I was going to talk about this research I´ve been doing for the last three years or so, about rare macroeconomic disasters and how they relate to financialn markets,

and I actually did start working on this in 2005, and the topic has become a lot more current than it was when I started working on this, so it´s applicablen to this ongoing financial macroeconomic crisis that we´ve been experiencing;

but I didn´t actually start working on that topic in response to this situation.

I´m gonna try to sketch some of the underlying theory and then particularly talk about this sort of long-term data from many countries that we´ven constructed, in order to try to implement these ideas and then talk about some of the results and some of the work that´s more current.

So let me get started with that, so initially when people were thinking about the large macroeconomic disaster events and how that might map into some of then financial puzzles, people focused on the Great Depression, of the early 1930s, and that was particularly from the U.S. context, because that was the main disaster event that you saw in then data, if you look in the United States.

But there wasn´t enough information there to really figure out something about how often these events occur and how big are they? Are they important enoughn to explain some of the patterns we see in the financial data?.

So, the first thing that we did, really, was try to expand to a broader set of countries and experiences. So the kind of data that I´m gonna be looking at,n are basically 100-year-periods on nature macroeconomic variables, such as gross domestic product and consumption, and then it turns out there are about 30 countries from which you can assemblen data like that.

So it´s gonna be like 30 countries, 100 years; about three thousand annual observations, and that´ll turn out to be big enough that we´ll have a sample ofn these rare disaster events, like the Great Depression in the U.S., if one good example.

So that we can infer something about the probability of these events and how big they are. Julio observed a few minutes ago that I was simply an actuary,n figuring out the probabilities of disasters, like floods. I don't know whether he meant this as a complimentary or...

I´m sure this will come out in the discussion, but to some extent, I think that´s right, I´m trying to figure out how often these things occur and how bign they are; and at least initially, I´m really not explaining why these events occur.

And for some purposes it may not be necessary to do that, but for other purposes that would be central and important.

So, the underlying background; there was this famous paper in 1985 by Mehra and Prescott, which established the term Equity Premium Puzzle, which raised then question: Why was it that the average return on risky assets, like stocks, are much higher than the average return on comparably low-risk securities? Perhaps short-term treasury bills are forn something like that.

So it is an established fact in the data, particularly over the long-term, that there´s a major gap between the average return on stocks and low-riskn securities; it´s something on the order of perhaps 7% per year.

And they were trying to explain that, because the usual view was that you couldn´t account for that within the usual kind of macro model that economists liken to use.

So they raise this as a puzzle in the 1985-paper; and then, in fact they argued that really, they couldn´t explain it, it remained a puzzle in terms of then results from their paper.

By the way, Prescott told me he originally submitted this paper in 1979, and the paper was rejected several times by, for example, the Journal of Politicaln Economy, and it was even rejected by the Journal of Monetary Economics, where it was ultimately published in 1985 on the second try. So, I try to tell this to students that get discouraged whenn they submit a paper and it gets rejected.

Sometimes rejected papers end up being important in classics so, Prescott was quite bitter that his paper was delayed by five or six years because ofn this.

Anyway, this Equity Premium Puzzle is raised there and then Tom Rietz in a paper three years later. So maybe you can explain this by the probability ofn really large disaster events, which you don't see very often in the data, and he focused on the U.S., and he mainly talked about the Depression, and he said "well you see this one thing, butn then you don´t see other cases".

But he argued that maybe the probability of this kind of large terrible event is enough to explain the Equity Premium Puzzle.

So I haven´t known Rietz, probably this came from his PhD thesis at the University of Iowa; and Mehra and Prescott have a rejoinder to his paper in the 1988n Journal of Monetary Economics, saying that basically this is a stupid idea, and then they make some arguments about why this sort of rare disaster explanation.

But the Equity Premium Puzzle can´t be right, if we go back and read their rejoinder, I think their arguments are pretty silly. But I think they persuaded an lot of people at the time.

This is me reproducing the facts, because I wasn´t really involved in this contemporaneously. I think they persuaded a lot of people that this idea was notn very promising.

And then, there wasn´t a lot of follow-up research on the Reitz idea, about the rare disasters being a way of thinking about some of these famous financialn market puzzles.

So, I only came to this work relatively recently, ´cause I didn´t really work in finance. So, I only really started thinking about this issue in 2005, and In thought that this rare disaster idea might be promising and then I looked at the Rietz-paper, and I thought what he argued made a lot of sense; I thought the Mehra and Prescott complaintsn didn´t make a lot of sense, so

anyway, I´ve been pursuing this line of research since 2005, and the first paper I published in 2006 on this topic, was in the Quarterly Journal ofn Economics. And that´s the work I´m gonna try to summarize and then talk about, this is still an ongoing major line of research, so I´m gonna try to talk about that.

The claim is that this idea can explain the so-called Equity Premium Puzzle, as a related matter of why the risk-free rate of return is very low in the data,n which seems to be true.

And also the extend that it can perhaps explain things like why stock prices are so volatile, which is also true in the data, and a number of other puzzles.n I think there can be a common explanation for a series of these puzzles. I think that´s the argument.

I´m gonna basically use a standard model, which is not so different from Mehra and Prescott, but add to that the Reitz-type of rare disaster ideas,

and there could be two key parameters that appear here, that bring the rare disaster´s potential into what is otherwise a fairly standard kind of model.

There´s gonna be a probability of disaster which I call P, in the base line model I´ll think about that as a constant, depending on how you define disastersn it might end up being something like 3% per year, I´ll talk about the numbers.

More generally, it can, of course, vary over time, and I think that, to think about some of the recent events, I think you have to consider that then perceived disaster probability, probably went up, and that has some important implications for asset prices.

But in the base line model, I´ll think about the disaster probability as being some kind of constant, and I´m gonna try to infer what it is.

The other thing about disasters is how big they are; so first is how often do they happen, and second is how big they are.

So the parameter Bis gonna represent the size of a disaster. So, it would be like if Bis 0.3, it means that output went down by 30%, which is about the sizen of the U.S. Great Depression; B of 0.3 means you´re losing 30% of your output over a short period.

I´m not gonna think of Bas a constant, because some disasters are much bigger than others; and it turns out that actually, the biggest disaster events aren not the Great Depression, but rather the World Wars.

If you look outside the United States, for example, in Europe or in Japan, the biggest declines in GDP that you see from disasters are really World War IIn and World War I, is also quite large.

U.S. is pretty dominant in macroeconomics, and therefore, U.S. macroeconomists think of World War II as a time of very rapid growth, which was true for then U.S., but the U.S. is really an outline with respect to the world, and many other countries suffered greatly at that time.

And it actually ends up being the biggest set of disaster events, World War II.

Anyways, I´m gonna try to figure out from the data what´s P,and what the frequency distribution of disaster size is B,and that´s what that method I thinkn about being an actual...

you can think about floods here if you want, and trying to ensure floods and that´s fine.

I´m not gonna go through too much of the detail of the models, but some papers that are relevant are the ones that are indicated there.

My 2006-paper, turns out there are some short comings of the theoretical analysis, so I had to make some extension of that in a paper that came out recentlyn in the American Economic Review, the 2009-paper.

But I´m only gonna say a few words about the nature of that, I´m not gonna go into details unless people ask me questions about that.

The basic setting I´m gonna think about is this so-called Lucas-Tree Model. Lucas had this paper, Bob Lucas in 1978, which is a very simplified macron setting, where one which is very convenient for thinking about asset pricing, how do you price various claims,

like stocks or risk-free claims, and it´s a very convenient setting for thinking about that pricing, and then you can make extensions on that base linen model: you can make the model more resistant.

So, in equity claim, a stock certificate is gonna be a claim on some kind of stream of output and consumption, which in Lucas, comes from a tree. So there´sn a tree that spits off fruit, and the fruit is the gross domestic product and in the simple model all of it is consumed.

So the fruit that falls off the tree is both, GDP and consumption, in the simple model. And you can extend that model so you can allow for investment, butn the baseline model has all the GDP being consumed. But some extensions of that can allow for investments evidence that´s crucial for the basic asset pricing.

So there´s some disturbance that occurs in the probability P, that´s the stochastic part that´s gonna be important, that´s gonna be the rare disaster and youn can think about that as affecting the productivity of the trees in the Lucas´s model,

or in an alternative story, you can think about some disaster like 30% of the trees get wiped-out by bad weather; so that´s the flood andn hurricane-risk.

So that´s another version of the story which worked perfectly fine. So every now and then, you got these tress, they spit out fruit, but every now and then,n there´s a storm and it wipes out 30% of the trees, and that´s a simple way of thinking about the disaster.

So Pis how often that happens; Pis what fraction of the trees get wiped out, when there is one of these disasters.

The alternative that the main disasters that we isolate in the data, there´re really three types that we found; one is wars, which turn out to be then biggest, World War II, especially.

A second class of events, is financial crisis, such as the U.S. Great Depression or great depressions in other countries in the early 1930s; but you can alson think of the Asian financial crisis, the Latin-American debt crisis of the 1980s, and some other events like that.

The third kind of event we found, which you can think about in principle, is like a disease epidemic, and people worry now about the flu, for example. Turnsn out that the Great Influenza epidemic of 1918 and 1920 looks like its another one of the disasters, which affected multiple countries.

But that´s the only case we have that´s like that, like a shock to health, or you can imagine a natural disaster like a typhoon or a hurricane, but none ofn that seems to be big enough in the data to show up in macroeconomic terms as a crisis, given the way we define how big a crisis has to be to call it one of these rare disasters.

Just a question, is it against the potential growth of that specific year or is it against the growth of the past years´ GDP?

In terms of what a disaster is?

Yeah, the measurement of the disaster.

So there are two things here; one is what the theory is, I´ll sketch that and of course have a simple theory; and secondly, how to relate that to the data,n which are more complicated. So, I´m gonna talk about how we measure the disasters in the data, so let me hold off on that for a moment.

First, I´ll talk about what the theoretical framework is. It won´t exactly coincide with the way we measure the disasters and then in some ongoing work we´ren trying to get more accurate in regard to that.

So let me say first about, a little bit about the model, how we model disaster events. There´s a very simple way of doing that, so we can actually get assetn pricing formulas in a very clear and simple way.

One of the things that matters is preferences, and particularly attitudes toward risks are gonna be important for this asset pricing, or for the equityn premium,

you know, how big the difference between the required return is on average on stocks, versus risk-free claim, it´s certainly gonna depend on how averse ton risk individuals are through the economy.

So this, like the Lucas model or like Mehra and Prescott, is a so-called representative agent, or representative consumer model, and in simple saying, then utility of this representative agent is given by this formula.

So this is a pretty standard approach, there are two key parameters that appear here, let me just indicate what they are. Utility is gonna be an expectationn of something involving current and future consumption.

So C is the consumption that appears for this representative consumer; grow is gonna be a rate of time preference, that´s not very important in thisn model.

So grow is positive, it makes you discount future expected utils relevant to current ones, that doesn´t really make much difference in this model. What´sn more important is this kind of curvature parameter, which is gonna be describing the attitude toward risk, toward the extent of risk aversion.

That´s this parameter gamma here and a higher gamma is going to mean that you are more averse to risk. A kind of estimating gamma, from the results, turnedn out a typical non-averse amount three, in order to fit the data, in terms of the risk premium, but I'll talk more about how it enters into the results, but basically a higher gamma is gonnan mean more aversion to risk.

This is a very standard way of writing this utility function. Now there´s a problem here, which is why I had to modify the results, the framework in thisn 2009-paper, and this is what I´m not gonna discuss in great detail, because in this standard approach the parameter gamma really functions in two ways.

So, one is gonna to tell you how averse to risk people are, and a higher gamma means more risk aversion; but gamma also tells you how you feel aboutn substituting consumption over time, consuming today versus tomorrow.

And if gamma is really high it means that you don't really like to have an irregular pattern of consumption. You don't like the idea of giving up consumptionn today to get more consumption tomorrow, so that´s something, about saving behavior is gonna relate to that.

And the problem with this standard formulation is that this parameter does double duty, tells you about both of these things, and conceptually they´re reallyn very different.

Attitude toward risk versus consuming different amounts in different points in time, conceptually, should be completely different. There´s no reason why theyn should be tied together.

Yet the standard formulation, links the two in a very rigid manner and what turns out to be the case, is if you think about gamma being free, about riskn aversion, which I´ll talk more how that matters,

then, it means that some other parameter about how willing you are to substitute over time, is gonna be one over that gamma, which it would be like onen third, and that´s gonna mean you´re not very willing to substitute over time.

But there´s no reason these two things should be linked together. And that was the problem that I was looking at in this 2009-paper, so I have gone to an different preference formulation, which allows you to distinguish these two attributes.

So I can talk a little bit later about where this comes in, and I´m not gonna worry about it too much right now. And it matters for some results and not forn other results.

People mostly like to stick with this simple formulation, because it is simple, but some of the implications of this model are problematic, that´s why In finally decided that I really can´t keep this forever.

But let me leave that aside, I´ll talk about that later, that´s in the 2009-paper.

This is about the special occasion of this Lucas-Tree economy, which I´ve already mentioned, the risky assets here you can think about as a claim on then consumption that comes from the tree, or put it on the path of gross domestic product itself,

so, I'm thinking about a stock market certificate, as a claim on some share of the gross domestic product or consumption, because they´re gonna move togethern in a simple framework, although in some other models they wouldn´t be tied together.

So, that´s what a risky asset is gonna be, it´s a claim on the path of future gross domestic product or consumption. And then that´s gonna to be related ton stock market return. So you can also think about a risk-free asset, which is really hard to find, risk-free assets in the world.

Short-term index U.S. treasury bonds seem to be pretty close now to some kind of risk-free clients, but there are a lot of problems with the details so it´sn not quite right.

You can have risk-free assets being priced in this economy, but those things are gonna be something created internally, it´s in zero net supply, there is non sort of risk-free, there´s no risk-free investment activity in this model.

It´s not like there is some physical thing you can put in that pumps out a risk-free return. So, the first risky claim on the trees, or on the path of then fruit from the trees, is gonna correspond to what the economy's capital is, so that´s basically a claim on the economy's capital, which are the trees, and the trees are a positive netn supply.

But this risk-free claim, or all kinds of derivatives that you could price within this model are gonna be things that are zero net supply, somebody is on onen side, somebody else is on the other side, there´s gonna be some difference.

Okay let me explain how I model in a particular disaster, which gets back to the question before; and I´ll try to relate that to how I measure the size ofn disasters in the data.

There was some assumption about the probabilistic process that generates GDP, which is the fruit from these Lucas Trees and that´s gonna be the same asn consumption in this first setting.

The preference in the log is gonna be the growth rate of GDP; a population is constant so this is the same as GDP per capita in the model.

So the growth rate of GDP has three terms on the right hand side, there´s some deterministic average growth being determined by some parameter g, and heren I´m not determining where that comes from.

In other work I try to pick about long land roads, but here I´m just taking it as it such is. So g is kind´a gonna determine the average rate of economicn growth.

There´s the term u,which is supposed to represent the usual kinds of business fluctuations or business cycles, that´s gonna be like what´s in the Mehra andn Prescott paper.

So, formally, u might have zero main and some variants sigma squared, and then normally you would try to get sigma square to match the sort of annualn fluctuations in GDP and consumption for the U.S., or some other economy. So that´s basically what Mehra and Prescott had,

and their main conclusion was that it was not enough to explain the Equity Premium; there wasn´t enough uncertainty in the economy to associate with then usual business cycle risk, which is the u.

So that´s the Mehra and Prescott term which I have; and then, v is the Rietz type of rare disaster term. So, the way I modeled that is here, with somen probability p a disaster occurs, and then, in the simple model output contracts by a proportion v at an instant of time.

So GDP is moving around and then all of a sudden it falls by 30%; that´s a disaster, where v is 0.3, so that´s a jump.

In a model the disaster occurs at a moment of time, more realistically, disasters are sprayed out over time and they take one, two, three, four years, orn something like that, and that´s why there are some disconnects between some simple model and the theory;

but I think this is not one of the bigger problem about these models.

But anyway, in the model, a disaster is a jump downward in the level, and that´s when v is positive by 0.3, GDP and consumption fall instantaneously byn 30%.

But v is not gonna be a constant, vis gonna have some frequency distribution because empirically, there is a lot of difference in the sizes, among differentn disaster events, so it would not fit the data at all the force vto be the same for all, it wouldn´t work.

And you couldn´t have like a reverse effect that would make a positive jump?

You could... in the data is not actually symmetric, the main big ups that you see in the data, are following on from previous big downs, especiallyn wars.

So in the data you sometimes see, like in World War II, a really big decrease, maybe 50%, but then it tends to be followed by higher than normal growth for an while; that´s not gonna be in this model, and the main big ups you see are in response to the previous big downs.

I think the biggest one-year positive GDP increase, in my data, is for France in 1946, at the end of World War II when the Germans finally left, and then reported GDP went up by more than 40% in one year.

You know, they weren´t working very hard when the Germans were forcing them to work, they turned down the lights, but that´s a very usual observation.

The second thing is, that for the asset pricing results it´s not symmetric so even if you had bonanzas and disasters that were symmetric, the way that theyn would enter into the pricing of the risky claims is very different.

What´s gonna be true is that in a disaster, when you are really suffering, you put a lot of weight on that because your margin utility is really high, whenn your output and consumption fell by 50%.

That gets a lot of weight; if you think in contrast about getting a 50% bonanza, that doesn´t have nearly as big an effect in the usual kind of utilityn formulation, because of what we´re assuming about how margin utility reacts to the level of consumption.

So actually if you make the thing more symmetric, and you allow for bonanzas as well as disasters, it wouldn't have that much impact on the asset prices,n it´s also like not true in the data that´s symmetric.

But this money income is the same in all the countries that you have studied?

Well, the set of countries that I´m studying is those that have long-term data going back at least before 1914, so that includes OECD countries and thenn another 15 or so countries in Latin America and Asia that have long-term data.

So, some of these are middle-income countries which are pretty subject to the kind of shocks that you are talking about. But the main limitation on the datan is this long-term perspective.

I´ll talk about the data momentarily, but it´s basically a data requirement that limits the sample. Now if you want it to study just the period since 1950 orn something, you´d have a lot more countries, but that has very different implications about disasters. And we can discuss that later.

You could add another few more countries, I think you could add Russia and Turkey if you really worked on it hard, for example, but you can´t add too manyn more countries; maybe two or three more, I think about the limit in terms of the long-term...

This is the model of how GDP and consumption evolve, and the part that´s gonna matter in terms of the results, the equity premium is the disaster part, then Reitz part; not the normal business fluctuation, which is like Mehra and Prescott. So we start with the fact that the pand the vare gonna be quite important.

This is just a form of the gstar as the expected rate of economic growth in this model, it´s not that important but it depends on g. There´s some funnyn curvature effect because the thing isn´t linear involving sigma square.

You see a typical sigma is like 0.02 per year, if you´re trying to match particular recent business fluctuations, it´s about 0.02 plus or minus 2% per yearn in terms of the growth rate.

So that´s roughly what fits some of the recent observations. We have sigma 0.02 and sigma square is some 0.0004 per capita, and this is basically zero; itn does not matter.

This is the effect that Julio just mentioned because they have only disasters and no bonanzas. This is the arrow size of disasters multiplied by how oftenn they occur, p; and this is a negative, because I only have bad events that are large, I don´t have offsetting bonanzas. If it was symmetric, this would be gone, actually.

What´s the difference between u and u(t+1)?

That´s just describing the statistical process that generates the normal business fluctuation, the assumption here on the simple model is that these aren independently, identically distributed so that, if output goes up this year, it doesn´t tell you whether output is gonna go up or down next year,

these are independent and they always have the same distribution; and the distribution is characterized as being normal with a zero main and some customn variants, which is the sigma square.

This is a very simple model of the normal business fluctuations, and actually none of that part matters quantitatively, just as in Mehra and Prescott,n because they don´t explain the equity remium in their paper.

As well-know, going back to Lucas, in this simple model, you have a very simple formula to pricing assets, which comes from the consumer's first ordern optimization condition for consuming at one point in time versus another.

So if you go back in utility, the way to get the first order commission is you can ask a simple question, suppose I´m a time tand I have some path where I´mn consuming today and I´m making some plans about the future, but subject to what the shocks turn out to be in terms of their realizations.

Given some plan I could perturb that plan; I could consume a little bit less that say today, I could reduce ctimes t a little bit, I could reduce consumptionn Cta little bit.

And then, I could plan on consuming more next period, t+1. Suppose the rate of return that I have on some asset is capital Rt, this is a gross return, so ifn the real interest rate for example on a risk-free reclaim is 0.02 per year, then the gross return over one year is 1.02.

So one thing I can do, is I can reduce consumption Ct by one unit, and I could raise consumption C (t+1) next year by capital R units, which would be 1.02 inn that simple example.

And that wouldn´t affect anything about the rest of my plan; I could consume a little bit less today, a little bit more tomorrow, how much depending on whatn the return is.

And if I´m really optimizing that kind of perturbation shouldn´t make me better off. So the first order commission says that, that kind of switch doesn´tn affect expected utility, otherwise you´re not optimizing.

So that´s a simple way to get the first order commissions and that turns out to be this formula. This thing here is the margin utility from consuming an little bit more or less today, which depends on the form which is to see the one minus gamma, and just take the derivative of that.

So if gamma is high, that´s gonna determine how much this margin utility moves around with Ct. But anyway, this is how much utility I give up by consuming an little bit less today.

And this is how much I think I´m gonna get, what the expectation is of how much I´m gonna be getting next period because I can consume more at t+1.

So parallel Rtis the gross return on any asset that I called, and that´s gonna tell me how much I can raise consumption in t+1. And then, this term C(t+1) inn a minus damage, how much value the utils that I get next period. That´s the margin utility of consumption next period.

So Rt tells me how much extra consumption I get next period, like 1.02, and the C(t+1) minus gamma tells me how much I value that in terms of utils, becausen it´s the margin utility.

But I don´t know what all this stuff is, right now in time t, so that´s why I have an expectation. So this is the expectation formed of time t, of thisn object, now, in general I don´t know either the rate of return or the future level of consumption.

So it´s two things together that you are forming an expectation of, and then because there´s some time preference you might be discounting this whole futuren junk, by one over one plus some ROE thing, but that´s too important.

But let me forget about gross star, but the idea is that this is some time preference, so that if future utils are not worth as much as current ones, youn discount the future utils by one over one plus ROE; but that´s not too important.

This is the level of consumption that maximizes the utility in two periods or is it like..?

Suppose I have a candidate path for consumption, so that means Ct today. Today, I picked deterministically how much to consume, that´s Ct, and then I haven some kind of plan on which I´m gonna be consuming C(t+1), C(t+2), but it may not be deterministic so I don't know exactly what it´s gonna be.

But start out with some plan like that, and then I´m asking the question; is that an optimum plan? This is the condition that has to hold between t and t+1,n in order for the whole plan to be optimum.

So this is just one perturbation from some proposed plan, lowering consumption today and raising it the next period, so this has to hold. Then you could alson look at shifting between tand t+2, or whatever.

But this is not a two-period model, this is a model that starts today and goes out into the indefinite future. And then this first order condition has ton hold, and the important thing about this from the stand point of asset pricing, which is well-known since Lucas, is that this holds for any kind of asset you think about, capital Rt.

So capital Rtcould represent the risk-free return, that maybe it´s 0.02 or 2% per year and capital Rtis 1.02, but it also holds, if you think about holding,n the stocks, and then capital Rtis the gross return on the stocks, which might be subject to a lot of uncertainty but that would be inside the expectation rate.

But this holds for every kind of asset you might think about in this economy, some complicated set of derivatives or falling short on stocks or, anything,n this has to hold; and then, instead of this being a first order condition for determining the path of consumption, you can turn this around and ask what does the equilibrium rate of return haven to look like on every type of asset?

As you can use this to price various assets, or equivalently, to determine what the expected return has to be on various types of assets, which are then capital Rts.

So I´m gonna use this to think about stocks, which are claims on these Lucas Trees and risk-free claims, but you can apply this to any kind of financialn instrument in this ROE.

I don´t know, the ROE star might be different from what I wrote as ROE, because I wanted as more complicated preference relation, which I mentioned, thisn entangles risk-aversion from substitution of consumption over time. And it turns out there´s some other parameter which is not quite ROE.

But let me forget about that, I don't really want to get into that. You can think about this gross star as being ROE, which is true in the standardn formulation of preferences, and this is just a rate of time preferences.

More important is that the model, as I mentioned, could be used to price various assets, or equivalently figure out what the expected return has to be; thisn turned out the be the two formulas that I´m most gonna rely on.

The first one is the expected rate of return in equilibrium, on these equity claims, or stock market claims. Turns out this is what the expected rate ofn return has to be, and RFis the rate of return in a risk-free asset in equilibrium.

It´s gonna have to be, so I´m gonna get some formulas, let me just say one thing that´s important, if you think particularly about the recent events, I thinkn one way to think about that is that disaster probability P probably went up.

The amount of uncertainty that people perceived in the economy probably went up in one way or another, so you can model that as a higher p, it could be an higher sigma, but Pprobably more important, but one thing that´s true that you get from this, is more uncertainty that´s gonna reduce the rate of return, on risk-free claims.

And in fact you look at the rate of return on U.S., kind of like short-term assets specially; the rates of return have gone way down basically towards zero.n And that´s a way of thinking about that within this model, so that´s consistent with this form of the risk-free rate.

In terms of the Mehra and Prescott, what I´m gonna focus on is the difference between the expected return on stocks and the risk-free rate, which in then model is this Re minus Rf.

This formula is easier to interpret so let me try to interpret this formula. The first part is pretty much what Mehra and Prescott have for the equityn premium, the difference between these two average rates of return.

So think about a long run average of stock returns, and a risk-free rate and the difference between the two. So gamma sigma square sort of makes sensen because gamma is the coefficient of relative risk aversion.

It´s how averse people are to risk, so a bigger gamma is gonna mean a bigger equity premium, that makes sense, because we´re gonna demand a bigger premium ton hold the equity claim.

And sigma square makes sense too, because that is the amount of uncertainty and the more the uncertainty you would think the bigger the equity premium.

The problem with the first term is quantity, it does not fit, that´s what Mehra and Prescott said. So again if you have sigma being something like 0.02 pern year, so sigma square is 0.0004,

you´re trying to explain equity premium which is something like 0.05, 0.06, 0.07, and the number in sigma square is 0.0004 and then as observed by Mehra andn Prescott, well no part is multiplied by gamma by 100, but if you think about what risk aversion of 100 means, it´s completely ridiculous.

Even risk aversion coefficient of 6 or 7 means as, Paul said in a conference a few days ago, that means you won´t get out of bed in the morning; toon risky.

So I think in order to fit the thing properly, you have to get it working for a gamma that´s kind of reasonable, and I´m trying to estimate that, I thinkn something like 2, 3, or 4 is the kind of right number, something on that border, but if you have gamma being, say, 3 and sigma square is 0.00004, you only get 0.001, which is off by about 40 orn something.

It´s off by an order of magnitude through numbers and that was the content of the Mehra and Prescott 1985 paper, that was like nine years old; but forn realistic parameters you don´t explain the equity premium, so that´s the first term.

The second term is the Reitz, rare disaster stuff, which depends critically on two features which were in my actuarial role of trying to figure out what doesn rto be reasonable; that disaster probability p, and something about the distribution of disaster sizes, which is the b, and the parameter gamma, about how averse people are to risk, is gonna ton enter importantly in that term.

I try to think about this term in the brackets, in fact it´s easier to rearrange the terms into this one, it can show out directly that this is the same asn what´s up there. This term is actually intuitive, let me try to explain why it looks like this.

So you got this here, 1 -b, here, remember bis how much you lose in the disaster, so bis 0.3. It means you are down to 70% of your normal consumptions, son 1-b is 0.7.

So alternately, you can think about 1 -b is a ratio of disaster consumption to normal consumption; so bis 0.3, you´re consuming at 70% of your normaln level.

The question is: How much do you care about that? How much do you need a premium to be suffering to have only 70% of normal consumption?

Well, that´s what gamma tells you; it tells you how much you need to be compensated for that.

So, insted of thinking about this as 1-b to the minus gamma, think about 1/1 -b, that´s the ratio of normal disaster consumption raised to the power plusn gamma ;so let me take bequals 0.5 because then I can do the algorithm in my head.

So in 1/1-b is 2; so that means the ratio of normal to disaster consumption is 2 if bis 0.5. You´re supposed to take that to the power of gamma to figure outn how much people care about that.

I´m suggesting that gamma is something like 3, so two to the third power is 8; so a rough magnitude for this terms would be by eight, which is fairlyn big.

But you see putting gamma of 100, it´s completely ridiculous results, right? So, I don´t know... putting gamma of two, then you get four; putting gamma of 3,n and you get 8.

That´s where gamma is gonna be important, and indirectly, and particularly what we´re doing currently, estimate what gamma has to be, in order to fit then financial market data.

So this is gonna be 8 or something, that´s gonna be how much the margin utility is higher, in a disaster compared to normal times, like 8.

This is 1; 1 is the normal thing, this is how much, it´s 8 - 1 is how much the margin utility is higher, so that´s 7, so it´s higher in that amount.

And bout here tells you, if you´re thinking about holding a risky claim, which is a claim like on the economy´s capital, which is like a stock market, an certificate in this model.

Btells you how badly that does in a disaster. So bis 0.5, so it means you lose half of the day of what your stock market claim during the disaster.

So that´s why you care so much about this. You are losing half of your stuff just at the time when you margin utility is really high; because you´ren multiplying this by 7.

Alright, so you´re gonna be losing here, just at the time when you´re suffering, which is why you need a big premium to hold this stuff. It has all the wrongn characteristics. It does really badly in the bad times, so things like gold as an alternative.

Gold might be an asset which usually does badly, but in a big disaster it does well. So that´s why you might be willing to hold gold; because it does well,n exactly in the time when you care the most,like during the Great Depression.

It´s a little tricky though, because like the U.S. government changed the price of gold in response to the depression; so, there some complications; let men forget about that.

Ok, so I think the interpretation of why this appears like it does with respect to b, is because it´s made very clear with this way of writing.

So you have this term and you multiply the whole thing by the probability per year of disaster p. That´s the way a disaster curve enters and what I´m tryingn to estimate from the long history of data from many countries, is what is sort of a typical p, which I think about as a single number into an average disaster probability and what is a typicaln probability frequency distribution for the Ps.

I mean that´s gonna appear there, and if it has the right magnitude it might match with observed equity premium, so that´s where the theoretical term isn supposed to be.

Can you tell us a little bit about the Eterm, the expectations; what were the assumptions on the expectations, that doesn´t really matter to the...

Oh, for sure. What else could I do? What the problem of rational expectations apply to rare disasters usually, is you don´t have a sample with a large numbern of realizations, so you can figure out what rational expectations are.

You know, often you have this kind of passel-problem idea, you think there might be something terrible to happen, that currency might devalue by a pack ofn three, but you don't have a whole set of data with realizations that you can use to sort of pin down distribution.

So I´m gonna assume that these expectations that appear here are rational; but then, see what that corresponds to, I´m gonna need this large set of data forn many countries, so I have a really large number of disaster realizations, and then I´m gonna assume that these various expectations or moments, correspond to what the frequency distribution isn in the data.

So, that´s the whole point of the sampling of all these data. Of course you might think people are learning about this stuff or something; systematicallyn pessimistic or whatever, but I don't know how to deal with that, so first, I wanna know what the rational answer would be.

There might be a sort of central point, even though you might not wanna accept it. And you can see in the way this turns out if you look at the first one. Son one thing you need to know about is what the distribution of plooks like.

And that´s what I´m gonna get from the data, but then what it is, depends on gamma in terms of the moment. So, if you pick different gammas, you´re gonna getn different answers for the terms in brackets or for this one.

So I´m gonna end up asking the question, given what I get from the data for disaster probability p; and for the distribution of the bs, I´m gonna ask whatn gamma do I need to make this thing fit?

Because I´m gonna pretend I know what the Web site is. And then if I get a gamma that´s crazy like a hundred, I think you´ll say that I failed; and if I getn a gamma that might be reasonable in other grounds, I´m gonna claim that it´s gonna be something between 2 and 4, then it might be more of a success.

Now, what are we doing in this working paper, which I also distributed, which I´m sure you´ve all read in great detail; instead of taking the frequencyn distribution of the disasters, in terms of the observed, how many were there of each size, try to fit that with a parameterized formula, a form of the distribution function for disaster sizen and then you could compute the expectations given there, based on just one or two parameters.

I'll say a little bit more about this later, but we´re taking a somewhat of an alternative approach to estimating these things, they´re still rationaln expectations but it´s based on assumed form of the frequency distribution for the bs;

while at the same time I´d see there were 10 of these, there were 20% and there were 15, they were 25%.

That´s the first approach, which is where the calibration comes from for disaster probability and disaster sizes. Of course macroeconomists like to calibraten all sorts of stuff, in terms of business cycle models.

I think we´ll have some idea of what parameters they like to go on, but I sometimes forget because people don't have any idea about disaster probabilitiesn and sizes.

So I sort of get to set what I think is reasonable, but these are not data that people have been using, so it´s not like there is an established set of factsn about reasonable numbers for disaster probability and sizes.

But it´s like what people might do in terms of some business cycle analysis applied to this weird disaster.

So p and b are gonna be calibrated here from the long-term experience for a large number of countries, basically as many as have the long-term data in then world.

So in my 2006-paper I used this Angus Madison long-term GDP data set. I don't know whether you´re familiar with that but Madison is very well-known, hen doesn´t really, in most cases, assemble the GDP numbers himself,

he looks around at all these countries, and he sort of tries to put together everything that´s available and then reports these in whole series of books, andn he has a Web site now that he updates, and I think he´s 87 years old, or something. I have never actually met him.

So, of course most people don't know anything about long-term national accounts data, even in their own country. I find that if you talk to somebody in then United States, and you ask him about the GDP data going back to 1870, they´ll probably say "I have have no idea".

But there are experts in each country, which have spent their lives sort of looking into long-term national accounts, so we know the experts now, some ofn whom are alive in about 36 countries.

In the U.S. I think you would´ve said that the main expert was originally Kuznets and then Kendrick and a few other people.

Anyway, so Madison has put this all together, and the usual approach from economists who wanna use this long-term data is they take the Madison data onn faith, because they don´t have any legal alternative.

So that was what I did in my 2006-paper. I basically looked at the long-term data that Madison had prepared, it was only for gross domestic product, real GDPn per capita actually, is what comes from Madison.

There´s a big selection issue that comes up as to what time series of data are available for a country. Because often, the crises are when the data are notn available, particularly when there are wars, and in the U.S. is actually true for the Civil War.

The Civil War data are terrible, especially for the south which undoubtedly underwent a rare disaster; but the data are no good.

So it´s endogenously the case that in the disasters, which I typically care about, you probably don´t have the data and that´s a tremendous bias download forn the disaster probability, don't even look at where you have the data, you´re almost, for sure, gonna underestimate the probability of disaster.

So I kind of knew what the main disasters were for the world, if you think about the 20th century it´s World War I, Great Depression, World War II, and thenn there are some postwar stuff.

So the sample selection criteria I established early on, was to look only at countries that had data going back at least for before World War I. I didn´tn wanna look at a place like Turkey, which has good data from 1923,

and systematically the reason that data start there, is because the Ottoman Empire disappeared during World War I and the Republic started in 1923, but it´sn obviously systematic that the data are nonexistent from 1914.

So to try to adjust for that, I only included countries that had data annually going back at least from 1914, which is the year when World War In started.

There still are selection issues, but I think this lessens it. Actually the U.S. is a good case ´cause the U.S. data people usually use starts in 1869.

And it starts in 1869 because of the civil war, so that´s actually the initiative I think they´re missing the crisis for the U.S. for the Civil War years,n but the data is not so good.

Anyway, I adopted the criteria, and I´m only gonna include countries that have pretty much full series back to at least 1914, on an annual basis. A fewn countries were computed when you can figure out how big the disaster is, even though you don´t have the annual data, but basically that was the criteria; and I did that with the Madison numbersn initially.

Wouldn´t it have been easier to eliminate war? Because war is a total distortion of everything, it´is not anything, it´s totally abnormal, it´s not then market, its other stuff.

Well, part of the mission here is to explain the equity premium, so people holding things like equity claims are gonna worry about what happens to theirn assets in this circumstance, in the war circumstance.

Now you might argue that other things also get destroyed during wars, like treasury bills or whatever, so there could be some issues there.

But from an outsider standpoint about risk, I don't think you wanna eliminate from the data the kind of biggest risks that occurred, and some of those aren wars.

Couldn´t you create a so-called normal eliminating war, and then compare that with the War I?

Well, we´ve done that. So we have the results later where you consider only disasters that are not associated with wars, so I can give you the results onn that.

And this work I´ve been doing recently about stock market crashes also distinguishing non-wars from wars, and there are some big differences. But in thisn baseline setting I´ve included all the disaster events, including the wars.

That´s why they say that in new years they do all this?

Well, yeah, I mean like estimating the data for the war years is often the most challenging. Because I started with the Madison data, that was in 2006, thenn our idea in doing more work was that, well in the theory of asset pricing what enters its consumption, not GDP; and Madison, only provides GDP, so our idea in terms of the data was:

"Well maybe we can expand on the Madison numbers, or at least to subset his coverage with long-term data to have something closer to consumption."

It turns out personal consumer expenditure is about as good as you can do in most cases. You can´t really distinguish consumer durables from non-durables inn a lot of... but anyway it´s closer to consumption.

So, we started this second project where we would expand the Madison data to have something closer to consumption, and then I´m doing that jointly with myn PhD student at Harvard, José Osua, who´s from Mexico.

So, José is terrific with these data, and then the problem was that when we started really looking into the data I was just using Madison, as many people do,n we found out Madison does a lot of ridiculous things.

For example, for example, I´ll give you some examples, we care particularly about wartime disasters, so Belgium is a country that did badly in World War IIn and World War I, but he reports data for Belgium during the World Wars and when you look at the notes, you go through it enough; he doesn´t have the data for Belgium during the wars.

So he says: "I´m gonna assume that Belgium behaved the same way as France during the World Wars." Those are his numbers for Belgium during the World War,n there totatlly ridiculous, I mean, whether it´s true or not that they behaved similarly, I mean, you wanna have separate data to get more information.

So you take Mexico, which got José really exercised. So, even I, with my limited knowledge of Mexican history, know that the biggest crisis must´ve been then Revolution Civil War period between 1910 and 1920.

And you look at the Madison data for Mexico on that period, you look at it online, it´s smooth, there´s no disaster. And what does Madison say? Well, "I gotn the data from so and so," and then eventually you find so and so and what does the guy say?

He says, "I don't have any data for this period." So he just interrelates between 1910 and 1920, and that´s why it´s smooth, that´s what Madison did forn us.

So clearly missing what is the biggest disaster event for Mexico. And eventually, José did enough work to fill in the missing data for Mexico on GDP andn consumer spending, so we can include that in the sample.

But the Madison numbers are ridiculous. But, there are a lot of cases like that, where you start looking at Madison and it´s really... I mean you´d be bettern off leaving gaps where he doesn´t have the data. The problem is he hates gaps, so if he doesn´t have the data he makes it up.

Anyway, given this, we had to also redo the GDP numbers. So we´ve also redone the GDP numbers in many cases, not all the cases.

So the data set that what we have now is for the long-term results, GDP and consumer spending. And it turns out that we have 36 countries for which we cann do GDP back from at least 1914; and 24 countries where we can do consumer spending, and that´s where we stand at the moment...

These are thirty-nine countries which we´re working with, we have some chance of getting the long-term data. These are the thirty-nine countries, andn these are, José doesn´t like it when I say this; but these are real OECD countries, not counting the recent members like Mexico and South Korea.

So these are the long-term OECD countries, they´re twenty-one. And this is another eighteen countries from Latin America and Asia, which have long-termn data. So these are thirty-nine, we have some chance of getting this long-term data.

There´s more information on GDP than on consumer expenditure, so the starting here varies between the two for a given country, but these are the basicn data for this.

These data are available on my Web site at Harvard, and there´s a discussion there about how the data were estimated and what the sources are for eachn country; because it´s different for every country in terms of how we would procede.

Some cases are some very good long-term studies that have been done, like a recent one from Spain, for example, that´s very good. Chile, there´ve beenn some very good recent studies.

In other cases it´s more complicated, particularly for wartime periods, where you often have to do different things. But anyway, that´s all in the Webn site.

Let me just bring out a couple of properties, about the growth rates of consumption or GDP. So, when I say consumption is personal consumer expenditure,n so if you look on the right side for the real per capita GDP, for example, the upper root has twenty-one OECD countries with long-term data.

In the full sample we go back as far as 1870, but some countries start later. So, 1870-2006 is the full time period. And the numbers here mean, that then average growth rate of real per capita GDP among the twenty-one OECD, is about 2% per year, 0.02; with an average standard deviation, annual standard deviation of 0.05 something.

But that incorporates three sub periods, which are very different in terms of the history. So you can bring it up into Pre-World War I, 1870 and 1913; then inter-war period, including the two World Wars; and then the Post World War II period, starting in 1948.

So the period starting in 1948, the average growth rate of per capita GDP is 2.9%, the U.S. is already about 2.0, maybe a little more than that, these aren always per capita numbers.

And the average standard deviation of the recent period is 0.028. So, that´s like the sigma kind of number, but I used 0.02 when I was talking before,n which is what it is for the U.S., but this is the average over the twenty-one OECD.

But if you look at the middle period, you can see with respect of disasters for OECD countries; all the action is in this middle period, because thatn includes World War I, the Great Depression, and World War II.

So in that period the standard deviation is 0.088, we´ve got all these disasters, so those are really a big deal. If you look at the early period,n Pre-World War I, this was a point made by Christina Romer in her PhD pieces a long time ago. That period is not too much more volatile than the recent one, except for OECD countries.

She was only looking at the U.S., but this is also true. So you have 0.037 as the standard deviation in Pre-World WarI, for the OECD; and 0.028 in then recent period, so it´s not much different and probably the whole difference is based on the fact that you have better measurement in the recent period.

It doesn´t look like the Post-World War II is more volatile than the Pre-World War I, for the OECD countries. But the middle period is really a smalln choice.

And then, when you do the whole thing, of course you´re putting it all together. When you put it all together, the data do not like the idea of beingn normally distributed, because of these disasters realizations in the middle, that you get this big tail out to the right with the disasters.

And that´s what makes the whole distribution look like it´s not normally distributed. And that matters a lot for the asset pricing, in terms of this kindn of, that tail as the usual description. Anyway that´s just some discussion.

You can also look at the non-OECD countries, but there´s less data in terms of the long-term. These are just some pictures, the twenty-four countriesn where I have a long-term data on both, GDP and consumption, and these are six examples of the long-term data.

Of course, U.S. economists usually focus on this; worst yet, they usually focus on Post-World War II, and we can see how boring those data are. Incrediblyn calm at least until 2006, so 2008, 2009 is gonna change some of these pictures, remarkably enough; but the U.S. is unbelievably tranquil in the Post-World War II period, basically.

We can see the Great Depression for the U.S. around 1930. The blue graph for GDP is a decline by 29%, in per capita GDP, that´s between 1929 and 1933.n That´s the sort of peak to draw a proportional decline of 29%; and consumption it´s a bit less, it´s like 20 to 21%, but that´s the red...

We might see around 1920, there´s actually a second major contraction for the U.S., it has a bottom in 1921, and that´s about 20 percent, depending onn which 15 to 20%.

We think that that´s the great influence after 1918 to 20, but that´s a conjecture; but that´s the second biggest negative for the U. S. And you can seen also World War II, if you look at the blue graph, there´s a tremendously positive growth rate during World War II and only the U.S. really looks like that. But then people sort of generalizen like wars are good for capitalistic communism, the Chinese always say that.

It´s ridiculous in the whole sample, but it´s true, the U.S. grew like crazy, and actually has a big reduction GDP after the war, which is then demobilization. Consumption doesn´t do anything, but GDP does fall substantially in 1946-47. But basically, a tremendous reduction in the government part.

So you wouldn´t call it a recession in an economic sense, but there is a decline in GDP. Those are the U.S. data. The other thing that´s true about then U.S. data, is that if you take a ruler and you stick it on the first part of the graph, it amazingly enough, fits with the subsequent stuff at the end.

Making it look like there´s a trend line in per capita GDP, which is like a magnet. And that would be really exciting, except the U.S. is the only countryn for which that´s true in the whole set.

So, because of this focus on the U.S., I think people often might think about models with a fixed trend and always tending to come back to it. But itn doesn´t fit well with the rest of them. I think the U.S. is a terrific outline. For the other countries you can see more what the crises looked like.

Germany is really good because it has all the crises. So you can see the first part is World War I. We can actually also see there´s another bign contraction with the hyperinflation in 1923. This was exciting for me because my PhD thesis was about hyperinflation in Germany and elsewhere, and at the time I couldn´t get data on GDP thatn were any good, and I didn´t know whether.

So the big question is if you have this phenomenal event, like a hyperinflation, does it have implications for the real economy, and the data weren´t veryn good at the time, but we have better data now, and it turns that there´s a contraction of about 15% in real GDP per capita in 1923, which is the extreme mirror of the hyperinflation.

The depression you can also see for Germany, but it´s not as big as in some other countries, actually like the U.S. And finally, you have World War II,n which has a tremendous decline but only toward the end of the war. It´s often been said that during World War II, Germany was actually doing quite well in terms of the economy, GDP growth.

And that´s true, it´s only in 1944, 45 and 46 we see the massive decline. And that´s a decline by 50 to 60%. I should´ve said that the scale is such thatn a movement of 0.1 on this vertical is about a 10% movement. So you can roughly get what the proportionate declines are from looking at the scale. In Spain the big event is the Spanish Civil Warn rather than World WarII.

In this graph, Australia, is this like the trend or sequence for consumption and GDP in the 1940s?

There are some problems going back in the long-term in terms of the relative levels, you really can´t have a long-term trend that would be different. Thisn is for the U.S., I think there are some issues that the spread between the two, but try to get it in terms of levels that are comparable, so you can think about GDP per capita versus consumern spending per capita, but the rest of the mission is going back for the U.S.

One part of that is that the government sector is much smaller earlier, than these later. It´s also true that the foreign sector is different because then U.S. was borrowing a lot from the earlier period. But some of that I don´t quite understand, so I think there are some similar issues with the Australians and I don´t know how seriously ton take...

Should maybe the last century for Japan, I expect the..., I don´t know, there is a little dip there?

There´s something there involving..., I´ve forgotten which war, there´s a... Japan is involved in a number of wars in this period over here...

Russia in 1905.

Yeah, there is the Russian and there´s Chinese, I´m forgetting the dates...

57

But this is earlier... There´s something in the 1890s and then...

No, but I meant at the end of the twentieth century.

You mean the depression in Japan.

No, but the depression in Japan, quote, unquote, is really a slowdown in the growth rate, it´s not a contraction in the level, like you see with thesen other kinds of disaster events.

So you can see that with Japan, I mean the growth rate is very strong here, until the end of the 1970s, and then completely flattens out, so that´s whatn people often mean when they talk about depression or stagnation in Japan.

It´s not a sharp decrease in the level, or in fact, it´s not a decrease in the level, it´s a reduction in the growth rate, that can be important. Somen other analyses, there´s a work by ... and... for example ,they think about the long-term growth rate what I call g, and a reduction in g is something that can add important effects in thingsn like stock prices.

So for Japan, I think we´ve got a model like that, that gwent down, a trend break in terms of a lower growth rate at some point over the long-term of thisn true event.

I don´t think about that as a depression the way I´m defining it, but it certainly can have major economic implications. The long run growth rate pern capita used to be 4 to 5% per year, then it becomes 1% per year, and that makes a big difference, I don´t call that a disaster.

People are talking about stock marketing

Right, but you can get that from a reduction in g, by say 4 percentage point downward can do exactly housing prices and stock prices.

By the way, that´s why this preferences resulted, that I discussed earlier when I was in, because suppose you have a big reduction in the growth rate,n intuitively you think the stock prices should go down,

if the growth rate goes down, and similarly if uncertainty goes up you would think the stock prices should go down, which I think is right. But to getn those answers within this model, it doesn´t work in standard formulation, that´s a main reason I shifted to a preference relation of this sort I discussed earlier.

In order to get those facts correct, you can´t get it within the standard preference formulation, if you also want to fit the equity premium, which meansn you need a coefficient gamma of about three,

you get the wrong sign with respect of how stock prices react, to the growth rate or to uncertainty, you get the wrong signs for both of them. But not inn this revised, that was actually the main reason, as I discussed in the 2009-paper model, of why I made these changes. But the work by ...and ... is also related to this, it´s the samen point

So you can certainly get a stock market crash, buy you won´t get a macroeconomic depression of the sort that I´m discussing from a reduction in then growth.

Some people also talk about New Zealand and Switzerland in several parts of these Post-War periods in a way of analysis to the Japanese slowdown, they seen similar kinds of patterns where the growth rates seem to have, at some point permanently been decreased, and some people call those depressions.

Some of this is just terminology, but the question is what´s the implication of a reduction in the growth rate versus a disaster in a sense that you know,n in a short period you lost 30% of your actions, so you can look at both of them.

Do you think from the data that some countries like France, Japan, and Germany are kind of approaching some kind of some breaks in their growthn rates?

I´m doing some other work with Emi Nakamura and John Steinson who are at Columbia, where we are trying to use all the data, not just the disasters.

I´ve focused on disasters because I´m a negative person, I guess. So we´re using all the data in their analysis, so part of that is the sort of long-runn growth rate, and why do you get these sort of breaks where the growth rate goes up or down.

So that work might answer the question that you just made but I cant do it from what I have been doing here. So we do have to allow for breaks in then growth rate, except for the U.S.

The U.S. is crazy, there´s nothing. But for the other countries, basically every other country needs a break at some point in the growth rate.

I mean you can see that even for Spain, you can´t inflict that constant growth rate, that pattern, it´s very different before World War II, and then theren is a big boom, and then there seems to be another slowdown.

But let me discuss, there are two kinds of data that I´m using in this project; one is a macro aggregates, which are the GDP and consumption data, whichn I´ve just been discussing.

So the other data that enter in are something about financial rates of return, various classes of securities, so I particularly focused on stock returns,n and short-term bill returns, which are some approximation to a risk-free rate, it´s not really risk-free but maybe closer than the stocks.

As a try to indicate on the national account´s data, we´ve done a lot of our own construction of the long-term numbers, along with using other peoplesn numbers, where they look revived;

but on the asset returns we´ve mostly used the data that´ve been constructed by other people, except in a few cases, more recently.

But the biggest source we have for these rates of return data are these things called global financial data.

I don´t know whether the Marroquín subscribes to that, but you should subscribe to that. It´s a wonderful data resource; it´s very valuable bothn contemporaneously, in terms of looking at stock prices and rates of return, and various securities for lots of countries in pretty good detail, but also for the long-term history. It has both,n long-term data where available, and also the contemporaneous situation.

We´ve used it mostly for looking at stock returns, and bill returns, but it has a lot of other data. They also have consumer price indexes, and keepingn that elsewhere, but it´s convenient there, because we always look at rates of return in real terms, but you look at real rates of return for holding stocks, or bills or bonds.

So Harvard for example subscribed to this, it was very convenient.

So we´ve solved that in some cases with some other sources, a party to go back earlier in some cases, because in some cases there are earlier datan available than the reported by global financial data,

and in some few cases now such as Mexico, and Brazil, we can get longer-term series that are not reported by global financial data by basicallyn constructing it ourselves or by consulting with some people who were doing that.

We are kind of trying to get the long-term data for Brazil, looks like it is feasible, but I´m talking about the stock return.

So these are three types of series we´ve constructed for as long a period as is available for each country, which tends to be a shorter period usually,n than the macro data.

So for stocks, conceptually what you really want is this total return index, which includes both, price changes and dividends.

So, Global Financial Data reports them, but not always, so for some cases we have to estimate dividend deals, in order to get conceptually total returnn for holding stocks; and I say, we always deflate that by consumer pricing, that´s to get real rates of return from holding stocks.

But certainly not for the short-term bills, usually this is an allocate of sigma treasury bills but sometimes is something else, which is may be close ton that depending on, particularly if you go back far in time, you dont see that kind of security.

They had one error which was a Madison-type mistake, which I convinced them to get rid off, because I looked earlier at Global Financial Data, that theyn had the long-term data for Canada on treasury bills, and it turned out that for a certain period, they just assumed that it was the same as the U.S., because they did not have the Canadiann data.

That was a Madison type of thing, I convinced them to leave it blank for Canada where data was missing,

but they dont tend to do that in the same extent these days.

We´ve also included the measured long-term rate returns, excuse me, a realized rate returns for holding long-term government bonds, usually ten yearn maturity, those data are not as much available as the other two, and we havn´t really used the long-term data that is available.

In terms of the equity premium, what I´m focusing on is the difference between the long-term real returns on stocks and these bills. The bills are notn really risk-free, but there´s gonna be some counterpart or approximation to the risk-free return.

These are what the returns look like for countries that have long-term data on both, stocks and bills, here; and bonds and bills, here. So these aren countries that enter here because they have long-term information on these financial terms.

Let me just look here at these averages, so these are the averages over the countries with the long-term stock data, on the annual real rate of return. Son 0.08 means that the average real rate of return holding stocks on these, I think it´s 17 countries, is about 8.1% per year, this is above inflation, it´s a pretty high average real rate ofn return.

This is the average annual standard deviation of these returns, as it´s well-known stock returns are high on average, but enormously volatile. So then volatility is like 0.24 per year, so you get 0.08 plus or minus 0.24, so it´s an enourmous volatility.

This is the average real bill return, it´s about 1% per year; so that includes some countries like Germany that are in hyperinflation where the realn return on the bills is dead.

So that´s just an example of it not really being close to risk-free. But anyway, the average real return is pretty low, about 1%, risk-free rate, I dontn know, it might be, in that order, it might be a little bit less. I should say in the theoretical model there´s no reason for the risk-fee rate to be positive, it could be negative in the model,n and in the world it looks like it´s pretty low.

One thing that comes up in some of the analysis that probably we won´t discuss is, whether the rate of return is bigger than the growth rate.

If you ever do any growth theory kind of analysis, one of the basic comissions is that in the long-run the real rate of return is supposed to be largern than the growth rate.

And the question is: is that true in the data? Well, it is true for average stock returns, it´s not true for something that´s closer to a risk-freen rate,

a long-term average per capita growth rate is something like 2%, 2 and a-half percent, per year. If you look at the level of GDP, it´s more like 3%, orn something like that.

The growth rate in the long-term seems to be higher than the risk-free rate, but lower than the average rate return on stocks, and actually the model isn completely consistent with that kind of pattern.

For the model to make sense, it does have to be true that the expected return on these claims on capital, which is something like this, has to be biggern than the expected growth rate on the long run,

but the risk-free rate can be very low, it could even be negative. And it looks like in the data it´s maybe a little bit possible.

So these are what the return data look like; now if you look at the difference between those two, is shown here for the seventeen countries, averagen return stocks is 8.1, on bills it´s a little bit less than 1%. The difference between the two is the counterpart in the long-term equity premium about 70% per year.

So to think about that as being stable, then, the challenge for the model is to explain why the equity premium is about 0.07; 7% per year in realn terms.

Why do you prefer to use the arithmetic average instead of the geometric one?

That´s what a verge is naturally, if you think about the first order condition, in terms of consuming now versus consuming in the future,

you really have to think about what it is the expected level of stuff a year later. In a discreet model, that´s gonna look like 1 plus the rate of returnn and the gross return.

And that result works even as you let the period shrink and become all the current small.

There´s a good discussion of this, I think Merha and Prescott have the survey paper, in a recent financial economic sample.

They have a whole section on this, I think that what they say is correct, the geometric return is basically missing the piece that looks like the one halfn sigma square,

in terms of the formerly expected growth rate of gstar, it´s the same thing here with respect of rates of return,

there is a term that looks like, if it were normally distributed, the difference between the two concepts would look like one-half sigma square, at leastn if you let the period a bit small.

So if the rates of return are, the average returns are not too high and not too volatile, then it doesn´t make much difference.

So the growth rates of GDP, where sigma was like 0.02, didn´t really matter. But the sigma for stocks is like 0.24, think about that as being sigma, andn then sigma squared, the half of that, is what the difference is, roughly, between the geometric and the arithmetic and that turns out to be about 0.02 per year, so it´s not negligible.

So instead of getting 8% you have about 6 if you do the geometric and I think that this is just incorrect.

So, I´m told I need to stop for break.

Initial credits

Introduction

Macroeconomic disasters

Equity premium puzzle, Rajnish Mehra and Edward C. Prescott

Rare disasters

Risk-free rate of return

Rare disasters approach

Lucas-tree model of rates of return

GDP and C data

Long-term data from 39 countries

Returns data

Final credits