swab on absolutely everybody in the village at different stages of the outbreak. And we can compare that with official case numbers, and those data all point to the fact that we are nowhere near the Gupta [the Oxford analysis] scenario in terms of the extent of the infection.”

Paul Hunter, professor in medicine at the University of East Anglia, said that the simple model “assumes complete mixing of the population,” which is “almost always wrong” at a country level. “We do not all have an equal random chance of meeting every other person in the UK.” He said that reproduction number was a “very clumsy” measure of how disease spreads, which is likely to change over time. He also criticised the researchers’ assumption that only a very small proportion of the population was at risk of being admitted to hospital because of the disease. “This is a big assumption and it is far too early in the epidemic to know what this value is,” he said.

Market Implications

If the Oxford results were true, we might reasonably expect significantly less economic disruption than many in the market have been expecting, in the UK, in Italy, and quite possibly across Europe and in the US as well. Given the historic reduction in economic activity that appears to be occurring, such a result would have significant implications for equity prices, bond yields, credit spreads, commodity prices, etc.

To be clear, I do believe the number of infections in the UK, the rest of Europe, and the US are very likely to be significantly higher than reported, particularly since there has been relatively little testing in these regions.

But the results of the comprehensive tests in Italian villages, mentioned by Dr Ferguson, suggest that the infection rates are likely significantly lower than suggested in the Oxford paper.

As I result, I continue to expect that the peak infection rates in the US and the UK are still weeks if not months ahead, with much higher infection rates becoming clear in coming weeks. As a consequence, I expect public policy measures will become more disruptive for the economy rather than less disruptive. And in that scenario, it seems quite unlikely that we've already seen the lows in the S&P 500 and in Treasury yields.

Empirical Infection Curves

In Time-Varying Parameters in Virus Transmission Models, I illustrated the effect that non-pharmaceutical interventions could have on the infection curve in a basic SIR virus transmission model. In particular, it's difficult to estimate the parameters of these models when there's reason to believe they're varying over time. And for that reason, it's useful to consider the empirical curves, when available.

For example, the graph below shows confirmed cases, recovered cases, infected cases, and deceased cases in recent months in China.

Source: John Hopkins

Note that none of the curves appear as simple as our basic SIR model would predict. Of course, there are numerous reasons for this. For example, the basic SIR may be too simple to capture dynamics in a real-world epidemic. In addition, Chinese authorities intentionally changed the basis on which some of the data was reported in mid-February. And important for our discussion: interventions on the part of authorities in China almost surely changed the dynamics of viral transmission within China. In particular, lowering the reproduction number, R0, is an important tactic in all efforts to restrict viral transmission.

Notwithstanding, the basic characteristics of viral transmission are clear in these reported figures. The 'infected' curve starts with near-exponential growth; it then reaches an inflection point followed by a peak, after which the number of people infected with the virus declines.

The next graph shows similar curves for South Korea.

Source: John Hopkins

Here again, note the initial growth, followed by an inflection point, followed by a peak, followed by a decline. South Korea appears to have rigorously followed WHO advice for people to isolate and for authorities to engage in intensive testing and contact tracing. And these curves give the impression that the South Koreans have made notable progress in recovering from the infection.

In contrast, the figures for Italy are shown in the next graph.

Source: John Hopkins

The 'infected' curve for Italy is no longer growing at a near-exponential rate, but neither has it reached an inflection point. As a result, it appears too early to forecast when Italians might experience the peak and the number of people who will be infected at the peak.

The same is true of Spain.

Source: John Hopkins

The reported transmission curves for the UK are shown in the next graph below.

Source: John Hopkins

Unfortunately, the number of reported infections in the UK is still exhibiting near-exponential growth. More specifically, simply fitting an exponential curve to the 'infected' curve in the UK still provides quite a good fit, as seen in the graph below.

Source: John Hopkins; author. The blue line is the actual number of active COVID-19 infections reported daily in the UK since these first topped 100, on 5-Mar-20. The dotted red line is the exponential curve of best fit through these points. The fact that this exponential curve still fits the data quite closely suggests the UK infection is still in its early stages of transmission. 

How about the US? The confirmed, infected, recovered, and deceased curves are shown in the graph below.

From this graph, it's difficult to tell whether the US infection is still at the initial near-exponential stage or whether it's beginning to bend toward an inflection point. The next graph below, with a fitted exponential curve, confirms that the US still appears to be at the near-exponential stage of growth. As with the UK, there's no basis in this data for predicting the timing or severity of peak infection in the US.

Source: John Hopkins; author. The blue line is the actual number of active COVID-19 infections reported daily in the US since these first topped 100, on 3-Mar-20. The dotted red line is the exponential curve of best fit through these points. The fact that this exponential curve still fits the data quite closely suggests the US infection is still in its early stages of transmission.

Even when circumstances are changing and the parameters governing the evolution of the infection are changing, it's useful to analyze the empirical data, when available -- particularly to look for

• the period of near-exponential growth in the infection curve
• the subsequent inflection point, when the rate of growth stops increasing
• the peak of the infection curve
• the subsequent decline in the incidence of infection.

Market implications

Of course, our interest isn't in epidemiology per se. We're ultimately interested in assessing the likely effect of the pandemic on economic growth, inflation, employment, and the financial markets.

With that in mind, what might we have learned from this exercise? A few things:

• China and South Korea appear to have turned the tide on their epidemics.
• The virus still appears to be spreading rapidly in Europe -- eg, in Italy and Spain, where the rate of infection is still increasing, though not at a near-exponential rate.
• In the US and the UK, the rate of infection still appears to be in the initial, near-exponential stage.

Equity markets have reacted favorably to news of fiscal stimulus in the US, the UK, and elsewhere -- and perhaps also to President Trump's insistence that US workers should return to work en masse by Easter (April 12). But given the still rapid rate at which the virus appears to be spreading in the US, it seems more likely that more American workers will be in isolation in mid-April than there are now. 

In fact, if the number of infections were to continue to grow at the current exponential rate between now and Easter, the total number of reported infections in the US would be on the order of 15 million.

Of course, it's quite possible that the infection curve will begin to move away from this near-exponential growth between now and then. It's even possible that the infection curve in the US will have reached an inflection point between now and Easter, in which case the number of reported infections may be well below 15 million.

But to put this in context, the peak number of active infections in China was reported to be about 58,000. Even if the number of infections in the US peaked at, say, 5 million, the number of infected would be roughly 86 times the peak number of infected people in China.

In Italy, 12% of all detected COVID-19 cases were admitted to the intensive care unit. If that percentage were observed in the US, and if the peak number of infections in the US were 'only' 5 million, that would still point to a simultaneous demand for 600,000 ICU beds in the US -- far outstripping the available supply, estimated to be on the order of 100,000.

I should stress that these figures are all just attempts to produce first-order approximations of the eventual numbers. But they're broadly consistent with figures produced elsewhere. For example, The Harvard Global Health Institute produced an online report, in which ICU beds in the US were estimated at just over 87,000. The number of patients requiring hospitalization for COVID-19 over time was predicted to be 10 million, and the number of patients requiring an ICU bed was predicted to be just over 2 million. These Harvard figures aren't for a single point in time, as were my attempts to predict peak demand for ICU beds. But these sorts of figures at least put the problem into broad perspective.

All in all, what are the conclusions from analyses of this sort?

1. The rate of infection in the US is still increasing at a near-exponential rate.
2. Peak infection is still well in the future in the US, with the peak infection rate many times greater than the infection rate currently reported.
3. The situation at Easter will almost surely be far worse than it is currently.
4. Under these circumstances, it is difficult to see droves of US employees returning to their workplaces, admonitions by the President notwithstanding. More likely, an even larger percentage of the US population will be under strict lock-down orders at Easter.

The rally in the S&P yesterday and today, from a low of 2174 (June S&P futures) at the start of the week to 2498 now (up by nearly 15%) is impressive. But my sense is that with peak infection well ahead of us, it's quite unlikely that we've seen the lows in the June S&P futures -- particularly if the number of peak infections is even close to the estimates consistent with the analysis discussed here.

Time-Varying Parameters in Virus Transmission Models

In Modeling the Incidence of COVID-19 Over Time, I mentioned some of the models I follow for the purpose of tracking the spread of the SARS-CoV-2 virus. These models are particularly useful in projecting the incidence of infection, recovery, death, etc., given a set of parameter values describing characteristics such a the average number of contacts per person per unit time, the probability of a contact resulting in an infection, and the typical duration of an infection.

In practice, most governments have introduced public policy measures designed to slow or reverse the spread of the virus, including social distancing, testing, and contact tracing. 

From the perspective of a transmission model, these non-pharmaceutical interventions are intended primarily to reduce the reproduction number, R0 of the pandemic. And to the extent these interventions are successful, we need to reflect that in our models.

For example, the graph below shows two infection curves for a hypothetical virus, in a population the size of the UK (67.9 million people). The blue curve assumes:

1. the average person has 20 contacts per day
2. the probability of a contact resulting in an infection is 0.01
3. the duration of the infection is 28 days.

With these three assumptions, the reproduction number, R0, is 5.6. The maximum number of infections under this scenario is 34.9 million people (51.4% of the poplulation), reached 114 days into the epidemic. Infections then decrease as individuals move from being infected to being removed (ie, immune or deceased).

The blue curve shows the trajectory of infections when the reproduction number, R0, is held constant at 5.6. The orange line shows the alternative trajectory in the event measures are introduced to reduce R0 to 0.7 -- in this case when the number of infected people first exceeds 10 million.

Now let's imagine that public policy interventions are introduced as the number of infections first reaches 10 million. In particular, let's assume these interventions reduce the average number of contacts per person per day from 20 to 5 and that the probability of a contact resulting in an infection declines from 0.01 to 0.005.

In this case, the reproduction number decreases from 5.6 to 0.7, with the result that that trajectory of the virus changes dramatically, from a rapid rate of increase to a moderate rate of decline, as illustrated by the orange line in the graph.

Now consider a country like the UK or the US, for which public policy measures have been changing nearly every day in recent weeks. These continual changes in policy measures create a situation in which R0 can't be assumed to be constant over the period of analysis one might otherwise use to estimate the parameters of the model.

As the experience of Imperial's MRC Centre illustrates, a model with inappropriate parameter values may actually be worse than useless in the event it supports policy measures that do more harm than good. So in that case, what good are these models -- particularly for financial analysts wishing to consider implications for the macroeconomy and financial markets?

From my perspective, these sorts of models offer two types of benefits. First, they help us understand the reasons that viral infection curves observed in practice are characterized by a period of near-exponential growth, followed by an inflection point, followed by a maximum, followed by a steady decline. Second, they provide a framework we can use to produce first-order approximations of viral transmission under various public policy initiatives. These projections needed to be treated with caution, due to the combination of model error, parameter estimation error, and data errors. (For example, who really knows the actual current number of infectious persons in the UK?)

And speaking of infection curves observed in practice, it's time to turn to some actual data...

More on Imperial's Modeling

In Modeling the Incidence of COVID-19 Over Time, I mentioned a few of the transmission models I'm using and/or following during the pandemic, with particular focus on the individual-based simulation model used by the MRC Centre for Global Infectious Disease Analysis at Imperial College.

The Guardian published an article today under the headline, Britain had a head start on Covid-19, but our leaders squandered it, written by Professor Devi Sridhar, chair of global public health at the University of Edinburgh. It's worth quoting from Professor Sridhar's article [emphasis added]:

"On 17 March, Imperial College released a study noting that it had revised the model the government had been using, and stating that suppressing the virus was in fact the best way to avoid a vast number of people dying. The earlier model did not include the ICU data shared in the Lancet on 24 January. Instead, it was similar, but much later information from Italy, that changed their recommendation."

In other words, the MRC Centre's model was calibrated to less representative data than was available at the time.

What are the implications for policy and outcomes? Professor Sridhar offers some thoughts on that as well [again, emphasis added]:

"We had a choice early on in the UK’s trajectory to go down the South Korean path of mass testing, isolating carriers of the virus (50% of whom are asymptomatic), tracing all contacts to ensure they isolate as well, and at the same time taking soft measures to delay the spread. Instead, we watched and waited, and whether it was academic navel-gazing, political infighting, a sense of British exceptionalism, or a deliberate choice to minimise economic disruption over saving lives, we have ended up in a position where we are now closer to the Italy scenario than anticipated, and are faced with taking more and more drastic measures."

Modeling can be difficult, particularly when parameter values need to be estimated with limited data. Over time, I imagine a number of interesting articles will be written about the modeling efforts at Imperial and their effect on policy. Until then, it's useful to appreciate the wide range of forecasting errors we face when trying to model the transmission of a new virus.

Modeling the Incidence of COVID-19 Over Time

I've been following the efforts of various groups charged with modeling the spread of the SARS-CoV-2 virus, particularly the MRC Centre for Global Infectious Disease Analysis at Imperial College, given that the UK Government appears to be paying particular attention to this group's efforts.

According to MRC's Report 9: Impact of non-pharmaceutical interventions (NPIs) to reduce COVID-19 mortality and healthcare demand, Imperial's MRC Centre is using an individual-based, simulation transmission model -- apparently a modification of the work discussed in the article, Modeling targeted layered containment of an influenza pandemic in the United States.

It's clear from Report 9 that the MRC Centre has considered quite a number of factors in their model, including heterogeneity of US and UK populations. On the other hand, the authors dramatically revised their policy recommendations in Report 9, as they updated parameters of the model. In fact, it's worth quoting a few paragraphs from their report [emphasis added]:

Perhaps our most significant conclusion is that mitigation is unlikely to be feasible without emergency surge capacity limits of the UK and US healthcare systems being exceeded many times over. In the most effective mitigation strategy examined, which leads to a single, relatively short epidemic (case isolation, household quarantine and social distancing of the elderly), the surge limits for both general ward and ICU beds would be exceeded by at least 8-fold under the more optimistic scenario for critical care requirements that we examined. In addition, even if all patients were able to be treated, we predict there would still be in the order of 250,000 deaths in GB, and 1.1-1.2 million in the US.

In the UK, this conclusion has only been reached in the last few days, with the refinement of estimates of likely ICU demand due to COVID-19 based on experience in Italy and the UK (previous planning estimates assumed half the demand now estimated) and with the NHS providing increasing certainty around the limits of hospital surge capacity.

We therefore conclude that epidemic suppression is the only viable strategy at the current time. The social and economic effects of the measures which are needed to achieve this policy goal will be profound. Many countries have adopted such measures already, but even those countries at an earlier stage of their epidemic (such as the UK) will need to do so imminently.

In switching from a mitigation strategy to a suppression strategy, the MRC Centre gives the impression that it's model isn't particularly robust to the values of its input parameters.

I'm not sufficiently familiar with the details of their model to offer a reasoned critique. On the other hand, switching from a mitigation strategy to a suppression strategy is a very significant change, and I come away from reading their research reports with the impression that their model isn't as robust as they thought.

George Box is attributed with the phrase, "All models are wrong; some are useful." With this in mind, my concern with the relatively sophisticated efforts of Imperial's MRC Centre is that their model may be too nuanced and sophisticated to be genuinely useful in practice.

In my own efforts to model the incidence of the virus, I've used the model that I understand is the simplest of the deterministic models used in epidemiology: the susceptible-infected-removed (SIR) model. In particular, it captures the characteristics of the infection curve that appear to be most important:

• an initial period of near-exponential growth
• followed by an inflection point, when the second derivative is zero
• following by a peak in infections, when the first derivative is zero
• following by a decline in the number of infections.

The SIR model has a small number of parameters, particularly the basic reproduction number, R0, of the epidemic, which can be used to evaluate various scenarios for transmission.

In the basic SIR model, individuals move from being susceptible to being infected to being removed (recovered, immunized, or deceased), according to a series of ordinary differential equations. The rate at which individuals move from being susceptible to infected is determined by the reproduction number, R0, while the rate at which individuals move from being infected to removed is determined by a second parameter. For more on this model, see Compartmental models in epidemiology, and A Contribution to the Mathematical Theory of Epidemics.  

I'm not suggesting that this simple model is the best one can use to model the COVID-19 pandemic. Surely, practicing epidemiologists have models that are more useful for this purpose. But for the purposes of a financial analyst or an investment strategist trying to capture the key characteristics of viral transmission, the basic SIR model seems to provide a useful trade-off between parsimony and robustness.

For readers interested in going beyond this basic model, an online Epidemic Calculator using a Susceptible-Exposed-Infected-Removed (SEIR) model is available online.

A very well-done online COVID simulator has been produced by Martin Eichner, Stefan Brockmann, and Markus Schwehm.

While these models vary in sophistication, they all capture the basic characteristics of viral transmission discussed above. For the purpose of financial analysts, a useful check when using such models is to compare the results produced by various models, to assess the extent to which relevant inferences are robust to the choice of model.

A pop-up blog to discuss financial markets during the COVID-19 pandemic

I've been asked by friends and colleagues for my views on various financial markets in light of the COVID-19 pandemic, and I thought a pop-up blog might be easier to manage than a series of emails.

I anticipate most readers will be friends and colleagues in the industry, but for those who would appreciate some biographic information, feel free to check my LinkedIn profile.

You'll note that I have no expertise in the fields of infectious diseases, epidemiology, etc. At the same time, it seems one must take some views on the evolution of the pandemic as one considers the macroeconomy and financial markets. So my general approach is to read widely among presumably authoritative sources and whenever necessary do my best to choose judiciously between conflicting medical analyses. I welcome any and all comments that improve on these sources.

Those of you who have received my emails will be familiar with the frameworks I'm using to assess the situation. But for those who haven't received these emails, I'll summarize these in the first few posts.

Otherwise, feel free to comment or email if there are specific topics you'd like me to address.

And in the meantime, good luck to us all.