Financial sophistication and portfolio choice:Asset allocation over the life-cycle, and the

impact of macroeconomic outcomes

Zefeng Chen and Timothy Mok

Abstract

Empirical evidence suggests that households acquire a deeper ap-preciation of risk as they age; they hold more diversified portfolios ofrisky assets and face less volatile returns on wealth as they get older.Motivated by this evidence, we augment an otherwise standard life-cycle model of portfolio choice to allow for the accumulation of suchfinancial sophistication. Our model is able to account for the observedflat age profile of the conditional risky share, a fact which has hith-erto confounded existing models; it also generates realistic age profilesfor risky asset market participation rates, financial wealth and wages.Moreover, it also implies that households who have experienced morebenign macroeconomic outcomes are more willing to take financialrisks.

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1 Introduction

Over their lifetimes, every household must confront the proverbial issue ofportfolio choice: how much, if any, of their savings should they invest inrisky assets (i.e. equities and mutual funds)? According to the Survey ofConsumer Finances (SCF), the fraction of households that participate inrisky asset markets rises from 40% among households aged 26-30 to 60%among households aged 61-65. Conditional on participation, households,regardless of age group, on average allocate about 47% of their savings torisky assets, with the remaining fraction held in bonds and cash equivalents.

Traditional life-cycles models of portfolio choice, such as Merton (1971)and Cocco, Gomes, and Maenhout (2005), are unable to rationalize suchobserved behavior. These models imply that households should, at all pointsover their life-cycle, participate in risky asset markets; that is, householdsshould always invest some fraction of their savings in risky assets. They alsoentail that the fraction households allocate to risky assets should decreasewith age. More recent variants of these traditional models, which incorporatetransaction costs, are able to explain the rising age profile of participationin risky asset markets but continue to struggle to account for the flat ageprofile of the conditional risky share. In addition, they tend to predict anunrealistic accumulation of financial wealth (see Ball (2008) for a discusstionof this last point or Guiso and Sodini (2011) for more in-depth review).

In this paper, we introduce a framework that addresses the above gapbetween data and theory. Chiefly, we augment an otherwise standard life-cycle model of portfolio choice to allow for the accumulation of financialsophistication, which we associate with a capacity to understand risk anddiversification. Households, in the spirit of Ben Porath (1967), can chooseto invest in their level of financial sophistication at the expense of currentlabor income; more financially sophisticated households, in turn, face lessvolatile returns on the risky asset. Secondarily, we also allow for per-periodtransaction costs, as in Vissing-Jorgensen (2002) and Alan (2006).

Our framework extends the existing literature on portfolio choice over thelife-cycle in two main ways. First, it relaxes the implication that the risk-return profile of a given household’s portfolio is entirely determined by therisky share, an implication that follows from the assumption that householdsconstruct their portfolios from only two underlying assets – a risk-free bondand the market portfolio. Instead, it posits that this profile is determinedboth by the risky share and the household’s level of financial sophistication.

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Second, in contrast to previous studies that focused on exogenous risk, be it inthe form of rare disasters (Ball (2008), Fagereng, Gottlieb, and Guiso (2017))or age-dependent labor-market uncertainty (Chang, Hong, and Karabarbou-nis (2017)), it emphasizes endogenous risk: the portfolio risk households facedepends on their accumulated level of financial sophistication, itself an en-dogenous decision.

To be sure, there is strong empirical evidence that supports these exten-sions. Numerous surveys have uncovered a general lack of financial sophisti-cation among U.S. households and have concluded that this lack of financialsophistication is driven largely by households’ inability to appreciate risk anddiversification. For example, Lusardi and Mitchell (2013) find that roughlyhalf of U.S. households incorrectly believe that holding a single stock is asrisky as holding a stock market index fund. We reinforce this survey evidenceby documenting a distinct life-cycle profile in portfolio diversification amongU.S. households: older households typically hold more diversified portfolios.We also establish that older households, compared to their younger coun-terparts, face a similar mean return on wealth but experience significantlyless volatile returns in the cross-section. Taken together, this evidence sug-gests that households accumulate financial sophistication and hence face lessportfolio risk over their life-cycle, as is the case in our framework.

Crucially, our framework is able to provide a decent account, both qual-itatively and quantitatively, for the age profiles of the risky asset marketparticipation rate, the mean conditional risky share, the mean level of fi-nancial wealth, and the mean hourly wages among households aged 26-65.In particular, it generates a flat profile of roughly 40% for the conditionalrisky share. Loosely speaking, the accumulation of financial sophisticationand consequent reduction in portfolio risk incentivizes households, ceterisperibhus, to allocate a larger fraction of their savings to the risky asset overtheir life-cycle; this incentive offsets the traditional motive that householdshave to reduce the risky share as the discounted value of their future laborincome, relative to wealth, decreases over the life-cycle.

An auxiliary, but nonetheless important, implication of our framework isthe households who have experienced benign macroeconomic outcomes (i.e.a positive shock to labor income or asset returns), especially recent ones,are more willing to take financial risk. These households tend to participatemore in risky asset markets and allocate a larger fraction of theirs savingsto the risky asset. This implication is consistent, at least qualitatively, withthe findings in Malmendier and Nagel (2011) who detail a similar pattern of

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behavior among U.S. households. Pertinently, while Malmendier and Nagel(2011) interpret such behavior through a behavioral or psychological lens, weshow that it can also be understood through a rational framework. Expe-rience of benign macroeconomic outcomes induce households to accumulatefinancial sophistication at a higher rate and this higher level of financialsophistication, in turn, emboldens them to take more financial risk.

The rest of the paper is structured as follows. Section 2 presents empiricalevidence that households accumulate financial sophistication as they age; thishigher level of financial sophistication induces them to hold a more diversifiedportfolio and reduces their portfolio risk. Section 3 incorporates this evidenceinto a life-cycle model of portfolio choice. Section 4 calibrates our model tomatch the profiles of participation, conditional risky share, financial wealth,and wages in the data; it also discusses the role of our main mechanism – theaccumulation of financial sophistication – that allows the model to generatea rising age profile of participation and a flat age profile of the conditionalrisky share. Section 5 analyzes the impact of macroeconomic outcomes onfinancial risk taking. Section 6 concludes.

2 Empirical evidence

In this section, we present three empirical findings: 1) more financially so-phisticated households hold more diversified portfolios, 2) households holdmore diversified portfolios as they age, and 3) while their portfolios are morediversified, the mean return remains relatively constant. Together, thesethree findings paint a portrait of a household that acquires a more refinedunderstanding of risk and diversification as he ages and, consequently, a im-proved ability to hedge idiosyncratic portfolio risk. This household stands instark contrast to the perfectly financially sophisticated household generallyposited in traditional models of portfolio choice over the life-cycle. Rectifyingthis discrepancy, as we will show in later sections, better aligns the model’spredictions to the data.

2.1 Portfolio diversification and financial sophistica-tion

The increased intricacy of financial markets has spawned a large literatureinvestigating just how well-equipped households are to make complex finan-

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cial decisions. Numerous studies, through the use of surveys, have focusedon characterizing households’ financial sophistication along three dimensions:the capacity to do calculations related to compound interest and returns, thecapacity to understand inflation, and the capacity to understand diversifica-tion. In general, these studies have found that while households, broadly,grasp the concepts of interest and inflation, they do not comprehend the no-tion of diversification. For example, in the U.S., roughly 70% of householdsanswered questions pertaining to interest and inflation correctly but only50% got the question on diversification right.1

What distinguishes financially sophisticated households from their finan-cially unsophisticated counterparts, then, appears to be their understandingof risk and diversification. Several empirical studies drive home this distinc-tion further. Guiso and Jappelli (2009), using data that details the portfoliochoices, financial sophistication, and demographic characteristics of a sam-ple of Italian households, find that more financially sophisticated householdshold better diversified portfolios. Abreu and Mendes (2010) and Gaudecker(2015) report similar findings for samples of Portuguese and Dutch house-holds respectively.

Financial sophistication, or rather, the lack of it, explains why house-holds hold under-diversified portfolios. Crafting a well-diversified portfoliois by no means a trivial task. At the most primitive level, it requires anappreciation that asset returns are not perfectly correlated and that hold-ing multiple stocks, either individually or through a mutual fund, reducesthe return volatility of the overall portfolio. The existing evidence stronglysuggests that many households simply do not even posses this basic appre-ciation.

2.2 Portfolio diversification over the life-cycle

Is the capacity to understand risk and diversification an innate quality, ora a quality that can be nurtured over the life-cycle? Using data from theSurvey of Consumer Finances (SCF), we document an increase in the degreeof diversification of households’ portfolios over the life-cycle that favors thelatter interpretation. One cravat with with our analysis in the subsection isthat it is constrained to focus on households’ retail portfolios – the portion ofthe portfolio that excludes holdings in retirement accounts – as the SCF does

1See Lusardi et al (2014)

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not provide any indication of the diversification of these retirement accounts.As such, we are unable to analyze the portfolio diversification of the half ofstock market participants who hold only retirement accounts. Nonetheless,for households who hold a retail portfolio, this portfolio represents representsa significant proportion – roughly 60% – of their total equity holdings.

We begin with snapshots of the typical household’s portfolio at two stagesof the life-cycle. Households aged 25-35 invest 33% of their retail portfolio inmutual funds, and another 19% in the stock of their employer; the remaining48% of their portfolio is split between 4 stocks.2 In contrast, while householdsaged 65-75 allocate a similar fraction of their portfolio to mutual funds 34%they allocate a significantly smaller fraction 8% to the stock of the employerand also allocate the remaining portion of their portfolio among 9 morestocks. The typical household, then, bears less idiosyncratic risk as he getsolder. As a rule of thumb, Statman (1987) estimates that a portfolio with 4stocks faces a volatility that is 1.28 times as large as one with 8 stocks.

Table 1 provides more details on how households’ portfolio compositionsvary over the life-cycle. As they age, households, both on average and acrossthe entire distribution, hold more stocks and allocate a smaller fraction oftheir portfolio to their employer’s stock which, ceteris peribhus, increasesthe degree of diversification of their portfolios. Moreover, less householdshold grossly under-diversified portfolios. Between ages 25-35 and ages 65-75,the fraction of households whose portfolio consists of a single stocks fallsdramatically from 43% to 28% and the fraction who allocate more than two-thirds of their portfolio to their employer’s stock decreases from 21% to 7%.However, not all households fully appreciate the benefits of diversification assome still hold under-diversified portfolios, even as they get older. In general,the life-cycle trends in households’ portfolio composition is unaffected by theyshare they allocate to mutual funds, which remains relatively constant acrossage groups.

Pertinently, the increased diversification of households’ portfolios over thelife-cycle is consistent with survey evidence on the life-cycle profile of the levelof financial sophistication. Hung, Parker, and Yoong (2009) compare variousmeasures of financial sophistication for U.S. households and find that all ofthem are strongly increasing with age. This finding further lends support to

2The above statistics are calculated based on the pooled sample of households aged25-35 across successive waves of the SCF. Analyzing each wave of the SCF separatelywould not qualitatively change our results.

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the interpretation that the capacity to understand risk and diversification isakin to a kind of capital that households accumulate over the life-cycle.

2.3 Portfolio diversification v.s. expected returns

We have thus far conceived financial sophistication as an increased capac-ity to understand risk and diversification and have shown that householdsseem to acquire this capacity as the age, as reflected in the increased degreeof diversification in their portfolios over the life-cycle. A natural alterna-tive conception of financial sophistication would have it associated with anenhance ability to extract a higher expected return. This alternative con-ception has has come to dominate the nascent literature that incorporatesfinancial sophistication in life-cycle models (see, for example, Jappelli andPadula (2011), Lusardi et al. (2013), and, to a lesser extent, McKay (2013)).

Our approach to disentangle these two different conceptions is to considerthe distribution of returns on wealth of a cross-section of younger (agedbetween 25 and 44) and older (aged between 45 and 64) households. Bythe first conception, we should see that the distribution for older householdsis more concentrated about its mean that that for younger households; themeans of the two distributions should be roughly identical. By the secondconception, we should see a shift in the entire distribution, with that for olderhouseholds centered at a higher mean; the shapes of the two distributionsshould be roughly identical.

We construct the returns on household wealth using data from the PanelStudy of Income Dynamics (PSID), which measures the evolution of income,wealth, and, since 1999, aggregate consumption, for a panel of householdsover their life-cycle. Let Rw

i,t,t+2 be the gross return of the household i’swealth between periods t and t + 2 (the PSID is administered biennially).We estimate this return as follows

Rwi,t,t+2 =

wealthi,t+2 − savingsi,t+2 − savingsi,t+1

wealthi,t,

where savingsi,t = laborincomei,t − consumptioni,t (see the data appendixfor more details).

Figure 1 plots the estimated kernel density of the distribution of the re-turns on wealth for younger and older households in the 2005 wave of thePSID. We see that both distributions have roughly identical means but thatfor the older households is more concentrated about its mean. (Plots for other

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waves of the PSID reveal similar patterns are, in the interest of brevity, rel-egated to the appendix.) To further emphasize this point, Table 2 comparesthe mean return and the cross-sectional variance of returns experienced byyounger and older households across different years.3 The p-values, listedin parenthesis, indicate that, in general, the mean return on wealth amongyounger and older households is not statistically different; older households,however, experience less volatile returns in the cross section.4

The above evidence, coupled with our findings in the previous subsec-tions, tells a coherent story. As they age, households develop an increasedunderstanding of risk. This understanding induces them to hold a morediversified portfolio. Consequently, they experience less volatile returns.

3 Model

Motivated by the empirical evidence in the preceding section, we augment atraditional model of portfolio choice over the life-cycle to allow for, amongother aspects, the accumulation of financial sophistication. Our model willserve as the theoretical framework through which to understand the im-portance of financial sophistication in explaining households’ stock marketparticipation and portfolio composition over the life-cycle, and how theirfinancial risk taking responds to macroeconomic outcomes.

3.1 Households and preferences

Time is discrete and denoted by t. Our model economy consists of a largenumber of households, indexed by i, that differ in terms of their initial en-dowments and the resolution of uncertainty over their life-cycle. For ease ofnotation, we will suppress the index i in what follows.

3In the table, a positive value of the difference in means indicates that younger house-holds experienced a higher mean return; a ratio greater than 1 indicates that youngerhouseholds experienced greater return volatility.

4This finding is robust to various observable household characteristics. In unreportedresults, we repeated the above analysis conditional on stock market participation, educa-tion levels (with and without a college degree), and income levels (above and below medianincome) and found that the mean returns experienced by younger and older householdsare not statistically different but that younger households face a higher cross-sectionalvolatility of returns.

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Each household lives from period t = 0 to period t = T and has Epstein-Zin utility Vt over the non-negative consumption stream {ct} defined recur-sively by

V1− 1

σt = c

1− 1σ

t + βEt[V 1−γt+1

] 1− 1σ

1−γ ,

with VT = φ(aT + a), where aT is the terminal wealth of the household.β reflects the household’s rate of time preference, γ his risk aversion, andσ his elasticity of intertemporal substitution. As in De Nardi (2004), theparameters φ and a control, respectively, for the strength of the household’sbequest motive and the extent to which bequests are luxury goods.5

The life-cycle of each household is divided into two stages: working lifefrom period t = 0 to period t = Tr < T and retirement. During each periodof his working life, the household chooses consumption, ct, and the fractions,ωt and 1−ωt, of his savings to invest in the risky and safe assets. In addition,he decides how to allocate his time among doing productive work, an activitythat generates labor income, and accumulating financial sophistication andhuman capital, activities that affect future asset returns and labor income;we denote the fraction of time spent in each of these activities by 1−If,t−Ih,t,If,t, and Ih,t. During returement, the household only makes the former setof choices.

We next describe how financial sophistication and human capital affectfuture asset returns and labor income.

3.2 Financial sophistication and asset returns

In the spirit of Ben-Porath (1967), each household can accumulate financialsophistication during his working-life according to the production function

Ft+1 = Ft(1− δF ) + πF IαIff,t F

αFt , t ∈ [0, Tr],

where Ft is the level of financial sophistication in period t.6 The above func-tional form for the production function captures the intuition that the accu-mulation of financial sophistication depends both on new investments madeand the existing level of financial sophistication, with the relative importance

5De Nardi, Mariacristina. 2004. “Wealth Inequality and Intergenerational Links.?Review of Economic Studies, 71: 743?768.

6Ben-Porath, Yoram, “The Production of Human Capital and the Life Cycle of Earn-ings,? Journal of Political Economy, August 1967, 75 (4), 352?365.

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of each channel determined by the weights αIf and αF . The parameter δF al-lows for the depreciation of financial sophistication. Financial sophisticationremains constant after retirement: Ft = FTr for t > Tr.

Given his level of financial sophistication, the household has access to arisky asset that generates a gross return

R + ζt(Ft),

withζt(Ft) ∼ N (0, σ̄(1 + ψF−η

t )),

where σ̄ represents the market volatility (i.e. households with perfect finan-cial sophistication, Ft−1 →∞, hold the market portfolio). The interpretationhere is that financial sophistication is associated with access to a better di-versification technology. The parameter ψ determines the difference in returnvolatility experienced by perfectly sophisticated and less sophisticated house-holds. Moreover, for η > 0, the reduction in volatility is decreasing in thelevel of financial sophistication. To the extent that households already have abasic understanding of the benefits of diversification, a deeper understandingof such benefits is unlikely to provide a huge incremental benefit. As a casein point, the returns on a portfolio with ten stocks is likely far less volatilethan the returns on one with a single stock, but the returns on a portfoliowith a hundred stocks are unlikely to be much less volatile than the returnson one with ten.

To access the risky asset market, households have to pay a per-periodfixed cost of p. Alternatively, he can choose to invest his savings in the safeasset that guarantees a time-invariant gross return Rf and whose access isnot subject to per-period participation costs. Regardless of the assets heinvests in, the household faces short sale constraints which limit ωt ∈ [0, 1].

3.3 Human capital and labor income

Analogously, each household can accumulate human during his working-lifeaccording to the production function

Ht = Ht−1(1− δH) + πHIαIhh,t H

αHt−1, t ∈ [0, Tr],

where Ht is the level of human capital in period t. As is the case with financialsophistication, human capital remains constant after retirement: Ht = HTr

for t > Tr.

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The household’s level of human capital, along with the fraction of timespent doing productive work, jointly determine the non-stochastic componentof his labor income

Yt = hHt−1(1− Ih,t − If,t) exp(εt + νt), t ∈ [0, Tr]. (1)

The parameter h represents hours worked which, in turn, imbues Ht−1 withthe interpretation as the effective hourly wage. Uncertainty over labor incomeis governed by the permanent shock εt which follows an AR(1) process

εt = ρεt−1 + ut,

with ut ∼ N (0, σu), and the transitory shock νt ∼ N (0, σν). We assume thatut and νt are uncorrelated and distributed independently across householdsand time.

Retirement benefits are a fraction λ of the non-stochastic component ofthe household’s labor income at retirement

Yt = λhHTr , t ∈ (Tr, T ]. (2)

Such a specification, while an oversimplification of the retirement benefitshouseholds receive in reality, is, nonetheless, a good approximation. It alsofacilitates the computation of the model as it precludes the introduction ofan additional state variable.

3.4 The optimization problem

Define the state variable St := {at, εt, Ht, Ft}, where at, as in Deaton (1991),is a measure of cash-on-hand. Formally, we can write the household’s problemduring his working-life (i.e. t ≤ Tr) recursively as

Vt(St) = maxct,ωt,Ih,t,If,t

{(c1− 1

σt + βEt

[(Vt+1(St+1))

1−γ] 1− 1σ

1−γ

) 1

1− 1σ

}s.t.

at+1 = (at − ct − pI(ωt > 0)) ((1− ωt)Rf + ωt (R + ζt(Ft))) + Yt+1

Ft+1 = Ft(1− δF ) + πF IαIff,t F

αFt

Ht+1 = Ht(1− δH) + πHIαIhh,t H

αHt

εt+1 = ρεt + ut+1,

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where Yt is given by (1). The recursive formulation of the household’s prob-lem during retirement is almost identical except that the state variables{εt, Ht, Ft} remain constant and the household optimizes only over {ct, ωt}.We omit it in the interest of brevity.

3.5 Solution method

We solve the household’s optimization problem by backward induction. Giventhe specification of the value function in the final period T , we computethe policy functions for the penultimate period using the Bellman equation.As is common practice, we apply the methodologies of Tauchen (1986) andRouwenhorst (1995) to discretize the transition matrices of the respectiveshocks in order to compute the expectation in the Bellman equation. Theabove process, repeated until t = 0, yields the complete set of policy func-tions over the life-cycle. Finally, to calculate aggregate statistics for ourmodel economy, we simulate the life-cycle for large number of households,each subject to a particular series of idiosyncratic shocks.

4 Portfolio choice over the life-cycle

As alluded to in the introduction, traditional models of portfolio choice overthe life-cycle have difficulty explaining the life-cycle profiles of aggregate stockmarket participation rates as well as the mean portfolio risky share amongstock market participants. In this section, we show that our model, withreasonably calibrated parameter values, is able to jointly account for the life-cycle profiles of participation and the mean conditional (on participation)risky share, as well as those of wealth and wages. We discuss the importantrole that financial sophistication plays in this account.

4.1 The benchmark model

We calibrate our model to target the life-cycle profiles of four series in thedata: the aggregate stock market participation rate, the mean conditionalrisky share, the mean level of financial wealth, and the mean hourly wage.7

7Data for the first two series is taken from the waves of PSID from 1999 to 2013; datafor the last two series is taken from the waves of the SCF over the same time period. Pleaserefer to the data appendix for more details regarding our data construction process.

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Our calibration is carried out in two steps. First, we fix parameters thatcan be cleanly identified without the explicit use of our model. Second, wecalibrate the remaining parameters to best match the data moments.

4.1.1 Parameterization

Each period in our model represents one year and we assume households atperiod t = 0 have an age of 25. Households work for Tr = 41 years and retireat the age of 66, as per the eligibility requirements for full Social Securitybenefits; they live for a total of T = 55 years. Over their lifetimes, householdscan save at the gross risk-free rate of Rf = 1.016 or invest in risky assetswith an equity premium of µ = 0.06. Perfectly financially sophisticatedhouseholds face a volatility of σ̄ = 0.17, in line with historical volatility ofthe market portfolio.

With regards to their labor income, households face uncertainty charac-terized by ρ = 0.95, σu = 0.025, and σν = 0.04, which is consistent withthe estimates in labor economics literature (see Carroll et al. (2014) for areview of past empirical studies). They work for h = 2500 hours annually, asis the case with the households in the PSID. Upon entering retirement, theyreceive a fraction λ = 0.35 of their labor income at retirement as benefits, asthey roughly would under the current Social Security scheme.8

For convenience, we collect our parameterization in Panel A of Table 3.

4.1.2 Calibration

Panel B of Table 3 lists the calibrated values of our remaining parameters.Our calibrated values of the coefficient of relative risk aversion and the elas-ticity of intertemporal substitution is within the range generally used in thefinance literature where Epstein-Zin preferences are common (see, for exam-ple, Bansal and Yaron (2004)) while our discount factor is consistent withestimates in the life-cycle savings literature (see, for example, Cagetti (2003)).Our specification of the human capital production function is comparable tothat in Fan, Seshadri, and Taber (2015); any difference in parameter values isdue to differences in the estimated wage profile. We differ a discussion of theparameters governing financial sophistication to the end of this subsection.

Using the policy functions derived from the above parameters, we simulatethe life-cycle profiles of stock market participation, risky asset allocation, and

8https://www.nasi.org/learn/socialsecurity/benefits-compare-earnings

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wealth and human capital accumulation for a large panel of households. Theirinitial levels of wealth, human capital and the shock to permanent incomewere estimated from the pooled sample of households aged 25 from the PSID;their initial level of financial sophistication was normalized to 1. Figure 2compares the model-generated profiles to the data. We focus on the periodof the life-cycle prior to retirement as our model does not capture manyimportant aspects of retirement, such as health shocks and and the relatedpurchase of insurance, that impact savings and asset allocation decisions(Lockwood (2017)).

The central point is that our parsimonious model is able to account forthe relatively flat profile of the mean conditional risky share. Explainingthis flat profile has, hitherto, posed a challenge for models of portfolio choiceover the life-cycle. Traditional workhorse models, such as Merton (1971) andCocco et al. (2005), predict a strongly decreasing conditional risky share ashouseholds age. Recent variants which incorporate a small probability of adisaster state, as in Fagereng et al. (2017), predict a distinctly hump-shapedprofile.

Importantly, our model is able to account for flat profile of the meanconditional risky share while jointly matching the profiles of stock marketparticipation and wealth accumulation reasonably well. To date, models ofportfolio choice, in their narrow focus on explaining the participation rateor the risky share, have tended to imply an unrealistic wealth accumulationprofile. For example, Alan (2006), through the introduction of fixed partic-ipation costs, is able to account for the life-cycle profile of the participationrate fairly precisely but her calibration implies an implausibly low accumu-lation of wealth; she also makes no mention of the conditional risky share.At the other extreme, Gomes and Michaelides (2005) assume an excessiveaccumulation of wealth in order to better match the conditional risky shareduring the later part of the life cycle; their model, contrary to the data, alsopredicts a decreasing profile of the conditional risky share.

Finally, our model predicts a plausible accumulation of financial sophis-tication. Figure 3 depicts the life-cycle profile of the mean level of financialsophistication among all households as well as among stock market partici-pants. As expected, stock market participants are generally more financiallysophisticated. Moreover, the above rate of accumulation, coupled with ourcalibrated values of ψ and η, implies that households aged 25 to 45 face arisky asset volatility that is 1.22 higher than that faced by households aged45 to 65, consistent with the comparison in Table 2. This evidence lends

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some credibility to our calibrated financial sophistication parameters whichdo not have a natural counterpart in the data.

4.2 The role of financial sophistication

Overall, our benchmark model provides a coherent account of the life-cycleprofiles of participation, asset allocation, wealth, and human capital. Wenow discuss the importance of our main mechanism – the accumulation offinancial sophistication – in terms of its impact on the conditional risky share(i.e. the intensive margin) and the participation decision (i.e. the extensivemargin).

4.2.1 Intensive margin

Figure 4 plots the household’s optimal portfolio risky share, as a function oftotal wealth, at different points over the life-cycle, conditional on various lev-els of financial sophistication.9 For perspective, low, medium and high levelsof financial sophistication correspond to the levels of financial sophisticationthat the typical household has at the ages of 25, 45, and 65 respectively.In all panels, a portfolio risky share of zero reflects non-participation in therisky asset market.

Holding fixed the level of financial sophistication, the optimal portfoliorisky share decreases with wealth and age. To understand this trend, considera household at period T − 1. During period T , he receives non-stochasticretirement benefits that act as a substitute for the risk-free asset. The lesswealth this household has, relative to his retirement benefits, the more hetilts his portfolio towards the risky asset as he has a relatively larger risk-free position from his retirement benefits. More generally, since innovationsto labor income and risky asset returns are uncorrelated, the household’sdiscounted future stream of labor income closely mimics a position in therisk-free asset and, thus, households with a higher discounted future streamof labor income, relative to current wealth, tend to allocate a higher fractionof their portfolio to the risky asset. Ceteris paribus, these households areprecisely the less wealthy and younger households.

9In all panels, we fix the permanent shock to labor income at a neutral level (i.e.εt = 0) and set the level of human capital as Ht = 17. The qualitative features of thepolicy functions do not depend on these parameters.

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Varying the level of financial sophistication, the optimal portfolio riskyshare increases with financial sophistication. An increase a household’s levelof financial sophistication reduces the return volatility of the risky asset.Given that the household is risk adverse, this reduction increases the attrac-tiveness of the risky asset and incentivizes the household to tilt his portfoliotowards it.

Overall, then, the household is subject to two opposing inclinations duringhis working life. On the one hand, he is disposed to shift his portfolio awayfrom the risky asset as he gets older and accumulates wealth. On the other,he is induced to shift his portfolio towards the risky asset as he accumulatesfinancial sophistication.10 In our calibration, the second inclination slightlydominates and generates a slight hump shape in the aggregate portfolio riskyshare, as in the data. After retirement, the household is only subject to thefirst inclination and thus shifts his portfolio away from the risky asset. Thelack of a counteracting force in our model explains this divergence with thedata.

4.2.2 Extensive margin

The inclusion of a per-period participation cost introduces a wealth-participationthreshold (Figure 4): at low levels of wealth, households do not invest in therisky asset as the benefit of earning the equity premium does not outweigh theparticipation costs. For a given level of financial sophistication, this wealthparticipation threshold increases with age and decreases with the level ofhuman capital: households that are younger and who have higher humancapital have a strong incentive to shift their portfolio towards the risky assetsince they have a larger discounted stream of future labor income relative towealth. In the absence of the accumulation of financial sophistication, thesetwo effects interact to generate a very pronounced hump-shaped life-cycleprofile of aggregate stock market participation. Younger households quickly

10It should be noted that the accumulation of human capital over the life-cycle alsocontributes, though to a lesser degree, to this force. As the household accumulates humancapital and hence increases his discounted future stream of labor income relative to hiscash-on-hand, he re-balances his portfolio towards the risky asset. This effect is morepronounced in the early years of the life-cycle, between the ages of 25 to 30, where thehourly wage profile is steep. It is almost absent in the later pert of the life-cycle wherethe hourly wage profile is significantly flatter; during these years, the impact of financialsophistication is the chief driver of the above force. So as not to deluge the reader withexcessive detail, we omit this aspect of the analysis from the main text.

16

accumulate wealth and human capital that pushes them above the threshold;older households decumulate wealth and thus exit the risky asset market.

Allowing for the accumulation of financial sophistication impacts the life-cycle profile of aggregate stock market participation in two main ways. First,and most clearly, it reduces aggregate participation. The lack of financialsophistication increases the wealth-participation threshold and thus discour-ages participation in general. Second, and more subtly, it decreases the rateof entry into the risky asset market. Younger households take longer to accu-mulate the wealth and financial sophistication needed to push them beyondthe wealth-participation threshold.

To illustrate the latter impact, Figure 5 plots the policy functions forconsumption and investments in financial sophistication for less wealthy (i.e.those below the wealth participation threshold) households aged 25 with var-ious levels of financial sophistication. Compared to more financially sophis-ticated households, less financially sophisticated households consume moreand, hence, save less due to their relatively poorer prospects in the risky assetmarket. Concomitantly, they also also invest less in financial sophistication,at least for lower wealth levels, due to their decreased likelihood of participa-tion in the risky asset market.11 These two effects are mutually reinforcing:a lack of financial sophistication induces less wealth accumulation which, inturn, induces less accumulation of financial sophistication. Together, theyincrease the expected time it takes for a young, financially unsophisticatedhousehold to cross the wealth-participation threshold.

5 Portfolio choice and the impact of macroe-

conomic outcomes

A recent strand of the literature (see Guiso et al. (2004), Osili and Paul-son (2008), Malmendier and Nagel (2011), among others) has focused on thelink between personal experiences and financial risk-taking. Most relevantto the present discussion, Malmendier and Nagel (2011) find that householdswho have experienced bad macroeconomic outcomes are less likely to partic-ipate in the stock market and invest a smaller fraction of their portfolio in

11Holding all other state variables fixed, the wealth-participation threshold is decreasingin the level of financial sophistication. For a given level of wealth, less financially sophisti-cated households are further away from their threshold and thus less likely to participate;they thus have little incentive to invest in financial sophistication.

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stocks when they do. In addition, the authors also observe that more recentmacroeconomic experiences have a greater impact on households’ financialrisk-taking. That is, households that have lived through a boom followed bya depression are less likely to take financial risk compared to households thatlived through a depression followed by a boom precisely because the morerecent experience dominates.

Thus far, these two facts have been given a behavioral interpretation. Per-sonal experience affects belief formation and, hence, alters risk preferencesas individuals do not incorporate all existing information when making de-cisions. In this section, we propose an alternative framework through whichto understand these facts: macroeconomic outcomes impact the accumula-tion of financial sophistication and, therefore, influences financial risk-taking.Specifically, we show that our model, which assumes households are perfectlyrational and have stable risk preferences, is able to reproduce, at least qual-itatively, the above facts.

5.1 Boom and depression babies

To illustrate the impact of macroeconomic outcomes on financial risk-taking,we simulate the life-cycle profile of the portfolio risky share for two householdsthat have similar initial levels of wealth and human capital but differ in termsof the resolution of uncertainty. The first household, which we will refer toas the boom baby, experiences a positive shock to his permanent incomeover the first three periods of his life-cycle while the second household, thedepression baby, experiences a negative shock to his permanent income overthe corresponding period.12 For simplicity, all remaining shocks are set totheir mean value of zero.

Figure 6 plots the life-cycle profile of the portfolio risky share for thesehouseholds based on the policy functions from our benchmark model; we setthe initial levels of wealth and human capital at $50,000 and 15 respectively

12Our interpretation of macroeconomic outcomes differs slightly from that in Mal-mendier and Nagel (2011) who associate them with stock market returns. We adoptthis slightly different interpretation as it allows us to say something about the extensivemargin. In our model, stock market returns do not affect the decisions of non-participants;shocks to labor income, as we shall see, do. This slight difference in interpretation shouldnot be too troubling for the reader: stock market returns are indeed correlated with unem-ployment and, hence, labor income. In any case, our mechanism would still be operationalunder the alternative interpretation.

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and positive and negative shocks at 1.5 standard deviations above and belowtheir mean. It also plots the life-cycle profiles of financial sophistication,human capital and wealth. The top left panel of Figure 6 shows that theboom baby participates more in the stock market – as evidenced by thelonger period in which his portfolio risky share is greater than zero – andinvests a larger fraction of his portfolio in the risky asset when he does.

More importantly, Figure 6 reveals that the increased financial risk takingof the boom baby is entirely due to his higher level of financial sophistication.The bottom panels depict that, compared to the depression baby, the boombaby has a higher level of wealth, owing to his higher labor income in theinitial periods, and a (very) marginally lower level of human capital. Hadboth households had the same level of financial sophistication, as they wouldin traditional models of portfolio choice over the life cycle, the depressionbaby would invest a larger fraction of his portfolio in the risky asset as hewould have had a higher ratio of discounted future labor income comparedto wealth (see Figure 4). Therefore, it is strictly because of his greateraccumulation of financial sophistication that the boom baby is more inclinedto take financial risk.

Two effects incentivize the boom baby to accumulate a higher level offinancial sophistication. First, a positive shock to his permanent incomeincreases his ratio of expected discounted future labor income to currentwealth and thus induces him to tilt his portfolio towards stocks when hedoes, in fact, decide to participate in the stock market; this desire to allocatea larger fraction of his portfolio to stocks increases the benefits of investingin financial sophistication. Second, his higher wealth increases his absolutebenefit of participating in the risky asset market and, in turn, his benefitof investing in financial sophistication. The first effect explains the initialdivergence between the financial sophistication of the boom and depressionbabies while the second accounts for the continued divergence over the life-cycle (see the top right panel of Figure 6).

The above simulation exercise highlights how the interaction betweenmacroeconomic outcomes and the incentives to accumulate financial sophis-tication drives variation in the level of financial risk-taking. Overall, it re-produces the stylized fact that households who lived through a depressionare less likely to participate in the stock market and invest a smaller fractionof their portfolio in stocks when they do compared to households that livedthrough a boom. We now discuss why more recent macroeconomic eventsseem to have a larger impact on this financial risk-taking behavior.

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5.2 Recent macroeconomic experiences

Consider, as before, two households with similar initial levels of wealth andhuman capital. Now, instead, suppose that the first household experiences apositive shock to his labor income in the first two periods followed by nega-tive shocks the next two while the second household experiences a negativeshock to his labor income in the first two periods and positive shocks in thenext two; all remaining shocks are set at their mean value of zero. That is,while both households experience the same number of good and bad macroe-conomic outcomes, the second experiences relatively better recent outcomes.

Figure ?? plots the life-cycle profile of the portfolio risky share, financialsophistication, wealth, and human capital for these households; the initiallevels of wealth and human capital and magnitude of the shocks are as inour previous simulation. The top left panel shows that the second household(depicted by the solid line), who experienced a good macroeconomic outcomemore recently, participates more in the risky asset market and, largely, investsa larger fraction of his portfolio in the risky asset when he does. The otherthree panels highlight that the difference in the households’ financial risk-taking is driven primarily by their accumulation of financial sophistication:the second household invests more in the risky asset despite having a higherlevel of wealth and a similar level of human capital.

To understand the above result, recall that the accumulation of financialsophistication proceeds according to

Ft+1 = Ft + πF IαIff,t F

αFt .

Two channels amplify the impact of recent macroeconomic outcomes on theaccumulation of financial sophistication. First, more recent macroeconomicoutcomes generally occur after the household has accumulated a higher levelof wealth. Given that investment in financial sophistication, If,t, is a highlyconvex, upward-sloping function of wealth, more recent differences in macroe-conomic outcomes, and hence wealth, lead to more pronounced differences ininvestment in financial sophistication. Second, more recent macroeconomicoutcomes also occur after the household has accumulated a higher level offinancial sophistication. This higher level of financial sophistication furthermagnifies and differences in investments in financial sophistication.

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6 Conclusion

Existing life-cycle models of portfolio choice always, albeit implicitly, assumethat households are perfectly financially sophisticated; households in thosemodels hold an efficient portfolio of risky assets (i.e. the market portfolio)and hedge away all idiosyncratic risk. This assumption, as we have shown,is a gross abstraction from reality. Households, unable to fully appreciatethe concept of risk, typically hold an under-diversified portfolio. Relevantly,too, older households, compared to their younger counterparts, hold a morediversified portfolio and consequently experience less volatile returns in thecross-section.

This paper breaks away from the above assumption. Incorporating theendogenous accumulation of financial sophistication, our calibrated model isable to to account for two sets of stylized facts. First, it is able to generate a(quantitatively accurate) flat age profile of the conditional risky share whilealso matching, reasonably well, the age profiles for participation, wealth, andwages. Second, it is able to qualitatively reproduce the fact that householdswho have experienced benign macroeconomic outcomes are more willing totake financial risk. Our findings suggest that any complete theory of house-holds’ portfolio choice should consider the endogenous accumulation of in-vestment skill – which we have here termed as financial sophistication – andnot take as given that households are perfectly able to appreciate and managerisk.

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Ben-Porath, Y.. 1967. The Production of Human Capital and the LifeCycle of Earnings. Journal of Political Economy, 75(4), 352-365.

Carroll ,C., J. Slacalek and K. Tokuoka. 2014. “Buffer-stock Saving in aKrusell-Smith World.” Working Paper 1633, European Central Bank.

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Figure 1: Distribution of return on wealth between 2003 to 2005 for youngerand older households

0.5

11.

5De

nsity

-1 0 1 2Return

Young Old

Cross-sectional Distribution of Return

This figure plots the estimated kernel density for the return on wealth expe-rienced by younger and older households.

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Table 1: Portfolio composition over the life-cycle

Panel A: Number of stocksDist. over no. of stocks

Age group Mean 1 2-3 4-9 > 1025-34 4.03 .43 .27 .20 .1035-44 4.67 .36 .30 .20 .1345-54 6.18 .32 .29 .23 .1655-64 7.71 .31 .26 .23 .2165-74 9.44 .28 .24 .23 .26

Panel B: Share allocated to employer stockDist. over share

Age group Mean 0% 1-33% 33-67% > 67%25-34 .28 .53 .10 .16 .2135-44 .27 .52 .10 .21 .1745-54 .25 .57 .10 .17 .1655-64 .18 .66 .11 .10 .1365-74 .12 .77 .08 .08 .07

Panel C: Share allocated to mutual fundsDist. over share

Age group Mean 0% 1-33% 33%-67% > 67%25-34 .33 .55 .05 .05 .3035-44 .36 .51 .06 .06 .3345-54 .39 .50 .06 .07 .3655-64 .36 .50 .06 .07 .3265-74 .34 .51 .09 .07 .32

This table provides a detailed breakdown of three measures of households’financial portfolio composition: the number of stocks held, the share allocatedto the stock of their employer, and the share allocated to mutual funds. Thesecond column lists the mean of the corresponding measure for each agegroup. The last four column give a sense of how the measure us distributedacross households in a given age group; for example, 42% of households aged25-34 hold only a single stock in their portfolio. Data is from the SCFsbetween 1999-2013.

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Table 2: Comparison of cross-sectional return moments amongyounger and older households

Mean VariancePeriod Diff (%) p-value Ratio p-value

1999 - 2001 0.1 0.96 1.75 0.002001 - 2003 -0.5 0.84 1.34 0.002003 - 2005 1.3 0.54 1.27 0.002005 - 2007 0.1 0.95 1.30 0.002007 - 2009 -2.5 0.21 1.21 0.012009 - 2011 -5.1 0.02 1.39 0.002011 - 2013 2.4 0.33 1.61 0.00

The table compares the mean and variance of the return on wealth betweenyounger and older households. A positive difference indicates that youngerhouseholds experience a higher mean return over the relevant period; a ra-tio greater than 1 indicates the younger households experienced a greatervariance. p-values are for the tests of equal sample mean and variance re-spectively. Data is from the PSID between 1999-2013.

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Table 3: Parameter values for the benchmark model.

Parameter ValuePanel A: Preset parameters

Life-cycleLength of life-span T 55Length of working life Tr 41Asset returnsRisk-free return Rf 1.016Equity premium µ 0.06Volatility of market portfolio σ̄ 0.17Labor incomePersistence of permanent shock ρ 0.95Variance of permanent shock σε 0.025Variance of transitory shock σν 0.04Annual hours worked h 2500Replacement ratio λ 0.35

Panel B: Calibrated parametersPreferencesDiscount factor β 0.944Risk aversion γ 8Elasticity of intertemporal substitution σ 2Weight of bequest motive φ 20Financial sophisticationProduction function: Elasticity of F factor αF 0.55Production function: Elasticity of If factor αIf 0.55Production function: Productivity πF 1Diversification technology: Volatility wedge ψ 0.85Diversification technology: Elasticity η 0.70Participation costs p 1500Human capitalProduction function: Elasticity of H factor αH 0.287Production function: Elasticity of Ih factor αIh 0.06Production function: Productivity πH 2Production function: Depreciation δH 0.16

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Figure 2: Caliberation results for the benchmark model.

30 40 50 600

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cipa

tion

Rat

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30 40 50 600

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hare

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modeldata

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30 40 50 6015

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Figure 3: Accumulation of financial sopistication

25 30 35 40 45 50 55 601

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Figure 4: Policy functions for the risky share

0 2 4 6 8x 105

0

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shar

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0 2 4 6 8x 105

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0

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wealth

shar

e

This figure plots the the policy functions for the risky share at different pointsover the life-cycle, conditional on various levels of financial sophistication.The level of human capital is fixed at the median level and the permanentincome shock is fixed at its mean of zero.

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Figure 5: Policy functions for consumption and investment in financialsophistication at age 25

0 2 4 6 8 10 12 14 16 18x 104

0

1

2

3

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6x 104 Policy function for consumption

asset

cons

umpt

ion

Low FHigh F

0 2 4 6 8 10 12 14 16 18x 104

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1x 10−3 Policy function for investment of F

asset

IF

Low FHigh F

The left and right panels plot the policy functions for consumption and in-vestment in financial sophistication at various levels levels of financial sophis-tication. The level of human capital is fixed at its median and the permanentincome shock is fixed at its mean of zero.

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Figure 6: Simulation of boom and depression babies

20 40 60 800

0.2

0.4

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shar

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20 40 60 801

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20 40 60 800

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boom babydep baby

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Figure 7: Simulation of households with recent good shock and recent badshock

20 40 60 800

0.2

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shar

e

20 40 60 801

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recent bad shockrecent good shock

33