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Bridging the classroom gap between asset pricing and business cycle theory.

Abstract The tools presented in the standard undergraduate economics and finance curricula are insufficient for explaining the complex dynamics of the modern U.S. economy. For instance, students of macroeconomics are not provided with a satisfactory framework for assessing how a financial shock may reverberate through the real economy, as it did during the Great Recession of 2008-2009. Similarly, students of finance are left with little guidance as to the origins of two key inputs into asset pricing models, namely cash flows and discount rates. In this article we present a unified macro-financial model to bridge the gap between the typical undergraduate treatments of asset pricing and business cycle theory. The Dynamic Empirical Macroeconomic (DyEM) model we introduce here offers several innovations, the most important of which is expanding the role of the interest rate to include term and default risk premia. We combine these elements to construct a series of discount rates that are critical for a range of present discounted value calculations. Moreover, we link economic activity to earnings growth in order to facilitate macro-based equity pricing. The paper concludes with an illustration of how the DyEM model may be used in the classroom via a cooperative learning exercise centered on the Great Recession.

Keywords Dynamic Empirical Macroeconomic (DyEM) model * Great Recession * Asset pricing * Business cycle theory * Macro-finance modeling * Economic education

JEL Classification A22 * E30 * E44

Introduction

All economic models are wrong. This is true of the simplest textbook models as well as those on the frontiers of research. Models, by design, are abstractions of reality, intended to highlight a particular aspect of the economy. Unfortunately, numerous abstractions have compounded, causing the standard undergraduate treatment of macro and financial economics to become increasingly disconnected from the modern practice of these disciplines.

Students of finance, for instance, are left with lingering questions about the origins of two key inputs into their asset pricing formulas, namely earnings and discount rates. Meanwhile, students of macroeconomics are left without the tools to answer a simple, yet terribly relevant, question: How does the economy respond to a financial shock?

The importance of these questions is timeless, but is increasingly relevant since the Great Recession of 2007-2009. A central criticism of macroeconomics during the Great Recession was that policy makers and the economists that advised them did not pay sufficient attention to the role of the financial markets in determining aggregate levels of output, employment, and prices. This oversight led to a paradigm shift in not only the frontier of macroeconomic and financial research, but also economic education. As suggested by Blinder (2015), the undergraduate economics curriculum needs to be overhauled so that students can grasp the causes of the Great Recession, and the associated remedial policies.

The goal of this article is to introduce a unified macro-financial framework to answer these questions. By bridging the gap between asset pricing and business cycle theory in a cohesive manner, we offer students and instructors a single model that is applicable across disparate courses. Learning theory suggests (see Klein 2005, among others) that when students make connections across topics and courses, they attain a deeper understanding of the material.

Our macro-financial model, which is detailed in the next section, is motivated by several shortcomings of the standard textbook treatment of macroeconomic theory at the undergraduate level. The first of these shortcomings is that the standard constructs, such as the investment saving/liquidity preference monetary supply (IS/LM) model, provide unrealistic depictions of the role of interest rates in the macroeconomy. A single rate is assumed to drive all lending and borrowing decisions, without regard to default risk or term premia. In the DyEM structure we introduce a systematic risk factor, as well as idiosyncratic measures of risk for specific types of firms. Moreover, we detail a term structure of interest rates. These term and default risks feed into the pricing of financial assets as well as aggregate economic activity.

A second shortcoming is that the standard models generally are static. Consider constructs such as aggregate demand/aggregate supply (AS/AD), IS/LM, and the like, which often are introduced as models of the business cycle. The notion of a cycle connotes time variation and dynamics. Yet these models are not well suited for illustrating dynamics since they are static by design. The instructor might introduce the concepts of short run and long run, but can provide little guidance as to how the economy transitions from one to the other. Frontier research is cast in a dynamic setting, with a focus on growth rates, rather than levels, of key endogenous variables. Authors such as Mankiw (2013) and Mishkin (2011) are notable pedagogical exceptions, in that they each offer a chapter to dynamic models in their undergraduate textbooks. Our presentation here extends their dynamic frameworks to include not only inflation, which is the dynamic counterpart to the price level, but also economic growth, which is the dynamic counterpart to real gross domestic product (GDP). This extension supports our attempt to better reflect the frontier of research as well as to connect economic activity to asset pricing, wherein asset prices are built upon profit growth, which itself it driven by growth in the economy.

A third shortcoming is that the standard presentations are primarily theoretical. A typical presentation begins with the theoretical foundation, followed by visualization, and sometimes ends with a mathematical derivation of equilibrium conditions. The models, however, are not applied. In fact, de Araujo et al. (2013) find that none of the major intermediate macroeconomic textbooks discuss calibration or estimation. (1) Distancing the student from applications could jeopardize student interest, and hence their dedication to the material. The Dynamic Empirical Macroeconomic (DyEM) model can be calibrated to reflect current economic activity or can be augmented to highlight various schools of macroeconomic thought. We advocate the use of Impulse Response Functions (IRFs) to illustrate the dynamics of the key endogenous variables in the system. Moreover, we provide a sample of an Excel spreadsheet that the instructor may use to illustrate how different parameter settings can generate differing macro-financial impacts.

A fourth shortcoming in the standard presentation of macroeconomic models surrounds the link between output, unemployment, and inflation. For example, textbooks typically introduce the Phillips Curve as the trade-off between inflation and unemployment, visually depicting the curve with inflation on the vertical and unemployment on the horizontal axes, respectively. However, most mathematical formulations replace unemployment (and the associated gap) with output (and the associated gap). Most likely this substitution is prompted by the common technique used in frontier research, which itself is justified by a simple Okun's Law to connect these two metrics. However, skipping over this critical step in the modeling framework hinders the student's ability to connect the theory to the math. Similarly, most texts describe the Federal Reserve's dual mandate as stable prices and maximum employment. However, associated Taylor-type rules are formulated with inflation and output, not inflation and unemployment. Again, this disconnect may hinder student's ability to connect theory to the math. We easily remedy this shortcoming by including a very simple Okun's Law representation early within our model.

Lastly, the few dynamic models currently available in the undergraduate curriculum focus exclusively on adaptive expectations for price setting. Only one, to the best of our knowledge, allows agents to use a wider information set than the series of interest when forming their expectations. (2) The obvious benefit of this restriction is mathematical ease. However, the cost of the shortcut is that students are not able to relate class material to current economic events. For instance, economic agents might update their inflation expectations after a Consumer Price Index (CPI) release, or after a speech from the Federal Reserve Chair. Purely adaptive expectations are not capable of modeling such realistic behavior. We introduce a very broad specification that allows the instructor to augment purely adaptive expectations in order to illustrate myriad reactions by agents.

We detail the Dynamic Empirical Macroeconomic (DyEM) model. The complete model offers rich dynamics, the individual elements of which are built upon standard theories that are within the grasp of students of all levels. Meanwhile, a formal mathematical treatment of the complete DyEM is manageable only by the more advanced students. However, the model can be simplified in multiple directions, putting it in the reach of intermediate and beginner students as well. We connect this macro-financial model to asset pricing. Specifically, we illustrate how changes in the output gap and interest rates can impact bond and stock prices. We describe a cooperative learning exercise that the instructor can use to explain the macro-financial connections seen during the Great Recession.

DyEM

In this section we introduce a 10-equation DyEM model. The DyEM model builds off of the basic New Keynesian framework outlined by Clarida et al. (1999) and Woodford (2003), which is a standard for frontier macroeconomic research. Onto that framework we graft two key elements of the financial markets, namely default and term risks. These simple additions permit the model to reflect myriad aspects of the economy that are overlooked by standard textbook presentations. Notable exceptions include Blanchard and Johnson (2012), which outlines the qualitative effects of changes in the external finance premium and term spread on a standard IS-LM model, but fail to account for the liquidity premium. Similarly, Jones (2011) and Mishkin (2011) both include a risk premium in an investment saving-monetary policy (IS-MP) model, but lack the liquidity premium and connection to the equity market. (3)

Table 1 provides the complete set of equations for the full DyEM model, and Table 2 defines the model variables and parameters. The primary endogenous variables include [y.sub.t] = the annualized growth in real GDP, [[pi].sub.t] = the annualized rate of inflation, [i.sup.(1).sub.t] = the short-maturity (i.e., 1-yr) nominal risk-free rate of interest, and [[??].sup.(10).sub.t] = the long-maturity (i.e., 10-yr) real risky rate of interest. All parameters {[alpha], [gamma], b, [[theta].sub.[pi]], [[theta].sub.U R], [w.sub.j], [delta], [phi], [PHI]} are defined as constants, and are not restricted to the positive domain, unless otherwise stated. The variables {[[bar.y].sub.t], [[bar.r].sub.t], [[epsilon].sub.t], [[upsilon].sub.t], [[??].sub.t], [bar.U [R.sub.t]], [i.sup.(1)*.sub.t], [[omega].sub.t], [[eta].sub.t], [l.sup.(m).sub.t], [[bar.n].sub.t}, are set exogenously, and intended to capture potentially time varying shocks to the economy.

The model begins with the dynamic IS curve, which captures the demand for goods and services, and consists of three components. First, demand grows at its natural rate [[bar.y].sub.t]. Growth may deviate from its natural level due to an interest rate gap, which is defined as the difference between the time t risky, long-maturity real interest rate [[??].sup.(10).sub.t] and the risky natural long-maturity real interest rate [[??].sup.(10).sub.t]. The standard motivation applies: as interest rates rise above their natural level, the cost of borrowing is higher, which curtails demand for interest sensitive activities, such as mortgage-financed home purchases. One innovation in this dynamic IS curve is that the rates used here are long-maturity, as opposed to the standard models which are agnostic as to the duration. The superscript (10) is meant to indicate a 10-year maturity as the proxy for long-maturity rates, but this is easily adapted. Note however, that many residential and commercial loan rates are anchored to the 10-year US Treasury, and as such, this 10-yr horizon is in fact a plausible assumption for the U.S. A second innovation of the dynamic IS specification is that it includes risk, recognizing that most private borrowing/lending does not occur at the risk-free level. This innovation permits the DyEM model to subsume the external finance premium of Bernanke and Gertler (1989) and others. For notational purposes, the tilde superscript is used to differentiate risky rates from risk-free rates. Also notice that output is cast in terms of growth rates, rather than levels. This alteration not only makes the model fully dynamic, but also allows for an easier extension to asset pricing, wherein earnings growth is connected to output growth. Lastly, the exogenous variable [[epsilon].sub.t] represents demand shocks.

The Fisher Equation is a standard representation, converting nominal [[??].sup.(m).sub.t] to real rates [[??].sup.(m).sub.t] by adjusting for inflation expectations [E.sub.t][[[pi].sub.t+m]. Notice that the Fisher Equation in Table 1 is written for a risky long-maturity asset, but can be applied to any other nominal rate. Note also, that the horizon of the inflation expectations may be extended to horizons other than one-step ahead.

The Phillips Curve also is a standard representation, capturing the relationship between inflation [[pi].sub.t] and unemployment U [R.sub.t]. This New Keynesian-style Phillips Curve comes from the optimal price setting condition from monopolistically competitive firms affected by some nominal rigidity (i.e., sticky prices or sticky wages). Specifically, [[bar.U] [R.sub.t]] is the natural unemployment rate (i.e., NAIRU), corresponding to the natural rate of output [[bar.y].sub.t], permitting us to refer to (U [R.sub.t ] - [[bar.U] [R.sub.t]]) as the unemployment rate gap. Note also that v, is an exogenously determined adverse supply side shock. A simple innovation in this representation of the Phillips Curve is that it is cast in terms of the unemployment rate gap, rather than the output gap ([y.sub.t] - [[bar.y].sub.t)]. As mentioned in the introduction, casting in this way connects the typical theoretical underpinnings with the mathematical representation, thereby supporting student learning.

The monetary policy rule (MPR) is built upon a standard Taylor-type rule. The risk free short maturity nominal rate [i.sup.(1).sub.t] is set at its implied natural level [[bar.r].sup.(1).sub.t] which is determined exogenously, along with an inflation adjustment n,. The rate [i.sup.(1).sub.t], however, may deviate from this natural level depending upon the inflation gap ([[pi].sub.t] - [[??].sub.t]), where [[??].sub.t] denotes the monetary authority's target rate of inflation, and the unemployment rate gap. One innovation in this representation of the MPR is that it is cast in terms of the unemployment rate gap, rather than the output gap. Again, casting in this way connects the typical theoretical underpinnings with the mathematical representation, thereby supporting student learning. This MPR has two additional innovations. First, the parameter b controls a discretionary override to this rules-based monetary policy. If b is calibrated to 1, then the MPR follows the traditional Taylor-type rule just described. If, however, b is calibrated to 0, then MPR can reflect a purely discretionary policy wherein the central bank sets the short term rates at some target level [i.sup.(1)*], which is determined exogenously. A second innovation is the Max operator, which permits the MPR to reflect the zero lower bound on short-term nominal rates. (4) As discussed in the introduction, these simple additions permit the DyEM to more closely reflect the current U.S. macro environment.

Short-term inflation expectations are set adaptively, where the weights w, and backward looking horizon J, are set exogenously. A subtle innovation of this representation of inflation expectations is that the parameter a permits an override on the adaptive expectations. For instance, calibrating a > 0, suggests that agents in this economy are setting their inflation expectations more than what a simple adaptive expectation might suggest. Having this flexibility is useful to reflect the credibility of a central bank to influence these expectations through nontraditional tools such as forward guidance.

Long-term inflation expectations (i.e., those for horizons greater than one period) are set as a weighted average of the expectation of rates one period ahead [E.sub.t][[[pi].sub.t+1]] with the target inflation rate [[??].sub.t]. This weighting scheme is similar to that of Alpanda et al. (2013), wherein inflation expectations are set as a weighted average of the current inflation rate [[pi].sub.1] and next period's target rate [[??].sub.t+1]: [E.sub.t][[pi].sub.t+1] = w[[pi].sub.t] + (1 - w)[[??].sub.t+1]. Our weighting scheme is driven by the parameter [PHI]. By restricting [PHI] between 0 and 1 inclusive, we get a stable long-run equilibrium, wherein long-term inflation expectations default to the target level.

The Okun's Law is a standard representation of the trade-off between output [y.sub.t] and unemployment U [R.sub.t]. The exogenously determined parameter cot can be set to capture dislocations in this relationship. For example, [omega] > 0 might be used to reflect a scenario where a growing economy prompts an increase in the labor force participation rate, which actually increases, rather than decreases, the unemployment rate.

The risk-adjusted rates specification introduces an exogenously determined default risk premium [[eta].sub.t]. This simple innovation permits the DyEM structure to recognize that private borrowing/lending rates differ from the rates at which the Federal government can borrow. As [eta] rises, the default premium rises. We apply this same risk adjustment to all interest rates, whether they be nominal or real. For natural interest rates, we adjust by the concomitant level of risk [[bar.[eta]].sub.t]. For example, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The liquidity premium theory (LPT) is standard in fixed income asset pricing. Using one year as our proxy for the short maturity asset, the rates on all other maturities can be determined by the expectation of future short-term rates plus some exogenously determined term premium. Notice that we capture this term premium with the notation [l.sup.(m).sub.t] which is standard, and usually meant to signal that the premia is linked to liquidity differences by maturity. Specifically, the liquidity premium is assumed to rise with maturity, which is predicated on the stylized fact that liquidity of U.S. Treasury securities rises as maturity lengthens. Although this liquidity-based characterization is subsumed by our setting, the term [l.sup.(m)] may in fact capture any factor that causes rates to differ from the expectation of future short term rates. Notice also that the long maturity real natural rate of interest [[bar.r].sup.(10).sub.t] can be found by setting all rate expectations to [[bar.r].sup.(1).sub.t] and adding on the 10-yr term premia.

The LPT relies upon expected short rates, which are defined in the short rate expectations specification. For a given horizon h, the expectation of the nominal short rate deviates between that rate's current level [i.sup.(1).sub.t] and its natural level [[??].sub.t] + [[bar.r].sup.(1).sub.t]. The exogenously determined parameter 0 [less than or equal to] [phi] [less than or equal to] 1 sets the pace at which expectations move toward their natural level. For instance, a calibrated value of [phi] = 0.25 implies that the one step-ahead expectation of the short rate (i.e., h = 1), is built 25 % from its current level and 75 % of its long-run level.

Notice that the system is stable, in the sense that it produces consistent long-run equilibria. Specifically, in the long run, all shocks {[epsilon], [upsilon], [omega], [eta] - [[bar.[eta]].sub.t]} are set at zero. Moreover, the restriction on [phi] forces the short rate expectations to their natural level [[??].sub.t] + [[bar.r].sup.(1).sub.t] as t [right arrow] [infinity]. Likewise, [[PHI].sup.tm] [right arrow] 0 as t [right arrow] [infinity], implying that long-term inflation expectations go to the target level, nt. In the long run, all interest rates become free of excess default risk, meaning that r] is equal to its natural level of [bar.[eta].sub.t]. Notice, however, that at their natural levels, interest rates remain subject to term premia. Long maturity risky real rates [[??].sup.(10)] then equal their natural level [[??].sup.(10).sub.t] forcing output growth to its natural level. Likewise, unemployment proceeds to its natural level and inflation reaches its target rate. In summary, in the long run [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

With these short- and long-run equilibria defined, we can induce various additional aspects of the economy. The user of the model can easily infer the impact of the business cycle upon the national debt/deficit, the labor market, and myriad other aspects of the economy. The focus of this paper is asset pricing, and as such, the next section induces the impact of business cycle movements bond upon fixed income and equity prices.

Asset Pricing

In this section, we provide an overview of standard asset pricing structures and their connection to the DyEM model. Definitions of the additional variables and parameters required to conduct this macro-based asset pricing are provided in Table 3. For fixed income pricing we rely upon the liquidity premium theory, and give special consideration to how the expectation of future interest rates are determined. Meanwhile, for equity markets, we follow the Gordon growth model. In either case, the price of the assets is driven by the present discounted value of the expected future stream of earnings.

The Fixed-Income Market

Consider a simple fixed income structure such as an (m)-period discount bond:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where P [DV.sup.(m).sub.t] is the present discounted value, and FV is the face value of the bond. Recall that discount bonds issue no coupons, and return only the face value upon maturity. The focus on a discount bond simplifies the calculations, but our approach can easily be extended to coupon bonds and the like.

The denominator of the pricing equation above is built upon the (m)-period nominal interest rate [i.sup.(m)]. Any of the expected discount rates in the series above can be interpreted as the time (t) expectation of [i.sup.(m)] for period (t + j), otherwise known as an implied forward rate. Note that [E.sub.t] [i.sup.(m).sub.t]] = [i.sup.(m).sub.t] since the discount rate is known at time t. By using this sequence of implied forward rates, the pricing structure above can provide much richer dynamics than a simple yield-to-maturity calculation wherein a single discount rate is used for all periods.

Upon issuance of the security, the face value and maturity are fixed. Uncertainty is centered upon the discount rate, which is dependent upon the risk profile and maturity of the asset. For instance, let us set m = 10 and consider a 10-year corporate bond. The associated discount rate would be written as [[??].sup.(10).sub.t], which could be decomposed as

[[??].sup.(10).sub.t] = [i.sup.(10).sub.t] + [[eta].sub.t].

where [i.sup.(10)] is the 10-year risk-free interest rate, and [eta] is a systematic default risk premium. As the perception of default risk rises exogenously, the required compensation for assuming that risk rises, which is reflected in a higher [eta], and leads to higher interest rates and potentially slower economic activity. Notice that this systematic risk premium could be customized for specific corporate debt issuers, but we cast generally for ease of exposition.

The long-term risk-free rate [i.sup.(10)] is assumed to abide by a liquidity premium theory wherein the long-maturity nominal rate is equal to the average of the future expected short-maturity nominal interest rates plus an exogenous term (i.e., liquidity) premium:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As is standard in this literature, this term premium is determined exogenously and is assumed to rise with maturity. In order to embed this structure inside the PDV calculation above, we can employ a law of iterated expectations wherein [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The expectations of the future short-maturity nominal interest rate are based on a combination of the current short-maturity nominal rate and its nominal natural level:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We restrict the parameter [phi] to be between 0 and 1 inclusive, giving us two boundaries for the expected value of the short-term rate: (1) the short-term rate is expected to stay at its current level (i.e., [phi] = 1), or (2) the short-term rate is expected to achieve its natural level (i.e., [phi] = 0). As the horizon of the expectation extends along with the maturity of the asset, [phi] approaches 0, thereby pushing expectations to their natural level.

We can easily extend the pricing structure in Eq. 1 to risky fixed income assets. For instance, uncertainty might surround the payment of cash flows (i.e., FV), in which case a probability weighted average of these payouts could be embedded inside the discounting equation. Any time risk is introduced, we can also consider mispricings, such as under/overvaluation wherein the market price deviates from the PDV. We illustrate these elements of risky asset pricing via equities in the next subsection.

Equities

The Gordon growth model of equity pricing relates the price of an asset to the level (and growth) of dividends. Specifically, the PDV of asset i (P [DV.sup.i.sub.t]) via a standard Gordon growth model can be written as:

P [DV.sup.i.sub.t] = [E.sub.t][[D.sup.i.sub.t+1]]/[k.sup.i.sub.t] - [G.sup.i.sub.t],

where [E.sub.t][[D.sup.i.sub.t+1] is this period's expectation of next period's dividend, [k.sup.i.sub.t] is the asset-specific discount rate, and [G.sup.i.sub.t] is the natural (i.e., long-run) nominal growth rate of dividends for asset i.

A natural question a student might ask is: how are dividends determined? A casual, but unsatisfactory, answer might relate dividends to earnings, but that then begs the question: how are earnings determined? In the following we answer these questions by connecting the level and growth of dividends, as well as the discount rate, to the macroeconomy via the DyEM model.

We assume the dividends for company i are derived from earnings via:

[E.sub.t] [D.sup.i.sub.t+1] = [E.sub.t][E P [S.sup.i.sub.t+1]](1 - [c.sup.i])

where E P [S.sup.i.sub.t] is the current level of earnings per share (EPS) in dollars for company i, and [c.sup.i], which is set exogenously, is the amount of retained earnings as a percentage of EPS. If [c.sup.i] = 1 then all earnings are retained, forcing dividends, and ultimately the PDV of the asset, to zero. Obviously, there are companies with publicly traded equity that do not issue dividends. The presentation here can be extended to accommodate such circumstances. Multi-stage dividend discount models with a time varying fraction of retained earnings, wherein ct is time varying, can capture investor expectations that a company with no dividends today might issue dividends at some point in the future.

The growth in earnings is linked to the growth rate of the economy [y.sub.t]. Notice, however, that y is a real growth rate, whereas corporate earnings are nominal. We can convert from real to nominal with a simple Fisher specification. Moreover, we note that earnings growth rates may vary across companies, industries, and sectors of the economy. Earnings of some companies may comove with the growth rate of the economy (i.e., are pro-cyclical, such as industrials), whereas other companies have earnings that do not vary with the growth rate of the economy (e.g., utilities). We connect earnings to the business cycle through a simple exogenous term [[beta].sup.i], which is the degree to which firm i 's earnings comove with the macroeconomy. Specifically, we write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where we set short-term output growth expectations analogously to short-term inflation expectations. Specifically, [E.sub.t] [[y.sub.t+1]] = (1 + e)[[summation].sup.J.sub.j=-0] [w.sub.j][y.sub.t-j], where the parameter e acts as an override on purely adaptive expectations. When e > 0, agents expect higher output growth in the next period than purely adaptive expectations would suggest. Conversely, setting e < 0 implies agents expect lower output growth relative to purely adaptive expectations. The long-run growth rate of dividends is connected to the natural rate of growth in the economy, adjusted for each asset's cyclically: [G.sup.i.sub.t] = ([??] + [[??].sub.t])(1 + [[beta].sup.i]).

The discount rate in the Gordon growth model can be decomposed into a risk-free component and a risk premium. Often this discount rate is captured by the CAPM implied rate of return. We follow a similar inspiration and decompose the discount rate as:

[k.sup.i.sub.t] = [i.sup.(m).sub.t] + [[beta].sup.i][[E.sub.t] [R.sup.M.sub.t+m]] - [i.sup.(m).sub.t]]

where [i.sup.(m)] serves as the risk-free rate. Note that this is a nominal rate. Also note that the maturity of the discount rate is set to match the holding period horizon of m periods. The second term in the equation above quantifies the risk premium. As is typical in a CAPM specification, [E.sub.t][[R.sup.M.sub.t+m]] - [i.sup.(m).sub.t] captures the equity risk premium, where [E.sub.t][[R.sup.(M).sub.t+m]] represents the expected return on the broad equity market over the next m periods. We link the market expected return to the economy by setting [E.sub.t][[R.sup.M.sub.t+m]] = [[PI].sup.m.sub.h=0] [E.sub.t][(1 + [y.sub.t+h] + [[pi].sub.t+h]) - 1. This expected market return depends upon m future expectations of output growth and inflation. Inflation expectations, both short- and long-term, are defined in Table 1. As we outline above, short-term output growth expectations are defined adaptively with an allowance for a non-standard override of pure adaptive expectations. We define long-term output growth expectations analogously to the long-term inflation expectations. Specifically, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all maturities greater than 1 (i.e., m > 1), where the parameter 4> controls the weighting of agents' expectations between the current and the natural, long-term level of output growth. As is also typical in a CAPM specification, the level of risk for a particular asset within a portfolio (i.e., idiosyncratic risk) augments the market-wide (i.e., systematic) risk through the use of [beta], where a larger [beta] signals higher risk, and thus a higher discount rate on that asset.

This setting can connect to numerous aspects of asset pricing and modern portfolio theory. For example, we can examine the level of total risk for a given asset with a CAPM framework as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [[sigma].sup.2.sub.M] is the level of systematic risk and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the level of idiosyncratic risk for asset i, which is assumed uncorrelated with the systematic level of risk. We can further connect the systematic level of variability to the economy by setting [[sigma].sup.2.sub.M] = [[lambda].sub.1] + [[lambda].sub.2]|[y.sub.t] - [[bar.y].sub.t], where [[lambda].sub.1] and [[lambda].sub.2] are exogenous constants. The implication of this specification is that the equity market exhibits more volatility during periods of expansion and contraction.

In addition, asset returns, whether they be for the market [R.sup.M] or for a specific asset [R.sup.i], can be computed directly upon PDV's: [R.sup.i.sub.t+m] = ln ([PDV.sup.i.sub.t+m]/[PDV.sup.i.sub.t]). This implies that the assets are fairly priced in both periods. We can easily introduce a degree of mispricing: [P.sup.i.sub.t]= (1 + [d.sub.t])[PDV.sup.i.sub.t], where [d.sub.t] is an exogenous variable that controls under(over)-valuation. For instance, if d > 0, the asset's price is above its PDV, implying an overvaluation. Mispricing naturally leads to discussions of market efficiency. Notice that agents' expectations over the endogenous variables (output growth, inflation, interest rates, etc.) are not necessarily model-consistent, suggesting a potentially (weak-form) inefficient market.

Bringing the Great Recession into the Classroom

In this section we describe how the DyEM model can be used in the classroom to explain and explore the Great Recession.

A Simple DyEM for the Great Recession

Recessions may easily be triggered by supply or demand side shocks, [upsilon] or [epsilon], respectively. The ensuing changes in output, inflation, and interest rates, then may induce changes in the financial markets. Arguably, the Great Recession stands apart from most modern recessions in that the causality was the reverse: a collapse in the financial markets triggered an economic collapse. This financial trigger could come in many forms. For instance, one could begin with investors and economic agents underpricing the level of risk in the economy. This mispricing generates an increase in asset prices and economic activity, which might be referred to as a bubble. At some point, investors and economic agents reassess the price of risk. As risk premia rise, asset prices drop and economic activity falls, which might be referred to as a bust.

This mispricing and repricing of risk is easily accommodated within the DyEM model by exogenous changes in either the systematic default risk [eta] or the maturity specific term premia [l.sup.(m)] For ease of exposition, we focus here only on changes in default risk.

Our recommendation to the implementing instructor is to begin with an introduction to the complete DyEM model. The theoretical underpinnings of each equation of the model provide an opportunity for the instructor to explore in depth the major aspects of most undergraduate macroeconomics courses: activity, labor markets, inflation, etc.

For those instructors who introduce various macro models in a historical context (e.g. Froyen 2012), notice that the natural levels of activity embedded inside the DyEM structure correspond to those achieved under a classical-type setting. Moreover, as the instructor alters the calibration of the model, various economic philosophies are emphasized. For example, a dominance of supply side shocks v may be supportive of an RBC-type setting. Likewise, a b = 0 reflects discretionary monetary policy. (5)

For those instructors who wish to have their students solve these models analytically, we suggest a slight modification to ensure tractability. Specifically, by setting b = 1 and eliminating the Max operator, the MPR defaults to a standard Taylor-type rule. By setting a = 1 and J = 0, short-term inflation expectations are set myopically with a single-period adaptation. By setting [PHI] = 1, long-term inflation expectations are set at the current inflation rate [[pi].sub.t]. With [gamma] = 1 and [omega] = 0, the unemployment rate gap defaults to the output gap. Finally, by setting [phi] = 1, short rate expectations are set at the current rate.

The ensuing system of equations is captured in Table 4. All variables and parameters are as defined in Table 2, with the addition of [[theta].sub.y], which is simply -[gamma][[theta].sub.U R], and is defined as the central bank's weight on the output gap in the monetary policy rule. By solving this system, we find the dynamic aggregate supply (DAS and demand (DAD) curves as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The resulting equilibria are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

A Cooperative Learning Exercise

In this section we describe a cooperative learning technique that the instructor can use to supplement traditional instructional methods. Broadly speaking, cooperative learning consists of actions where students share knowledge among themselves. As Kagan and Kagan (1994) suggests, this type of knowledge sharing is shown to improve high order thinking skills, enhance interest in the subject, increase student morale and motivation, and promote higher quality interaction with peers and faculty. Kagan and Kagan (1994) identify the key elements to structuring a successful cooperative learning exercise as (i) individual accountability, (ii) equal participation, and (iii) simultaneous interaction. The following is a variant of the commonly known "send-a-problem" exercise, which adheres to this structure.

The target audience for the exercise is undergraduate students in either intermediate/advanced macroeconomics or finance courses. The technique we outline here is useful to illustrate a wide array of topics within these courses, and so may be utilized several times in the same course with only slight variation. The exercise described in the following is customized for a class of 40 students, but can be scaled up or down rather easily. In this setting, we recommend splitting the class of 40 into five groups of eight students.

As with all cooperative learning exercises, the instructor must prepare the activity and associated materials. First, the learning objective must be identified. In order to solidify this example, we offer the case where the instructor is attempting to illustrate the differing impacts of financial shocks, specifically with the purpose of highlighting the Great Recession. Depending upon the level of mathematical training among the students, the instructor may explicitly calibrate the model, or may focus simply on the direction of change and the intuition behind the reactions to this shock. In either case, the instructor can use impulse response functions (IRFs) as a visual device to summarize the dynamic effects of these shocks upon the variables, which summarize both the macroeconomy and financial markets.

The setup for the exercise consists of the instructor printing out five sheets of paper. Each piece of paper contains the evolution of shocks in the top left. Moreover, the paper contains blank IRFs (i.e., only titles and axis labels). The reason why they are blank is that the students will use these axes to provide a visual summary of their answers. (See Fig. 1 for an example.) Notice that there are six IRFs: one for the shock of interest: [[eta].sub.t], and five blank IRF's for the endogenous variables of interest: [y.sub.t], [[pi].sub.t], [i.sup.(1).sub.t], [i.sup.(10).sub.t], and the percent return in the equity market. For each variable's IRF, the horizontal axis denotes the time period (t = -5, -4, ..., 25) while the vertical axis indicates the level in percentage terms.

Each piece of paper is unique, either by varying the source, intensity, or duration of the shocks, and/or by the specific financial asset under consideration. The instructor then places each of the five sheets of paper into separate manila envelopes, and hands one envelope to each group.

The groups then have 10 minutes to draw the impact of these shocks upon the macroeconomic variables of interest (e.g., [y.sub.t], [[pi].sub.t], [i,sup.(1).sub.t], [i.sup.(10).sub.t]). At the end of the 10 minutes, the group places their completed IRF drawings back into the envelope. The groups switch envelopes among each other. The groups then have another 10 minutes to sketch out the impact of the new shock, as indicated by the IRF drawings of the group that previously had this envelope, upon the financial market asset (e.g., equity returns). At the end of this second round, each team should have an envelope with five completed IRFs.

While the teams are working, the instructor can monitor progress by simply glancing at the IRFs that are being completed by each group. Moreover, the instructor can encourage individual accountability and equal participation by assigning group members to specific roles. For example, a natural assignment might be to identify "macroeconomic experts" and "financial experts," the first of which would take the lead during the first round, and the second of which would take the lead during the second round of the exercise.

The exercise closes with a full-class debrief wherein each group is asked to open their envelope, describe the shock and its impact on the macroeconomy and financial markets. The instructor leads discussion by verifying the analysis and comparing and contrasting each group's conclusions.

Table 5 outlines the suggested calibration for the in-class exercise. Note that this calibration is adaptable to any preferred economy or model specification outlined above. Additionally, many of these parameters are considered to be slow-moving, such as the natural level of output growth or the central bank's target inflation rate. Shocks to these medium-to-long-term variables can also be considered in the reduced form framework.

In Fig. 1, we provide a sample of the blank worksheet. We capture the Great Recession through changes in the systematic default risk premium [[eta].sub.t]. Specifically, default risk begins at its long-run level of fy in periods -5 to -3. In period -2, that risk is severely underpriced, causing [[eta].sub.t] - [[eta].sub.t] to go negative. In period -1, the default risk remains low. As investors recognize this underpricing, [eta] spikes in periods 0 and 1, returning to its natural level in period 2.

In Fig. 2, we illustrate the completed worksheet for the calibration detailed in Table 5. To facilitate implementation, we provide an Excel template of this reduced form DyEM model. (6) Notice, however, that the students need not have the calibration to complete the assignment. They could simply sketch directions of change in each variable. Similarly, a more in-depth homework assignment could include specific calibrations that capture different beliefs about the macroeconomy.

For those instructors that wish to illustrate these concepts mathematically, the partial derivatives of each endogenous variable with respect to y would be an informative exercise. For instance, [partial derivative][y.sup.*.sub.t]/[partial derivative][[eta].sub.t] = [alpha]/1 + [alpha][[theta].sub.y] + [delta][alpha][[theta].sub.[pi]]. Notice that this partial derivative is negative, implying that as [eta] falls, output rises.

Similarly, for those instructors that wish to emphasize the visual representation, the DyEM can be captured in a dynamic AS/AD setting. In the aforementioned Excel template, we provide the ability for the user to visualize the DAS and DAD curves for several periods surrounding the shock.

Finally, the implementing instructor can highlight various aspects of the economy by reinstating some of the elements of the complete DyEM model detailed in Table 1. For instance, if [theta] is no longer set to 1, the liquidity premium theory resumes its rich dynamics. The implementing instructor could depict a risk-free yield curve. In the aforementioned exercise, the DyEM model produces an inverted yield curve, which, as per Estrella and Mishkin (1996), is a well-known predictor of the impending recession.

Conclusion

The DyEM model presented in this paper provides undergraduate students with a unified macro-financial framework. The DyEM model stands apart from the current textbook treatments of macroeconomic models in that it explicitly includes term and risk premia in its presentation of interest rates. This innovation allows instructors to display the linkages and propagation of shocks between the macroeconomy and financial markets.

The rich dynamics afforded by the complete DyEM system bring the student closer to the frontiers of macroeconomic research, and allow them to implement the theroretical models they encounter in the classroom. The DyEM structure is flexible enough to be customized through well-chosen calibrations, and is therefore within the reach of beginners as well as advanced students.

For classroom purposes, we presented in this article a reduced form DyEM model that is capable of capturing the dynamics of the US economy during the Great Recession. We accompany this tractable system with a cooperative learning exercise wherein students are tasked with explaining how a financial shock could reverberate to the real economy, as it did during 2008-2009.

The DyEM structure can be modified and extended in many directions. For example, Aguilar and Soques (2015), a companion piece to this article, introduces an open-economy DyEM model by incorporating an interest parity condition.

DOI 10.1007/s11294-015-9546-8

References

Aguilar, M., & Soques, D. (2015). Open DyEM: A cohesive open-economy macro-financial model for the undergraduate level. Working Paper.

Alpanda, S., Honig, A., & Woglom, G. (2013). Extending the textbook dynamic AD-AS framework with flexible inflation expectations, optimal policy response to demand changes, and the zero-bound on the nominal interest rate. Modern Economy, 4(3), 145-160.

Bernanke, B., & Gertler, M. (1989). Agency costs, net worth and business fluctuations. American Economic Review, 79(1), 14-31.

Blanchard, O., & Johnson, D. (2012). Macroeconomics, 6th ed. Boston: Pearson.

Blinder, A. (2015). What did we learn from the financial crisis, the great recession, and the pathetic recovery? Journal of Economic Education, 6(2), 135-149.

Clarida, R., Gali, J., & Gertler, M. (1999). The science of monetary policy: A new Keynesian perspective. Journal of Economic Literature, 37(4), 1661-1707.

de Araujo, P., O'Sullivan, R., & Simpson, N.B. (2013). What should be taught in intermediate macroeconomics? Journal of Economic Education, 41(4), 74-90.

Estrella, A., & Mishkin, F.S. (1996). The yield curve as a predictor of U.S. recessions. In Current Issues in Economics and Finance, (Vol. 2 pp. 1-7): Federal Reserve Bank of New York. (Jun), 1996.

Froyen, R. (2012). Macroeconomics: Theories and Policies, 10th ed. Boston: Pearson.

Jones, C. (2011). Macroeconomics, 2nd ed. New York: Norton.

Kagan, S., & Kagan, M. (1994). The structural approach: Six keys to cooperative learning. In Sharan, S. (Ed.) Handbook of Cooperative Learning Methods (pp. 115-136). Westport: Greenwood Press.

Klein, J.T. (2005). Integrative learning and interdisciplinary studies. Peer Review, 7(4), 8-10.

Mankiw, G. (2013). Macroeconomics, 8th ed. New York: Worth.

Mishkin, F. (2011). Macroeconomics: Policy and Practice, 1st ed. Boston: Pearson.

Woodford, M. (2003). Interest and prices: Foundations of a theory of monetary policy. Princeton: Princeton University Press.

Mike Aguilar [1] * Daniel Soques [1]

Published online: 12 October 2015

The authors would like to thank Ben Horlick for research assistance.

[mail] Mike Aguilar

maguilar@email.unc.edu

Daniel Soques

soques@unc.edu

[1] Economics Department, University of North Carolina, Chapel Hill, NC, USA

(1) Mishkin (2011) is a notable, recent exception. Mankiw (2013) briefly mentions calibration without much detail.

(2) Alpanda et al. (2013) augment the standard adaptive expectations to allow for expectations of next period's inflation to be a weighted average of last period's inflation and the Federal Reserve's target rate.

(3) See de Araujo et al. (2013) for an overview of the various models offered by different textbooks.

(4) This innovation is similar to that of Alpanda et al. (2013), which incorporates a zero lower bound within a dynamic AS/AD framework.

(5) Note that although well chosen calibrations can reflect various schools of economic thought, those schools may not necessarily advocate the dynamic modeling strategy of the DyEM.

(6) The Excel template can be found at the corresponding author's website: www.unc.edu/~maguilar.

Table 1 DyEM--equations

Dynamic IS                 [y.sub.t] = [[bar.y].sub.t] - [alpha]
                             ([[??].sup.(10).sub.t] -
                             [[??].sup.10.sub.t]) +
                             [[member of].sub.t]
Fisher equation            [r.sup.(m).sub.t] = [i.sup.(m).sub.t] -
                             [E.sub.t][[[pi].sub.t+m]]
Phillips curve             [[pi].sub.t] = [E.sub.t-1][[[pi].sub.t]] -
                             [delta] (U [R.sub.t] - [bar.U]
                             [[bar.R].sub.t]) + [[upsilon].sub.t]
Monetary policy rule       [i.sup.(1).sub.t] = Max [b
                             ([[bar.r].sup.(1).sub.t] + [[pi].sub.t]
                             + [[theta].sub.[pi]] ([[pi].sub.t] -
                             [[??].sub.t]) - [[theta].sub.U R] (U
                             [R.sub.t] - [[bar.UR].sub.t])) + (1 - b)
                             [I.sup.(1)] *. 0]
Short term inflation       [E.sub.t] [[[pi].sub.t+1]] = (1 + a)
  expectations               [[SIGMA].sup.J.sub.j] = [0.sup.w] j
                             [[pi].sub.t-j]
Long term inflation        [E.sub.t] [[[pi].sub.t+m]] =
  expectations               [[PHI].sup.tm] [E.sub.t][[[pi].sub.t+1]]
                             + (1 - [[PHI].sup.tm]) [[??].sub.t]
                             [for all]m >1
Okun's law                 U [R.sub.t] = [bar.UR].sub.t] - [gamma]
                             ([y.sub.t - [[bar.y].sub.t]) +
                             [[omega].sub.t]
Risk adjusted rates        [[??].sup.(m).sub.t] = [i.sup.(m).sub.t] +
                             [[eta].sub.t]; [[??].sup.(m).sub.t] =
                             [[bar.r].sup.(m).sub.t] +
                             [[bar.[eta]].sub.t]
Liquidity premium theory   [i.sup.(m).sub.t] = 1/m
                             [[SIGMA].sup.m-1.sub.h=0] [E.sub.t]
                             [[i.sup.(1).sub.t+h]] +
                             [l.sup.(m).sub.t] [for all]m > 1
Short rate expectations    [E.sub.t] [[I.sup.(1).sub.t+h]] =
                             [[theta].sup.th] [i.sup.(1).sub.t] +
                             (1 - [[theta].sup.th]) ([[??].sub.t] +
                             [[bar.r].sup.(1).sub.t])

Table 2 DyEM--variables and parameters

Variable/Parameter         Description

Endogenous variables

[g.sub.t]                  Real output growth
[[pi].sub.t]               Inflation rate
U [R.sub.t]                Unemployment rate
[i.sup.(m).sub.t]          Nominal risk-free (risky) interest rate of
  ([[??].sup.(m).sub.t])     maturity m
[r.sup.(m).sub.t]          Real risk-free (risky) interest rate of
  ([[??].sup.(m).sub.t])     maturity m
[[bar.r].sup.(m).sub.t]    Real risk-free (risky) natural interest
  ([[??].sup.(m).sub.t])     rate of maturity m

Exogenous variables

                           Demand-side shock
[[member of].sub.t]        Supply-side shock
[[upsilon].sub.t]          Deviations of Okun's Law
[[omega].sub.t]            Default risk premium
[[eta].sub.t]              Central bank's target inflation rate
[[??].sub.t]               Central bank's discretionary target
                             interest rate
[i.sup.(1).sub.t] *        Natural unemployment rate (NAIRU)
[[bar.y].sub.t]            Natural real output growth rate
[[bar.r].sup.(m).sub.t]    Natural risk-free interest rate of
                             maturity m
[[eta].sub.t]              Natural systematic default risk premium
[l.sup.(m).sub.t]          Term/liquidity premium

Exogenous parameters

[alpha]                    Demand sensitivity to interest rate
                             differential
[delta]                    Inflation sensitivity to output gap
[gamma]                    Sensitivity of unemployment to output gap
                             in Okun's Law
b                          Degree to which the central bank follows a
                             simple interest rate rule
[[theta].sub.[pi]]         Central bank's weight on inflation gap in
                             interest rate rule
[[theta].sub.U R]          Central bank's weight on unemployment gap
                             in interest rate rule
a                          Override of adaptive inflation
                             expectations
[w.sub.j]                  Weight on lagged j inflation level in
                             expectations formation
[PHI]                      Degree of backward-looking behavior in
                             long-term inflation expectations
[phi]                      Degree of backward looking behavior in
                             short-term interest rate expectations

Table 3 Macro-based asset pricing--variables and parameters

Variable/Parameter     Description

Endogenous variables

P D [V.sup.i.sub.t]    Present discounted value of asset i
[P.sup.i.sub.t]        Market price of asset i
[D.sup.i.sub.t]        Dividend of asset i
[k.sup.i.sub.t]        Discount rate for asset i
[R.sup.i.sub.t]        Return on asset i

Exogenous variables

E P [S.sup.i.sub.t]    Earnings per share of firm i
[G.sup.i.sub.t]        Natural nominal growth rate of dividends for
                         asset I

Exogenous parameters

FV                     Face value of fixed income instrument
e                      Override of purely adapative expecitations in
                         output growth expectations
d                      Controls over(under)valuation of price
                         relative to PDV
[c.sup.i]              Percentage of profits held as retained
                         earnings
[[beta].sup.i]         Degree of co-movement of firm earnings with
                         output growth

Table 4 Reduced form DyEM

Dynamic IS                 [y.sub.t] = [[bar.y].sub.t] - [alpha]
                             ([[??].sup.(10).sub.t] -
                             [[??].sup.10.sub.t]) +
                             [[member of].sub.t]
Fisher Equation            [[??].sup.(10).sub.t] =
                             [[??].sup.(10).sub.t] - [[pi].sub.t]
Phillips Curve             [[pi].sub.t] = [[pi].sub.t-1] + [delta]
                             ([y.sub.t] - [[bar.y].sub.t]) +
                             [[upsilon].sub.t]
Monetary Policy Rule       [i.sup.(1).sub.t] = [[bar.r].sup.(1).sub.t]
                             + [[pi].sub.t] + [[theta].sub.[pi]]
                             ([[pi].sub.t] - [[??].sub.t]) +
                             [[theta].sub.y] ([y.sub.t] -
                             [[bar.y].sub.t])
Risk-Adjusted Rates        [[??].sup.(10).sub.t] = [i.sup.(10).sub.t]
                             + [[eta].sub.t]
Liquidity Premium Theory   [i.sup.(m).sub.t] = [i.sup.(1).sub.t] +
                             [l.sup.(m).sub.t] [for all]m > 1

Table 5 Calibration for reduced form DyEM

Parameter               Description                              Value

[[??].sub.t]            Central bank's target inflation rate     2
[alpha]                 Demand sensitivity to interest rate      1
                          differential
[delta]                 Inflation sensitivity to output gap      0.5
[[theta].sub.[pi]]      Central bank's weight on inflation       0.5
                          gap in interest rate rule
[[theta].sub.[gamma]]   Central bank's weight on output          0.5
                          growth gap in interest rate rule
[[??].sub.t]            Natural (long-run) real output growth    2
                          rate
[[??].sup.(1).sub.t]    Natural short-maturity risk-free         2
                          interest rate
[PHI]                   Degree of backward-looking behavior      0.5
                          in long-term expectations formation
[[??].sub.t]            Natural systematic default risk          0.5
                          premium
[C.sup.i]               Percentage of profits held as            0.4
                          retained earnings
[[beta].sup.i]          Degree of comovement of firm earnings    0.1
                          with output growth
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Author:Aguilar, Mike; Soques, Daniel
Publication:International Advances in Economic Research
Article Type:Report
Geographic Code:1USA
Date:Nov 1, 2015
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