Simulating the Risk of Investment in Human Capital

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Simulating the Risk of Investment inHuman CapitalJoop Hartog a , Hans Van Ophem a & Simona Maria Bajdechi aa Tinbergen Institute , University of Amsterdam , The NetherlandsPublished online: 30 Jul 2007.

To cite this article: Joop Hartog , Hans Van Ophem & Simona Maria Bajdechi (2007)Simulating the Risk of Investment in Human Capital, Education Economics, 15:3, 259-275, DOI:10.1080/09645290701273434

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Education EconomicsVol. 15, No. 3, 259–275, September 2007

0964-5292 print/1469-5782 online/07/030259-17 © 2007 Taylor & FrancisDOI: 10.1080/09645290701273434

Simulating the Risk of Investment in Human Capital

JOOP HARTOG, HANS VAN OPHEM & SIMONA MARIA BAJDECHI

Tinbergen Institute, University of Amsterdam, The NetherlandsTaylor and Francis LtdCEDE_A_227250.sgm10.1080/09645290701273434Education Economics0964-5292 (print)/1469-5782 (online)Original Article2007Taylor & [email protected]

ABSTRACT The risk of investment in schooling has largely been ignored. We mimic theinvestment decision facing a student and simulate risky earnings profiles in alternativeoptions, with parameters taken from the very limited evidence. The distribution of rates ofreturn appears positively skewed. Our best estimate of ex ante risk in university educa-tion is a coefficient of variation of about 0.3, comparable with that in a randomly selectedfinancial portfolio with some 30 stocks. With risk attitudes varying by parental back-ground, this may be relevant for differences in schooling participation rates. Allowing forstochastic components in earnings also markedly affects expected returns.

KEY WORDS: Education; return; earnings dispersion; risk

Introduction

A remarkable omission in the analysis of investment in human capital is thefailure to account properly for risk. Theoretical and empirical analyses ofinvestment in physical capital and in financial portfolio’s squarely focus on therisk properties of the alternatives. Just as investors in physical and financialcapital, an investor in human capital will not only be interested in the expectedreturns but also in the corresponding risk. In fact, the perceived risk of theinvestment may well be a dominant concern in the decision-making process.Yet, the necessary theory is underdeveloped and the empirical analysis isvirtually non-existent.

In terms of ex ante risk, the first risk associated with human capital is that oneducational performance: how well will the individual do in school. As perfor-mance in school is not the same as performance in the labour market, the secondsource of risk is uncertainty about the relative position in the post-school earningsdistribution. A third source of risk is market risk. The value of an education, orassociated occupation, may shift over time in response to changes in technology,product-demand patterns or relative supply.

Surprisingly little is known empirically about the dispersion in returns toeducation. Even though heterogeneity among individuals and hence in theirreturns has been stressed in several contributions, such as Willis and Rosen (1979)and Card (1995), this has not led to a focus on variations in returns to human

Correspondence Address: Joop Hartog, Department of Economics, University of Amsterdam, Roetersstraat11, 1018 WB, Amsterdam, The Netherlands. Email: [email protected]

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260 J. Hartog et al.

capital investment. Bajdechi (2005) draws several conclusions from a search of theempirical literature. The Mincerian rate of return in one country may easily betwo to three times the return in another country; across countries, the coefficientof variation may be something like 0.5. Within countries, there is generally a fairamount of stability over several decades. Over time, within countries the differ-ences between the minimum and the maximum rate seem generally perhaps nomore than one-third of the minimum rate. On differences in returns betweenindividuals there is even less information. Available studies suggest a coefficientof variation between 0.4 and 0.6.

Variation in ex post rates of return is not the same as ex ante risk. In this paper,we assess ex ante risk with a simulation model that mimics the situation facing anindividual about to decide on investment in education. The model is simply thebasic human capital investment model that compares two future earningsstreams. In this way we augment information on average rates of return withinformation on its variance. For obvious reasons, a simulation model is the onlyfeasible approach: direct observations on variances in the rate of return are simplynot feasible.

The model has been constructed to focus on the essentials: the stochasticcomponent in earnings. Earnings after completing any given education are onlyrelated to years of schooling and experience, augmented by an error term. Foreach of two alternative educations from which an individual may choose, wesimulate a lifetime earnings profile and then solve for the internal rate of return.By repeating the simulations for a given set of parameter values, we generate adistribution of returns. Of course, actual returns will depend on a host of otherfactors such as ability, parental background, ethnicity, and so forth. Even thestochastic component may depend on these variables. We do not explicitlyconsider these influences, as they are subsumed in the parameter values. Weshow how conclusions depend on parameter values and the interval of valuesshould cover these effects. If individuals with a particular background and ethnic-ity can expect a particular return to experience, their experience slope parametershould be covered in our interval of parameter values.

It is important to point out limitations on the relevance of our exercise.Essentially in our model, risk is caught by the relative volatility of two earningsprofiles, say from high school or from university. Now suppose university earn-ings are completely determined by graduation and experience: the error term iszero. Then, increasing the variation in high school earnings increases the varianceof the internal rate of return. This reflects that the opportunity cost of the alterna-tive earnings stream (not going to university) becomes harder to predict and thereturn on the investment will become riskier. Yet by choosing university ratherthan high school, the individual can evade all uncertainty in future earnings(Eden, 1980) and this may be a strong argument in favour of a university educa-tion. Thus, while our measure properly reflects the uncertainty in the internal rateof return, the question is whether it is relevant to guide individuals’ choice ofeducation. For example, if individuals use a mean–variance utility function, theyevaluate perspectives on the differences in means and variances of the earnings ineach alternative in a dynamic context. This would be different from assessingperspectives on the basis of expected value and variance of the internal rateof return as we calculate here. The question relates to the wider question onwhat information individuals actually use when deciding on their education. Onthis question we know very little (see Dominitz and Manski (1996) for student

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Risk of Investment in Human Capital 261

perceptions on earnings uncertainty associated with schooling). There is also aquestion how much uncertainty can be resolved by buying insurance or investingin the stock market. On this, however, there is reason for scepticism. Davis andWillen (2000) show that the optimal combination of human and financial invest-ment may require quite unrealistic stock portfolio’s, Palacios-Huerta (2003) showsthat adding financial investment to human capital investment is no guarantee toimprove the risk–return properties of the portfolio. The discussion on whatmeasure of risk is most relevant for students’ schooling decisions has really justbegun. While our analysis and simulation give a picture of the ex ante distributionof the internal rate of return to education, we do not know whether this is whatstudents use to make their decisions. We offer our calculations as a quite naturalextension of the Mincer framework.

In the next section, we use a model with a log linear effect of experience onearnings and a single lifetime earnings shock to allow analytical solutions. In thethird section, we move to a quadratic experience profile and annual shocks inearnings, making simulation the only feasible approach. In the subsequent sectionwe discuss parameter selection, and the fifth section presents the simulations. Thefinal section concludes. We find that the ex ante distribution of internal rates ofreturn is skewed to the right and has a coefficient of variation of about 0.3. Inpassing we point to the substantial effect of different career growth by level ofeducation on the rate of return, an effect that is commonly overlooked.

Analytical Solutions

We will mimic the position of an individual that has to decide whether tocontinue education or not, and will assess ex ante risk of the investment as thedispersion in the rate of return. We will develop a simulation model that faith-fully follows the standard human capital model.

By definition, the internal rate of return to education is the rate of discount, δ,that equates the present values of lifetime earnings for two different educationallevels, s0 and s1; that is, the interest rate that solves the equation:

The earnings functions are f(s0, (t – s0)) and f(s1, (t – s1)), with s0, s1 years of school-ing and with t – s0, t – s1 years of work experience. T0 and T1 are the durations ofthe working life after graduation. Note that this is a quite general framework,although not without limitations. We compare two investments, of differentlengths, with a binding commitment up front. This might apply to different typesof education, possibly but not necessarily differing in length (e.g., three yearseducation in economics or four years education in law). It can be reduced to thebasic Mincer model by setting s0 = 0 and s1 at the relevant value for a particulareducation (high school, university), or at s1 = 1 to study marginal investments. Butit excludes the option value of education, a worthy target for future work.

Let us start simply by computing the internal rate of return to s1 rather than s0

years of schooling, for an infinitely lived individual (T = ∞), with potential earn-ings functions that include independent stochastic components u0 ∼ (0, σ0

2), u1 ∼(0, σ1

2). We assume only one lifetime shock. The amount of human capital

f S t S e dt f S t S e dtt

S

S Tt

S

S T

0 0 1 1

0

0 0

1

1 1

1, ( ) , ( ) ( )−( ) = −( )−+

−+

∫ ∫δ δ

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produced in school is unknown when entering but revealed at the predeterminedtime of leaving, and then determines annual earnings for the rest of the workinglife. The profiles differ in average returns to schooling years and to experience.For the moment we ignore the usual quadratic term in experience. The internalrate of return then follows from

Working out equation (2) we obtain:

If , we can rewrite equation (3) as:

Equation (4) is a generalization of the Mincer specification. With equal means inshock exponentials, identical experience profiles and minimum level of educationzero (s0 = 0), we obtain δ = β3: the internal rate of return equals the coefficient inthe earnings function. Generally, in equation (4), the internal rate of return is thereturn for selecting a longer education with a different reward per year of school-ing and per year of experience. The latter feature is routinely neglected. If highereducation brings more earnings growth, this boosts the returns to education.1

Note that β3 measures the average return per school year for s1 years of schoolingand β1 measures the average return per school year for s0 years of schooling. Inempirical earnings functions with dummies for different levels of education, aver-age returns for longer educations are often lower than for shorter education. Thisdepresses the internal rate of return δ.

Equation (4) has to be solved numerically for δ. If s0 = 0 or β1 = β3, δ will begiven by the transcendent equation:

The effect of different experience profiles (β2 ≠ β4) can be substantial. Suppose, thereturn to school years is 0.065; with equal experience growth terms this would bethe internal rate of return. Now, let the experience growth differ by one percentagepoint: β2 = 0.05 for s0 = 0 and β4 = 0.06 for s1 = 4. Then the internal rate of return δ= 0.160. If β2 = 0.01 for s0 = 0 and β4 = 0.015 for s1 = 4, then the internal rate of returnδ = 0.114. Both are substantially higher than in case of equal experience slopes.

In the more general case with u0, u1 normally distributed with means µ0, µ1,variances σ0

2, σ12 and correlation ρ, we can, as shown in Appendix 3, approximate

expectation and variance by:

E e dt E e e dts t s e

S

s t s t

S

tβ β µ β β µ δδ1 0 2 0 0

0

3 1 4 1 1

1

2+ − +∞

+ − + −∞

−=∫ ∫( ) ( ) ( )

E e ee

E e eeu s

su s

s

( ) ( ) ( )( )( )

( )( )

0 1 2 0

2 0

1 3 4 1

4 1

2 4

3β ββ δ

β ββ δ

δ β δ β−

−−

−

−=

−

E e E eu u0 1( ) = ( )

δβ β δ β

δ β=

−−

+−−

3 1 1 0

1 0

2

4

4s ss s

ln ( )

δ βδ βδ β

= +−−3

2

4

5ln ( )

Es s

δ µ µσ σ ρσ σ

β≈−

− ++ −

+

1 2

26

1 01 0

12

02

1 03 ( )

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Again this is easily recognized as a generalization from the standard Mincer casewith µ0 = µ1 = 0, σ0 = σ1 = 0. The expected return is positive in the differencebetween the shock means and negative in the correlation between the shocks. Theexpected return is negative in σ0 and positive in σ1 if σ0/σ1 > ρ. It is positive in σ0

and negative in σ1 if σ0/σ1 < ρ, which certainly holds if the correlation is negative.The variance of the rate of return is independent of the shock means and respondsto correlation and variances with the same signs as the expectation. Note that theexpected return responds to the variances of the shocks, which may come as aninitial surprise, but will also be intuitively clear upon reflection, as large variancesincrease the probability of large incomes in either alternative and this must affectreturns. With perfect (positive) correlation and identical shock variances, the vari-ance of the return is equal to zero: with equal shocks for both schooling options,the shocks have become irrelevant. If the standard deviations of the shock bothincrease by the same proportion, the expectation and variance of the rate of returnare unaffected. If the standard deviations increase by the same amount (but byless than 100% of the original levels), the expectation and variance of the returnboth increase.

A Framework for Simulation

If we add a quadratic experience term, as commonly estimated, and allow forannual shocks instead of a single lifetime shock, possibly correlated over time, thesolution can no longer be derived analytically. We must then resort to numericalsolutions. We will apply the model to simulations for individuals who may leaveschool after completing high school or continue their education in college. Theearnings functions are:

β1,HS, β1,C are the average rates of return to sHS and sC years of schooling, respec-tively. β2,HS, β3,HS and β2,C, β3,C determine the effects of experience and experiencesquared for an individual with sHS and sC years of schooling respectively. The errorsare generated by first-order autoregressive process (AR(1)) processes of the form:

We suppress the individual subscript as we only deal with the perspective of asingle individual. We assume that ηt,HS is i.i.d. N(0, σHS

2) and ηt,C is identicallyand independently distributed. N(0, σC

2). We study the case when ηt,HS and ηt,Care uncorrelated at any t, as well as the case when ηt,HS and ηt′, C correlate at ρHS, Cfor equal experience t = t′ and at zero otherwise.

The inter-temporal correlations are:

Vs s

eδ σ σ ρσ σ≈−

−( )+ −11 7

1 02

202

22

0 1

( )( )

ln t s t s ut

S twHS HS HS HS HS HS HS HS= + − + − +β β β1 2 3

2 8, , , ,( ) ( ) ( )

ln t s t s ut tSwC C C C C C C C, , , , ,( ) ( ) ( )= + − + − +β β β1 2 3

2 9

u uHS t HS HS t HS t, , , ( )= +−γ η1 10

u uC t C C t C t, , , ( )= +−γ η1 11

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It may seem that with this specification we only allow for transient shocks duringworking life and not for permanent shocks emanating from effectiveness ofhuman capital production in school. Alternatively, one might have expected aspecification with uncertain effectiveness of schooling reflected in a single life-time shock revealed upon completion of schooling, combined with annual earn-ings shocks (cf. Chen, 2001). However, only the lifetime shock from one schoolinglevel relative to the other is relevant. The shocks may be perfectly correlated indi-cating that the individual would do as well in one education as in the other, as ina model with hierarchical ability. Or they may be perfectly negatively correlatedreflecting perfect comparative advantage: being the best in one education wouldconcur with being the worst in the other. The essence of such cases can be caughtin the correlation between annual innovations η: comparative advantage wouldbe reflected in negative correlation and hierarchical ability in positive correlation.Our specification can therefore describe the options to a large extent. If ρHS, C = +1talent is something like a general ability that puts an individual in the sameperformance rank with every education he/she pursues, whereas at ρHS, C = −1two different educations completely reverse the individual’s standing.

We consider a working lifespan T of 40 years, independent of the length ofschooling. An individual record consists of 40 draws of the disturbance term ut,HS,used to predict earnings with sHS years of education for fixed values of βi,HS (I ∈{1, 2, 3}), and 40 draws of the disturbance term ut,C from an alternative distribu-tion with sC years of education, added to predicted earnings from the associatedβi,C (I ∈ {1, 2, 3}) for that education. For such an individual record, we solvenumerically for the internal rate of return δ. This process is repeated 100 000times, with 100 000 new sets of draws for the two earnings profiles. We thencalculate the mean and standard error of δ from the 100 000 runs. We repeat thisfor several sets of parameter values. As explained in Appendix 1, we rewrote thestochastic specification for easier computation.

Parameter Values

In Appendix 2 we scan the empirical literature for the possible magnitudes of ourparameters. In principle, all parameters are individual specific. However, we willnot provide for variation of all parameters, as some variations are not interestingfor our conclusions. For example, we will not vary the basic return to years ofschooling, as this will not yield any surprises. Thus, for the return to a year ofschooling we assume a rate of 0.065 throughout, without alternatives. Thisimplies that our benchmark internal rate of return is 0.065, the rate that wouldresult in a Mincer world. For the experience profile we take a linear term of 0.05and a quadratic term of −0.001 as our reference values. As an alternative, we setthe quadratic term for high school at −0.002, maintaining the college quadraticterm at −0.001; this means that the decline of earnings growth with experience forcollege education is half the decline for high school education.

The core parameter for our purpose is the variance of the residual. Our readingof the evidence indicates that residual earnings standard deviations are generallybetween 0.25 and 0.65; we take that as our range of variation, with the basic reference

ρ γ

ρ γ

u u

u u

HS t HS t HS

C t C t C

, ,

, ,

,

,

−

−

( ) =

( ) =

12

12

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value in the middle (0.45). In our simulations, we allow residual variance for collegeearnings to be both larger and smaller than for high school earnings. Increasing riskby level of education may empirically be a realistic case and certainly is theoreticallyinteresting as it provides a counterweight against the attraction of a fair rate ofreturn. For the persistence term γ, we use 0.6 as our preferred value; we will allowvariation to vary the relative weight of the innovation in the earnings process. Weare fairly confident that these are reasonable values, based upon our reading of theempirical literature.2 We are least confident about the correlation across educations,simply because there is no empirical evidence to guide us, in spite of all the empha-sis it gets in the self-selection literature. Willis and Rosen (1979), who started thisliterature, could not identify the correlation. Carneiro et al. (2003) provide probablythe first empirical evidence, and their results suggest modest positive correlationfor college compared with high school. We opt for 0.5 as our preferred value, butalso consider the extremes of −1 and + 1. We set sHS = 0 and sC = 4, thus calculatingthe return to university education after completing high school.

Simulation Results

The core results are presented in Table 1.The first row gives the reference case wedefined above: 0.6 persistence, 0.5 cross-education correlation, and identicalresidual correlations of 0.45. Moving from a risk-less world to stochastic earningsprofiles increases the expected rate of return from 0.065 to 0.071 in our benchmarkparameter set, and generates a standard deviation of 0.031 (i.e., a coefficient ofvariation just under 0.5). In Figure 1a we have graphed the entire frequencydistribution of 100 000 draws. Interestingly, the distribution is skewed to theright, with an elongated upper tail. This feature holds for all the simulations weran. With individuals generally not only caring for risk but also for skewness, thisis an interesting observation (cf. Hartog and Vijverberg, 2002). The degree ofskewness varies with the parameters. In Figure 1b we show the case with themost skewed distribution in our parameter set, obtained when we set the coeffi-cient of correlation at −1 rather than at our reference value of 0.5.

As anticipated from the linear single-shock case, differences in earnings profileshave a strong effect on expected returns. A percentage point difference in thelinear term boosts the return by almost three per cent, cutting the decline in earn-ings growth for college in half relative to high school boosts it by almost four percent. Changes in the experience profile barely affect the variance in the rate ofreturn.

The variances panel in Table 1 has three components: joint change in shockvariance, change in the low-education shock and change in the high-educationshock. Increasing the standard deviations in both earnings profiles simulta-neously, while maintaining equality, has a very small effect on expected returnsand a substantial effect on its variance. Both increase monotonically if the shockvariance increases; but while the increase in the expectation is a mere 18%, thevariance increases four-fold between the minimum and maximum values consid-ered. Thus, for the expected return the results are close to the zero effect found forthe linear single-shock case; for the variance they are much stronger. In thesecond block of the shock variance panel, the shock dispersion in the low educa-tion increases monotonically. The increase reduces the mean and increases thevariance of the rate of return, with a big discontinuity when the ratio of the stan-dard deviations starts to surpass the correlation (when the high school shock

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dispersion increases from 0.20 to 0.45). On either side of the discontinuity, theeffects are small. The discontinuity threshold reminds of the effects we found inthe linear single shock model. In the third block, the shock dispersion of the higheducation increases monotonically, but again without monotonic effects. Bothexpected return and variance increase with increasing dispersion of the higheducation shock on both sides of the discontinuity, where the discontinuity is

Table 1. Simulation results

β1,C β2,C β3,C β1,HS β2,HS β3,HS σuCσuHS γC γHS ρHS,C E(δ) σ(δ)

Reference case

0.065 0.05 −0.001 0.065 0.05 −0.001 0.45 0.45 0.6 0.6 0.50 0.071 0.031

Experience Slopes

−0.002 0.110 0.031

0.04 0.099 0.033

Variances

0.25 0.25 0.067 0.014

0.35 0.35 0.069 0.022

0.55 0.55 0.075 0.043

0.65 0.65 0.079 0.057

0.45 0.15 0.037 0.170

0.45 0.20 0.036 0.173

0.45 0.25 0.085 0.028

0.45 0.35 0.079 0.029

0.15 0.45 0.087 0.034

0.20 0.45 0.112 0.048

0.55 0.45 0.085 0.040

0.65 0.45 0.101 0.051

Persistence over Time

0.0 0.0 0.067 0.016

0.2 0.2 0.068 0.020

0.4 0.4 0.069 0.025

0.8 0.8 0.074 0.084

0.0 0.2 0.068 0.018

0.0 0.4 0.069 0.021

0.0 0.8 0.078 0.035

0.2 0.0 0.067 0.018

0.4 0.0 0.067 0.020

0.8 0.0 0.063 0.024

Correlation in Alternatives

−1.00 0.081 0.056

−0.75 0.080 0.054

−0.50 0.078 0.050

−0.25 0.077 0.046

−0.10 0.076 0.044

0.00 0.075 0.042

+0.10 0.074 0.040

+0.25 0.073 0.037

+0.75 0.069 0.024

+1.00 0.068 0.017

Note: Parameters have the reference value stated in the top row, unless a different value is stated.

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now defined by the difference in dispersions: when the high education dispersionceases to be smaller than the low education dispersion, both the mean and vari-ance of the return drop substantially. Clearly, the relation between shock disper-sions and the distribution of returns is more complicated than in the linear singleshock system but certainly not without regularities.

If correlations over time (persistence γi, i ∈ {HS, C}) increase jointly, expectedreturns go up but the dispersion increases non-negligibly. This reflects thatalthough the variance of u is itself unaffected (we constrain it to be constant), theconditional variance (conditional on the past draw) increases. If we only vary oneof the inter-temporal co-variances, the dispersion of the rate of return increases ineither case. But the expected return reacts in opposite ways, increasing with highschool correlation but decreasing with college correlation.

Figure 1. Distribution of internal rates of return. (a) ρ = 0.5, σuHS = σuC = 0.45, γHS = γC = 0.6; β1,HS = β1,C = 0.065, β2,HS = β2,C = 0.05, β3,HS = β3,C = −0.001. (b) ρ = −1, σuHS = σuC = 0.45, γHS = γC = 0.6; β1,HS =

β1,C = 0.065, β2,HS = β2,C = 0.05, β3,HS = β3,C = −0.001.

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Correlation across educations has a monotonic effect on expected return anddispersion. Both decline when the correlation increases from −1 to +1. But theeffect on dispersion is much stronger than on expected return. Positive correlationdampens stochastic differences, negative correlation widens them. With perfectpositive correlation, the standard deviation is about one-half that in our referencecase (correlation 0.5); with perfect negative correlation, the standard deviation isalmost double that in the reference case.

Now let us assess the likely magnitude of risk involved in investing in school-ing. In our basic Mincer case the internal rate of return is 0.065, with zero disper-sion. In what we consider a realistic case, college education would give access tosteeper experience profile (earnings growth slope of −0.001 instead of −0.002),annual shocks would have a dispersion of 0.45 for both educations, persistencewould be 0.60 in both educations and the shocks would correlate at 0.50. Thiswould generate an expected rate of return of 0.110, with standard deviation 0.031(coefficient of variation 0.28). Hence for the case of college versus high schooleducation, we consider a coefficient of variation in the rate of return of about 0.3our most reasonable guess. But given the uncertainty about parameter values, wehave to admit a wide range of possible outcomes. In our simulations, the standarddeviation lies between 0.014 and 0.173. The lowest value is obtained when theinnovations in the earnings both have minimum standard deviation (0.25), andthe highest value is obtained when the standard deviations in earnings shocks are0.45 and 0.20 for college education and high school education, respectively. In theformer case, at the lowest dispersion, the coefficient of variation is 0.2; in the lattercase, it is 4.80.

Conclusions

Ex post realizations of Mincerian rates of return to education show fairly widevariation across countries (up to double or triple in one country relative toanother, with coefficient of variation of perhaps 0.5), modest variation over timewithin countries (with a country’s maximum generally not more than one-thirdabove its minimum, in a time series) and coefficient of variation across individu-als within a country of perhaps 0.5. To the extent that the results also reflect indi-vidual heterogeneity, and individuals are better informed about their potential,individual risk may be smaller than reflected in these ex post realizations.

From our simulations of ex ante risk we conclude that a coefficient of variationof about 0.3 is a reasonable guess. As the relation between risk and return is at theheart of financial investment theory, we may turn to that literature for somebenchmark information. Fisher and Lorie (1970) gave an overview of returns toportfolio’s on the New York Stock Exchange. They calculated one-year meanreturns and standard deviations for randomly selected portfolios differing in size.All portfolios had a mean return of 28.2%. But with the portfolio size increasingfrom one to eight and then further to 32 and 128, the standard deviationdecreased: 41.0, to 14.4, to 7.1, to 3.4. With increasing diversification, the coeffi-cient of variation appears to drop from 1.45 to 0.12. These results suggest that, interms of risk, investment in a college education is similar to investing in the stockmarket with a portfolio of some 30 randomly selected stocks. We also found thatthe distribution of the internal rate of return is skewed to the right, with an elon-gated upper tail. This matches results found by Maier et al. (2004) in their focus onheterogeneity of returns to education.

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The expected rate of return is not independent from the earnings dispersions inthe alternatives. Hence, allowing for stochastic earnings is not something that onlyaffects the variance in the rate of return to education. This is shown analytically ina model with a single lifetime shock to earnings (upon graduation) and furtherdemonstrated in simulations for a model with annual shocks in earnings. For real-istic variations in parameter values, the expected rate of return can easily doublein magnitude. The variance in the rate of return is generally even more sensitive toparameter values than the expectation. If the standard deviations in the earningsinnovations increase from their joint low of 0.25 to their joint high of 0.65, the stan-dard deviation of the rate of return increases four-fold. If the correlation increasesfrom −1 to +1, the standard deviation increases more than three-fold.

Unintentionally, we have found substantial effects on the expected rate ofreturn. Differences in earnings growth rates for different educations can easilybring an extra four per cent return. While obvious, this effect is routinely over-looked. Less obvious, just introducing stochastic components in earnings profileshas a marked effect on the expected rate of return. When in the risk-less Mincerworld the rate of return would be 0.065, in our reference case it has an expectedvalue of 0.071. With increasing differences in shock distributions between thealternatives, the difference can easily increase to several percentage points.

We conclude that investment in human capital indeed carries a substantial riskand, therefore, risk aspects in human capital are worthy of further research. Sincewe also know that individuals are generally risk averse, in different degrees(Hartog et al., 2002), it is clear that an interesting agenda is waiting. It is wellknown that women are more risk averse than men, and that risk aversion tends tobe higher in poorer households. Thus, the results of substantial risk may beimportant to understand gender and social differences in schooling participationbehaviour, not only by level but also by type of education.

Acknowledgements

Comments by two referees are gratefully acknowledged.

Notes

1. The same point is made by Heckman et al. (1999, p 331).2. By taking the parameter values as we found them in the literature, without correction for selectiv-

ity or heterogeneity, we assume full ignorance on the position in future distributions. If individu-als have better information, their risk will be reduced. This may be reflected in variances near thelow end of the intervals, and possibly even lower (as the observed values would be biased).

References

Bajdechi, S. M. (2005) Essays on education and risk, PhD Dissertation, University of Amsterdam,Tinbergen Institute.

Becker, G. S. (1964; 1993) Human Capital, 3rd edn (Chicago, IL: University of Chicago Press).Blomquist, N. S. (1976) The distribution of lifetime income, a case study of Sweden, PhD Dissertation,

Princeton University, NJ.Brunello, G. and Comi, S. (2000) Education and earnings growth: evidence form twelve European

countries, IZA Discussion Paper 140 (Bonn).Card, D. (1995) Earnings, schooling and ability revisited, in: S. Polachek (Ed.) Research in Labor Economics,

pp. 23–28 (Greenwich, CT: JAI Press).Card, D. (1999) The causal effect of education on earnings, in: O. Ashenfelter and D. Card (Eds) Handbook

of Labor Economics, 3A, Chapter 30, pp. 1801–1863 (Amsterdam: North Holland).

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Carneiro, P. et al. (2003) Estimating distributions of treatment effects with an application to the returnsto schooling and measurement of the effects of uncertainty on college choice, NBER Working Paper9546. Boston: NBER.

Chen, S. (2001) Is investment in college education risky? Discussion paper, State University of NewYork,.

Davis, S. and Willen, P. (2000) Occupation-level income shocks and asset returns: their covariance andimplications for portfolio choice, NBER Working Paper 7905. Boston: NBER.

Dominitz, J. and Manski, C. F. (1996) Eliciting student expectations of the returns to schooling, Journalof Human Resources, 31(1), pp. 1–26.

Eden, B. (1980) Stochastic dominance in human capital, Journal of Political Economy, 88(1), pp. 135–145.Fagerlind, I. (1975) Formal Education and Adult Earnings (Stocholm: Almquist and Wiksell).Fisher, L. and Lorie, J. (1970) Some studies of variability of returns on investments in common stocks,

Journal of Business, 43(2), pp. 99–134.Hartog, J. et al. (2002) Linking measured risk aversion to individual characteristics, Kyklos, 55(1),

pp. 3–26.Hartog, J. and Vijverberg, W. (2002) Do wages really compensate for risk aversion and skewness affec-

tion? IZA Discussion Paper 426 (forthcoming in Labour Economics, Special Issue on Education andRisk). Bonn: IZA Institute for study of Labour.

Hause, J. C. (1972) Earnings profile: ability and schooling, Journal of Political Economy, 80(3II), S108–S138.Hause, J. C. (1980) The fine structure of earnings and on-the-job training hypothesis, Econometrica,

48(4), pp. 1013–1029.Heckman, J. et al. (2003) Fifty years of Mincer earnings regressions, IZA Discussion Paper 775. Bonn:

ZEW Centrum for Europaische Wirtschaftsforschung Mannheim.Heckman, J. et al. (1999) General equilibrium cost benefit analysis of education and tax policies, NBER

Working Paper 6881. Boston: NBER.Lillard, L. A. and Weiss, Y. (1979) Components of variation in panel earnings data: American Scientists

1960–70, Econometrica, 47(2), pp. 437–454.Maier, M. et al. (2004) Returns to education and individual heterogeneity, ZEW Discussion Paper 04-34

(Mannheim).Mincer, J. (1974) Schooling, Experience and Earnings (New York: Columbia University Press).Palacios-Huerta, I. (2003) An empirical analysis of the risk properties of human capital returns, American

Economic Review, 93(3), pp. 948–964.Webbink, D. and Hartog, J. (2004) Can students predict their starting salaries? Yes!, Economics of

Education Review, 23(2), pp. 103–113.Willis, R. J. and Rosen, S. (1979) Education and self-selection, Journal of Political Economy, 87(5),

pp. S7–S36.

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Appendix 1. The simulation procedure

Our simulation problem is stated for certain values of the variance in errors:, correlation between errors: ρHS, C, and inter-temporal correlations: γHS,

γC. Therefore, firstly we set the values of these parameters in a vector

Given this targeted structure, we construct the errors uHS,t and uc,t, t from 0 to40, as follows:

-at time t=0:

with ηHS,0, ηC,0 independent and normally distributed and ,respectively.

-at time t, 1 ≤ t ≤ 40,

where the innovations ηHS,t, ηC,t are independent, normally distributed N(0,σHS2)

and N(0,σC2) respectively. The variances of the innovations are obtained from

and . The correlation between the errors uHS,t

and uc,t can be set by controlling the correlation between the innovations ηHS,t,

ηC,t. In order to do this, the innovations are generated using four scalars λ1, λ2, λ3,

λ4, and three independent random variables ε1, t, ε2, t, ε3,t, normally distributed

, such that:

Taking variances in (I), (II), (III), and (IV) we write the equations system (wedropped the time subscripts):

where the correlation in innovations, let us denote it ρ(ηHS, ηC), satisfies

.

With the parameters initially set in the vector , we

want to find the values for that satisfy the

constraints in equations system (V).

σ σu uHS C,

p u u HS C HS CHS C= ( , , , , ).,σ σ ρ γ γ

u uHS HS C C, , , ,,0 0 0 0= =η η

N uHS( , )0 2σ N uC

( , )0 2σ

u u IHS t HS HS t HS t, , , ( )= +−γ η1

u u IIC t C C t C t, , , ( )= +−γ η1

σ γ σHS HS uHS

2 2 21= −( ) σ γ σC C uC

2 2 21= −( )

N ii

( , ), ,0 1 32σε =

η λ ε λ εHS t t t III, , , ( )= +1 1 2 2

η λ ε λ εC t t t IV, , , ( )= +3 2 4 3

σ γ σ σ γ σ λ σ λ σ

σ γ σ σ γ σ λ σ λ σ

ρ σ σ γ γ ρ σ σ λ λ σ

ε ε

ε ε

ε

u HS u HS HS u

u C u C C u

HS C u u HS C HS C u u

HS HS HS

C C C

HS C HS C

V

2 2 2 2 2 212 2

22 2

2 2 2 2 2 232 2

42 2

2 32

1 2

2 3

2

= + ≡ + +

= + ≡ + +

= +

( )

, ,

ρ η η σ σ λ λ σε( , )HS C HS C = 2 32

2

p u u HS C HS CHS C= ( , , , , ),σ σ ρ γ γ

σ λε ii jj, , , , , , , ,∈{ } ∈{ }1 2 3 1 2 3 4

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Hence, we have to solve an over-determined equations system that has three

equations and seven unknowns . We have to set the

four freedom degrees (for instance ) and solve the system for the

remaining three unknowns (λ2, λ3, λ4).

Appendix 2. Finding parameter values for the simulations

We have scanned the literature for our key parameters, with the aim of selectingparameter values in a representative range. In no way have we aimed for acomplete review of the empirical literature.

Rates of returns to schooling and experience

The Mincer parameters have been documented very often. Estimates of rates ofreturn to education have been controversial because they are based on ex-post real-izations and need not reflect structural parameters necessary for correct predic-tions. They are subject to selectivity effects from individuals’ schooling choices.However, it is not inconceivable that individuals deciding on extending or nottheir education use uncorrected, biased estimates simply because they do nothave more information than a researcher. A meta-analysis based on the collectedOLS estimated rates of return to schooling from the PURE project and supple-mented by a number of findings for the USA points to a range for return to school-ing from 4.5% to 9.5% with an average return of 6.5%. We take this as our basevalue. We note that it is lower than the value reported by Card (1995, 1999), rang-ing from 8 to 13%, but in fact the base return is not very essential to our results.

Using dummies for educational levels and accounting for the number of yearsusually required for completing a degree at each particular level is a way ofcomparing the returns to different educational levels. Obviously, the outcomevaries considerably depending on the number of years assigned to each educa-tional level. The PURE project (Table 4.5 page 76) reports for Finnish men a rate ofreturn to one year of upper secondary school at 0.071 and to one year of college at0.059. Chen (2001, Table 7, page 32) presents marginal returns to years of school-ing of about 4.5% for two-year college and 6.5% for four-year college educationrespectively. Thus, we have conflicting evidence on returns as a function of lengthof education.

Experience in Mincer specifications is seldom directly measured in typical datasets and is often proxied by age minus the age of leaving school. We read the maleaverage value of the experience coefficient to be 0.05 and the experience squaredcoefficient −0.001. The steeper experience/age profile for the higher educated iswell documented (Heckman et al., 2003; Brunello and Comi, 2004; Chen, 2001).

Parameters in the error structure

When the log of earnings is the dependent variable, the estimated standard devia-tion of the residual usually lies in the interval (0.25, 0.65). This assertion is basedon a causal inspection of empirical earnings functions studies, including Blomquist(1976), Fagerlind (1975), Hause (1972) and Mincer (1974).

Comparing characteristics of college graduates and high-school graduates,Chen (2001, Table 3, page 28) shows that the college graduates in her sample have

λ λ λ λ σ σ σε ε ε1 2 3 4 1 2 3, , , , , ,

λ σ σ σε ε ε1 1 2 3, , ,

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10% higher standard errors in average real hourly wage. Becker (1964; 1993) andChen (2001) use the coefficients of variation in earnings to measure earningsuncertainty for those who attended college vs. those who did not. Both find thatfour-year college graduates exceed high-school graduates in terms of variation inearnings. Chen (2001) defines risk for a given schooling choice as the variance inpermanent and transitory shocks conditional on schooling, individual character-istics and scholastic ability. The time invariant individual effect represents thepermanent shock whose variance is adjusted for selection bias. With this methodChen finds the risk, i.e. the variance of log earnings conditional on individualcharacteristics, of 0.385 for high-school graduates and of 0.449 for college gradu-ates (Table 8 page 33). For other results on earnings variance in relation to school-ing length, we refer to Table 3.

If the shocks of alternative educations’ paths are not independent we have toassume unrestricted covariances among the unobservable. As Willis and Rosen(1979) noted, there may be negative covariance among talent components. Plumb-ers (high-school graduates at most) may have very limited potential as highlyschooled lawyers, but by the same reasoning lawyers may have much lowerpotential as plumbers. This contrasts with single factor specifications (IQ) in theliterature that assume that the best lawyers would also be the best plumbers andwould imply strictly hierarchical sorting in the absence of financial constraints. Ineffect an IQ ability model constrains to large positive covariances in the unob-served ability components. Willis and Rosen only speculate on these matters: theycannot identify the relevant parameters.

Hause (1980) assumes that the covariance matrix of log earnings time series iscomposed of an individual-specific parameter related to the amount of ‘on-the-jobtraining’ received and of a variance component generated by a non-stationaryAR(1) process with time varying autoregressive parameters (ui,t=γt-1ui,t-1+εi,t), andinnovations independent across periods and individuals, with time varying vari-ances σtt. Hause obtains estimates of the variance of individual earnings profilesover six years for a sample of young Swedish males in their twenties with elemen-tary and intermediate school education. Hause omits the ‘On-the-job training’structure and finds (Table I, model 4, page 1023) a model that still fits the datafairly well. He finds an average AR parameter of 0.63 (asymptotic standard error0.4) and an average innovation variance of 0.045 (asymptotic standard error 0.4).

Lillard and Weiss (1979) provide a parameterization of an earnings functionthat incorporates three distinct aspects of the residual covariance structure overtime: individual differences in the level of earnings, individual differences in thegrowth of earnings and transitory but serially correlated differences. The sampleonly contains highly educated individuals: a few categories of American scien-tists. The parameter estimates are quite stable across different scientific fields

with an average γ of 0.52 and an average of 0.0072.They find that the residual

log earnings variance of chemistry scientists varies from 0.042 to 0.067. Maximumlikelihood estimates of the residual covariance structure for chemists include

γ=0.43 and =0.0191 (Table II, page 444). For the same category of scientists they

ignore the variance components in growth and level and obtain γ=0.88 and

=0.0133.

Carneiro et al. (2003) have developed a model that identifies the counterfactualdistribution of outcomes for ‘treated’ and ‘non-treated’ individuals. They apply

ση2

ση2

ση2

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the model to the decision to attend college or not. In their Table 8, they give theearnings decile probabilities for college attendance conditional on earnings decilefor high school attendance. Probabilities are fairly concentrated along the maindiagonal, which motivates our choice of a reference value for the disturbancecorrelation coefficient between high school and college of 0.5.

Ex post variability in returns is not the same as ex ante risk. Selectivity, exten-sively highlighted in the recent literature, is an important cause of deviations. It isan interesting question to what extent heterogeneity coincides with risk for theindividual investor. If the individuals themselves are imperfectly informed ontheir abilities, future efforts, job opportunities, etc., heterogeneity comes close toex ante risk. Observed residual earnings variance is an unknown mixture of heter-ogeneity and risk. The distinction between the two is actually not relevant for thestructure of our simulation model, as it can accommodate both foreseeable heter-ogeneity (variation between individuals) and risk. The distinction is mostly rele-vant when it comes to selecting the parameter values in the simulations. We takekey parameters from a survey of the empirical literature, without paying muchattention to this distinction. Without digging deeply into the question, let us notesimply that Webbink and Hartog (2004) found that freshman in university couldnot even predict at any acceptable level of reliability their starting salary fouryears ahead: the correlation between predicted and realized starting salary is 0.06.Hence, most of the ex post variance may very well reflect true ex ante ignoranceand hence risk for the individual.

Appendix 3. Deriving equations (6) and (7)

We are interested in the distribution of exp(u1-u0) if (u1-u0) is a random variablenormally distributed N(µ1-µ0, σ2

0+σ21-2ρσ0σ1). In order to derive our result, we

use a standard rule:If x is a continuous variable with pdf fx(x) and if y=g(x) is a continuous monotonicfunction of x, then the density of y follows: fy(y)=fx(g

−1(y))/g−1′(y)/(C.1.)where the term g−1(y) is the inverse of g and /g−1′(y)/ is the Jacobian of the trans-formation from x to y, in absolute value.

In our case, x is normally distributed N(µ,σ2) (µ=µ1-µ0, σ2=σ2

0+σ21−2ρσ0σ1) then

If exp(x)=y=g(x), then x= g−1(y)=ln(y) (C.3.)

The Jacobian is given by /

From (C.1.), (C.2.), (C.3.) and (C.4.) we get:

which corresponds with the probability distribution function of a lognormaldistributed variable. Hence, y = exp(x) is LN (µ, σ2) and its mean and variance aregiven by:

f x e Cx

x

( ) ( . .)

( )

=−

1

22

2

22

σ π

µσ

g ydxdy

d ydy y

Cx y

− = ==

↓=

1 14

'

ln

( )/ln

( . .)

f y eyy

y

( )

(ln )

=−

1

2

12

22

σ π

µσ

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If w=E(x) then E(h(x))≅h(w) and V(h(x)) ≅[h′(w)]2V(x), where h′ is the first deriva-tive of h with respect to x). In our case, h=ln and we can approximate E(ln(exp(x))by lnE(exp(x)) and V(ln(exp(x)) by (1/E(exp(x)))2V(exp(x)), therefore:

E x e

V x e e

(exp( ))

(exp( ))

=

= −( )+

+

µ σ

µ σ σ

2

2 2

2

2 1

E e e

V e e e

u u

u

( )

( )

1 01 0

02

22

0 1

1 0 1 0 02

22

0 1 02

22

0 1

2

2

2 2 2 1

− − ++ −

− − + + − + −

=

( ) = −( )µ µ

σ σ ρσ σ

µ µ µ σ σ ρσ σ σ σ ρσ σ

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Documents

Simulating the Risk of Investment in Human Capital