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3 Business Cycles, Asset Returns and Labor Income

3 Business Cycles, Asset Returns and Labor Income

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104



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.



106



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,



112



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



114



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.



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