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1 Institutions, Models and Empirics

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



12



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



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