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Schwartzman, Felipe. "How Can Consumption-Based Asset-Pricing Models Explain Low Interest Rates?." Economic Quarterly. Federal Reserve Bank of Richmond. 2014. HighBeam Research. 23 Oct. 2018 <https://www.highbeam.com>.
Schwartzman, Felipe. "How Can Consumption-Based Asset-Pricing Models Explain Low Interest Rates?." Economic Quarterly. 2014. HighBeam Research. (October 23, 2018). https://www.highbeam.com/doc/1G1-433590950.html
Schwartzman, Felipe. "How Can Consumption-Based Asset-Pricing Models Explain Low Interest Rates?." Economic Quarterly. Federal Reserve Bank of Richmond. 2014. Retrieved October 23, 2018 from HighBeam Research: https://www.highbeam.com/doc/1G1-433590950.html
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The Great Recession gave way to a period of very low short-term nominal and real interest rates. As the recovery proceeds and the Federal Reserve starts to decide the rhythm with which it intends to raise policy rates, one fundamental question is whether the low interest rates are just a symptom of a recessionary period (even if prolonged) in which the Federal Reserve chose to take a deliberately expansionary stance, or if they reflect longer-run fundamental forces that may not dissipate easily. In the latter case, optimal policy may warrant a slow increase of the policy interest rate, so that it remains low by historical standards even when inflation and the labor market are close to their long-run levels. Currently, Federal Open Market Committee members appear to forecast such a slow increase, as documented in the Summary of Economic Projections.
The purpose of this article is to use consumption-based asset-pricing models to gain some insight into the determinants of the "natural interest rate," that is, the interest rate that would prevail in the absence of nominal rigidities. Since this natural rate is not itself a function of central bank decisions, it can be used as a yardstick for the stance of monetary policy. In particular, in terms of modern monetary theory (Woodford 2003), one can say that the policy stance is expansionary if the interest rate is below the "natural rate of interest" and contractionary otherwise. (1) The question about the optimal pace of interest rate liftoff can thus be recast in terms of the speed with which the natural rate of interest is likely to increase.
Consumption-based asset-pricing models are a natural starting point for the discussion of the fundamental determinants of interest rates for macroeconomists since they share conventional assumptions of most workhorse macroeconomic models: rational expectations, frictionless asset markets, and a representative household. This contrasts with behavioral economics models, which emphasize departures from rational expectations, and with segmented markets models, in which asset prices are determined by only a subset of households. (2) While these alternatives are certainly worthy of further discussion, the purpose of this article is to provide a first look at the progress that one can make with this more familiar baseline. (3) I will review three main strands within the consumption-based asset-pricing literature: habit formation, long-term risk, and disaster risk. Rather than provide a comprehensive review of the literature within each of those strands, I will discuss some of the main ideas based on a small number of influential articles. (4) At the end of each section I include a short discussion of how the model could be used to explain low interest rates. Those discussions are meant to be illustrative rather than conclusive, in that they delimit promising areas for further research rather than provide a complete answer to how well consumption-based asset-pricing models can explain currently low interest rates.
As we will see in the models reviewed, interest rates can be low either because market participants expect consumption growth to be low, because they perceive consumption risk to be high, or because they have low risk tolerance. In contrast, equity risk premia do not depend on expected consumption growth. Hence, one can gain some insight into the driving force behind low interest rates by examining the behavior of the risk premium. The evolution over time in the two variables can be seen in Figure 1. It depicts the postwar values of the real interest rate, measured by the 30-day Treasury bill rate deflated by the consumer price index, and of the equity risk premium, both of which averaged over various five-year periods. (5) The five years since the onset of the Great Recession stand out not only because of the exceptionally low real rate of interest, but also because of a historically high equity risk premium. Given the models reviewed, the high risk premium suggests that low interest rates in the recent period are likely to be either a consequence of a perception that consumption risk is particularly high, or of very low risk tolerance.
The article is structured as follows: In the following section, I lay out the notation used in the article as well as common conventions, simplifications, and approximations. Each subsequent section discusses one variety of consumption-based asset-pricing models: the Mehra and Prescott (1985) benchmark, the recursive utility and long-run risk extensions of Weil (1989) and Bansal and Yaron (2004), the disaster-risk model of Rietz (1988) and Barro (2006), and Campbell and Cochrane's (1999) habit-formation models. The final section concludes.
1. NOTATION, CONVENTIONS, SIMPLIFICATIONS, AND APPROXIMATIONS
Assets are claims on streams of dividends. In particular, purchasing some asset, i, provides an economic agent with a stochastic stream of dividends [{[D.sup.i.sub.t+s]}.sup.[infinity].sub.0] for as long as the agent holds it. In consumption based asset-pricing models there are no liquidity constraints or other transaction costs, so agents can trade assets freely at each period. If the price of asset i is given by [P.sup.i.sub.t], then we can define its return between periods t and t + 1 as
[R.sub.i,t+1] [equivalent to] [[P.sub.i,t+1] + [D.sub.i,t+1]]/[R.sub.i,t+1] (1)
Asset pricing concerns itself either with determining the price-dividend ratio for an asset, [P.sup.i.sub.t]/[D.sup.i.sub.t], or its expected returns, Et [[R.sub.i,t+1]]. Typically, higher returns are associated with lower price-dividend ratios.
While the literature discusses the pricing of many kinds of assets, the three main ones are the risk-free asset, a market portfolio of equities, and total wealth. The risk-free asset (denoted by i = f) is exactly what the name implies: an asset that pays the same dividend in all states of nature. As an empirical matter, the asset-pricing literature identifies the risk-free asset with short-term Treasury bills. Thus, the predictions of the models under review for the risk-free rate are going to be the most relevant ones for the purpose of monetary policy analysis.
The market portfolio of equities (i = e) refers to a well-diversified portfolio of shares issued by firms and traded in stock markets with prices summarized by indices such as the S&P 500. This is, in turn, different from total wealth (i = w), which is a fictitious asset (in the sense that there are no formal markets for it) that pays out aggregate consumption as dividends. It includes equity, bonds, housing, and human capital. Oftentimes studies of equity pricing at first identify equity with the wealth portfolio and then in refinements treat the two as distinct. The distinction between equity and the wealth portfolio normally focuses on the fact that firms are leveraged, both because they issue bonds and because salaries are normally insulated from high-frequency fluctuations in output. Therefore, for any change increase in aggregate endowment, dividends should change by a greater amount. The simplest way of modeling this leverage is to assume that aggregate dividends on equity are a deterministic function of consumption, with [D.sup.e.sub.t] = [([D.sup.w.sub.t]).sup.[lambda]] = [C.sup.[lambda].sub.t], for some [R.sub.i,t+1] > 1.
One simplification used by the asset-pricing literature to obtain analytical results is to rely on log normality assumptions. If the log of asset returns is normally distributed, one can use the fact that for any normally distributed x, E [[e.sup.x]] = [e.sup.E[x]+1/2 Var[x]. Thus, if returns [lambda] are log-normally distributed,
ln (E [R.sub.i,t+1]) = E [[r.sub.i,t+1]] + 1/2 Var [[r.sub.i,t+1]],
where we use small letters to denote the natural logarithm.
A further simplification, used in disaster models, is the use of a continuous time formulation to study disaster risk. Denote by dt the length of a period of time. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the gross return per period of time of that asset. Suppose the return on some asset i is either [e.sup.[bar.r]dt] with probability [e.sup.-pdt] or (1 - b) [e.sup.[bar.r]dt] with probability 1 - [e.sup.[bar.r]dt]. Then
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Taking logs and dividing by dt yields
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Taking the limit as dt [right arrow] 0 and applying l'Hopital's rule,
E [[r.sub.i,t+1]] = [bar.r] - pb.
The continuous time approximation yields an intuitive expression for expected log returns. Those are equal to [bar.r], except that with probability p they fall by b.
Finally, a common approximation used in the analytical literature is to log-linearize equation (1) to obtain
[r.sub.i,t+1] = [rho][p.sub.i,t+1] + (1 - [rho]) [d.sub.i,t+1] - [p.sub.i,t],
where [rho] is the average P/P+D ratio and is typically calibrated to some value close to 1. Rearranging and iterating forward up to some time t + T with T > 0 yields
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The expression is useful in that it breaks down three different determinants of the price-dividend ratio. The first term on the right-hand side is a discounted sum of future dividends growth. The faster dividends are expected to grow, the more a portfolio that pays off the consumption good as dividends is worth. The second term is a discounted sum of returns. All else constant, if prices are low in spite of high dividend growth, then the returns will be high as prices catch up with dividends. The third term is a "bubble" term. In most asset-pricing applications, one assumes that the bubble term goes to zero almost as surely as T increases. Given the no-bubble condition,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The equation highlights that a high price-dividend ratio can forecast either a high growth in dividend payments or low future rates of returns. Taking expectations and rearranging,
[p.sub.i,t] - [d.sub.i,t] = [[infinity].summation over (s=0)] [[rho].sup.s][DELTA][D.sub.t+1+s] - [[infinity].summation over (s=0)] [[rho]. …
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