Financial Analysts Journal ©2006, CFA Institute Human ...corporate.morningstar.com/ib/documents/TargetMaturity/...January/February 2006 97 Financial Analysts Journal Volume 62 •

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Financial Analysts JournalVolume 62 • Number 1

©2006, CFA Institute

Human Capital, Asset Allocation, andLife Insurance

Peng Chen, CFA, Roger G. Ibbotson, Moshe A. Milevsky, and Kevin X. Zhu

Financial planners and advisors increasingly recognize that human capital must be taken intoaccount when building optimal portfolios for individual investors. But human capital is notsimply another pre-endowed asset class; it contains a unique mortality risk in the form of the lossof future income and wages in the event of the wage earner’s death. Life insurance hedges thismortality risk, so human capital affects both optimal asset allocation and demand for lifeinsurance. Yet, historically, asset allocation and life insurance decisions have been analyzedseparately. This article develops a unified framework based on human capital that enablesindividual investors to make these decisions jointly.

cademics and practitioners increasinglyrecognize that the risk and return charac-teristics of human capital, such as wageand salary profiles, should be taken into

account when building portfolios for individualinvestors. Merton (2003) pointed out the impor-tance of including the magnitude of human capital,its volatility, and its correlation with other assets inasset allocation decisions from the perspective ofpersonal risk management. The employees ofEnron Corporation and WorldCom sufferedextreme examples of this risk. Their labor incomeand their financial investments in the companiesprovided no diversification, and they were heavilyaffected by their companies’ collapses.

A unique aspect of an investor’s human capitalis mortality risk—that is, the family’s loss of humancapital in the event of the wage earner’s death. Lifeinsurance has long been used to hedge againstmortality risk. Typically, the greater the value ofthe human capital, the more life insurance the fam-ily demands. Intuitively, therefore, human capitalaffects not only optimal asset allocation but alsooptimal life insurance demand. These two impor-

tant financial decisions are generally analyzed sep-arately, however, in theory and practice. We foundfew references in either the risk/insurance litera-ture or the investment/finance literature to theimportance of considering these decisions jointlywithin the context of a life-cycle model of consump-tion and investment. In other words, popularinvestment and financial planning advice abouthow much life insurance one should carry is sel-dom framed in terms of the riskiness of one’shuman capital. And optimal asset allocation, whichhas only lately started to be framed in terms of therisk characteristics of human capital, is rarely inte-grated with life insurance decisions.

Motivated by the need to integrate these twodecisions, we merged these traditionally distinctlines of thought together in one framework. Weargue that these two decisions must be determinedjointly because they serve as risk substitutes whenviewed from the perspective of an individualinvestor’s portfolio. Life insurance is a perfecthedge for human capital in the event of deathbecause term life insurance and human capitalhave a negative 100 percent correlation with eachother in the “alive” (consumption) state versus“dead” (bequest) state. If insurance pays off at theend of the year, human capital does not, and viceversa. Thus, the combination of the two providesgreat diversification to an investor’s total portfo-lio. Figure 1 illustrates the types of decisions theinvestor faces, together with the variables thataffect the decisions.

The unified model we discuss is intended toprovide practical guidelines for optimal asset

Peng Chen, CFA, is managing director and chief invest-ment officer at Ibbotson Associates, Chicago. Roger G.Ibbotson is professor of finance at Yale School of Man-agement, New Haven, Connecticut. Moshe A.Milevsky is associate professor of finance at the Schu-lich School of Business at York University and execu-tive director of the Individual Finance and InsuranceDecisions Centre, Toronto. Kevin X. Zhu is seniorresearch consultant at Ibbotson Associates, Chicago.

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allocation and life insurance decisions for individ-ual investors in their preretirement years (accu-mulation stage).1

Human Capital and Financial CapitalAn investor’s total wealth consists of two parts.One is readily tradable financial assets; the other ishuman capital. Human capital is defined here asthe present value of an investor’s future laborincome.2 Economic theory predicts that investorsmake asset allocation and life insurance purchasedecisions to maximize their lifetime utilities ofwealth and consumption. Both of these decisionsare closely linked to human capital because,although human capital is not readily tradable, it isoften the single largest asset an investor has.

Typically, young investors have far morehuman capital than financial capital because theyhave more years to work than older workers andhave had fewer years to save and accumulate finan-cial wealth. Conversely, older investors tend tohave more financial capital than human capital.Figure 2 illustrates the patterns of financial capitaland human capital over an investor’s working (pre-retirement) years from age 25 to age 65.

Financial Asset Allocation and HumanCapital. The changing mix of financial capital andhuman capital over a life cycle affects allocations ofindividuals’ financial assets. In the late 1960s, econ-omists established models that implied that indi-viduals should optimally maintain constantportfolio weights throughout their lives (Samuel-son 1969; Merton 1969). Those models assumed,however, that investors have no labor income (i.e.,

no human capital). When labor income is includedin the model of portfolio choice, individuals opti-mally change their allocations of financial assets ina pattern related to the life cycle. In other words,optimal asset allocation depends on the risk–returncharacteristics of assets and the flexibility of theindividual’s labor income (such as how much orhow long the investor works). In our model, theinvestor adjusts the financial portfolio to compen-sate for nontradable risk exposures in human cap-ital (Merton 1971; Bodie, Merton, and Samuelson1992; Heaton and Lucas 1997; Jagannathan andKocherlakota 1996; Campbell and Viceira 2002).The key theoretical implications of the models thatinclude labor income are as follows: (1) Younginvestors will invest more in stocks than olderinvestors; (2) investors with safe labor income (thussafe human capital) will invest more of their finan-cial portfolio in stocks; (3) investors with laborincome highly correlated with stock markets willinvest their financial assets in less risky assets; and

Figure 1. Relationships among Human Capital, Asset Allocation, and Life Insurance

HumanCapital

Age

Labor Income

Mortality Estimates

Bequest Preferences

Savings Initial Wealth

Risk Aversion

Correlation between Human Capital and Financial Markets

Capital Market Assumptions

FinancialWealth

AssetAllocationDecision

LifeInsuranceDecision

Figure 2. Expected Financial Capital and Human Capital over the Working-Life Cycle

Capital

Human Capital

Financial Capital

25 6535 454030 50 6055

Age

Human Capital, Asset Allocation, and Life Insurance


(4) the investor’s ability to adjust his or her laborsupply (i.e., higher flexibility) also increases theinvestor’s allocation to stocks.

Empirical studies show, however, that mostinvestors do not, considering the risk of theirhuman capital, efficiently diversify their financialportfolios. Benartzi (2001) and Benartzi and Thaler(2001) showed that many investors use primitivemethods to determine asset allocations and manyof them invest very heavily in the stock of thecompany they work for.3

Life Insurance and Human Capital. Manyresearchers (e.g., Samuelson; Merton 1969) havepointed out that the lifetime consumption and port-folio decision models need to be expanded to takeinto account lifetime uncertainty (or mortality risk).Yaari’s 1965 article is considered the first classicalpaper on this topic. Yaari pointed out ways of usinglife insurance and life annuities to insure againstlifetime uncertainty. He also derived conditionsunder which consumers would fully insure againstlifetime uncertainty.4

Theoretical studies show a clear link betweenthe demand for life insurance and the uncertaintyof human capital. Campbell (1980) argued that formost households, labor income uncertainty domi-nates the uncertainty of financial capital income. Hefurther developed solutions, based on human cap-ital uncertainty, for the optimal amount of insur-ance a household should purchase.5 Smith andBuser (1983) used mean–variance analysis to modellife insurance demand in a portfolio context. Theyderived optimal insurance demand and the optimalallocation between risky and risk-free assets. Theyfound that the optimal amount of insurancedepends on two components: the expected value ofhuman capital and the risk–return characteristics ofthe insurance contract. Addressing the uncertain-ties and inadequacies of an individual’s humancapital, Ostaszewski (2003) went further by statingthat life insurance is the business of human capitalsecuritization. Empirical studies of life insuranceadequacy, however (e.g., Auerbach and Kotlikoff1991), have shown that underinsurance is preva-lent. Gokhale and Kotlikoff (2002) argued that ques-tionable financial advice, inertia, and theunpleasantness of thinking about one’s death arethe likely causes of underinsurance.

The ModelTo merge asset allocation and human capital withoptimal demand for life insurance, we need a solidunderstanding of the actuarial factors that affect thepricing of a life insurance contract. Note that,

although there are a number of life insurance prod-uct variations—term life, whole life, universallife—each worthy of its own financial analysis, wefocus exclusively on the most fundamental type—namely, the one-year, renewable term policy.6 On abasic economic level, a one-year, renewable termpolicy premium is paid at the beginning of theyear—or on the individual’s birthday—and pro-tects the human capital of the insured for the dura-tion of the year. (If the insured person dies withinthat year, the insurance company pays the facevalue of the policy to the beneficiaries soon after thedeath or prior to the end of the year.) For the nextyear, the contract is guaranteed to start anew, withnew premium payments made and protectionreceived; hence, the word “renewable.”

In this general overview of how to think aboutthe joint determination of optimal asset allocationand prudent life insurance holdings, we assumethere are two asset classes. The investor can allocatefinancial wealth between a risk-free asset and arisky asset (i.e., a U.S. government bond and astock). Also, the investor can purchase a term lifeinsurance contract that is renewable each period.The investor is assumed to make asset allocationand insurance purchase decisions and receive laborincome at the beginning of each period. The inves-tor’s objective is to maximize overall utility, whichincludes utility from the investor’s alive state andthe investor’s dead state.

The optimization problem is to determine theamount of life insurance (the face value of lifeinsurance—that is, the death benefit) together withthe allocation to risky assets to maximize the end-year utility of total wealth (human capital plusfinancial wealth) weighted by the alive and deadstates subject to certain budget constraints.

The optimization problem in the model can beexpressed as

(1)

where θx = amount of life insuranceαx = allocation to the risky assetD = relative strength of the utility of be-

quest, as explained in Appendix C= subjective probabilities of death at

the end of year x + 1 conditional onbeing alive at age x

= subjective probability of survivalWx+1 = wealth level at age x + 1, as ex-

plained in Appendix CHx+t = human capital

max ( )( ),θ αx x

E D q U W H

D q U W

x alive x x

x dead x

( ) + +

+

− − +( )⎡⎣ ⎤⎦

+ ( )

1 1 1 1

1 ++( )⎤⎦θx ,

qx

1 qx–



and Ualive(⋅) and Udead(⋅) are the utility functionsassociated with the alive and dead states. Appen-dix C contains a detailed specification of the modeland optimization problem.

The model was inspired by Campbell and bySmith and Buser, but we extended their frameworkin a number of important directions. First, welinked the asset allocation decision with the lifeinsurance purchase decision into one frameworkby incorporating human capital. Second, we specif-ically took into consideration the impact of thebequest motive—that is, the investor’s desire toleave a bequest to heirs—on asset allocation and lifeinsurance.7 Third, we explicitly modeled the vola-tility of labor income and its correlation with finan-cial market volatility. Fourth, we also modeled theinvestor’s subjective survival probability.

Human capital is the central component thatlinks the insurance and the investment decisions.An investor’s human capital can be viewed as a“stock” if its correlation with a given financial mar-ket subindex is high and the volatility of the per-son’s labor income is high. It can be viewed as a“bond” if both the correlation and the volatility arelow. In between these two extremes, human capitalis a diversified portfolio of stocks and bonds, plusidiosyncratic risk.8 We are quite cognizant of thedifficulties involved in calibrating these variables—as pointed out by Davis and Willen (2000); we relyon some of Davis and Willen’s parameters for ournumerical examples.

The model has several important implications.First, it clearly shows that asset allocation decisionsand life insurance decisions both affect an inves-tor’s overall utility; thus, these decisions should bemade jointly.9 The model also shows that humancapital is the central factor. The effects of humancapital on asset allocation and life insurance deci-sions are generally consistent with the existing lit-erature (e.g., Campbell and Viceira; Campbell).One of our major enhancements is the explicit mod-eling of correlation between the shocks to laborincome and financial market returns. The correla-tion between income and risky asset returns playsan important role in both decisions. All else beingequal, as the correlation term between shocks toincome and risky assets increases, the optimal allo-cation to risky assets should decline and so shouldthe optimal quantity of life insurance. Although thefirst decision might be intuitive from a portfoliotheory perspective, we provide precise analyticguidance on how it should be implemented. Fur-thermore, and contrary to intuition, we show thata higher correlation with any given subindexshould reduce the demand for life insurance. Thereason is that the higher the correlation, the higher

the discount rate used to compute human capitalbased on future income. A higher discount rateimplies a lower valuation of human capital—thus,less insurance demand.

The second implication is that the asset alloca-tion decision affects well-being in both the alivestate and the dead (bequest) state whereas the lifeinsurance decision affects primarily the dead state.Bequest preference is arguably the most importantfactor other than human capital when evaluatinglife insurance demand.10 Investors who weightbequest more (higher D) are likely to purchasemore life insurance.

A unique aspect of our model is the consider-ation of subjective survival probability ; themodel shows the intuitive result that investors withlow subjective survival probability will tend to buymore life insurance. This adverse-selection problemis well documented in the insurance literature.11

Other implications are consistent with theexisting literature. For example, our model impliesthat (everything else being equal) the greater thefinancial wealth, the lower the life insurancedemand. More financial wealth also indicates amore conservative portfolio when human capitalresembles a “bond.” When human capital resem-bles a “stock,” more financial wealth indicates amore aggressive portfolio. Naturally, risk tolerancealso has a strong impact on the asset allocationdecision. We found that investors with less risktolerance will invest conservatively and buy morelife insurance.

These implications are illustrated in the casestudies presented in the next section.

Case StudiesTo understand the predictions of the model, weanalyzed the optimal asset allocation decision andthe optimal life insurance coverage for five differ-ent cases. We solved the decision problem via sim-ulation; the detailed solving process is presented inAppendix C.

For all five cases, we assumed the investor caninvest in two asset classes. Table 1 provides thecapital market assumptions used for the cases. Wealso assumed that the investor is male. His prefer-ence for leaving a bequest is one-fourth of his

Table 1. Capital Market Return Assumptions

Asset/Inflation

CompoundAnnual Return

(geometric mean)Risk

(standard deviation)

Risk-free (bonds) 5% —Risky (stocks) 9 20%Inflation 3 —

1 q–( )



preference for consumption in the live state; thatis, 1 – D = 0.8 and D = 0.2. He is agnostic about hisrelative health status (i.e., his subjective survivalprobability is equal to the objective actuarial sur-vival probability). His income is expected to growwith inflation, and the volatility of the growth rateis 5 percent.12 His real annual income is $50,000,and he saves 10 percent each year. He expects toreceive a pension of $10,000 each year (in today’sdollars) when he retires at age 65. His currentfinancial wealth is $50,000. The investor isassumed to follow constant relative risk aversion(CRRA) utility with a risk-aversion coefficient ofγ. Finally, he rebalances his financial portfolio andrenews his term life insurance contract annually.13

These assumptions remain the same for all cases.Other parameters, such as initial wealth, will bespecified in each case.

Case #1: Human Capital, Financial AssetAllocation, and Life Insurance Demand overLifetime. In this case, we assumed that theinvestor has moderate risk aversion (a CRRA, γ, of4). Also, the correlation between shocks to theinvestor’s income and the market return of therisky asset is 0.20.14 For a given age, the amountof insurance this investor should purchase can bedetermined by his consumption/bequest prefer-ence, risk tolerance, and financial wealth. Hisexpected financial wealth, human capital, and thederived optimal insurance demand over the

investor’s life from age 25 to age 65 are presentedin Figure 3.

Several results of modeling this investor’s sit-uation are worth noting. First, human capital grad-ually decreases as the investor gets older and hisremaining number of working years becomesfewer. Second, the amount of his financial capitalincreases as he ages as a result of growth of hisexisting financial wealth and the additional savingshe makes each year. The allocation to risky assetsdecreases as the investor ages because of thedynamic between human capital and financialwealth over time. Finally, the investor’s demandfor insurance decreases as the investor ages. Thisresult is not surprising because the primary driverof insurance demand is human capital, so thedecrease in human capital reduces insurancedemand.

Case #2: Strength of the Bequest Motive.This case shows the impact of the bequest motiveon the optimal decisions about asset allocation andinsurance. In the case, we assumed the investor is45 and has an accumulated financial wealth of$500,000. The investor has a moderate CRRA coef-ficient of 4. The optimal allocations to the risk-freeasset and insurance for various bequest preferencesare presented in Figure 4.

In this case, insurance demand increases as thebequest motive strengthens (i.e., as D gets larger).This result is expected because an investor with a

Figure 3. Case #1: Human Capital, Insurance Demand, and Financial AssetAllocation over the Life Cycle

Insurance Amount ($ thousands) Allocation to Risk-Free Asset (%)

Human CapitalAllocation to Risk-Free Asset

Face Value of TermLife Insurance

1,000

500

400

300

200

900

800

700

600

100

0

100

50

40

30

20

90

80

70

60

10

025 6535 45 55 6030 40 50

Age



strong bequest motive is highly concerned abouthis or her heirs and has an incentive to purchase alarge amount of insurance to hedge the loss ofhuman capital. In contrast, Figure 4 shows almostno change in the proportional allocation to the risk-free asset at different strengths of bequest motive.This result indicates that asset allocation is prima-rily determined by risk tolerance, returns on therisk-free and risky assets, and human capital. Thiscase shows that the bequest motive has a strongeffect on insurance demand but little effect on opti-mal asset allocation.15

Case #3: Effect of Risk Tolerance. In thiscase, we again assumed the investor is age 45 andhas accumulated financial wealth of $500,000. Theinvestor has a moderate bequest preference level(i.e., D = 0.2). The optimal allocations to the risk-free asset and the optimal insurance demands forthis investor for various risk-aversion levels arepresented in Figure 5.

As expected, allocation to the risk-free assetincreases with the investor’s risk-aversion level—the classic result in financial economics. Actually,the optimal portfolio is 100 percent in stocks for

Figure 4. Case #2: Insurance Demand and Asset Allocation by Strength of Bequest Preference

Insurance Amount ($ thousands)

600

500

400

300

200

100

0

Allocation to Risk-Free Asset (%)

100

80

60

40

20

0

Bequest Preference, D

Proportional Allocation to Risk-Free AssetFace Value of Term Life Insurance Policy

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45

Figure 5. Case #3: Insurance Demand and Asset Allocation by Level of Risk Aversion


450

400

300

200

100

350

250

150

50

0


100

80

90

60

40

20

70

50

30

10

0

Risk Aversion, D


1.0 1.5 2.0 2.5 3.0 3.5 4.5 5.04.0 5.5 6.0



risk-aversion levels less than 2.5. The optimalamount of life insurance follows a similar pattern:Optimal insurance demand increases with riskaversion. For an investor with moderate risk aver-sion (a CRRA coefficient of 4), optimal insurancedemand is about $290,000, which is roughly sixtimes the investor’s current income of $50,000.16

Naturally, conservative investors should investmore in risk-free assets and buy more life insurancethan aggressive investors.

Case #4: Financial Wealth. For this case, weheld the investor’s age at 45 and the risk preferenceand the bequest preference at moderate levels (aCRRA coefficient of 4 and bequest level of 0.2). Theoptimal asset allocations to the risk-free asset andthe optimal insurance demands for various levelsof financial wealth are presented in Figure 6.

First, Figure 6 shows that the optimal alloca-tion to the risk-free asset increases with initialwealth. This outcome may seem inconsistent withthe CRRA utility function because the CRRA utilityfunction implies that the optimal asset allocationdoes not change with the amount of wealth theinvestor has. Note, however, that wealth hereincludes both financial wealth and human capital.In fact, this case is a classic example of the effect ofhuman capital on optimal asset allocation. Anincrease in financial wealth not only increases totalwealth but also reduces the percentage of totalwealth represented by human capital. In this case,human capital is less risky than the risky asset.17

When initial wealth is low, human capital domi-

nates total wealth and asset allocation. As a result,to achieve the target asset allocation of a moderateinvestor—say, an allocation of 60 percent in therisk-free asset and 40 percent in the risky asset—theclosest allocation is 100 percent in the risky assetbecause human capital is illiquid. With an increasein initial wealth, the asset allocation graduallyadjusts until it approaches the asset allocation amoderately risk averse investor desires.

Second, Figure 6 shows that optimal insurancedemand decreases with financial wealth. Thisresult can be intuitively explained through the sub-stitution effects of financial wealth and life insur-ance. In other words, with a large amount of wealthin hand, one has less demand for insurance becausethe loss of human capital will have much lessimpact on the well-being of one’s heirs. The optimalamount of life insurance decreases from more than$400,000 when the investor has little financialwealth to almost 0 when the investor has $1.5 mil-lion in financial assets.

In summary, for an investor whose humancapital is less risky than the stock market, the moresubstantial the investor’s financial assets are, themore conservative optimal asset allocation is andthe smaller life insurance demand is.

Case #5: Correlation between Shocks toWage Growth Rate and Stock Returns. In thiscase, we want to evaluate the life insurance andasset allocation decisions for investors with highlycorrelated income and human capital. This kind ofcorrelation can happen when an investor’s income

Figure 6. Case #4: Insurance Demand and Asset Allocation by Level ofFinancial Wealth


450

400

300

200

100

350

250

150

50

0


100

80

90

60

40

20

70

50

30

10

0

Current Financial Wealth ($ thousands)


50 250 500 750 1,000 1,250 1,500



is closely linked to the stock performance of thecompany where the investor works or when theinvestor’s compensation is highly influenced by thefinancial market (e.g., the investor works in thefinancial industry).

Again, the investor’s age is 45 and he has amoderate risk preference and bequest preference.Optimal asset allocation to the risk-free asset andinsurance demand for various levels of financialwealth in this situation are presented in Figure 7.

As the correlation between income and stockmarket return increases, optimal asset allocationbecomes more conservative (i.e., more allocation ismade to the risk-free asset). One way to look at thereason is that a higher correlation between humancapital and the stock market results in less diversi-fication—thus, higher risk—in this investor’s totalportfolio (human capital plus financial capital). Toreduce this risk, the investor will invest more finan-cial wealth in the risk-free asset.

Optimal insurance demand decreases as thecorrelation increases. The reason lies with the pur-pose of life insurance. Because life insurance ispurchased to protect the fruits of human capital forthe family, as the correlation between the riskyasset and the investor’s income flow increases, theex ante value of human capital to the survivingfamily decreases. This decreased value of humancapital induces lower demand for insurance. Also,less money spent on life insurance indirectlyincreases the amount of financial wealth the inves-tor can invest, so the investor can invest more inrisk-free assets to reduce the risk associated withhis total wealth.18

In summary, as wage income and stock marketreturns become more correlated, optimal asset allo-cation becomes more conservative and the demandfor life insurance falls.

ConclusionWe expanded on the Mertonian idea that humancapital is a shadow asset class that is worth muchmore than financial capital early in life and that ithas unique risk and return characteristics. Humancapital—even though it is not traded and is highlyilliquid—should be treated as part of a person’sendowed wealth that must be protected, diversi-fied, and hedged.

We developed a unified human capital–basedframework to help individual investors with lifeinsurance and asset allocation decisions. And wepresented five case studies to demonstrate the opti-mal decisions in different scenarios. The modelprovided several key results: • Investors need to make asset allocation deci-

sions and life insurance decisions jointly.• The magnitude of human capital, its volatility,

and its correlation with other assets signifi-cantly affect the two decisions over the life cycle.

• Bequest preferences and a person’s subjectivesurvival probability have significant effects onthe person’s demand for insurance but little influ-ence on the person’s optimal asset allocation.

• Conservative investors should invest rela-tively more in risk-free assets and buy morelife insurance. We demonstrated that the correlation between

human capital and financial capital (i.e., whether

Figure 7. Case #5: Insurance Demand and Asset Allocation by CorrelationLevel


330

310

270

290

250


22

18

20

14

16

12

Correlation between Wage Growth Rate and Risky Asset Return


−0.50 −0.25 0 0.25 0.50 0.75 0.90



the person’s human capital resembles a bond or astock) has a noticeable and immediate impact on aperson’s demand for life insurance as well as theusual portfolio considerations. The decisionsregarding how much life insurance a person needsand where the person should invest cannot bemade separately. Rather, they are different aspectsof the same problem. For instance, a person whoseincome relies heavily on commissions should con-sider his human capital “stock-like” because theincome is highly correlated with the market. Thischaracteristic results in great uncertainty in hishuman capital, so he should purchase less insur-ance and invest more financial wealth in bonds.Conversely, the human capital of a tenured univer-sity professor is more “bond-like,” so she shouldpurchase more insurance and invest financialwealth in stocks.

Obviously, more research remains to be doneto make the joint decision making suitable for prac-tical applications. One possible direction would beto model the various types of life insurance andtheir unique tax-sheltering aspects within a unifiedasset allocation framework. For example, whole lifeinsurance (and other forms of variable life insur-ance) could be viewed as a hedge against possiblechanges in systematic population mortality rates, soit might coexist in an optimal portfolio with short-term life insurance. Another direction would be todiverge from the traditional expected utility modelsby examining other methods, such as minimizingshortfall probabilities, to determine the appropriateasset allocation/life insurance decision.

Appendix A. Human CapitalLet the symbol ht denote the random (real, after-tax)wage or salary that a person will receive during thediscrete time period (or year) t; then, in general, theexpected discounted value of this wage flow atcurrent time t0 is represented mathematically by

(A1)

where n is the number of wage periods over whichwe are discounting, r is the relevant risk-free dis-count rate, and v is a (subjective) parameter thatcaptures illiquidity plus any other potential riskpremium associated with one’s human capital. InEquation A1, the expectation E[ht] in the numeratorconverts the random wage into a scalar. Note that,in addition to expectations (under a physical, real-world measure) in Equation 1, the denominator’sv, which accounts for all broadly defined risks,obviously reduces the t0 value of the expressionDVHC accordingly.

And depending on the investor’s specific joband profession, he or she might be expected to earnthe same exact E[ht] in each time period t, yet therandom shocks to wages, ht − E[ht], might have verydifferent statistical characteristics vis-à-vis the mar-ket portfolio; thus, each profession or job wouldinduce a distinct “risk premium” value for v, whichwould then lead to a lower discounted expectedvalue of human capital. Therefore, because we arediscounting with an explicit risk premium, we feeljustified in also using the term “financial economicvalue of human capital” to describe DVHC.

Note that when we focus on the correlation orcovariance between human capital and other macro-economic or financial factors, we are, of course, refer-ring to the correlation between shocks ht − E[ht] andshocks to or innovations in the return-generatingprocess in the market. This correlation can induce a(quite complicated) dependency structure betweenDVHC in Equation A1 and the dynamic evolution ofthe investor’s financial portfolio.19

Appendix B. One-Year, Renewable Term Life Pricing MechanismThe one-year, renewable term policy premium ispaid at the beginning of the year—or on the indi-vidual’s birthday—and protects the human capitalof the insured for the duration of the year. If theinsured person dies within that year, the insurancecompany pays the face value to the beneficiariessoon after the death or prior to the end of the year.If the insured does not die, the contract is guaran-teed to start anew the next year, with new premiumpayments to be made and protection received.

The policy premium is obviously an increasingfunction of the desired face value, and the two arerelated by the simple formula:

(B1)

that is, premium P is calculated by multiplying thedesired face value, θ, by the probability of death, q,and then discounting by the interest rate factor, 1 + r.

The theory behind Equation B1 is the well-known law of large numbers, which guarantees thatprobabilities become percentages when individu-als are aggregated. Note the implicit assumption inEquation B1 that, although death can occur at anytime during the year (or term), the premium pay-ments are made at the beginning of the year and theface values are paid at the end of the year. From theinsurance company’s perspective, all of the premi-ums received from the group of N individuals withthe same age (i.e., probability of death q) and withcontracts having face value θ are commingled and

DVHCE h

r vt

tt

n

=+ +=

∑ [ ]

( ),

11

Pq

r=

+1θ;



invested in an insurance reserve earning rate of inter-est r, so at the end of the year, PN(1 + r) is parti-tioned among the qN beneficiaries.

No savings component or investment compo-nent is embedded in the premium defined by Equa-tion B1. Rather, at the end of the year, the survivorslose any claim to the pool of accumulated premiumsbecause all funds go directly to the beneficiaries.

As the individual ages and probability of deathincreases (denoted by qx), the same exact faceamount (face value) of life insurance, θ, will costmore and will induce a higher premium, Px, as perEquation B1. In practice, the actual premium isloaded by an additional factor, denoted by 1 + λ toaccount for commissions, transaction costs, andprofit margins; so, the actual amount paid by theinsured is closer to P(1 + λ), but the underlyingpricing relationship, driven by the law of largenumbers, remains the same.

Also, from the perspective of traditional finan-cial planning, the individual conducts a budgetinganalysis to determine his or her life insurancedemands (i.e., the amount the surviving family andbeneficiaries need to replace the lost wages inpresent-value terms). That quantity would be takenas the required face value in Equation B1, whichwould then lead to a premium. Alternatively, onecan think of a budget for life insurance purchases,in which case, the face value would be determinedby Equation B1.

In our model and the discussion, we “solve”for the optimal age-varying amount of life insur-ance, denoted by θx, which then induces an age-varying policy payment, Px, which maximizes thewelfare of the family by taking into account riskpreferences and attitudes toward bequest.

Appendix C. Model Specification: Optimal Asset Allocation and Insurance DemandsIn this model, the investor is currently age x andwill retire at age Y. The term “retirement” issimply meant to indicate that the human capitalincome flow is terminated and the pension phasebegins. The assumption is that the financial port-folio will be rebalanced annually and that the lifeinsurance—which is assumed to be of the one-year term variety—will be renewed annually.Taxes are not considered in the model.

The investor would like to know how much(i.e., the face value of term life) insurance he shouldpurchase and what fraction of his financial wealthhe should invest in risky assets (stock).

In the model, an investor determines theamount of life insurance demand, θx—the face

value of life insurance (that is, the death benefit)—together with allocation αx to risky assets to maxi-mize end-year utility of his total wealth (humancapital plus financial wealth) weighted by the“alive” and “dead” states. The optimization prob-lem is expressed as

(C1)

(which is the same as Equation 1 in the text), subjectto the following budget constraint:

(C2)

where e is the exponent, 2.7182,

(C3)

and0 ≤ αx ≤ 1. (C4)

The symbols, notations, and terminology used inthe optimal problem are as follows:θx = amount of life insurance.αx = allocation to risky assets.D = relative strength of the utility of bequest.

Individuals with no utility of bequest willhave D = 0.

qx = objective probability of death at the end of theyear x + 1 conditional on being alive at agex. This probability is determined by a givenpopulation (i.e., mortality table).

= subjective probability of death at the end ofthe year x + 1 conditional on being alive atage x; denotes the subjective probabil-ity of survival. The subjective probability ofdeath may be different from the objectiveprobability. In other words, a person mightbelieve he or she is healthier (or less healthy)than average. This belief would affect ex-pected utility but not the pricing of the lifeinsurance, which is based on an objectivepopulation survival probability.

λ = fees and expenses (i.e., actuarial and insur-ance loading) imposed and charged on atypical life insurance policy.

Wt = financial wealth at time t. The market hastwo assets, one risky and one risk free. Thisassumption is consistent with the two-fundseparation theorem of traditional portfoliotheory. Of course, the approach could beexpanded to multiple asset classes.

max ( )( ),θ αx x

E D q U W H

D q U W

x alive x x

x dead x

( ) + +

+

− − +( )⎡⎣

+ ( )

1 1 1 1

,11 +( )⎤⎦θx

W W h q e C

e

x x x x xr

x

x

Z

x

f

S S S S

+−

− +

= + − +( ) −⎡⎣⎢

⎤⎦⎥

× + −

1

1

2

1

12

λ θ

α αμ σ σ

,( )eerf

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

θ θλ0 1

≤ ≤+ −

+xx x x

r

x

W h C e

q

f( )

( ),

qx

1 qx–



rf = return on the risk-free asset.S = value of the risky asset. This value follows a

discrete version of a geometric Brownianmotion:

(C5)

where μS is the expected return, σS is thestandard deviation of return of the riskyasset, ZS,t is an independent random vari-able, and ZS,t ~ N(0,1).

ht = labor income. In our numerical cases, weassumed that ht follows a discrete stochasticprocess specified byht+1 = ht exp(μh + σhZh,t+1), (C6)

where, ht > 0; μh and σh are, respectively, theannual growth rate and the annual standarddeviation of the income process; Zh,t is anindependent random variable; and Zh,t ~N(0,1).

Based on Equation C6, for a person atage x, income at age x + t is determined by

(C7)

The correlation between shocks to laborincome and the return of the risky asset isρ, and

(C8)

where Z is a standard Brownian motionindependent of ZS; that is,cor(ZS,Zh) = ρ. (C9)

Ht = present value of future income from age t +1 to death. Income after retirement is thepayment from pensions. Based on EquationC7, for a person at age x + t, the present valueof future income from age x + t + 1 to deathis determined by

(C10)

where ηh is the risk premium (discount rate)for the income process and captures the mar-ket risk of income; ζh is a discount factor inhuman capital evaluation to account for theilliquidity risk associated with one’s job. Inthe numerical examples, we assumed a 4percent discount rate per year.20 Based onthe CAPM, ηh can be evaluated by

(C11)

Furthermore, the expected value of Ht,E(Hx+t), is defined as the human capital aperson has at age x + t + 1.

Ct = consumption in year t. For simplicity, Ct isassumed to equal C (i.e., constant consump-tion over time).

Equation C3 requires the cost (or price) of theterm insurance policy to be less than the amount ofthe investor’s current financial wealth, and there isa minimum insurance amount, θ0 > 0, an investoris required to purchase to have minimum protec-tion from the loss of human capital.

The power utility function (CRRA) is used inour numerical examples. The functional form of theutility function is

(C12)

for x > 0 and γ ≠ 1 and

U(x) = ln(x) (C13)for x > 0 and γ = 1. The power utility function isused in the examples for both Ualive(⋅) and Udead(⋅),which are the utility functions associated with,respectively, the alive and dead states.21

We solved the problem via simulation. We firstsimulated the values of the risky asset by usingEquation C5. Then, we simulated Zh from EquationC8 to take into account the correlation betweenincome change and the return of the financial mar-ket. Finally, we used Equation C6 to generateincome over the same period. Human capital, Hx+t,is calculated by using Equations C7 and C10. Ifwealth level at age x + 1 is less than 0, we set thewealth equal to 0; that is, we assumed the investorhas no remaining financial wealth. We simulatedthis process N times. The objective function wasevaluated by using

(C14)

and

(C15)

In the numerical examples, we set N equal to 20,000.

This article qualifies for 1 PD credit.

S S Zt t S S S S t+ += − +⎛⎝⎜

⎞⎠⎟

12

11

2exp ,,μ σ σ

h h Zx t x h h h kk

t

+=

= +( )⎡

⎣⎢⎢

⎤

⎦⎥⎥∏exp .,μ σ

1

Z Z Zh S= + −ρ ρ1 2 ,

H

h j t r

x t

x j f h hj t

Y x

+

+= +

−= − − + +⎡⎣ ⎤⎦{ }∑ exp ( )( ) ,η ζ

1

η μ

ρ μσσ

hh S

SS

r

Sr h

S

Z Z

Ze

e

f

f

= − −( )⎡⎣⎢

⎤⎦⎥

= − −( )⎡⎣⎢

⎤⎦⎥

cov

var

( , )

( )1

1 ..

U xx( ) =−

−1

1

γ

γ

11 1

1N

U W n H nalive x xn

N

+ +=

+⎡⎣ ⎤⎦∑ ( ) ( )

11

1N

U W ndead x xn

N

+=

+⎡⎣ ⎤⎦∑ ( ) .θ



Notes1. How much an investor should consume or save is an impor-

tant decision that is also frequently tied to the concept ofhuman capital. This article focuses on the asset allocationand life insurance decisions, however, so we have simpli-fied the model by assuming that the investor has alreadydecided how much to consume/save. And for a model thatoptimizes consumption in addition to insurance and assetallocation, see Huang, Milevsky, and Wang (2005).

2. The term “human capital” can convey a number of different—at times, conflicting—concepts in the literature. We definehuman capital here to be the financial economic value of allfuture wages, which is a scalar quantity and depends on anumber of subjective or market equilibrium factors. Appen-dix A provides a detailed explanation of human capital andthe discounted present value of future salaries.

3. Heaton and Lucas (2000) showed that wealthy householdswith high and variable business income invest less in thestock market than other, similar wealthy households, whichis consistent with the theoretical prediction.

4. Like Yaari, Fischer (1973) pointed out that these earliermodels either dealt with an infinite horizon or took the dateof death to be known with certainty.

5. Economides (1982) argued that in a corrected model, Camp-bell’s approach underestimates the optimal amount ofinsurance coverage. Our model takes this correction intoconsideration.

6. Appendix B provides a description of the pricing mecha-nism of the one-year, renewable term life insurance policy.We believe that all other types of life insurance policies arefinancial combinations of term life insurance with invest-ment accounts, added tax benefits, and embeddedoptions—although investigating this aspect is beyond thescope of this article.

7. Bernheim (1991) and Zietz (2003) showed that the bequestmotive has a significant impact on life insurance demand.

8. Note that when we make statements such as: “This person’shuman capital is 40 percent long-term bonds, 30 percentfinancial services, and 30 percent utilities,” we mean that theunpredictable shocks to future wages have a given correla-tion structure with these particular subindices. Thus, forexample, the human capital of a tenured university profes-sor could be considered to be a 100 percent real-return(inflation-linked) bond because shocks to wages—if thereare any—would not be linked to any financial subindex.

9. The only scenarios in which asset allocation and life insur-ance decisions are not linked are when the investor derivesher or his utility 100 percent from consumption or 100percent from bequest. Both are extreme scenarios, espe-cially the 100 percent from bequest.

10. A well-designed questionnaire could help elicit an individ-ual’s attitude toward the importance of bequest, eventhough a precise estimate may be hard to obtain.

11. Actuarial mortality tables can be taken as a starting pointfor this factor. Life insurance is already priced to take intoaccount adverse selection.

12. The salary growth rate and the volatility were chosenmainly to show the implications of the model. They are notnecessarily representative.

13. The mortality and insurance fees and expenses imposed forhandling the insurance contract were assumed to be 12.5percent.

14. Davis and Willen estimated the correlation between laborincome and equity market returns by using the U.S. Depart-ment of Labor’s Current Occupation Survey. They foundthat the correlation between equity returns and laborincome typically lies in the interval from –0.10 to 0.20.

15. In our model, subjective survival probability has a similarimpact on optimal insurance need and asset allocation asthe bequest motive does. When subjective survival proba-bility is high, the investor will buy less insurance.

16. This result is close to the typical recommendation made byfinancial planners (i.e., purchase a term life insurance policythat has a face value four to seven times one’s currentincome). See, for example, Todd (2004).

17. In this case, income has a real growth rate of 0 percent anda standard deviation of 5 percent, yet the expected realreturn on stock is 8 percent with a standard deviation of 20percent.

18. See Case #3 for a detailed discussion of the wealth impact.19. For a more rigorous and mathematically satisfying treat-

ment of the ongoing interaction between human capital andmarket returns as it pertains to the purchase of life insur-ance in a continuous-time framework, see Huang,Milevsky, and Wang (2005).

20. The 4 percent discount rate translates into about a 25 per-cent discount on the overall present value of human capitalfor a 45-year-old person with 20 years of future salary. This25 percent discount is consistent with empirical evidenceon the discount factor between restricted stocks and theirunrestricted counterparts (see Amihud and Mendelson1991). Also, Longstaff (2002) reported that the liquiditypremium for the longer-maturity U.S. T-bond is 10–15 per-cent of the value of the bond.

21. Stutzer (2004) pointed out the difficulties in applyingexpected utility theories in practice and proposed using theminimization of short-fall probability as an alternative. Wefollowed the traditional expected utility model of linkingasset allocation and portfolio decisions to individual riskaversion.

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