|
...sections (see figure 6). Demography cannot explain everything, nor should it. The long-term trends equity prices over this period coincided not only with demographic trends but also with runs of luck: the 1970s and early 1980s saw mainly negative shocks (oil shortages, bursts of high inflation followed by restrictive monetary policy, leading to unemployment and low productivity), whereas the 1990s were characterized by aggregate shocks that were mainly positive (low inflation and energy prices, rapid technological progress resulting in low unemployment and high productivity). We thus add to the demographic model of the previous section the possibility of random shocks to income, to study the combined effect of demographic and business cycle fluctuations for asset prices.
[FIGURE 6 OMITTED]
Once uncertainty is introduced, risky equity and the riskless bond cease to be perfect substitutes. Equity must earn a risk premium relative to the bond to induce agents to hold it, and the model permits us to study the effect of the changing demographic structure on the risk premium.
The certainty model of the previous section showed that the qualitative results of the simplest model, with three-period-lived agents, exogenous dividends, and no bequests, are robust to the introduction of more-realistic features. We thus revert to this simplest model, adding the possibility of random wages and dividends, to study the combined effect for asset prices of demographic and business cycle fluctuations.
Risk Structure
We model the risk structure of the economy by assuming that the wage and the dividends on equity are subject to shocks. We use a highly simplified structure, assuming that at each date there are four possible states of nature (shocks): [s.sub.1], high wages, high dividends; [s.sub.2], high wages, low dividends; [s.sub.3], low wages, high dividends; and [s.sub.4], low wages, low dividends. Given the nature of the risks and the very extended length of time represented by a period (twenty years), we have chosen not to invoke a Markov structure, but rather to assume that the shocks are independent and identically distributed (i.i.d.). To reflect the fact that aggregate income and dividends are positively correlated, we assume that [s.sub.1] and [s.sub.4] are more likely (probability 0.4 each) than s2 and s3 (probability 0.1 each). This gives rise to a correlation coefficient between dividends and wages of 0.6.
Figure 2 shows that the maximum variability of the real annual wage income of the 45-54 cohort is about 4 percent: in the recession of 1990-91 the mean wage (in 1999 dollars) of this cohort fell from $65,000 to $60,000, a variability of (2.5/62.5) = 0.04; the variability of the wage income of the 25-34 cohort is somewhat lower. To take into account that some periods, such as 1970-83, experienced a sequence of negative shocks, in the calibration we increase the coefficient of variation of the wage income of the middle-aged to 20 percent and that of the young to 15 percent. Since the fluctuations of real (generalized) dividends are of the same order as those of wages, we take a coefficient of variation of 19 percent for dividends. This leads to a coefficient of variation of about 16 percent for aggregate income. In short, we assume four possible shocks with probabilities (0.4, 0.1, 0.1, 0.4), and wage income and dividends across the four states given by [w.sup.y] = (2.3, 2.3, 1.7, 1.7), [w.sup.m] = (3.6, 3.6, 2.4, 2.4), and D = (74, 50, 74, 50).
Equilibrium
Since the financial markets in the model are incomplete--each date-event is followed by four possible income-dividend shocks, and agents can trade only two securities (equity and the bond)--the equilibrium cannot be solved, as in the previous section, in terms of the consumption variables with a single present-value budget constraint for each agent. We need to explicitly introduce the asset trades, portfolio optimization, and market-clearing asset prices. Let [z.sub.t] = ([z.sup.y.sub.t], [z.sup.m.sub.t]) = ([z.sup.y.sub.b,t], [z.sup.y.sub.e,t], [z.sup.m.sub.b,t], [z.sup.m.sub.e,t]) denote the lifetime portfolio of an agent born at date t, namely, the holdings of the bond and equity [z.sup.y.sub.t] = ([z.sup.y.sub.b,t], [z.sup.y.sub.e,t]) in youth and [z.sup.m.sub.t] = ([z.sup.m.sub.b,t], [z.sup.m.sub.e,t]) in middle age. Let [c.sub.t] = ([c.sup.y.sub.t], [c.sup.m.sub.t], [c.sup.r.sub.t]) denote the agent's lifetime consumption in youth, middle age, and retirement. Both [z.sub.t] and [c.sub.t] are stochastic, depending on the past history of shocks and on the shocks to wages and dividends during the agent's lifetime. The agent's consumption and portfolio holdings must satisfy the agent's budget constraints in each state, given by
(17) [c.sup.y.sub.t] = [w.sup.y.sub.t] - [q.sup.t][z.sup.y.sub.t] [c.sup.m.sub.t] = [w.sup.y.sub.t+1] + [V.sub.t+1][z.sup.y.sub.t] - [q.sub.t+1] [z.sup.m.sub.t] [c.sup.r.sub.t] = [V.sup.t+2][z.sup.m.sub.t],
where [q.sub.t] = ([q.sup.b.sub.t], [q.sup.e.sub.t]) denotes the vector of bond and equity prices at date t, and [V.sub.t+1] = [1, [D.sub.t+1] + [q.sup.e.sub.t+1]] denotes the payoffs of the bond and equity at date t + 1. An equilibrium on the bond and equity markets is then a sequence [([z.sub.t], [q.sub.t]).sub.t [greater than or equal to] 0] of portfolios and prices such that the representative agent born at date t maximizes lifetime expected utility in equation 1, subject to the budget equations 17, and such that the bond and equity markets clear at each date t [greater than or equal to] for each state
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]
Our objective is to study how the alternating cohort sizes of young and middle-aged influence the equilibrium on the financial markets. In view of the alternating cohort structure and the assumption that the wage income and dividends are i.i.d., it is natural to look for a stationary equilibrium of the economy: in appendix B we define such an equilibrium and explain how it can be calculated.
Calibration Results
To study the properties of the equilibrium trajectories, we consider an economy with cohort sizes (N, n) = (79, 52) and risk aversion parameter [alpha] = 4. The characteristics of equity prices and interest rates on equilibrium trajectories are shown in table 4, and the characteristics of the consumption and portfolio strategies in table 5. A less detailed description is given in appendix table C1 for an economy with a smaller variation in cohort sizes (N, n) = (79, 69), calibrated to the sizes of the cohorts born over the periods 1945-64 and 1965-84, for three different parameters of risk aversion ([alpha] = 2, 4, 6).
As explained in appendix B, in order to find a Markov equilibrium, an endogenous state variable--the portfolio income that the middle-aged bring over from their youth--needs to be added to the exogenous state (k, s), where k is the population pyramid state (k = 1, 2, depending on whether the period is even or odd), and s is one of the four income-dividend shocks. Along every path, each pyramid-shock state (k, s) will occur infinitely often: in table 4 the standard deviations of the prices (the numbers in parentheses) about their means (the numbers not in parentheses) are given for each pyramid-shock state (k, s), averaged over all paths. An interesting feature of the equilibrium trajectories is that the standard deviations are very small, meaning that prices essentially depend only on the exogenous state (k, s). Thus the average values of the equity price ([q.sup.e]) and of the interest rate ([r.sup.an]) in the different states (k, s) give a rather precise description of the prices on the equilibrium trajectories. Table 4 also shows the price-dividend ratio for each state, which we have divided by 2 to make it comparable with the more familiar PE ratio, commonly used for evaluating the level of prices on the stock market.
A new variable that enters when uncertainty is introduced is the equity premium, namely, the amount by which the expected return on equity exceeds the return on bonds. The (annualized) equity premium is calculated on a trajectory as
r[p.sup.an] = average([r.sup.an.sub.e] - [r.sup.an]),
where
[r.sup.an.sub.e,t] = [([[q.sup.e.sub.t+1] + [D.sub.t+1]]/[q.sup.e.sub.t]).sup.1/20] - 1
is the (annualized) ex post rate of return on equity at date t. The ex ante equity premium is thus defined as the mean ex post equity premium and is given in table 4. The high variance of the ex post equity premium, even for a given pyramid-shock state (k, s), is natural, since the realized equity premium is large when a favorable state follows state s, and is small when an unfavorable state follows state s.
As is well known, the ex ante risk premia predicted by standard rational expectations models are significantly smaller than those obtained ex post from the data, at least for the United States. Several approaches have been proposed to obtain models with larger risk premia. One is to take into account the fact that agents face individual risks, which make their consumption significantly more variable than aggregate consumption. We cannot take into account individual risks without unduly complicating the model; to compensate, we have been generous in the calibration with the aggregate risk. Other solutions involve entering as constraints some observed deviations of the behavior of agents from that predicted by the model. One prediction of the model is that agents make use of all the available instruments to redistribute income and share risks. However, even though the proportion of U.S. households investing in the stock market has increased significantly over the last fifty years, (33) it still remains less than 50 percent. To take this into account, we solve for the equilibrium under the restriction that 50 percent of the agents in any cohort do not trade on the equity market and restrict their financial transactions to the bond market (case B in tables 4 and 5).
An alternative approach, recently proposed by Constantinides, Donaldson, and Mehra, is to impose a borrowing constraint on the young: (34) as shown in table 5, without such a constraint, the young typically borrow and use much of the proceeds to invest in the equity market, to take advantage of the equity premium. As Constantinides, Donaldson, and Mehra argue, this is not especially realistic. Although young agents can and do borrow significantly to buy houses (which serve as collateral), they do not typically borrow to invest in the stock market. The simplest way of preventing the young from taking leveraged positions on the equity market is to impose a borrowing constraint. Such a constraint on the young decreases the demand for the risky security and tends to increase the risk premium. However, in the simple model that we study, preventing every young agent from borrowing closes the bond market, and the interest rate is no longer well defined. To avoid this, while studying the effect on prices of reducing the demand for equity by the young, we solve for the equilibrium assuming that 90 percent of the young face borrowing constraints and the remaining 10 percent are unconstrained (case C in tables 4 and 5). In addition to the intrinsic interest and potentially greater realism of these two cases with restricted participation, they are also useful for checking the robustness of...
NOTE: All illustrations and photos
have been removed from this article.
More articles from Brookings Papers on Economic Activity
What does the public know about economic policy, and how does it know ..., March 22, 2004
Comments and discussion.(Demography and the Long-Run Predictability of..., March 22, 2004
Looking for additional articles?
Search our database of over 3 million articles.
Looking for more in-depth information on this industry?
Search our complete database of Industry & Market reports by text, subject, publication
name or publication date.
About Goliath
Whether you're looking for sales prospects, competitive information, company
analysis or best practices in managing your organization,
Goliath can help you meet your business needs.
Our extensive business information databases empower business
professionals with both the breadth and depth of credible,
authoritative information they need to support their business
goals. Whether it be strategic planning, sales prospecting,
company research or defining management best practices -
Goliath is your leading source for accurate information.
|
%20%20&position=1)