3 Business Cycles, Asset Returns and Labor Income
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6 Asset Accumulation and Portfolio Decisions with Time Varying Asset Returns. . .
facts concerning asset returns and employment and labor income.12 As to the
financial market it holds as a stylized fact that firms’ cash flows are highly volatile
but usually procyclical. Also nominal interest rates are procyclical. The risk premia
for bonds of different maturity are rising with the maturity of bonds (upward sloping
yield curve in the expansion, because of expected high interest rates) and shrinking
at the onset of a recession (expected lower interest rates). The risk premium on
private bonds (private debt) and the spread between yields of corporate and Treasury
bills (with 6 months maturity) tend to decline during expansions and increase during
recessions.13 This is reflecting the time varying default risk. Stock prices and returns
tend to exhibit a risk premium exceeding the returns from bonds, but here volatility
is higher. Although overall the stock prices are positively related to expected growth
rates of the GDP in the long run the volatility of the stock price is often making the
strict relationship quite unclear in the short run. Often the stock prices anticipate
growth rate of output, but the impact of other variables on financial market volatility
often blurs the relationship between stock price and output growth. Thus, risky
returns from stock markets rarely indicate a clear cut correlation with some measure
of the business cycle.
On the other hand, labor market variables are more clearly related to some
measure of business cycles, for example output.14 Since capacity utilization is highly
correlated with the business cycle, so is employment and, lagging slightly, capacity
utilization (by roughly one quarter). Employment coincides with the fluctuation of
output, but it is less volatile than output. Most changes in total employment (total
hours worked) corresponds less to the average weekly hours worked, but to the
movement of the labor force in and out of employment. Although the real wages
are usually only slightly procyclically, total labor income, as it is used here in our
study is significantly procyclically. This results from the movement in and out of
employment. We thought the latter should be the appropriate measure for labor
income in the context of an asset accumulation model. Following our approach
in Semmler and Hsiao (2011) and Chap. 4, we want to stylize the returns and
labor income as low frequency movements.15 The data sources and the estimation
of the low frequency components of the risky returns and risk free returns are
reported in Hsiao and Semmler (2009). Here we report the estimated coefficients.
As demonstrated before, the mean of the returns for the short term interest rate, Rf ;t
and the equity return, Re;t are time dependent and can be formulated using the results
12
For the subsequent summary, see Altug and Labadie (2008), Stock and Watson (1999) and
Kauermann et al. (2011).
13
For details and empirical estimates on default spread, see Semmler (2011).
14
For a detailed study of the comovements of output and labor market variables, using the HPfilter, the BP-filter and Penalized Splines, see Kauermann et al. (2011), see also Stock and Watson
(1999) who use the BP filter. As in the other examples, the cyclical components of the labor market
variables come out more distinctively for the spline-filter than from the HP- and BP-filters.
15
For an example, see the Appendix.
6.3 Business Cycles, Asset Returns and Labor Income
105
in Hsiao and Semmler (2009).16 For the risk free rate we have, setting t0 D 0:
Rf ;t D 0:0021.t/ C 0:0521
2
X
C
ai sin
2
.t/ C bi cos
2
i
iD1
.t/
Á
;
(6.5)
i
with i and coefficients ai ; bi are given (Semmler and Hsiao 2011). Taking also
the estimated first two components of the equity returns—see Hsiao and Semmler
(2009)—the low frequency movement of equity returns can be represented by:
Re;t D
C
0:0046.t/ C 0:1259
2
X
ai sin
2
.t/ C bi cos
2
i
iD1
.t/
;
(6.6)
i
For the empirical estimate of the low frequency components of labor income we also
used monthly data. The data source for the labor income is the Bureau of Economic
Analysis (2008), Labor income is measured by wage and salary disbursement
1980.08–2008.06, seasonally adjusted. The nominal series was converted to real
labor income by using the CPI. A measure of how relevant it might be to include
labor income into a portfolio and asset accumulation model, is indicated by the share
of labor income (measured as labor and salary income) in total income. The share of
labor income in total income is roughly 70 % whereas the share form capital income
(interest, dividend and rent) is roughly 30 %, all before tax. We here followed the
procedure to estimate the low frequency components of labor income as reported in
Hsiao and Semmler (2009) and Chap. 2. In the estimation we could capture at least
six oscillations (periods of 25, 8, 6, 4, 3 and 2 years). The first two periods, the 25
and 8 years’ period estimation appeared to us as sufficient and significant. The result
of the coefficient estimates are as follows:
Lt D 92:8.t/ C 1402:3
C
2
X
iD1
16
ai sin
2
.t/ C bi cos
i
2
.t/
Á
;
(6.7)
i
Again we use here the well established fact that long run investors would respond to new
investment opportunities given by the time path of the (risk free) interest rate and the equity return.
Of course, a myopic risk averse investor, investing in a static Markowitz portfolio, would hold a
fixed fraction of bonds in his/her portfolio.
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6 Asset Accumulation and Portfolio Decisions with Time Varying Asset Returns. . .
Table 6.1 Coefficient estimates of the low frequency components of labor income
a1
16.244
a2
2.911
b1
101.879
b2
41.692
1
305.00
2
101.67
Hereby again, the first two terms of Lt represent the trend component of labor
income and the remaining ones the cyclical components. For details of the coefficients, see Table 6.1.
6.4 Dynamic Decisions on Asset Accumulation
As one can observe from Eqs. (6.3) and (6.4) if the probability of being employed
goes to zero, only the second part of the Euler equation (6.4) tends to hold. In our
set up we do consider labor income and we might treat investors that come closer to
retirement as characterized by a shorter expected life time and use 1= d as expected
life time. Yet, the larger d is (and the shorter the expected life time is) the smaller
d
d
the term .1
/ in Eq. (6.4) will be and thus the discount factor .1
/ will be
smaller and the discount rate larger. Since we are here considering wealth as being
affected by labor income, we can treat the investor looking at both labor income as
well retirement income and we study the investor’s time horizon effect by varying
d
the actual discount factor .1
/ : If d goes to zero the usual discount rate is
approached. Next we are employing our dynamic programming procedure again
to explore the effects of the investor’s time horizon on consumption decisions, asset
allocation, on the fate of wealth and the welfare of the investor.17 In studying this we
employ the above low frequency movements for equity and short term bond returns
of Eqs. (6.5) and (6.6). Yet we will vary the discount rates in order to study the
effect of the different time horizon across investors. Note that we here again allow
for time varying expected returns given by our estimated low frequency movements
of returns of Eqs. (6.5) and (6.6). We also allow for time varying labor income as
estimated by Eq. (6.7). The parameter of risk aversion is taken as D 0:8 since
this case gives us a growing wealth. Again we can refer here to the fact that long
run investors, with a long way to go to retirement, would exhibit a small discount
rate. As in our previous models we assume that the investor would respond to new
investment opportunities given by the time path of the (risk free) interest rate, the
equity return and labor income. On the other hand, a myopic investor, with an
infinite discount rate would behave like a static Markowitz portfolio investor, and
17
For details of a dynamic programming procedure applicable to the problem at hand, see Semmler
and Hsiao (2011).
6.4 Dynamic Decisions on Asset Accumulation
107
would hold a fixed fraction of equity and risk free assets in his/her portfolio. We will
again also explore the effect of the varying discount rate on consumption, Ct ; the
weight for the equity choice, ˛t ; the build up and the fate of wealth and the welfare
of the investor. We also want to compare our results from using DP with the results
obtained by Campbell and Viceira (2002, Chap. 6). In our example below here we
use a portfolio of two assets, equity and short term bonds. The return given by the
short term interest rate and the condition that there is also an exogenous stream of
labor income. Note that, in contrast to Campbell and Viceira (2002, Chap. 6) and
Viciera (2001), we allow for time varying expected returns given by our estimated
low frequency movements of returns, using the harmonic estimations of Eqs. (6.5)
and (6.6). Our DP problem can be stated as
Z
1
max
fC;˛g
e
ıt
0
P
s:t: W.t/
D ˛t Re;t Wt C .1
U.Ct /dt
˛t / Rf ;t Wt C Lt
xP .t/ D 1:
Ct
(6.8)
(6.9)
with Re;t , Rf ;t and Lt following a harmonic oscillations as defined above. In our
above DP problem as formulated above we have again taken instead of the control
Ct the control ct D Ct =Wt with 0 < ct < 1 and Ct D ct Wt : Note that we have
taken here again, in the DP algorithm, the time index xP .t/ D 1; as running index
which introduces a new dimension along which the consumption and portfolio
decisions. By doing so, we thus make Re;t , Rf ;t and Lt dependent on time. This
way we can observe what happens over time and along the state space, which is
here represented by wealth.18 But note that here in our case, in contrast to the static
portfolio of Markowitz type, the build-up of wealth is not only affected by the total
return (and the relative asset allocation) but also by the consumption and thus saving
rate. Moreover labor income is also affecting those two decisions. We start with our
results for D 0:8: As we can observe from Fig. 6.1, given our cyclically moving
returns for equity and risk free interest rate and starting with almost zero wealth,
18
Note that our above formulation is similar to the one by Blanchard (1985) who has introduced
instead of the term Re;t , a fixed insurance pay-out term received by the agents if he/she dies,
determined by the probability to die. But note that the insurance company has to get the payout term
in the Blanchard model from the asset market, which, in our case, it can pay out as risky return,
after deducting some transaction fee. Thus, our model can be read as the Blanchard model, but we
have time depending risk free and risky returns. Yet in both cases, in the Blanchard case as well as
in our case above, the discount rate, ı; needs to be viewed as being modified by the probability to
die, which is inversely related to the expected life time of the agent. If the probability to die goes
to zero, then one approaches again the usual infinite horizon optimization model.
108
6 Asset Accumulation and Portfolio Decisions with Time Varying Asset Returns. . .
65.000
52.000
39.000
t
26.000
13.000
0.000
5000.000
6300.000
7600.000
8900.000
10200.000
11500.000
W
Fig. 6.1 Long swings in asset build up for
D 0:8 and ı D 0:03
the wealth will be build up and then moves cyclically, as well as consumption ct
and ˛t : Yet, there is not only a cyclical asset allocation, shown through the cyclical
movement along the vertical axis, but there is also a trend in wealth build-up. As
the wealth is built up to 11,500 (horizontal axis) at the time period t D 52 (vertical
axis), a new cycle of wealth build-up starts. In other words, given our parameters
of low risk aversion and low discount rate, from the higher level of wealth a new
cycle of asset accumulation will commence. So overall, we will see a trend and
a cyclical component in asset accumulation. All together, not only is wealth built
up, but it is built up faster than for the same asset allocation mechanism with no
labor income. The results of a model of asset accumulation without labor income
are shown in Chap. 4. Note, that given the setup of the numerical algorithm here, our
dynamic programming algorithm, we cannot separate the effect of the consumption
(saving) decisions from the asset allocation decisions and their relative impact on
wealth accumulation. This is done in Chap. 4 where the two decision variables are
plotted jointly with wealth accumulation. The value function for the numerical result
depicted in Fig. 6.1 is shown in Fig. 6.2. It shows that the welfare is also moving
cyclically as time goes on, starting with t D 0. Welfare is first small and it is
build up over time and by increasing time the further welfare declines. At least
a new cycle will commence. In the next computation we increased the discount
rate to ı D 0:5: This is equivalent to still having labor income but getting closer
6.4 Dynamic Decisions on Asset Accumulation
V
109
955
950
945
940
935
930
925
920
915
910
5000
6000
7000
8000
W
70
60
9000
50
10000
40
30
11000
12000 0
Fig. 6.2 Value function for
20
10
t
D 0:8 and ı D 0:03
to retirement age. Figure 6.3 now, however, shows, as compared to Fig. 6.1, that
starting with high wealth, close to 11,500, wealth will decline in time. Yet, after
the cycle is completed, this time as shown in Fig. 6.3, the cycle starts at a lower
level and in the next cycle wealth decreases further as depicted in Fig. 6.1. This is
indicated by the fact that the new cyclical movement, with t D 0, the wealth build up
starts at a lower level (roughly 10,500, rather than 11,500) and in fact moves further
to the left, which overall means shrinking of wealth. Thus, in this case of a high
discount rate,19 here we have also a cyclical movement in wealth accumulation but
overall a downward trend in wealth emerges. Figure 6.4 shows the corresponding
welfare function for a discount rate of ı D 0:5 thus of 50 %. It shows that welfare
is much lower on average over the whole cycle of wealth build up. Figure 6.4 also
shows that if the wealth build up occurs (starting from tD0), then—by finishing the
cycle—with the declining wealth the welfare declines as well. Of course, in the next
cycle the variables would move cyclically but there would be a downward trend in
wealth accumulation and welfare. Next, we undertake an exercise where we have
no labor income but only asset income, and thus we assume that the investor is in
the retirement period with no labor income. This case is equivalent to the results of
the study in Semmler and Hsiao (2011) with high discount rate, see also Chap. 5.
We are thus studying here the case when an investor has low expected life time with
no labor income, but only asset income. We here again, use the same parameter
of risk aversion as before, D 0:8; and also the same high discount rate of the
investor, namely ı D 0:5. Here, we solve the model through our DP algorithm, too.
19
Note that we would have also a similar phenomenon of a cyclical wealth movement as well as a
downward trend in wealth accumulation with a high parameter of risk aversion, see Semmler and
Hsiao (2011). Both parameters mainly affect the wealth accumulation through consumption and
saving decisions.
110
6 Asset Accumulation and Portfolio Decisions with Time Varying Asset Returns. . .
65.000
52.000
39.000
t
26.000
13.000
0.000
5000.000
6300.000
7600.000
8900.000
10200.000
11500.000
W
D 0:8 and ı D 0:5
Fig. 6.3 Decline in assets for
64
62
60
58
V
56
54
52
50
5000
6000
7000
8000
W
70
60
8000
50
40
10000
30
11000
12000 0
Fig. 6.4 Value function for
20
t
10
D 0:8 and ı D 0:5
Figure 6.5 shows the results for our new case with no labor income, but a discount
rate of ı D 0:5 which means a discount rate of 50 %, which is quite large.
In Fig. 6.5 we have reported the trajectories of wealth for different initial
conditions of wealth. As we can observe, even if we start with very large wealth,
for example, initial wealth of roughly 350.000 or 290.000, the wealth is always
monotonically dissipating, since there is no labor income to add to income and
6.4 Dynamic Decisions on Asset Accumulation
111
65.000
52.000
39.000
t
26.000
13.000
0.000
0.000
70.000
140.000
210.000
280.000
350.000
W
Fig. 6.5 Shrinkage of wealth with no labor income, for
D 0:8 and ı D 0:5
30
25
20
15
V
10
5
0
0
50
100
150
W
70
60
200
50
40
250
30
300
20
350 0
t
10
Fig. 6.6 Value function with no labor income, for
D 0:8 and ı D 0:5
savings. In fact, as the graphs show all wealth eventually goes to zero, no matter
how large the wealth is. As the graphs also show, it will usually take roughly
a bit more than 40 periods to deplete the wealth. But this is quite a reasonable
result, since the wealth build-up is not fueled and high consumption (resulting from
the high discount rate) at retirement will reduce wealth. As in the case of higher
risk aversion in Chap. 5, the value function for a larger discount rate, see Fig. 6.6,
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6 Asset Accumulation and Portfolio Decisions with Time Varying Asset Returns. . .
becomes flatter. This is an observation that is similar to the results of Chap. 4 for
higher discount rates, where the welfare function also became flatter. We can also
see that the welfare, in whatever cycle wealth moves, will be on average lower.
Note that our three cases might seem extreme in the sense that in the first two cases
there was the labor income and in the third case no labor income. Yet, if we allow
for a lower labor income in the second case and a high discount rate (indicating
a closeness to the retirement period), the wealth would shrink more over time and
finally gives results as shown by Fig. 6.5, where wealth shrinks more rapidly. Of
course, there could exist other cases of a low labor income, a low discount rate, as
well as a small risk aversion parameter, where the wealth could be still increasing.
We have not numerically explored those intermediate cases, but the result should be
obvious from the cases described above.
6.5 Wealth Disparities
The previous two chapters and so far this one, have indicated some features of
wealth accumulation that can result in substantial disparities in wealth for different
types of investors. Disparity of wealth have become a recent research topic, see for
example Piketty (2014), Wolff (2013), Kumhof et al. (2015), and Milanovic (2010),
to name a few. So far, we have not specifically discussed the issue of wealth disparity
in our variants of wealth accumulation models. Now we can do so, since we are also
allowing for labor income. Looking at our dynamic decision model, allowing for
labor income and leveraging, we can write a model such as (6.10) in terms of net
worth
Z 1
max
e ıt U.Ct /dt
fCt ;˛t >1g 0
P
s:t: W.t/
D ˛t Re;t Wt C .1
˛t / Rf ;t Wt C Lt
Ct
(6.10)
If we allow for leverage this means we will have ˛t > 1; wealth Wt is then expressed
as net worth and .˛t 1/ Rf ;t Wt is the interest payment on debt, whereby .˛t 1/Wt
is the leveraging.20 Equation (6.10) also tells us that with ˛t > 1; assets with higher
returns Re;t in some periods can be leveraged up though (possibly a lower) interest
will have to be paid as cost of the leveraging up of returns. On the other hand,
higher returns can be obtained and wealth can grow faster when extensive leveraging
is feasible.21 Now, we can summarize the factors affecting wealth disparities. The
last chapters have already provided us factors of considering this issue. Chapters 4
20
See Stein (2012) and Brunnermeier and Sannikov (2014).
For details of the distributional effects of high leveraging, see Brunnermeier and Sannikov
(2014).
21
6.6 Conclusions
113
and 5 demonstrate that disparities in wealth accumulation can emerge due to higher
returns, lower risk aversion, longer time horizon and lower discount rates.22 The
current chapter has introduced the additional factor of labor income. Now, by
looking at Eqs. (6.8) and (6.10)—beside asset income—the non-asset income, the
labor income could be large. Finally, wealth is likely to accumulated faster. This
is accelerated if consumption can stay below optimal consumption23 and thus the
saving rate can increase.24 This all may make wealth growing in the long run, rather
than falling. Moreover, for another group of financial market agents the returns and
the saving rates may be rather low, thus their wealth would grow at a lower rate. In
addition, if we allow some agents to face no borrowing constraints, ˛t > 1; as in
model (6.10), and a group of agents with strict borrowing constraints, ˛t < 1; then
the first type of agents would also accumulate assets faster then the second type of
agents, a case which in particular Brunnermeier and Sannikov (2014) have studied
in their model.
6.6 Conclusions
We have estimated and employed low frequency components of asset returns
and labor income and have used a numerical procedure to evaluate dynamic
consumption and asset allocation decisions. We use actual US time series data
to estimate the low frequency components of asset returns and labor income. We
employ harmonic estimations to estimate the underlying low frequency components.
After fitting the actual US time series data to low frequency components, our
numerical procedure is used to solve for dynamic consumption and asset allocation
decisions. We follow here Campbell and Viceira (2002) and explore dynamic asset
accumulation and asset allocation decisions for time varying returns, varying risk
aversion and varying time horizon across investors. We here also explore the role
of initial conditions of wealth and labor income, adding to wealth accumulation. As
discussed in this chapter our method appears to be more accurate than the method
proposed by Campbell and Viceira (2002). The optimal saving decisions, the asset
allocation, welfare of investors as well as the fate of the wealth of the investors can
be explored without linearization techniques. The impact of varying risk aversion
and time horizon across investors on the dynamics of asset allocation and the build
up of wealth can be traced. We can observe that there are cyclical movements
in wealth accumulation as well as upward and downward trends, depending on
22
A case that is concentrated on in Carroll et al. (2014).
Consumption might have an upper limit for holders of large wealth and thus the saving rate may
be higher.
24
See the Kaldor and Pasinetti debate on savings rate and asset accumulation in the 1970s, and a
summary of diverse studies on this issue in Nell and Semmler (1991). Differential in saving rates
is also an assumption that some econo-physicists work with.
23
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6 Asset Accumulation and Portfolio Decisions with Time Varying Asset Returns. . .
movements of the returns, the risk aversion, discount rates, the size of labor income
and the saving rate resulting from those factors. We have demonstrated what might
be the forces creating disparities in wealth accumulation. These are not only higher
asset returns, low risk aversion and low discount rates, but also higher labor income,
higher saving rates, and better excess to leveraging.