Yale ICF Working Paper No May 2005


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1 Yale ICF Working Paper No May 2005 HUMAN CAPITAL, AET ALLOCATION, AND LIFE INURANCE Roger G. Ibbotson, Yale cool of Management, Yale University Peng Cen, Ibbotson Associates Mose Milevsky, culic cool of Business, York University Xingnong Zu, Ibbotson Associates Tis paper can be downloaded witout carge from te ocial cience Researc Network Electronic Paper Collection: ttp://ssrn.com/abstract=723167
2 Human Capital, Asset Allocation, and Life Insurance Peng Cen, Roger G. Ibbotson, Mose A. Milevsky, Xingnong K. Zu 1 Version: January 25, 2005 Abstract Financial planners and advisors ave recently started to recognize tat uman capital must be taken into account wen building optimal portfolios for individual investors. But uman capital is not just anoter preendowed asset class tat must be included as part of te portfolio frontier. An investor s uman capital contains a unique mortality risk, wic is te loss of all future income and wages in te unfortunate event of premature deat. However, life insurance in its various guises and incarnations can edge against tis mortality risk. Tus, uman capital affects bot te optimal asset allocation and te optimal demand for life insurance. Yet istorically, asset allocation and life insurance decisions ave consistently been analyzed separately bot in teory and practice. In tis paper, we develop a unified framework based on uman capital in order to enable individual investors to make bot decisions jointly. We investigate te impact of te magnitude of uman capital, its volatility, and its correlation wit oter assets as well as bequest preferences and subjective survival probabilities on te optimal portfolio of life insurance and traditional asset classes. We do tis troug five case studies tat implement our model. Indeed, our analysis validates some intuitive rules of tumb but provides additional results tat are not immediately obvious. 1 Peng Cen is Cief Investment Officer at Ibbotson Associates, Cicago. Roger G. Ibbotson is professor of finance at Yale cool of Management, New Haven, Connecticut. Mose Milevsky is Eecutive Director, Te IFID Centre &Associate Professor of Finance at culic cool of Business, York University, Toronto, Canda. Xingnong Zu is enior Researc Consultant at Ibbotson Associates Cicago. Page 1 of 20
3 1. Introduction Tere is growing recognition amongst academics and practitioners tat te risk and return caracteristics of uman capital suc as wage and salary profiles sould be taken into account wen building portfolios for individual investors. Merton (2003) points out te importance of including te magnitude of uman capital, its volatility, and its correlation wit oter assets into asset allocation decisions from a personal risk management perspective. Te employees of Enron and WorldCom suffered an etreme eample of tis risk. Teir labor income and teir financial investment in te company provided no diversification, and tey were eavily impacted by teir company s collapse. 2 One unique aspect of an investor s uman capital is mortality risk, te loss of uman capital in te unfortunate event of premature deat. Life insurance as long been used to edge against mortality risk. Typically, te greater te value of uman capital, te more life insurance te family demands. Intuitively, uman capital not only affects te optimal asset allocation, but also te optimal life insurance demand. However, tese two important financial decisions te demand for life insurance and te optimal asset allocation ave consistently been analyzed separately in teory and practice. We find few references in eiter te risk and insurance literature or te investment and finance literature on te importance of considering tese decisions jointly, witin te contet of a life cycle model of consumption and investment. In oter words, popular investment and financial planning advice regarding ow muc life insurance one sould acquire is seldom framed in terms of te riskiness of one s uman capital. And, conversely, te optimal asset allocation decision is only lately being framed in terms of risk caracteristics of uman capital, and rarely is it integrated wit life insurance decisions. Motivated by te need to integrate tese two decisions, our paper merges tese traditionally distinct lines of tougt togeter in one framework. We argue tat tese two decisions must be determined jointly since tey serve as risk substitutes wen viewed from an individual investor s portfolio perspective. Life insurance is a perfect edge for uman capital in te event of deat; i.e., term life insurance and uman capital ave a negative 100 percent correlation wit eac oter in te live vs. dead states. If one pays off at te end of te year te oter does not, and vice versa. Tus, te combination of te two provides great diversification to an investor s total portfolio. Te diagram below illustrates te types of decisions te investor faces, along wit te variables tat impact te decisions. 2 Benartzi (2001) sowed tat many investors invest eavily into te stock of te company tey work for. Page 2 of 20
4 Diagram 1: Human Capital, Asset Allocation, and Life Insurance Human Capital Financial Wealt avings  Age  Initial Wealt  Labor Income  Risk Aversion  Correlation between Human Capital & Financial Market Life Insurance Decision Asset Allocation Decision  Mortality Estimates  Capital Market Assumptions  Bequest Preference We develop a unified model to provide practical guidelines for developing te optimal asset allocation and life insurance decisions for individual investors in teir preretirement years (accumulation stage). 3 Te remainder of tis paper is organized as follows. Te net section reviews te eisting literature of asset allocation and uman capital as well as te literature on te demand for life insurance over te uman life cycle. ection 3 presents our model and te critical variables. ection 4 provides a number of case studies or idealized scenarios under wic we illustrate various model allocations depending on income, age, and tolerance for financial risk. ection 5 provides summary and concluding remarks. 2. Human Capital and Financial Capital An investor s total wealt consists of two parts. One is readily tradable financial assets; te oter is uman capital. Human capital is defined as te present value of an investor s future labor income. 4 Economic teory predicts tat investors make asset allocation and life insurance purcase decisions to maimize teir lifetime utilities of wealt and consumption. Bot of tese decisions are closely linked to uman capital. Altoug uman capital is not readily tradable, it is often te single largest asset an investor as. Typically, younger investors ave far more uman capital tan financial capital. Tis is because younger investors ave more years to work and tey ave ad few years to save and accumulate financial wealt. Conversely, older investors tend to ave more financial capital tan uman 3 How muc an investor sould consume or save is anoter important decision tat is frequently tied to te concept of uman capital. In tis paper, we are concentrating only on te asset allocation and life insurance decisions; terefore, we simplify our model by assuming tat te investor as already decided ow muc e will consume/save. Our numerical cases assume tat te investor saves a constant 10 percent of teir salary eac year. 4 Te term uman capital often conveys a number of different and at times conflicting concepts in te literature. In tis paper, we define uman capital to be te financial economic value of all future wages, wic is a scalar quantity and dependent on a number of subjective or market equilibrium factors. Te appendi provides a detailed eplanation of uman capital and discounted present value of future salaries. Page 3 of 20
5 capital, since tey ave fewer years aead to work but ave accumulated financial capital over a long career. Cart 1 illustrates te amounts of financial capital and uman capital over an investor s working years (preretirement) from age 25 to age 65. Wen te investor is young, is uman capital far outweigs is financial capital. As te investor gets older, te investor will continue to make savings contributions and, wit te returns from te eisting financial portfolio, te amount of financial capital will increase. Cart 1: Epected Financial Capital and Human Capital over te Life Cycle Cart 1: Financial Capital and Human Capital Over Te Life Cycle $ Age Financial Capital Human Capital 2.1 Financial Asset Allocation and Human Capital Te canging mi of financial capital and uman capital over te life cycle impacts financial asset allocation. In te late 1960s, economists establised models tat implied tat individuals sould optimally maintain constant portfolio weigts trougout teir life cycle (amuelson 1969, Merton 1969). Tose models assumed investors ave no labor income (i.e., uman capital). Wen labor income is included in te portfolio coice model, individuals will optimally cange teir allocation of financial assets in a pattern related to te life cycle. In oter words, te optimal asset allocation depends on te riskreturn caracteristics and te fleibility of te labor income (suc as ow muc or ow long te investor works). In our model, te investor adjusts te financial portfolio to compensate for te nontradable uman capital risk eposures (e.g., Merton (1971), Bodie, Merton, and amuelson (1992), Heaton and Lucas (1997), Jaganatan and Kocerlacota (1998), and Campbell and Viceira (2002)). Te key teoretical implications are: 1) young investors will invest more in stocks tan older investors; 2) investors wit safe labor income (tus safe uman capital) will invest more of teir financial portfolio into stocks; 3) investors wit labor income igly correlated wit stock markets will invest teir financial assets into less risky assets; and 4) te ability to adjust labor supply (i.e., iger fleibility) also increases an investor s allocation toward stocks. However, empirical studies sow tat most investors do not efficiently diversify teir financial portfolio considering te risk of teir uman capital. Benartzi (2001) and Benartzi and Taler (2001) sowed tat many investors use primitive Page 4 of 20
6 metods to determine te asset allocation and many of tem invest very eavily into te stock of te company tey work for Life Insurance and Human Capital Many researcers ave pointed out tat te lifetime consumption and portfolio decision models 6 need to be epanded to take into account lifetime uncertainty (or mortality risk). Yaari (1965) is considered te first classical paper on tis topic. Yarri pointed out ways of utilizing life insurance and life annuities to insure against lifetime uncertainty. He also derived conditions under wic consumers would fully insure against lifetime uncertainty. 7 Teoretical studies sow a clear link between te demand for life insurance and te uncertainty of uman capital. Campbell (1980) argues tat for most ouseolds labor income uncertainty dominates financial capital income uncertainty. He furter developed solutions to te optimal amount of insurance a ouseold sould purcase based on uman capital uncertainty. 8 Buser and mit (1983) model life insurance demand in a portfolio contet using meanvariance analysis. Tey derive te optimal insurance demand and te optimal allocation between risky and riskfree assets. Tey find tat te optimal amount of insurance depends on two components: te epected value of uman capital and te riskreturn caracteristics of te insurance contract. Ostaszewski (2003) furter states tat life insurance is te business of uman capital securitization addressing te uncertainties and inadequacies of an individual s uman capital. On te oter and, empirical studies on life insurance adequacy ave sown tat underinsurance is prevalent. 9 Gokale and Kotlikoff (2002) argue tat questionable financial advice, inertia, and te unpleasantness of tinking about one s deat are te likely causes. 3. Description of te Model In order to merge asset allocation and uman capital wit te optimal demand for life insurance, we need to ave a solid understanding of te actuarial factors tat impact te pricing of a life insurance contract. Note tat altoug tere are a number of life insurance product variations suc as term life, wole life, or universal life eac worty of teir own financial analysis we will focus eclusively on te most fundamental type: namely te oneyear renewable term policy. 10 On a basic economic level, a oneyear renewable term policy premium is paid at te beginning of te year or on te individual s birtday and protects te uman capital of te insured for te duration of te year. If te insured person dies witin tat year, te insurance company pays te face value to te beneficiaries, soon after 5 Heaton and Lucas (2000) sowed tat wealty ouseolds wit ig and variable business income invest less in te stock market tan oter similar wealty ouseolds, wic is consistent wit te teoretical prediction. 6 E.g., amuelson (1969), Merton (1969). 7 Like Yaari, Fiscer (1973) also pointed out tat tese earlier models eiter dealt wit an infinite orizon or took te date of deat to be known wit certainty. 8 Economides (1982) argued tat under a corrected model, Campbell s approac underestimated te optimal amount of insurance coverage. Our model takes tis correction into consideration. 9 E.g., Auerbac and Kotlikoff (1991). 10 Oneyear renewable term life insurance is used trougout tis paper. Te appendi provides a description of te pricing mecanism of te insurance policy. It is beyond te scope of tis paper, but we believe tat all oter types of life insurance policies are financial combinations of term life insurance wit investment accounts, added ta benefits, and embedded options. Page 5 of 20
7 te deat or prior to te end of te year. Net year te contract is guaranteed to start anew wit new premium payments made and protection received; ence te word renewable. In following part of tis section we provide a general overview on ow to tink about te joint determination of te optimal asset allocation and prudent life insurance oldings. 11 We assume tere are two asset classes. Te investor can allocate financial wealt between a riskfree and a risky asset (i.e., bond and stock). Also, te investor can purcase a term life insurance contract tat is renewable eac period. Te investor s objective is to maimize te overall utility, wic includes utility from te alive state and te dead state. Te investor decides te life insurance demand (te face value of a term life insurance) and te asset allocation between riskfree and risky asset. 12 Te optimization problem is epressed in detail wit equation (4) in te appendi. Te model is inspired by Campbell (1980) and Buser and mit (1983). We etend teir framework in a number of important directions. First, we link te asset allocation decision wit te life insurance purcase decision into one framework by incorporating uman capital. econd, we specifically take into consideration te impact of te bequest motive on asset allocation and life insurance. 13 Tird, we eplicitly model te volatility of labor income and its correlation wit te financial market. Fourt, we also model one s subjective survival probability. Human capital is te central component tat links bot decisions. An investor s uman capital can be viewed as a stock if bot te correlation to a given financial market subinde and te volatility of te labor income are ig. It can be viewed as a bond if bot te correlation and te volatility are low. In between tese two etremes, uman capital is a diversified portfolio of stocks and bonds, plus idiosyncratic risk. 14 We are quite cognizant of te difficulties involved in calibrating tese variables as pointed out by David and Willen (2000) and we rely on some of teir parameters for our case numerical eamples in te following section. Tere are several important implications from te model. First, te model clearly sows tat bot asset allocation and life insurance decisions affect an investor s overall utility, and tey sould be made jointly. 15 Te model also sows tat uman capital is te central factor. Te impacts of uman capital on asset allocation and life insurance decisions are mostly consistent wit te eisting literature (e.g., Campbell and Viceira (2002) and Campbell 11 Te appendi contains a more detailed specification, wic is te basis of our numerical eamples and case studies in te subsequent section. 12 We assume tat te investor makes asset allocation and insurance purcase decisions at te start of eac period. Labor income is also received at te beginning. 13 Berneim (1991) and Zietz (2003) sow tat te bequest motive as a significant impact on life insurance demand. 14 Note tat wen we make statements suc as: Tis person s uman capital is 40% longterm bonds, 30% financial services, and 30% utilities, we mean tat te unpredictable socks to future wages ave a given correlation structure wit te named subindices. Tus, for eample, a tenured university professor could be considered to be a 100% realreturn (inflation linked) bond, since socks to wages if tere are any would not be linked to any financial subinde. 15 Te only scenarios in wic te asset allocation and life insurance decisions are not linked are wen te investor derives is/er utility 100% from consumption or 100% from bequest. Bot are etreme scenarios, especially te 100% from bequest. Page 6 of 20
8 (1980)). One of our major enancements is te eplicit modeling of correlation between te socks to labor income and financial market returns. Te correlation between income and risky asset returns plays a very important role in bot decisions. All else being equal, as te correlation term between socks to income and risky asset increases, te optimal allocation to risky assets will decline and so will te optimal quantity of life insurance. Wile te former result migt be intuitive from a portfolio teory perspective, we provide precise analytic guidance on ow tis sould be implemented. Furtermore, and contrary to intuition, we sow tat a iger correlation wit any given subinde reduces te demand for life insurance. Tis is because te iger te correlation, te iger te discount rate used to compute uman capital based on future income. A iger discount rate implies a lower uman capital valuation; tus, less insurance demand. econd, te asset allocation decision affects wellbeing in bot te alive consumption state and te dead bequest state, wile te life insurance decision mostly affects te bequest state. Bequest preference is arguably te most important factor oter tan uman capital wen evaluating te life insurance demand. 16 Investors wo weigt bequest more (iger D) are likely to purcase more life insurance. Anoter unique aspect of our model is te consideration of subjective survival probability (1 q ); it can be seen intuitively tat investors wit low subjective survival probability will tend to buy more life insurance. Tis adverse selection problem is welldocumented in te insurance literature. 17 Oter implications are consistent wit te eisting literature. For eample, our model implies tat everyting else being equal, te greater te financial wealt, te lower te life insurance demand. More financial wealt also indicates a more conservative portfolio wen uman capital is more like a bond. Wen uman capital is more like a stock, more financial wealt indicates more aggressive portfolios. Naturally, risk tolerance also as a strong impact on te asset allocation decision. We find tat investors wit less risk tolerance will invest conservatively and buy more life insurance. Tese implications will be furter illustrated in te case studies presented in te net section. 4. Case tudies To understand te predictions of te model, we analyze te optimal asset allocation decision and te optimal life insurance coverage for five different cases. We solve te problem via simulation; te detailed solving process is presented in te appendi. We assume tere are two asset classes in wic te investor can invest is/er financial asset. Table 1 provides te capital market assumptions used in all five cases. We also assume tat te investor is male. His preference toward bequest is onefourt of is preference toward consumption in te live state, 1 D = 0. 8 and D = 0.2. He is agnostic about is relative ealt status, i.e., is subjective survival probability is equal to te objective actuarial survival probability. His income is epected to grow wit inflation, and te volatility of te 16 A welldesigned questionnaire could elp elicit te individual s attitude towards bequest, even toug a precise estimate may be ard to obtain. 17 Te actuarial mortality tables can be taken as a starting point. Life insurance is already priced to take into account te adverse selection. Page 7 of 20
9 growt rate is 5 percent. 18 His real annual income is $50,000, and e saves 10 percent eac year. He epects to receive a pension of $10,000 eac year (in today s dollars) wen e retires at age 65. His current financial wealt is $50,000. Te investor is assumed to follow te constant relative risk aversion (CRRA) utility wit a risk aversion coefficient (γ). Finally, te financial portfolio is assumed to be rebalanced and te term life insurance contract is renewed annually. 19 Tese assumptions remain te same for all cases. Oter parameters suc as initial wealt will be specified in eac case. Table 1. Capital Market Return Assumptions Compounded Annual Return Risk (tandard Deviation) RiskFree (Bonds) 5%  Risky (tocks) 9% 20% Inflation 3%  Case #1: Human capital, financial asset allocation, and life insurance demand over lifetime In tis case, we assume tat te investor as a moderate risk aversion (relative risk aversion of 4). Also, te correlation between te investor s income and te market return (risky asset) is assumed to be For a given age, te amount of insurance te investor sould purcase can be determined by is consumption/bequest preference, risk tolerance, and financial wealt. His epected financial wealt, uman capital, and te derived optimal insurance demand over te investor s life (from age 25 to 65) are presented in Cart 2. Cart 2: Human Capital, Insurance Demand, and Financial Asset Allocation over te Life Cart 2: Human Capital, Insurance Need, and Asset Allocation Over Te Life $1,000,000 $900,000 $800,000 $700,000 $600,000 $500,000 $400,000 $300,000 $200,000 $100,000 $ Age 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% Human Capital Face Value of Term Life Insurance Proportional Allocation to Risk Free Asset everal results are wort noting. First, uman capital gradually decreases as te investor gets older and te remaining number of working years gets smaller. econd, te amount of 18 Te salary growt rate and te volatility are cosen mainly to sow te implications of te model. Tey are not necessarily representative. 19 Te mortality and insurance loading is assumed to be 12.5%. 20 Davis and Willen (2000) estimated te correlation between labor income and equity market returns using te Current Occupation urvey. Tey find tat correlation between equity returns and labor income typically lies in te interval from 0.10 to Page 8 of 20
10 financial capital increases as te investor ages; tis is te result of growt of eisting financial wealt and additional savings te investor makes eac year. Te allocation to risky asset decreases as te investor ages. Tis result is due to te dynamic between uman capital and financial wealt over time. Wen an investor is young, te investor s total wealt is dominated by uman capital. ince uman capital in tis case is less risky tan te financial risky asset, young investors will invest more financial wealt into risky assets to offset te impact of uman capital on te overall asset allocation. As te investor gets older, te allocation to risky assets is reduced as uman capital gets smaller. Finally, te insurance demand decreases as te investor ages. Tis is not surprising, as te primary driver of te insurance demand is te uman capital. Te decrease in te uman capital reduces te insurance demand. In te following cases, we will vary te investor s preference of bequest, risk preference, and eisting financial wealt to illustrate teir impact on te investor s optimal asset allocation and life insurance purcases. Case #2: trengt of bequest motive Tis case sows te impact of bequest motives on te optimal decisions on asset allocation and insurance demand. In te case, we assume te investor is at age 45 and as an accumulated financial wealt of $500,000. Te investor as a moderate risk aversion coefficient of 4. Te optimal allocations to te riskfree asset and te optimal insurance demands across various bequest levels are presented in Cart 3. Cart 3: Optimal Insurance Demand and Asset Allocation across trengt of Bequest Cart 3: Insurance Need, Asset Allocation vs. Bequest Motive (D) $600,000 $500,000 $400,000 $300,000 $200,000 $100,000 $ Bequest Motive (D) 100% 80% 60% 40% 20% 0% Face Value of Term Life Insurance Policy ($) Proportional Allocation to Risk Free Asset (%) It can be seen tat te insurance demand increases as te bequest motive gets stronger, i.e., te D gets larger. Tis result is epected because an investor wit a stronger bequest motive is more concerned about is/er eirs and as te incentive to purcase a larger amount of insurance to edge te loss of uman capital. On te oter and, tere is almost no cange in te proportional allocation to riskfree asset at different strengts of bequest motive. Tis indicates tat te asset allocation is primarily determined by risk tolerance, returns on risk Page 9 of 20
11 free and risky assets, and uman capital. Tis case sows tat bequest motive as a strong impact on insurance demand, but little impact on optimal asset allocation. 21 Case #3: Risk tolerance Te purpose of tis case is to sow te impact of te different degrees of risk aversion on te optimal decisions on asset allocation and insurance demand. In tis case, we again assume te investor is at age 45 and as accumulated a financial wealt of $500,000. Te investor as a moderate bequest level, i.e., D=0.2. Te optimal allocations to riskfree asset and te optimal insurance demands across various risk aversion levels are presented in Cart 4. Cart 4: Optimal Insurance Demand and Asset Allocation at Different Risk Aversion Levels Cart 4: Insurance Need, Asset Allocation vs. Risk Aversion $450,000 $400,000 $350,000 $300,000 $250,000 $200,000 $150,000 $100,000 $50,000 $0 Low 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% Hig Risk Aversion Face Face Value Value of of Term Term Life Life Insurance Insurance Policy Policy ($) ($) Proportional Allocation to Risk Free Asset (%) As epected, te allocation to te riskfree asset increases wit te investor s risk aversion level. Tis is te classical result in financial economics. Actually, te optimal portfolio is 100 percent in stocks for risk aversion levels less tan 2.5. Te optimal amount of life insurance as a very similar pattern. Te optimal insurance demand increases wit risk aversion. For a moderate investor (a CRRA risk aversion coefficient 4), te optimal insurance demand is about $290,000, wic is rougly si times te current income of $50, Terefore, conservative investors sould invest more in riskfree assets and buy more life insurance, compared to aggressive investors. Case #4: Financial wealt Te purpose of tis case is to sow te impact of te different amounts of current financial wealt on te optimal asset allocation and insurance demand. We old te investor s age at 21 In tis model, subjective survival probability as similar impact on te optimal insurance need and asset allocation as te bequest motive (D). Wen subjective survival probability is ig, te investor will buy less insurance. 22 Tis result is very close to te typical recommendation by financial planners; i.e., purcase a term life insurance policy tat as a face value four to seven times one s current income. ee, for eample, Todd (2004). Page 10 of 20
12 45 and te risk preference and te bequest motive at te moderate levels (a CRRA risk aversion coefficient 4 and bequest level 0.2). Te optimal asset allocations to riskfree asset and te optimal insurance demands for various financial wealt levels are presented in Cart 5. Cart 5: Optimal Insurance Demand and Asset Allocation at Different Financial Wealt Levels Cart 5: Insurance Need, Asset Allocation vs. Current Wealt $450,000 $400,000 $350,000 $300,000 $250,000 $200,000 $150,000 $100,000 $50,000 $0 $50,000 $250,000 $500,000 $750,000 $1,000,000 $1,250,000 $1,500, % 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% Current Financial Wealt Face Value of Term Life Insurance Policy ($) Proportional Allocation to Risk Free Asset (%) First, it can be seen tat te optimal allocation to te riskfree asset increases wit te initial wealt. Tis may seem inconsistent wit te CRRA utility function, since te CRRA utility function implies te optimal asset allocation does not cange wit te amount of wealt te investor as. However, we need to note tat te wealt includes bot financial wealt and uman capital. In fact, tis is a classic eample of te impact of uman capital on te optimal asset allocation. An increase in financial wealt not only increases te total wealt, but also reduces te percentage of total wealt represented by uman capital. In tis case, uman capital is less risky tan te risky asset. 23 Wen te initial wealt is low, te uman capital dominates te total wealt and te allocation. As a result, to acieve te target asset allocation of a moderate investor, say an allocation of 60 percent riskfree asset and 40 percent risky asset, te closest allocation is to invest 100 percent financial wealt in te risky asset, since te uman capital is illiquid. Wit te increase in te initial wealt, te asset allocation is gradually adjusted to approac te target asset allocation a moderate riskaverse investor desires. econd, te optimal insurance demand decreases wit financial wealt. Tis result can be intuitively eplained troug te substitution effects between financial wealt and life insurance. In oter words, wit a large amount of wealt in and, one as less demand for insurance, since te loss of uman capital as a muc lower impact on te wellbeing of one s eirs. Te optimal amount of life insurance decreases from over $400,000, wen te 23 In tis case, te income as a real growt rate of 0% and a standard deviation of 5%, yet te epected real return on stock is 8% and te standard deviation is 20%. Page 11 of 20
13 investor as little financial wealt, to almost zero, wen te investor as $1.5 million in financial assets. In summary, for a typical investor wose uman capital is less risky compared to te stock market, te optimal asset allocation is more conservative and te life insurance demand is smaller for investors wit more financial assets. Case #5: Correlation between wage growt rate and stock returns In tis case, we eamine te impact of te correlation between te socks to wage income and te risky asset returns. In particular, we want to evaluate te life insurance and asset allocation decision for investors wit igly correlated income and uman capital. Tis can appen wen te investor s income is closely linked to is employer s company stock performance, or were te investor s compensation is igly influenced by te financial market (e.g., te investor works in te financial industry). Again, we old te investor s age at 45 and te risk preference and te bequest motive at te moderate level. Te optimal asset allocations to te riskfree asset and te optimal insurance demands for various financial wealt levels are presented in Cart 6 below. Cart 6: Optimal Insurance Demand and Asset Allocation at Different Correlation Levels Cart 6: Insurance Need, Asset Allocation vs. Correlation $330,000 $310,000 $290,000 $270,000 $250, Correlation between wage growt rate and risky asset return 22% 20% 18% 16% 14% 12% Face Value of Term Life Insurance Policy ($) Proportional Allocation to Risk Free Asset (%) Te optimal allocation becomes more conservative (i.e., more allocation to riskfree asset) as te income and stock market return become more correlated. One way to look at tis is tat a iger correlation between te uman capital and te stock market results in less diversification, tus a iger risk of te total portfolio (uman capital plus financial capital). To reduce tis risk, an investor will invest more financial wealt in te riskfree asset. Te optimal insurance demand decreases as te correlation increases. Life insurance is purcased to protect uman capital for te family and loved ones. As te correlation between te risky (stock) asset and te income flow increases, te e ante value of te uman capital to te surviving family becomes lower. Terefore, tis lower uman capital valuation induces a Page 12 of 20
14 lower demand for insurance. Also, less money spent on life insurance also indirectly increases te amount of financial wealt te investor can invest. Tis also allows te investor to invest more in riskfree assets to reduce te risk associated wit te total wealt. 24 In summary, te optimal asset allocation becomes more conservative and te amount of life insurance becomes less, as wage income and te stock market returns become more correlated. 5. ummary and Conclusions In tis paper we ave epanded on te Mertonian idea tat uman capital is a sadow asset class tat is wort muc more tan financial capital early in life, and tat it also as unique risk and return caracteristics. Human capital even toug it is not traded and igly illiquid sould be treated as part of te endowed wealt tat must be protected, diversified and edged. We ave demonstrated tat te correlation between uman capital and financial capital, i.e., weter you are closer to a bond or a stock, as a noticeable and immediate impact on te demand for life insurance as well as te usual portfolio considerations. Our main argument is tat te two decisions How muc life insurance do I need? And were sould I invest my money? cannot be solved in isolation. Rater, tey are different aspects of te same problem. For instance, a person wose income eavily relies on commissions sould consider is uman capital closer to a stock since te income is igly correlated wit te market, wic results in great uncertainty in is uman capital. Consequently, e sould purcase less insurance and invest more financial wealt in bonds. Conversely, a tenured university professor wo considers er uman capital closer to a bond, purcases more insurance, and invests more financial wealt in stocks. We develop a unified uman capitalbased framework to elp individual investors wit bot decisions. Tere are several key results: 1) investors need to make asset allocation decisions and life insurance decisions jointly; 2) te magnitude of uman capital, its volatility, and its correlation wit oter assets ave a significant impact on te two decisions over te life cycle; 3) bequest preferences and te subjective survival probability ave a significant impact on insurance demand, but little influence on optimal asset allocation; and 4) conservative investors sould invest more in riskfree assets and buy more life insurance. We presented five case studies to demonstrate te optimal decisions under different scenarios. Obviously, we ave only traced out a roug sketc of te complete picture. More researc remains to be done in order to make tese decisions more suitable in practice. One possible net step along tis olistic integration is to model te various competing types of life insurance as well as teir unique taseltered aspects witin a unified asset allocation framework. Wole life insurance as well as oter forms of variable life insurance can be viewed as a edge against possible canges in systematic population mortality rates and ence migt coeist in an optimal portfolio wit sortterm life insurance. Anoter direction is to diverge from te traditional epected utility models to include oter metods, suc as minimizing sortfall probabilities, to determine te appropriate asset allocation and life insurance decision. 24 ee Case #3 for a detailed discussion on te wealt impact. Page 13 of 20
15 Appendi In tis appendi we describe uman capital, te basic pricing mecanism of life insurance, and most importantly te detailed model underlying te numerical results and eamples we provided. 1) Human Capital If we let te symbol w(i) denote te random (real, afterta) wage or salary tat a person will receive during time period or year i, ten te discounted value of tis income flow at te current time zero would be represented matematically by: n E[ w( i)] FHC : =, (1) i i= 1 (1 + r + v) were te epectation in te numerator converts te random variables into a scalar. Note tat in addition to taking epectations (under a pysical real world measure), te denominator in equation (1) contains te term v, wic accounts for risk, illiquidity, and oter subjective factors tat obviously reduce te timezero value of te epression FHC. And, depending on te investor s education and profession, e/se migt be epected to earn te same eact E[ w( i)] at eac time period i, te random socks to wages: w( i) E[ w( i)] migt ave very different statistical caracteristics vis a vis te market portfolio, and tus eac of tese professions would induce a distinct risk premium value for v in equation (1) and tus lead to a lower or iger financial economic value for teir specific uman capital. In addition, te v in equation (1) would capture illiquidity and oter market imperfections tat will furter increase te denominator and reduce te value of FHC. Likewise, in te ensuing discussion wen we focus attention on te correlation or covariance between uman capital and oter macroeconomic or financial factors, we are of course referring to te correlation between socks w( i) E[ w( i)] and socks or innovations to te return generating process in te market. Tis will induce a (quite complicated) dependence structure between FHC in equation (1) and te investor s financial portfolio. 2) Te Oneyear Renewable Term Life Insurance Pricing Mecanism Te oneyear renewable term policy premium is paid at te beginning of te year or on te individual s birtday and protects te uman capital of te insured for te duration of te year. If te insured person dies witin tat year, te insurance company pays te face value to te beneficiaries, soon after te deat or prior to te end of te year. Net year te contract is guaranteed to start anew wit new premium payments made and protection received; ence te word renewable. Te policy premium is obviously an increasing function of te desired face value, and te two are related by te simple formula: q P = θ, (2) ( 1 + r) Te premium P is calculated by multiplying te desired face valueθ by te probability of deat q, and ten discounting by te interest rate factor (1+r). Te teory beind equation (2) is te wellknown law of large numbers, wic guarantees tat probabilities become Page 14 of 20
16 percentages wen individuals are aggregated. Note te implicit assumption in equation (2) is tat altoug deat can occur at any time during te year (or term), te premium payments are made at te beginning of te year and te face values are paid at te end of te year. From te insurance company s perspective, all of te premiums received from te group of N individuals wit te same age (i.e., probability of deat q) and face value θ, are comingled and invested in an insurance reserve earning a rate of interest r, so tat at te end of te year PN ( 1+ r) is partitioned amongst te qn beneficiaries. Tere is no savings component or investment component embedded witin te premium defined by equation (2). Rater, at te end of te year te survivors lose any claim to te pool of accumulated premiums, since all funds go directly to te beneficiaries. As te individual ages and te probability of deat increases (denoted by ), te same eact face amount of (face value) life insurance θ will cost more and will induce a iger premium P, as per equation (2). Note tat in practice, te actual premium is loaded by an additional factor denoted by ( 1 + λ) to account for commissions, transaction costs, and profit margins and so te actual amount paid by te insured is closer to P ( 1 + λ), but te underlying pricing relationsip driven by te law of large numbers remains te same. Also, from a traditional financial planning perspective, te individual conducts a budgeting analysis to determine is or er life insurance demands, i.e., te amount te surviving family and beneficiaries need to replace te lost wages in present value terms. Tat quantity would be taken as te required face value in equation (2), wic would ten lead to a premium. Alternatively, one can tink of a budget for life insurance purcases, and te face value would be determined by equation (2). In our model and te ensuing discussion we will solve for te optimal agevarying amount of life insurance denoted by θ wic ten induces an agevarying policy payment P wic maimizes te welfare of te family by taking into account its risk preferences and attitudes toward bequest. 3) Model pecification Optimal Asset Allocation and Insurance Demands We assume tat te investor is currently age and will retire at age Y. Te term retirement is simply meant to indicate tat te uman capital income flow is terminated and te pension pase begins. We furter assume tat te financial portfolio will be rebalanced annually and tat te life insurance wic is assumed to be of te oneyear term variety will be renewed annually as well. We do not consider ta in our models. Te investor would like to know ow muc (i.e., te face value of term life) insurance e sould purcase and wat fraction of is financial wealt sould be invested in a risky asset (stock). In te model, an investor determines te amount of life insurance demand, θ te face value of life insurance, a.k.a. te deat benefit togeter wit te allocation q α to risky asset to maimize te end year utility of total wealt (uman capital plus financial wealt) weigted by te alive and dead states. Te optimization problem can be epressed as: Page 15 of 20
17 {(1 D) (1 q ) U [ W + H ] + D ( q ) U [ W + θ ]} ma E alive dead + 1 (3) { θ, α } subject to te budget constraint: 1 2 r µ σ + σ Z f 2 rf [ W + ( 1+ λ) q θ e C ] α e + (1 e W + 1 = α ) θ rf ( W + C ) e 0 θ, (5) (1 + λ) q and 0 α 1. (6) Equation (5) requires te cost (or price) of te term insurance policy to be less tan te amount of current financial wealt te client as, and tere is a minimum insurance amount ( θ 0 > 0 ) an investor is required to purcase in order to ave a minimum protection from te loss of uman capital. Te symbols, notations, and terminology used in te optimal problem are eplained below. (4) θ α D λ q q W t denotes te amount of life insurance. denotes te allocation to risky asset. denotes te relative strengt of te utility of bequest. Individuals wit no utility of bequest will ave D = 0. denotes te objective probabilities of deat at te end of te year +1 conditional on being alive at age. Tis probability is determined by a given population, i.e., mortality table. denotes te subjective probabilities of deat at te end of te year +1 conditional on being alive at age. (1 q ) denotes te subjective probability of survival. ubjective probability of deat may be different from te objective probability. In oter words, a person migt believe e or se is ealtier (or less ealty) tan average. Tis would impact te epected utility, but not te pricing of te life insurance, wic is based on objective population survival probability. denotes te fees and epenses (i.e., actuarial and insurance loading) imposed and carged on a typical life insurance policy. denotes te financial wealt at time t. We assume tere are two assets in te market, one risky and one riskfree. Tis is consistent wit te twofund separation teorem tat is consistent wit traditional portfolio teory. Of course, tis can always be epanded to multiple asset classes. Te return on te riskfree asset is denoted by r f. Te value, t, of te risky asset follows a discrete version of a Geometric Brownian Motion. 1 2 t+ 1 = t ep{ µ σ + σ Z, t+ 1). (7) 2 Were, µ is te epected return and σ is te standard deviation of te return of te risky asset. Z, is an independent random variable and ~ (0,1). t Z, t N Page 16 of 20
18 t denotes te labor income. In our numerical cases, we assume tat te income follows a discrete stocastic process specified by ep{ µ σ Z 1}. (8) Were, > 0. t+ 1 = t +, t+ t µ and σ are te annual growt rate and te annual standard is an independent random variable deviation of te income process. Z, t and Z, t ~ N(0,1). Based on equation (8), for a person at age, is income at age +t is determined by: t + t = ep{ µ + σ Z, k}. (9) k= 1 We furter assume tat correlation between te labor income innovation and te return of risky asset is ρ and Z 2 = ρz + 1 ρ Z (10) Were, Z is a standard Brownian motion independent of Z. Tat is, Corr Z, Z ) = ρ. (11) ( H t denotes te present value of future income from age t + 1 to deat. Te income after retirement is te payment from pensions. Based on equation (9), for a person at age +t, te present value of future income from age + t + 1 to te deat is determined by: H Y +[ + t = + j ep{ ( j t)( rf + + j = t 1 η ζ )}], (12) Were, η is te risk premium (discount rate) for te income process and captures te market risk of income. ζ is a discount factor in te uman capital evaluation to account for te illiquidity risk associated wit one s job. We assumed a 4 percent discount rate per year. 25 Based on CAPM, te η can be evaluated by Cov( Z, Z ) r r σ f f η = [ µ ( e 1) ] = ρ[ µ ( e 1)]. (13) Var( Z ) σ Furtermore, we regard te epected value of H t, i.e., E [ H + t ], as te uman capital one as at age + t +1. C t denotes te consumption at year t. For simplicity, we assume tat C t = C, i.e., te constant consumption over time. Te power utility function (CRRA) is used in our numerical results. Te function form of te utility function we used is, t 25 Te 4 percent discount rate translates into about a 25 percent discount on te overall present value of uman capital for a 45 year old wit 20 years future salary. Tis 25 percent discount is consistent wit empirical evidence on te discount factor between restricted stocks and teir unrestricted counterparts (e.g., Amiud & Mendelson (1991)). Also, Longstaff (2002) reported tat te liquidity premium for te longermaturity Treasury bond is 10 to 15 percent of te value of te bond. Page 17 of 20
19 1 γ U ( ) = (14) 1 γ for > 0 and γ 1, and U ( ) = ln( ) (15) for > 0 and γ = 1. We use te power utility function for bot U alive ( ) and U dead ( ), 26 wic are te utility functions associated wit te alive and dead states, respectively. We solve te problem via simulation. We first simulate te values of te risky asset using equation (7). Ten, we simulate Z troug equation (10) to take into account te correlation between te income innovation and te return of financial market. Finally, we use equation (8) to generate income over te same period. Human capital, H + t, is calculated using equations (9) and (12). If wealt level at age +1 is less tan zero, we set te wealt equal to zero. Tat is, one does not ave any remaining financial wealt. We simulate tis process N times. Te objective function is evaluated by: 1 N U [ W + 1 ( n) + H ( n)] (16) n = 1 alive + 1 N In te numerical eamples, we set N = N = U + + n dead [ W n 1 1 ( ) θ ] (17) N 26 tutzer (2004) pointed out te difficulties in applying te epected utility teories in practice and proposed te use of minimizing sortfall probability as an alternative approac. In tis paper, we coose to follow te traditional epected utility model of linking asset allocation and portfolio decisions to individual risk aversion. Page 18 of 20
20 References Ameriks, Jon and tepen Zeldes (2001), How Do ouseold Portfolio ares Vary wit Age? Working Paper, Columbia University. Amiud, Yakov and Haim Mendelson (1991), Liquidity, Asset Prices and financial Policy, Financial Analysts Journal, 47, Auerbac, Alan J. and Laurence Kotlikoff (1991), Life Insurance Inadequacy Evidence From a ample of Older Widows, National Bureau of Economic Researc, Inc, NBER Working Papers: Benartzi, lomo (2001), Ecessive Etrapolation and te Allocation of 401(k) Accounts to Company tock, Journal of Finance 56, Benartzi, lomo and Ricard H. Taler (2001), Naive Diversification trategies in Defined Contribution aving Plans, American Economic Review 91, Berneim, B. Douglas (1991), How trong Are Bequest Motives? Evidence Based on Estimates of te Demand for Life Insurance and Annuities, Journal of Political Economy 99, Bodie, Z., R.C. Merton and W.F. amuelson (1992), Labor supply fleibility and portfolio coice in a life cycle model, Journal of Economic Dynamics and Control, Vol. 16, pp Campbell, Jon and Luis Viceira (2002), trategic Asset Allocation Portfolio Coice for Longterm Investors," Oford University Press. Campbell, R.A. (1980), Te Demand for Life Insurance: An Application of te Economics of Uncertainty, Te Journal of Finance, Vol. 35, No. 5, pp Davis, tepen J. and Paul Willen (2000), OccupationLevel Income ocks and Asset Returns: Teir Covariance and Implications for Portfolio Coice, University of Cicago Graduate cool of Business, Working Paper. Economides, Nicolas (1982), Demand for Life Insurance: An Application of te Economics of Uncertainty: A Comment, Journal of Finance 37, Fiscer,. (1973), A Life Cycle Model of Life Insurance Purcases, International Economic Review, Vol. 14, No. 1, pp Gokale, Jagadees and L. J. Kotlikoff (2002), Te Adequacy of Life Insurance, Researc Dialogue, TIAACREF INTITUTE, tiaacrefinstitute.org, issue no. 72 July Goldsmit, A. (1983), Houseold Life Cycle Protection: Human Capital versus Life Insurance, Te Journal of Risk and Insurance, Vol. 50, No. 3, pp Heaton, Jon and Debora Lucas (1997), Market Frictions, avings Beavior, and Portfolio Coice, MacroeconomicDynamics. 1997; 1(1): Page 19 of 20
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