I will explore how alternative sources of uncertainty have an impact on asset valuation. I will show how decision theory, control theory and statistical theory provide valuable tools to model investor behavior and to reveal how uncertainty is reflected in security market prices. In intertemporal environments, risk-return tradeoffs depend on the payoff or investment horizon. To study these tradeoffs, I will construct pricing counterparts to impulse response functions. Recall that impulse response functions measure the importance of next-period shocks for future values of a time series. The asset-pricing counterparts are shock elasticities which measure the expected contribution to a cash flow from a shock in the next period shock and the price of that shock.

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Thank you, it’s very nice to have the opportunity to open this session. And to be here in Lindau and to be able to share some economic thinking with all of you. I look forward to conversations later in the day and throughout the week. So what I thought I’d do today is talk about revealing ways to characterise how uncertainty is reflected in security markets. These methods are supposed to be revealing for a couple of reasons. One is they’re going to help us understand how models work. And how model ingredients can have important impacts. They will shed light on empirical challenges and often lead to puzzles. And what are the challenges in building better models going forward. I’m going to take a shortcut in the following sense. I’m going to feature a component of a dynamic stochastic economic model. And this component is going to be typically combined with other model ingredients. And this is going to be a key component. Because this is going to be a component that’s going to have important impacts on investment and capital allocation decisions and the like. But I find it pedagogically valuable and revealing to strip away a piece of a dynamic model without having to analyse the full model simultaneously. Of course that latter step is absolutely critical. And I don’t mean to say we can skip it. But rather I think this helps to understand things at a more general level. I want to get started here. I want to tell you about these models of asset valuation. And let me just lay down some language that I find useful here. As you think about what determines an asset. So let’s think about an asset as something that has a pay-off in the future. And this could be a pay-off that’s growing stochastically. It could be some macro-economic variables. It could be some type of technology process that’s growing or the like. And when I think about that is some type of stochastic growth process. It’s going to be a process such that it’s going to capture stochastic growth between zero and date t. If I take logarithms of this process that might have stationary increments, but it might allow it grow stochastically over time. And so at the same time I’m going to have another process. And this is going to be a key process for me, this so-called stochastic discount factor process which I’m here denoting S. And if I look at the process S(t) here - S(t) is I want to assign an asset price. Say if I go down to this formula right here. If I want to assign an asset price at day 0 to something that pays off G(t) at date t, I have to discount it but it’s uncertain. So I have to also make some type of risk adjustment. A convenient way to think about doing that is the so-called stochastic discount factor. I’m going to discount different realisations of that process in different ways. And that just is a very convenient way to encode so-called risk adjustments or adjustments for the fact that there’s exposure to uncertainty. And the so-called stochastic discount factor is going to be a really critical part of what I’m going to be talking about today. And what it reflects is it reflects investor preferences through so-called intertemporal marginal rate of substitution. So there’s this standard price theoretic insight for what we do is look through things like marginal rates of substitution. And we connect those to prices. And that has a direct extension to the theory of asset prices. And here are these intertemporal marginal rates of substitution which we want to compute. Now, where things get subtle, where things get more complicated is the fact that participation in security markets may change over time. They may be isolated in certain components of the economy. It may not involve everybody. Lots of recent models of financial market frictions. What they do is they segment markets. They isolate certain types of trades within sub-sectors of markets. There may be changes in terms of who the so-called marginal investors are for which these conditions work. In general, I have to combine this with a statement of market structure and corresponding prices. But it’s valuable to think about this stochastic discount factor channel as being an important one. There’s one approach to this that has been used very, very extensively inside the macro asset pricing literature. It builds on recursive utility theory that goes back to Tjalling Koopmans, more recently to Kreps and Porteus. And what this does is, it highlights the role of uncertainty. And how uncertainty about the future affects asset valuation. And so it allows us to do things like explore ways in which expectations and uncertainty about say future growth rates of the macro economy really show up in valuation. As I’ll show you in just a minute there’s this so-called forward-looking component to the recursive utility model which shows up. And this provides an additional channel where things like investor beliefs, investor guesses and speculations about the future matter, even if you’re looking at risk return trade-offs over very short investment horizons. Let me just write down the most commonly used utility recursion. And just talk you through some of the notation here. So think about consumption. There is some underling consumption process for the foreign investor. That says process C(t) here. I’m going to raise this to a power, this power, the parameter rho here. Rho^-1 is typically thought of as the elasticity of intertemporal substitution. But I’m also going to discount the future, the subjective rate of discount. But there’s this risk adjustment of a future continuation value that shows up right here. That’s really the important piece of this. And so I’m going to take a continuation value that tells me how we value future consumption plans. In this case from date t plus epsilon forward. That’s going to be adjusted with a risk adjustment. This shows up in the second formula here. There’s a gamma parameter here which is distinct from the rho parameter. And this allows there to have change investors risk aversion without doing things like changing intertemporal substitution. And this has been used a lot in empirical asset pricing literature. In fact this parameter gamma here is one if you look at lots of papers. It’s a parameter that often gets set to rather sizable amounts. Even though it’s typically called risk aversion. Now what’s going to happen is the continuation value, the continuation value is forward-looking. It depends on what you think is going to happen to your consumption processes going way off into the future here. In general, we’re going to have to compute this as part of the solution to a model. But it’s going to provide a potentially important contribution to valuation which we want to talk about here. I always tell my students I never do algebra in public, and I’m not going to do it today. Basically, if you look at the functional form I wrote down. It involves a couple of CES things and I just need to differentiate to form marginal rates of substitution. You should just do this in private. My advice is that you trust my calculations. These have been repeated by many people, so you don't have to trust me. But anyway, what this leads to is a formula for stochastic discount factor, for here an increment between date t and date t plus epsilon, that has the following piece: A pure subjective discount factor piece. A piece that involves the intertemporal trade-offs between consumption. And this elasticity of substitution parameter. But there’s this additional piece that involves the continuation value, V(t) plus epsilon. This is the future continuation value and that in turn depends on your guesses about what’s going to happen way off in other future dates as well. So beliefs, perceptions and the like about the future show up in this third term here. And that’s a term which is what people have used a lot in the empirical literature. The nice part of this is it gives us a structured way to enhance the impact of things like perceptions about the future. And even if I’m looking at how I’m doing valuation between date t and date t plus epsilon. Now, to build in a stochastic discount factor I have to start compounding these over different investment horizons. So I have to multiply them up together. Now the special case of this leads to something which is used in older literature, in which rho is equal to gamma. You illuminate this piece. And then the stochastic discount factor is just the consumption ratio. And this is what gave rise to the so-called equity premium puzzle in a way. Because the consumption channel alone was just not a very successful one in trying to understand asset prices. Now, if I take a special case of all this. I take the special case in which rho is equal to 1. It’s just pedagogically simple. This last term here - one can actually verify that this has a conditional expectation of 1. Because you end up raising V to a power of 1 minus gamma. And the nominator is just the conditional expectation of that. So this is a random variable with a conditional expectation 1. It’s positive. I multiply it up and I get what’s called a martingale. The best forecast of the future is the current value. It’s a positive martingale. And this martingale contributes to a so-called stochastic discount factor as it gets compounded over time. And this martingale is going to have a very durable impact for the stochastic discount factors. So as I look across different investment horizons, longer investment horizons, this piece is going to come to play a very prominent role in terms of risk pricing. A more general result has to do with this, what I think of as stochastic discount factor factorisation. And that’s given in this line here. So now I’m looking at the valuation between 0 and date t and how I do the stochastic discounting. There's a general depiction of this which has 3 components to it. The first one is a constant discount factor component under which rho plays a role of the yield on a long-term discount bond. There’s a second component, that’s this martingale component. I just gave you an example of a contributor to this in my previous slide. And this can behave like a martingale. And there’s this last piece which is typically built-up from a stationary process. So if I build models with balanced growth paths and the like, then this third piece is a function of some stationary mark-off state. So there’s a sense of what dominates over the long horizons has to do with these first 2 terms and in terms of valuation. In time series we often think about things like permanent and transitory shocks and the like. And this has been a useful way to think about properties of time series. This is a bit of a valuation counterpart to this. This is in levels, not logs. It’s in products and that matters actually in important ways. But there’s a sense in which this term right here, how that behaves, really has big impacts on how I do valuation over longer and longer horizons. And these types of characterisations have been worked out in some stuff I’ve done, some stuff Alvarez and Jermann have done. And it has been used more recently in a variety of asset pricing papers. And this gives us a way to think about what dominates or what forces dominate asset valuation over longer horizons. So what can contribute to this martingale component? This piece that really lasts and is persistent in terms of its impact on valuation. There are 3 things that can show up here. One is macroeconomic growth. So macroeconomies on a stochastic growth trajectory. That alone can contribute to this martingale component to stochastic discount factors. But other stuff can show up here as well. And there are 2 other forces that are of interest. As I showed in that recursive utility model, I constructed a martingale contribution. And that had nothing to do with whether the consumption process was stationary or not. These continuation values coming out of these recursive utility models can contribute to this martingale component. So that's a second source for this. And the third source that I’ll be talking about later today is distorted beliefs. As we look at deviations, departures from the so-called rational expectations models, where investors have things figured out, it’s handy to have ways to think about how we might have investor beliefs that depart from that. And so there are differences between the econometricians model and investors model. And those also can contribute to these martingale components. I want to talk a little bit more generally, go beyond what happens over long horizons and what happens over short horizons. Before doing that though let me just summarise a couple of things that come out of the empirical asset pricing literature. One is that these increments in stochastic discount factors are known empirically to be highly volatile. And that’s going to mean that there’s this price channel in asset prices that’s known to be very prominent and is known to fluctuate over time in potentially important ways. Also there’s been more recent literature trying to characterise the quantitative importance of this martingale component. And there’s been recent work by Steve Ross and others that have tried to build an asset pricing theory where that is set to unity and degenerate. But for a lot of purposes I think it’s important that there’s some evidence that it could be quite prominent. We want to now go just beyond what happened over long investment horizons. What happens over the immediate investment horizons and generally you want to fill in what goes on in between. If you want to understand how valuation works. You want to understand, what models have to say about valuation. And here I want to draw insight or motivation from earlier literatures. The first one I’m just going to lift a quote out of Irving Fisher: according to the particular periods in the future to which it applies.’ So here I’m looking at ... I want to think about this notion that asset pricing, as I do risk return trade-offs and characterise, how investor’s prices to exposures change over different investment horizons. And there’s a second one, by Ragnar Frisch. I think of Ragnar Frisch as doing one of the first impulse response functions in macroeconomic analysis. He wrote this classic paper on the impulse problem, in which he talks about various ways to characterise it. And he says that one way would be to look what happens to a dynamic system if it were exposed to a stream of erratic shocks. And then to characterise what those shocks do. How does a shock today get transmitted over into the overall economy, over all the different future time periods? These impulse response methods have been demonstrably very, very successful in empirical macroeconomics in trying to characterise which shocks have the biggest impact on different macroeconomic variables. It’s become a common tool to use in empirical macroeconomics. I want to take that to a little bit different level. I want to take a different twist on it. Instead what I want to be doing is looking at a valuation counterpart to it. So an impulse response works as follows: I imagine a shock tomorrow and I trace through the consequences of that shock on a bunch of future time periods. So if I take a shock tomorrow: Imagine you’ve got a stochastic cash flow. It’s going to change that cash flow. So it’s going to change the uncertainty of that cash flow. Once I do that and I look at, it’s going to change a risk premium. And once I change a risk premium there’s going to be 2 forces to it. Well I’ve changed the exposure. The exposures now might be more risky along some dimensions. And I’m also going to change price. It’s going to also have implications for how that exposure gets priced. And so what I want to be doing is producing these pricing counterparts whereby I take different shocks that hit the macro economy. I look at their implications in the future. But I look at which shocks investors require the biggest compensations for. And that gives me this valuation counterpart. Now in contrast to Frisch we’re going to have to perturb stochastic paths, not deterministic ones, to make the asset valuation question much more interesting. It's a way to think about intertemporal characterisations of investor risk aversion if you like. So I want to study the consequence on the price today of changing exposure tomorrow on cash flows in the future. And I also want to represent consequences for prices. So by changing that exposure I’m going to change the valuation. And from that I can figure out what the corresponding price component is, because I look at the expected returns. We often talk about risk premia. And I find it very useful to take a risk premia and have 2 components to it. What is the exposure? If I change exposures I’m going to change a risk premia, because I’m more exposed to certain types of risk. But the other channel by which risk premia work is this price channel. Different shocks, different risk exposures get compensated in different ways. And it’s that price channel which I want to be focusing on. And it’s that price channel where the stochastic discount factor is really important. So let me just give you an example. It’s an example that’s been quite prominent in the macro asset pricing literature. It’s going be demonstratively too simple for a lot of purposes, but it's a good illustration. I’m going to introduce 2 state variables. I’m doing to build on some work by Bansal and Yaron. I’m going to build on 2 state variables. One is going to govern predictability of growth rates of the macro economy. And it’s going to be hit by shocks. It’s going to be like a so-called autoregressive process in time series. There’s going to be a second process that’s going to shift macroeconomic volatility. So there’ll be certain times in which volatility is high, certain ones low. It will be a persistent process. And I’m going to have 3 different shocks. I’m going to have a so-called permanent shock that hits this macroeconomy. I’m going to have a transitory shock and a stochastic volatility shock. Now as a macroeconomist you should hate these labels, they’re boring labels. And when you build a fully-fledged macro model it’s going to, along the way, have you think harder about what the underlying shocks are. This is just for way of illustrating a mechanism. So in principle we want better labels than these, and models provide those. What do we do when we take this out and compute these valuation counterparts to impulse response functions? What I want to do is, I want to compare a model where there’s that forward-looking piece coming from recursive utility to one in which it’s absent. So I want to go back to what’s called the power utility model where gamma and rho were the same, that I talked about before, to one in which I’m going to really let this gamma parameter, that is how I adjust, the continuation values, do all the work for me. And what am I be doing then? I’ve got these different shocks. I’ve got a so-called permanent shock. So the permanent shock works through the consumption process and it takes a while to build. A power utility model - everything is proportional to what goes on with consumption. And this red line tells you how the prices of different shocks work. I take the permanent price. These are like compensations per unit of risk, mean compensations per unit standard deviations, so-called sharp ratios from asset pricing. For the power utility model, it tracks how consumption behaves. And it’s given by this red line here. It starts off fairly modest and it builds. Now I’m going to take this recursive utility model with the unitary elasticity of substitution. But the same power as a power utility model has. It’s going to be like 8 or 9 here, I forget which. And that’s going to be this blue line. And the recursive utility model has its forward-looking component to it. You really care. There’s this piece where investors are caring about what’s going to happen in the future. That shows up in introducing this martingale component to stochastic discount factors. And there’s a very flat trajectory. And the key thing here is that I go to even shorter investment horizons, the recursive utility models assigning much, much bigger prices to things here than the power utility model. And then by design they eventually end up at the same point. So what this recursive utility model is doing, is it’s making these long-term considerations show up at even very short investment horizons. Take a temporary shock and look at the pricing implications. Again the red line tracks how consumption responds and makes a proportional adjustment. The blue line now is the transitory shock. This continuation value channel is very inconsequential for it, and the prices there are very tiny. So I take these 2 different models. They have very, very different implications depending upon the nature of the shocks. The third one is for stochastic volatility and again the recursive utility model gives these very flat trajectories because these martingale components end up being really prominent in those models. I’ve highlighted bands here as well. This is a model that has stochastic volatility. Stochastic volatility does 2 things here: If you take a shock to the volatility in the macro economy, exposure to that gets priced. That’s the bottom panel. That shock also shifts around the other prices. This band here gives you core tiles of how much those prices are moving around because of stochastic volatility. So stochastic volatility, as it gets introduced, has these 2 different roles in terms of prices. Now thinking about stochastic volatility for lots of time series work is just an added on separate shock. Of course, we need to think harder about what that really is and what drives that stochastic volatility. So if I look at a model like this, the question is, to what extent is this a success? It turns out this forward-looking mechanism, the way it works is investors have everything figured out. They know parameters of the model. They know how these processes evolve and everything. And so they’re given arguably statistical subtle components of the macro time series. They’re endowed with full confidence in them. And the question is where this confidence might come from. If you look at the primitive statistical evidence for some of this stuff - it's very, very modest and actually quite weak. And so there’s a sense in which a fairly large amount of confidence is built into these calculations on the part of investors. Also, as I indicated, stochastic volatility here is moving exogenously. And it does in lots of models. But really what you want to do is understand what’s driving that source of fluctuation and not just impose it from the outside. And finally, this imposes what I consider to be large risk aversion. The numbers I used here were like about 8 or 9. If you look at the macro finance literature it goes up to 20. There are papers, I’ve seen the numbers as high as 90 or 100 - at which point in time I have no idea what this parameter is. But it’s a parameter that’s used empirically a lot. And so the question is, are there other things that could be intimating what appears to be large risk aversion? So a recursive utility model has been handy. But I think the success requires some very important qualifications. So I think once we start thinking about relaxing investor confidence in their models, which I think is a very fruitful challenge or fruitful modelling approach and a fruitful endeavour, it's handy to think about these different notions of components to uncertainty. One is risk - so I think about risk. As you put a model on the table, you know all the parameters and everything. There are shocks that hit the model. And you take that model and that model generates out probabilities of all the future events. Now, maybe you don’t know parameters. Maybe there are multiple models on the table and the like. And so there’s a second source and that is how much confidence do we place in each of the different models. And there’s a third piece - I think that’s the hardest one to really consider. But it may be one that’s in many respects the most important piece. When we write down models, when we use them sensibly, we know they’re not perfect. And the models are demonstratively very, very simple. They abstract from various different things. This notion of how you use seemingly simple revealing models in ways that are sensible and acknowledge that they may be misspecified along some dimensions. And that’s perhaps the hardest thing to capture formally. Anyway, these things, these alternative sources of uncertainty, once they go beyond the usual risk story, they themselves can contribute to this martingale component, to these stochastic discount factors. Moreover, there are examples where they contribute to stochastic volatility. This belief channel that there’s uncertainty about the model channel itself can be a contributor to uncertainty in financial markets. So we might read in the press that markets look very risk-averse one day and more bold on other days. The question is what’s really driving this type of fluctuations. And it’s beyond having to impose stochastic volatility. So there’s a very rich literature, modern, coming out of statistics, coming out of control theory, coming from a variety of sources about how to address, confront different forms of uncertainty, and in ways that are tractable. There are axiomatic foundations to this. Myself, I find axiomatic treatments useful, but I find myself easily persuaded by lots of different axioms. So I have a hard time sorting them all out, but I follow them anyway. They provide tractable representations which is really critical in doing empirical work. And they often involve some type of recursive construction. What is missing but is important to be thinking about going forward, is that a fully fleshed-out theoretical justification for how we use potentially misspecified models in sensible ways. Whenever we push these new applications of decision theory, new parameters start popping up and start showing up. And what’s the right way to think about the source of parameters? So any time you enrich a model and add new parameters, you want to also want to think about, what a reasonable magnitude and the like. And that’s been an interesting challenge. And third, part of what the rational expectations paradigm did for us, is it gave us ways to say – suppose I want to look at long-term consequences of policy changes. And so I imagine one environment where investors have things figured out in one way. And another environment where they have it figured out in a different way. And work out policy conclusions. And that’s useful if I want to close down this channel of systematically fooling people. But once I start opening up this, there’s a clear discussion of, as you change environments, which parameters you import and which ones you need to modify. So there are some important challenges using this. But it’s very fertile ground to be thinking about. More generally this can imitate things like asset pricing under distorted beliefs. And so let me just go through this quickly as my time is running a bit short here. Suppose now what I do is I imagine that investor’s beliefs may differ from those coming out of the econometric model. So the notation here is to pick up the fact that investors are using a different expectation. And when they’re using a different expectation then I’m going to use a corresponding stochastic discount factor that goes along with it, S tilde. What’s convenient is to use these martingales to represent potential changes and beliefs. And that’s all that’s given here by this middle slide. And then what I can do is, I can rewrite the basic asset pricing equation under which the stochastic discount factor I was talking about before has these 2 components. One is this martingale component, now motivated by belief differences as well as a stochastic discount factor component, conditional on those beliefs. For instance, now I can produce a stochastic discount factor as distorted beliefs: martingale times risk preferences. And, as I talked about before, these martingale components might be quite important for valuation. They have long term consequences. This was one channel in order to get some action out of them. So M is this interpretation of likelihood ratios and that opens up possibilities of using statistical methods to quantify how big they are. I find it useful to think about, well, yes let’s introduce these different models from the table that investors might be considering. But it still may be useful to say, these are models which may be more reasonable if we leave them on the table. Even if it takes like many, many decades of data to sort them out rather than something which we can try to figure out very, very quickly. And the fact that I can have links to likelihood ratios and statistical criterion allows me to think about how hard models are to tell apart as for exploring these potential distortions and beliefs. I think it’s almost handy to use statistical methods to try to get some guide as for how big or small these things are. And there’s formal ways to do precisely this. So I find it useful then, as you're looking through these beliefs as a potential source of martingales, to think about, when historical evidence is very informative and when it’s not. And statistical methods are very valuable in doing exactly that. I think this can provide a certain type of discipline. If we allow beliefs to just be arbitrary then we can resolve asset pricing challenges and puzzles very, very quickly. But the question is, what kind of a resolution is that? So it’s handy to try to limit the belief distortions we consider motivated by more general models of uncertainty, by so-called animal spirits. In general, I think models with heterogeneous beliefs are important. But the belief heterogeneity may be more reasonable when it's beliefs that are hard to tell apart statistically. Subjective concerns about rare events. It also opens the door to thinking about models. Of which some investors are over confident, take their model too literally and other ones use them more cautiously. So there’s fertile ground to thinking through this. And again this martingale component to stochastic discount factors provides a very productive and instructive way to think about things. Just in closing here I find this factorisation to be useful. It’s useful to add structure and content to these belief distortions and not just let them be arbitrary. And I find it useful to explore how things like concerns about model misspecification might well contribute to these things. But there are other motivations one could consider as well. Let me just close by saying I find the type of tools I talked about to be very useful. Stochastic discount factors are a good way to frame empirical evidence. Their increments can be highly volatile. We need to have good stories for why that’s the case. They may have these martingale components that have these durable impacts to them. And it’s then useful to have theories that help us to think to those challenges. Thank you very much.

# Lars Peter Hansen (2014)

## Uncertainty and Valuation

# Lars Peter Hansen (2014)

## Uncertainty and Valuation

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