1 Institutions, Models and Empirics
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1.1 Institutions, Models and Empirics
3
Whatever institutional form is chosen, one must take account of labor income in
wealth accumulation and portfolio models. The suitable design and management of
portfolios that guarantee a sufficient retirement income for households with labor
income has also been at the center of recent debates on pension funds. Yet, portfolio
studies that include labor income are still rare. In their seminal work, Campbell and
Viceira (2002) devote two chapters to this issue, and this serves as an important
starting point for our study. In this direction we extend the modeling approach
to include not only asset income but also labor income in the dynamic decision
problem of asset accumulation and allocation.
In order to model the heterogeneity between generations, many researchers have
suggested overlapping generations models working with two periods4 : the first
period involves active labor market participation; the later period is for retirement.
Three generations models have also been used (see, e.g., Eggertsson and Mehrotra
2014). Those actually lead to life cycle models, which we will leave aside. However,
we could address those issues in the context of a dynamic decision approach in a
further step. For details of an overlapping generations model and its implication for
fund management, see Campbell and Viceira (2002, Chap. 7).
We follow here a procedure by Blanchard (1985) to convert an overlapping
generations model into a continuous time model. We stick to a continuous time
approach to avoid discrete time, two or three period models. This requires us to deal
with different time horizons at the different stages of agents’ lives: the time period
with primarily labor income and the period with primarily retirement income. To
deal with this problem of two time horizons, we employ a model with different
discount rates for the two periods as in Blanchard (1985).
We do not refer here to a saving and portfolio model for an individual investor.
If we had appropriate data for individual investors, we could also pursue, with our
method, an individual decision model, or life cycle model for an individual agent.
But this is not attempted in the first step of the research undertaken in this project.
We follow, to some extent, Merton (1971, 1973), Campbell and Viceira (2002,
Chap. 6) and Viciera (2001), but we depart from their assumptions that the expected
equity premium is a constant. In our model the equity premium will be time varying
and we also assume a time dependent risk free interest rate.
We start with an econometric harmonic fit of asset and labor incomes by
using spectral analysis. We use a Fourier transformation to decompose a function
(represented by time series data) into low frequency movements and residuals. We
employ actual time series data and estimate time variation of the data using the
harmonic fitting technique.5 We use US data, but financial and income data from
other countries could be employed as well. We employ low frequency movements
in asset and labor income in our dynamic decision approach to solve various model
specifications with Dynamic Programming.
4
5
See Kotlikoff and Burns (2005).
See Hsiao and Semmler (2009).
4
1 Introduction
1.2 Dynamic Programming as Solution Method
The Dynamic Programming (DP) algorithm has been employed in many areas of
economics and finance,6 using DP in our context has several advantages over other
methods. DP not only solves the dynamic decision model globally, it also lends itself
to extensions in which new market information becomes available. DP also helps to
simultaneously study the issue of accumulation and allocation of financial funds.
Other authors have already demonstrated the usefulness of dynamic programming
for dynamic decision making.7 The use of DP invokes the discussion on forwardlooking behavior of economic agents. This behavior accords with an individual who
invests current funds for some expected future outcome but, because of the long
time horizon, actually realized outcomes are uncertain. Similar forward-looking
decisions problems are present in the traditional static portfolio model. Regarding
the traditional model, Markowitz (2010) makes the following statement:
Judgment plays an essential role in the proper application of risk-return analysis for
individual and institutional portfolios. For example, the estimates of mean, variance, and
covariance of a mean variance analysis should be forward-looking rather than purely
historical. (Markowitz 2010: 7)
It is worth stressing that the use of DP to model forward-looking behavior of
individuals, households, and institutions, requires some methodological discussions.
The typical assumptions and postulates of DP are as follows:
• Marginal conditions, such as describing the balance between current costs and
future benefits, are instantaneously established (for example the Euler equation
in consumption and saving decisions)
• Information sets are a priori given for long time horizons, freely available and
fully used
• The decision maker can make smooth and continuous adjustments as the
environment changes
• The decisions are made under no income, liquidity, credit or other market
constraint
• The spillovers, externalities and contagion effects are negligible
• There are negligible macroeconomic feedback effects or propagation effects that
can significantly disturb the intertemporal arbitrage decision
• The decisions—responding to the realization of the state variables—can then be
made in nonlinear form at grid points of the state variables
6
See Grüne and Semmler (2004).
The many examples include dynamic choices over savings, occupation and job search, choices on
education and skills, investment in housing, health care choices, and insurance decisions. See Hall
(2010) and also the many examples in Grüne and Semmler (2004).
7
1.3 Previous Work
5
The use of the Dynamic Programming method thus presumes that none of the above
problems will significantly disturb dynamic decision making. Though in principle
one could claim that dynamic and forward-looking decision making is involved
in human behavior, particularly in economic decision making, but one should be
careful assuming away the above mentioned issues.
In our treatment of savings and asset allocation we will pay explicit attention to
the presumptions of the DP solution methodology. We will show that DP still gives
helpful answers to interesting questions of savings and portfolio decisions, such as
the role of risk aversion, discounting future outcomes, the role of initial condition on
wealth, constraints on the state and decision variables, and the evolution of income
and wealth arising from such decisions. This set up allows one also to study issues
of investor heterogeneity with respect to risk aversion, discount rates, initial wealth,
informational constraints and time horizons lengths on the paths of wealth and
inequality.8
1.3 Previous Work
Studying dynamic decision making in finance started with Merton (1971, 1973,
1990). More recently, seminal work has been undertaken to model dynamic
consumption and portfolio decisions. Originally, Merton (1971, 1973) provided a
general intertemporal framework for studying the decision problem of a long-term
investor who not only has to decide about savings but also of how to allocate
funds to different assets such as equity, bonds and cash. It is now increasingly
recognized that the static mean-variance framework of Markowitz needs to be
improved upon by extending it to a dynamic context that takes into account new
investment opportunities, different initial conditions, different risk aversion among
investors, different time horizons, and so on.
Much effort has been put forth to show that, under certain restrictive conditions,
the dynamic decision problem is the same as the static decision problem.9 Yet,
it is now well recognized that a more general dynamic framework is preferable.
However, there are many difficulties involved in obtaining closed-form solutions for
more general models. One must therefore employ numerical solution techniques to
solve for the consumption or saving paths and the dynamic asset allocation problem.
Important work on those issues has been presented by Campbell and Viceira
(1999, 2002). They use the assumption of log-normal distributions in consumption
and asset prices with the implication that the optimal consumption-wealth ratio—
or, equivalently, the saving-wealth ratio—does not vary too much. Using log-linear
8
A recent modification of the DP algorithm, making it useable for more complex decision making
problems, allows us to study those issues on a finite time horizon with informationally constrained
agents, (see Grüne et al. 2015 and Chap. 6 of this book).
9
See Campbell and Viceira (2002, Chap. 2).
6
1 Introduction
expansion of the consumption-wealth ratio around the mean, they show a link
between the myopic static decision problem and the dynamic decision problem (see
Campbell and Viceira, 2002, Chaps. 3–5). They solve a simplified model with time
varying bond returns but with a constant expected equity premium.10 In general,
models with time varying returns are difficult to solve analytically, and linearization
techniques as a solution method may not be quite accurate.11 This is likely to be the
case if returns and consumption-wealth ratios are too variable.12
If there is a predictable structure in equity (and bond) returns, and thus there
are time varying expected returns, then the dynamic decisions with respect to
consumption and portfolio weights need to respond to the time varying expected
returns.13 In some of our model variants we approximate the time varying expected
asset returns by the low frequency component of the returns.14 A further discussion
of these empirical issues is undertaken in Chap. 2.
Given such time varying returns, a buy and hold strategy for portfolio decisions
is surely not sufficient. Dynamic saving decisions, as well as a dynamic rebalancing
of a portfolio (following low frequency movements in investment opportunities
and returns), is needed in order to capture persistent changes in returns and to
avoid wealth and welfare losses. DP can be usefully applied here. It works with
flexible grid size, operates globally, and can solve for any point in the state space
simultaneously for both the consumption or saving decisions, as well as for time
varying portfolio weights. As our more solution technique shows, the consumptionwealth ratio can vary greatly but our solution remains sufficiently correct.15
10
See Campbell and Viceira (2002, Chap. 3).
For example, the log-linear expansion about the equilibrium consumption-wealth ratio is as
undertaken by Campbell and Viceira (2002, Chaps. 2–4).
12
In order to obtain an approximate solution of the model, Campbell and Viceira (2002:51)
presume that the consumption-wealth ratio is “not too” variable. However, they show that their
procedure loses serious accuracy with a parameter of risk aversion > 1. Moreover, Campbell
and Viceira use a model with a constant interest rate, see also Campbell (1993), Campbell and
Viceira (1999) and Campbell and Koo (1997). On the issue of the accuracy of first and second
order approximations in dynamic decision models, see Grüne and Semmler (2007).
13
There is much empirical evidence on time varying expected returns. For earlier work see
Campbell and Shiller (1989); for recent surveys, see Campbell and Viceira (1999) and Cochrane
(2006).
14
Recent theoretical research on asset pricing using loss aversion theory can give a sufficient
motivation for such an assumption on time varying expected asset returns following a low
frequency movement, for details see Grüne and Semmler (2008).
15
In Becker et al. (2007) the out-of-steady-state dynamics of second order approximations and
dynamic programming are compared. The errors from dynamic programming are much smaller do
not depend on the distance to the steady state.
11
1.4 Outline and Results
7
1.4 Outline and Results
As mentioned, dynamic decision making concerning saving and asset allocation
started with Merton, and has since been extensively developed. The most recent
seminal research work is by Campbell and Viceira (2002). Whereas they linearize,
we work with nonlinear modeling of the decision problem and deal with both saving
and asset allocation decisions.16 We here mostly deal with preferences characterized
by risk aversion.
More complex forms of preferences, such as loss aversion arising from prospect
theory, are to a great extent left aside.17 Dealing with loss aversion in preferences
requires a more complex solution method.18 The result obtained here on the application of Dynamic Programming to dynamic savings and portfolio decisions are
already significantly different from static portfolio theory. This will be demonstrated
in the subsequent chapters.
The book is organized as follows. Since we are mainly concerned with low
frequency decisions, Chap. 2 discusses the empirical results of low frequency studies
of asset returns. Among others, we show how one can obtain low frequency
periodic returns from asset price data using some harmonic regressions. Chapter 3
introduces portfolio models that take into account constraints such as social, ethical
or environmental constraints or restrictions on risk taking when modeling portfolio
decisions. Thus, the problem addressed here will be what are the best decisions
under such constraints.
Chapter 4 illustrates the dynamic decision problem of savings and asset allocation for artificial data in a model, first with one asset and constant returns and,
secondly, with two assets and periodic returns. We employ here also a stochastic
model version with mean reversion in returns. In Chap. 5 we pursue the same
modeling strategy by using actual data sets with the estimated low frequency
movements of asset returns. As a result, we use time varying returns that represent
stylized facts of low frequency movements in asset returns, and show that also
wealth accumulation and welfare may move cyclically. Chapter 6 introduces labor
income in savings and asset allocation decisions. We here again estimate the
periodic components in both asset and labor income in order to make proper savings
and portfolio decisions.
In all three Chaps. 4–6, we employ dynamic programming to study the impact of
the variation of risk aversion, asset returns and time horizon length on the paths of
savings (viz. consumption), asset allocation, wealth accumulation and welfare. We
study how the heterogeneity of investors can lead to cyclical movements in wealth
accumulation, and show under what conditions wealth is growing or shrinking,
which will add to an explanation of evolving wealth disparities.
16
Markowitz (2010) summarizes numerous older and recent approaches to portfolio decisions,
although he deals less with the issue of dynamic consumption or savings decisions.
17
For the application to loss aversion and portfolio theory, see Grüne and Semmler (2008).
18
For details of such algorithms, see Grüne and Semmler (2008).
8
1 Introduction
Since the previous chapters are in continuous time, Chap. 7 discusses the problem
of how to convert continuous time models into discrete time. Though it is a generic
problem for all asset price and portfolio models, it will be exemplified with respect
to short term interest rates. Building on those results, Chap. 8 discusses investment
decisions by allowing for inflation risk. Inflation risk will be introduced in asset
accumulation and allocation decisions. Savings and asset allocation decisions are
long-horizon decisions and expected inflation rates have to be properly taken into
account in dynamic decision making. We then consider inflation-adjusted asset
returns, such as inflation-indexed bonds.
The appendix provides a sketch of the dynamic programming algorithm, and
explains how it has been applied to dynamic decision problems.
Chapter 2
Forecasting and Low Frequency Movements
of Asset Returns
2.1 Introduction
In this chapter we provide an overview on forecasting asset returns and low
frequency movements in asset returns. Saving and asset allocation decision, usually
focus on low frequency movements in asset returns and how they are expected
to behave in the future. Thus, the prevailing consensus in the context of portfolio
theory, is of the view that the estimates of the mean, variance and covariance
should be forward looking rather than purely historically.1 Therefore, certain ways
of forecasting of asset returns is important for dynamic decision making. We here
first survey the empirical literature on forecasting asset returns, with an emphasis
on forecasting returns on aggregate stock price indices. Then we will review and
evaluate the literature that constructs low frequency movements in asset returns by
using harmonic estimations. This survey will help us to assess empirical results on
the measurements and time dependency of asset returns to be used in our dynamic
programming method in later chapters.
2.2 Limits on Forecasting Asset Returns
At time t the return on a stock price index is defined over period t and t C 1, however
at time t the return on a stock price index is not known but it can be forecasted.
Forecasting procedures on stock prices are contentious because of the suggested
processes that a stock price may follow. In this connection, forecasting stock returns
is a highly debatable issue and remains an open ended question. Fama (1965) argues
1
This is a point that also for example Markowitz (2010) stresses.
© Springer-Verlag Berlin Heidelberg 2016
C. Chiarella et al., Sustainable Asset Accumulation and Dynamic Portfolio
Decisions, Dynamic Modeling and Econometrics in Economics and Finance 18,
DOI 10.1007/978-3-662-49229-1_2
9
10
2 Forecasting and Low Frequency Movements of Asset Returns
that stock prices follow a random walk and he emphasizes that other methods of
describing and predicting stock prices are not credible. This is claimed based on
the argument that the random walk theory implies that successive price changes in
stocks are independent in an efficient market. Furthermore, Fama (1965) points out
that the investor has no knowledge of any analyses under which standard statistical
tools provide evidence of important dependence in series of successive stock returns.
The random walk hypothesis is controversial and is considered not to reflect
the dynamics of actual stock price data. In fact, Shiller (2014), by using CAPE
(cyclically adjusted price-earning ratios) contests that there is no forecastability
of asset returns. Also, for example, using a variance ratio test, Lo and MacKinlay
(1988) find that stock prices do not follow a random walk process. Instead they find
statistically significant positive serial correlation for stock returns over different time
frequencies and their results are robust to heteroscedasticity. In addition, the positive
correlation is significant for their entire sample period and all their subperiods.
Although the random walk hypothesis can be rejected, Lo and MacKinlay’s
(1988) evaluation procedure does not provide an alternative plausible framework
to characterize the data and this reinforces implicitly the difficulty associated with
forecasting stock returns. Another view is presented by Amini et al. (2010) who
note that large stock price changes exhibit reversals, however smaller price changes
are characterized by a tendency for price trends to continue and hence over a short
horizon stock returns can be predicted. These findings show evidence of short term
predictability of stock returns which is in contrast to the random walk hypothesis.
This is explored in the context of stock prices in the London Stock exchange and
not for the Standard and Poors (S&P) 500 index returns which are usually used.
Nevertheless, in recent times some research progresses in evaluating stock return
forecastability using univariate frameworks. Similarly, an important view in the
empirical work on time varying asset return is derived from the work of Campbell
and Shiller (1989) who transform an intertemporal asset pricing equation which in
turn relates an asset return and growth rate of a dividend payment to a dividend-price
ratio as follows:
" k Â
#
Ãi
Ãk
Â
X
1
1
Pt D Et
DtCi C Et
PtCk
(2.1)
1Cı
1Cı
TD1
with ı the discount rate, Pt the asset price, Dt dividend. By taking logs one can turn
Eq. (2.1) into the dividend-price ratio
dt
p t D Et
1
X
ˇ t .rtCi
dO
(2.2)
tD1
whereby dO is the growth rate of dividend payment and rt the asset return. Presume
for xt D dt pt then xt can usefully be employed as predictor variable. A simple
2.2 Limits on Forecasting Asset Returns
11
forecasting regression can be as follows: If we find j b j> 0 in
rtC1 D a C bxt C "tC1 ;
(2.3)
then we can say that Et .rtC1 / varies over time. The forecasting variable xt typically
may be correlated with a suggested business cycle variable. Along this line, using
a value weighted New York Stock Exchange (NYSE) composite stock price on
an annual frequency, Cochrane (2006) finds that excess returns/equity premium
(value weighted New York Stock Exchange less treasury bill rate) and the real
dividend growth on the value weighted NYSE are correlated. As a result, using an
alternative aggregate stock price measure, equity returns are predictable but exhibit
low predictability. Other studies evaluate predictability of the equity premium using
alternative frameworks. For example, Campbell and Viceira (1999) and Campbell
and Viceira (2002, Chap. 4) argue that predicted returns and thus new investment
opportunities are better captured when the dividend-price ratio is introduced into a
Vector Autoregression (VAR) of the following form:
2
3
r0;t
zt D 4rt r0;t 5
st
and
ztC1 D
0
C
1 zt
C vtC1
(2.4)
with st D dt pt the dividend-price ratio. Here, too the variable st is the exogenous
predictor variable, beside the risk-free interest rate r0;t and the equity premium
rt r0;t , in a VAR regression. Lettau and Ludvigson (2001, 2005) have produced
empirical evidence that the consumption-wealth ratio is a preferable empirical
predictor variable for time varying asset returns. Furthermore, they argue that
household consumption behavior aims at smoothing consumption over time and
is a good predictor of future asset returns and they use a regression of the following
form:
b
e
rtC1
D ˛ C ˇ Cayt C "tC1
b
(2.5)
with Cayt D Ct ˇOa at ˇOy yt , the latter being a regressor from log of consumption,
Ct , wealth, at , and labor income, yt .
Nevertheless, in an extensive analysis, Welch and Goyal (2008) examine the
predictability of stock returns, more specifically the predictability of the equity
premium (S&P 500 index return in excess of a risk free rate). Their analysis is
comprehensive from a variable perspective, horizon perspective, time perspective
and they bring variables up to date with the time at which they conduct their
analysis. In examining the predictability of stock returns some authors use predictive
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2 Forecasting and Low Frequency Movements of Asset Returns
regression models with predictors as follows: the dividend price ratio, dividend
yield, earnings price ratio, dividend payout ratio, stock variance, cross-sectional
premium, book-to-market ratio, net equity expansion, percent equity issuing, a set
of interest rate related independent variables, inflation and investment to capital
ratio. Further details on each of these variables and their relevant specifications are
provided by Welch and Goyal (2008). They conduct a systematic investigation of the
in-sample and out-of-sample performance of predictive regression models for the
equity premium. In general, Welch and Goyal (2008) find that most of the models
they examine, seem unstable or are even spurious. Moreover, they cannot identify
any model across all time frequencies that systematically has good in-sample and
out-of-sample performance. Based on these findings, they note that none of the
models in their analysis provide a valid basis for forecasting stock returns and none
of these models can serve as a basis for reliable investment advice. Similarly, Welch
and Goyal (2008) point out that no existing research has found a meaningful and
robust variable for forecasting stock returns.
Along this line and using a predictive regression framework, Zhou (2010) also
finds that stock returns (equity premium) are difficult to forecast in a model where
one of the ten predictors are the dividend-price ratio, earnings-price ratio, book-tomarket, T-bill rate, default yield spread, term spread, net-equity issuance, inflation,
long-term return, or stock variance. Ferreira and Santa-Clara (2011) use the sum of
the parts method under which they forecast separately the dividend-price ratio, the
earnings growth and the price-earnings growth components of stock market returns.
Their procedure exploits different time series properties of the components and this
results in better forecasting performance in comparison to predictive regression
models for forecasting stock market returns. For example, using monthly return
data they find that predictive regression models do not provide good forecasting
ability for stock market returns. Furthermore, using the simplest version of the sum
of the parts method, improves on the traditional predictive regressions however the
improvement is exhibited by an out-of-sample R2 D 0:0132. On the other hand,
using annual return data, they find substantially higher predictive ability because
they find an R2 D 0:132. Although Ferreira and Santa-Clara’s (2011) method
exhibits better forecasting performance, they note that predicting stock market
returns is inconclusive and remains an open question.
Updating from Welch and Goyal’s (2008) analysis, Rapach and Zhou (2013)
use 14 popular economic variables as potential forecasters of stock market returns.
The variables that Rapach and Zhou (2013) use are as follows: log dividendprice ratio, log dividend yield, log earnings-price ratio, log dividend-payout ratio,
stock variance, book-to-market ratio, net-equity expansion, T-bill rate, long-term
yield, long-term return, term spread, default yield spread, default return spread
and inflation. Using a multiple predictive regression that includes all 14 popular
economic variables, they find that this procedure has poor forecasting ability for the
excess stock market returns. Furthermore, Rapach and Zhou (2013) point out that
the best forecasting models can only explain a relatively small part of stock returns.
On this basis, they note that a lot of emphasis is on popular economic variables
as predictors whereas other variables such as options, micro structure measures of
2.2 Limits on Forecasting Asset Returns
13
liquidity and institutional trading variables such as trading volumes and money flows
for mutual and hedge funds are potential forecasters of stock returns.
Based on the outlined research, forecasting stock returns is a difficult task
and still remains an open question. Nevertheless, other problematic aspects are
uncovered as it is shown by Lettau and Nieuwerburgh (2008) that many of the
financial ratios used in linear regressions, result in incorrect or spurious regressions.
A number of causes have been uncovered why it is quite difficult to find time varying
expected asset returns through the standard linear prediction methods. The well
known causes are2 :
1. Financial ratios are extremely persistent and the possibility of the existence of a
unit root can often not be excluded.3
2. As already indicated above, financial variables have a poor out-of-sampleforecasting power, see Bossaert and Hillian (1999), but also Campbell and
Thompson (2008) for a different interpretation.
3. Related to the poor out-of-sample performance is the evidence of a significant
instability in forecasting relationship of asset returns and financial ratios as
discussed above. Rolling regression for example exhibit huge instability in the
regression, see Lettau and Nieuwerburgh (2008).
4. In addition to the above mentioned spurious regressions the asset returns have to
be as persistent as the financial ratios in order to obtain stable regression, Lettau
and Nieuwerburgh (2008, p. 2) conclude that researchers have to identify slow
moving factors that are primary determinants of risky assets.
What is therefore reasonable to presume is that underlying trends for the financial
ratios exhibit structural breaks, (one or several ones) or are shifting over time. This
may come from:
• increase (or slow down) of growth, for example from a change in permanent
technological innovations or
• changes of expected returns due to improved risk sharing or reduced or increasing
risk perception, changes in stock market participation, changes in tax rates or
changes in macroeconomic variables volatility.4
Further, we want to mention univariate mean reversion models that build on
stochastic differential equations with a Wiener process. For the interest rate process
for example a univariate mean reversion process typically reads as follows5 :
drt D Ä.Â
2
r/t dt C dW
(2.6)
For details see Lettau and Nieuwerburgh (2008).
See Ang and Bekaert (2007).
f
4
In the latter case, the decrease in the risk premium might arise from E.rti / rt D
. c/, with
. c/ the volatility of consumption growth.
5
For details of using such a Brownian motion for an asset return process, for example for equity
returns or interest rates, see Chap. 4. Moreover, Brownian motions are estimated in Chap. 7.
3