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Asset ManagementA Systematic Approach to Factor Investing$

Andrew Ang

Print publication date: 2014

Print ISBN-13: 9780199959327

Published to Oxford Scholarship Online: August 2014

DOI: 10.1093/acprof:oso/9780199959327.001.0001

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Investing over the Life Cycle

Investing over the Life Cycle

Chapter:
(p.149) Chapter 5 Investing over the Life Cycle
Source:
Asset Management
Author(s):

Andrew Ang

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199959327.003.0005

Abstract and Keywords

Labor income is an asset, and for young investors the value of labor income usually dominates the rest of their financial holdings. An investor’s mix of assets changes as her labor income evolves over her life cycle, and an investor whose labor income is bond-like should reduce his holdings of equities as retirement approaches. While economic theory suggests that annuities are ideal for retirees, few hold them.

Keywords:   labor income, human capital, retirement savings, life-cycle model, life-cycle asset allocation, consumption, intergenerational equality, health-care risk, pension underfunding, 4% rule, annuities

Chapter Summary

Labor income is an asset, and for young investors the value of labor income usually dominates the rest of their financial holdings. An investor’s mix of assets changes as her labor income evolves over her life cycle, and an investor whose labor income is bond-like should reduce his holdings of equities as retirement approaches. While economic theory suggests annuities are ideal for retirees, few hold them.

1. Employees’ Retirement System of Rhode Island

The State of Rhode Island and Providence Plantations, or Rhode Island for short, is the smallest of America’s fifty states.1 Although often overshadowed by its larger neighbors, Massachusetts and Connecticut, people in Rhode Island (there are only slightly more than a million of them) have a reputation for wanting to do their own thing and doing it first. Rhode Island was the first of the thirteen colonies to declare independence from Great Britain on May 4, 1776—beating the other colonies by two months. When President Abraham Lincoln asked the Union states to send soldiers to fight in the Civil War, Rhode Island was the first to enthusiastically respond. And in 2011, Rhode Island became the first state to undertake a regime-changing reform of its public pension plan.

The Employees’ Retirement System of Rhode Island (ERSRI) provides retirement, disability, and survivor benefits to public employees including state workers, school teachers, judges, and local uniformed personnel including police officers and firefighters. Like most public pension plans in the United States, it was grossly underfunded. In 2011, ERSRI had a funding ratio of 59% and held only $6.3 billion to meet liabilities of $10.8 billion. Each person in Rhode Island (p.150) would have needed to pay $4,230 to meet the shortfall.2 In addition, spending on pensions was crowding out other services. In 2002, 3 cents of each taxpayer dollar went to support pensions. This rose to 9 cents in 2009 and was projected to reach 20 cents in 2018.3 State employee contributions, however, had remained flat at 8.75%, and teachers’ contributions had also not changed from 9.5%.

An editorial in the Providence Journal summarized Rhode Island’s problem:4

Unless this unsustainable system is reformed, the state confronts a bleak future of brutally high taxes; deep cuts in government services and in assistance to our most needy and fragile citizens; the threatened loss of retirement security for our public employees; and the severe economic distress that will surely accompany these developments.

Rhode Island was not alone. Robert Novy-Marx and Joshua Rauh, finance professors at Rochester and Stanford, respectively, have shined a spotlight on the underfunding of our public pension plans. They estimate that by 2009, states had accrued $5.7 trillion of liabilities to their workers while assets in these plans totaled less than $2 trillion—a hole of $3.7 trillion. By way of context, consider that the outstanding publicly traded debt issued by states is about $1 trillion.5

Gina Raimondo, who took office as Rhode Island’s treasurer in 2010, believed that fixing the state’s pension system was essential to putting the state on a sound financial footing. Raimondo commissioned a report, Truth in Numbers: The Security and Sustainability of Rhode Island’s Retirement System, which was released in June 2011. The report showed that the large pension deficits were due to five factors:6

  1. 1. Insufficient contributions;

  2. 2. Raising benefits from the 1960s through the 1980s without corresponding increases in contributions;

  3. (p.151) 3. Non-optimal plan design, in some cases allowing workers to get paid more in retirement than while working;

  4. 4. Greater life expectancy; and

  5. 5. Low investment returns, with Rhode Island’s pension fund earning 2.3% from 2000 to 2010 while its projected return was 8.25% over the same period.

The pension deficits had accumulated over many years. “You’re paying for the sins of the past,” said Frank Karpinski, the Executive Director of ERSRI.7 Truth in Numbers succeeded in drawing attention to the pension problem. It also framed the problem: no matter which side of the issue, everyone was talking from the same set of notes.

Together with Gov. Lincoln Chaffee, Raimondo put together a special commission that included representatives from labor unions, business, and academia. This Pension Advisory Group held public meetings across the state from June to September 2011 and allowed the public to share their views. Presentations at the meetings highlighted the seriousness of the problem. “At the end of the day, no one could criticize the process. It was long, it was thorough, it was open, and it was transparent,” said Raimondo.8

Raimondo approached the process of pension reform by working backward. She first posed the question, “what does retirement security look like?” Targeting a replacement rate or ratio (the amount of income in retirement as a fraction of pre-retirement income) of 75% to 80%, Raimondo asked how the pension system should be designed to get there. In addition, she aimed to cut the unfunded liability in half immediately and to get to an 80% funded status in twenty years.

A key theme was intergenerational fairness. Under the current system, younger public employees were disadvantaged relative to older workers. Raimondo also wanted to design a plan in which risks were appropriately shared between the retirees and the state. She assured workers that reforms would not reduce accrued benefits.

“We are in the fight of our lives for the future of this state,” said Raimondo.9

2. Labor Income

Labor income is a form of wealth. It is not traded (the Thirteenth Amendment abolished slavery), making it very different from the financial assets that we hold in our portfolios. Much of our lifelong allotment of human capital is determined at birth, but individuals can augment their human capital through education, while (p.152) circumstances such as job loss or illness can reduce it. The riskiness of a person’s labor income—its volatility and how it correlates with financial returns—affects her asset allocation.

We take account of human capital by treating it like an asset.10 Suppose we denote the present value of an individual’s wages as H (for human capital). Human capital is the discounted value of a person’s lifelong earnings from work; it is the present value today that is equivalent to the wage profile of that individual. Human capital is risky because earnings from work can fluctuate—and in the most unfortunate case fall to and stay at zero.

We denote financial wealth as W. Total wealth is H+W, or, in other words, total wealth comprises both human capital and financial wealth.

2.1. Asset Allocation with No Labor Income

Consider an individual with no labor income and $1 million of financial assets. For simplicity, we’ll assume that there are only equities, which have an expected return of μ=10% and a volatility of σ=15%, and risk-free T-bills paying rf=2%. The investor has mean-variance utility, or equivalently has CRRA utility, with a risk aversion of γ=5. (Chapters 2 and 3 cover utility functions and mean-variance investing, respectively.11)

Our investor with only financial wealth holds w=62.5% in equities, which is obtained from the mean-variance investing formula:

(5.1)
w=1γμrfσ2=150.100.02(0.15)2=62.5%

(See equation (2.10) in chapter 2.) His asset allocation is then:

Hold 0.625 × $1 million = $625,000 in equities

$375,000 in T-bills

$1 million

2.2. Asset Allocation with Risk-Free Labor Income

Now contrast the asset allocation of an individual who has risk-free wages, like tenured professors at top universities (yes, I know, I’m lucky and very grateful) or federal judges. Her human capital is equivalent to her being endowed with risk-free assets. Let’s assume her wage flow is equivalent to human capital today of (p.153) H = $1 million and she also has financial wealth of W = $1 million. Her total wealth is thus H+W = $2 million. Like our first investor, we’ll assume her risk aversion is γ=5.

The optimal holding of risky equities with risk-free labor income is:

(5.2)
w=1γμrfσ21+HW.

Note that when human capital is zero, H=0, equation (5.2) is exactly the same as equation (5.1). Our investor with risk-free human capital tilts her portfolio toward stocks relative to an investor with only financial wealth. The proportion of equities held by the investor in this example is:

(5.3)
w=1γμrfσ21+HW=150.100.02(0.15)21+11=125%

The optimal holdings of our investor with labor income are:

Hold 1.25 × $1 million = $1,250,000 in equities

$250,000 in T-bills

$1 million

This is a much larger weight in equities than our first investor without labor income; the investor with risk-free labor income shorts the risk-free asset and holds a levered portfolio of 125% in equities. But it turns out that the combined portfolio has exactly the same 62.5% holdings in risky assets.

If we view the investor’s total wealth portfolio, we have:

Endowed with $1,000,000 in human capital

Hold 1.25 × $1 million = $1,250,000 in equities

$250,000 in T-bills

$2 million

As a fraction of total wealth, her equity allocation is 1,250,000/2 million = 62.5%, which is the same as our first investor without labor income.

The presence of risk-free labor income allows the investor to take on more risk in her financial portfolio. In fact, equation (5.2) levers up the standard equity position (from equation (5.1)), and the leverage ratio is the ratio of human capital to financial wealth, H/W.

Asset allocation must be viewed in the broader context of total wealth, which sums both financial and human capital wealth. Risk-free labor income is equivalent to being endowed with risk-free bonds. Your financial portfolio has to hold more equities to balance out your risk-free human capital. In economists’ jargon, the risk exposure in human capital wealth crowds out the risk-taking ability of our financial wealth.

(p.154) 2.3. Asset Allocation with Risky Labor Income

Let’s take an investor who has risky labor income. We need to consider whether her labor income looks like a stock or a bond. If you are reading this book, you probably work in a field related to finance, so your labor income is probably positively correlated with equities. Most of my MBA students have human capital that is highly correlated with the stock market.

A positive correlation between labor income and equity returns reduces the optimal allocation to risky assets in an investor’s financial portfolio. In this case, human capital is equivalent to already being endowed with equities. You wish to reduce your equity positions in your financial portfolio to offset your equity-like human capital. This undoes the equity risk added by your labor income. In short, when human capital acts like equities, you effectively already own equities and so you hold fewer equities in your financial portfolio.

What about those investors whose wages are negatively correlated with equity returns? These people actually earn more during stock market downturns. Some examples include bankruptcy attorneys, shrewd experts who can turn around poorly performing companies, and debt collectors. Human capital for these investors acts in the opposite way of equities, and so they have human capital that is anti-equities. If they are anti-equity in their human capital, they want to offset that by holding lots of equities in their financial portfolio. That is, their allocations to equity will be even more aggressively weighted toward equities than investors with risk-free labor income.

There are several important corollaries:

  1. 1. Investors with income that goes up and down with the stock market should hold more bonds (or other very low-risk assets) in their financial portfolios. These investors have equity-like human capital, so they need to take positions opposite to equities in their financial portfolios.

  2. 2. John Roberts, Antonin Scalia, Sonia Sotomayor, and all the other Justices of the Supreme Court should hold relatively aggressive positions in the stock market. Their incomes are like risk-free bonds, so their financial portfolios should take large, opposite positions in risky equities.

  3. 3. An investor should hold little or none of her employer’s stock. Her labor income is already exposed to her employer’s fortunes, and the last thing she should want is even more of the same risk in her savings. Her financial holdings should diversify away her employer’s risk reflected in her labor income. (See chapter 3.)

  4. 4. If you work in the United States, your financial portfolio should actually hold more overseas stocks than domestic stocks. Your income already comes from a U.S. source, so you need to hold more non-U.S. securities in your financial portfolio.12

(p.155) These are all normative statements, or statements about what investors should do (see chapter 2 for normative vs. positive economics). In reality, many do exactly the opposite. A few traders like to hold very risky assets in their personal portfolios, despite their labor income being highly dependent on the stock market. All too many investors hold outsized positions in their employers’ stock, and they hold home-biased portfolios with little international exposure. These investors load up on the same risks in their financial portfolios as their human capital is exposed to and fail to diversify away some of their human capital risk (see also chapter 4).

2.4. Labor Income Risk

Measuring labor income risk is tricky; we can’t observe the value of human capital because it is not traded.

The first economist to treat human capital as an asset and put a value on it was Gary Becker (1964), who won the Nobel Prize in 1992. Becker showed that education is an investment in human capital, rather than just a luxury or consumption expenditure. He also showed that investment in human capital was integral to economic growth. A large literature in labor economics measures human capital returns by computing the benefits to investments in human capital: labor economists compute extra wages earned by those who have gone to graduate school over those who have gone to college, over those with only high school educations, and over those who did not attend high school. Not surprisingly, wages increase with investment in education going from less than a high school diploma (median earnings of $471 per week in 2012 according to the Bureau of Labor Statistics) to bachelor degrees ($1,066) to professional degrees ($1,735). Interestingly, there is a drop to $1,624 for PhDs. (Someone should have told me before I sweated over mine.) As is well known, women earn less than men, and blacks earn less than whites.

Human capital is the present value of all future wage flows. The return on human capital is often computed as the growth in wages.13 (p.156) Palacios-Huerta (2003) computes human capital returns for various professions and genders. He finds white males with one to five years of experience and a college degree have annual human capital returns of 14.2% versus 10.5% for those with some college but no degree. While college graduates have higher human capital returns, they are more volatile: the standard deviation of human capital returns for white male college graduates is 11.3% versus 4.7% for those with post-college education. Thus, the raw Sharpe ratio (which I define as the expected return [not excess] divided by volatility; see chapter 2) of white college males with one to five years of work experience is 14.2%/11.3%=1.3, which is lower than the 2.2 raw Sharpe ratio of the same demographic group with post-college experience. In comparison, the S&P 500 has a raw Sharpe ratio of around 0.6. Education is a very good investment, on average.

In addition to the risk of your human capital varying over time, there is also large disparity in human capital across professions. Christiansen, Joensen, and Nielsen (2007) report the mean human capital return for doctors is over 25% with a standard deviation of 18%. Human capital returns for masters graduates in economics have a mean of 12% and a standard deviation of 16%. Christiansen, Joensen, and Nielsen find that some professions have significantly negative returns, like individuals with master degrees in education or music.

Wages can fall a lot during recessions.14 Guvenen, Song, and Ozkan (2012) study the income dynamics of millions of individuals using Social Security Administration records. They find that during recessions, big upward movements of wages become very unlikely while severe drops in income become very likely. The middle part of the distribution tends to be fairly constant compared to the movement tails. The falls in income for the most wealthy—the top 1% (probably you, dear reader)—have the highest labor income risk. During the Great Recession in 2008–2009, the top 1% experienced enormous income losses, which were 21% worse, on average, than the losses of those in the ninetieth percentile.

Computing human capital returns using the growth in wages, however, actually understates the riskiness of human capital. Wages for individuals are like dividends for stocks. Stock prices can soar and plummet while their dividends remain fairly constant. Measuring stock returns by changes in dividends would produce very poor approximations to actual stock returns. In valuing stocks, we need to use time-varying discount rates. Likewise, the discount rates on human capital are not constant.15 Human capital can be very risky.

(p.157) 2.5. Summary

Optimal asset allocation with labor income crucially depends on how risky your labor income is and how it correlates with equities. If your labor income is bond-like (lowly correlated with stock returns), treat it as such and hold more equities in your portfolio. Hold even more equities if your labor income acts the opposite way of equities. If your labor income is stock-like (highly correlated with stock returns), you will offset that equity risk by holding lots of bonds. Thus, paraphrasing Moshe Milevsky (2009), a pension finance expert at Canada’s York University, personal asset allocation depends on whether you are a stock or a bond.

In summary, all else being equal:

Bond-like labor income => Hold more equities.

Stock-like labor income => Hold more bonds.

In real life, people seem to act as our theory predicts. Guiso, Jappelli, and Terlizzese (1996) and Vissing-Jørgensen (2002) show that individuals with very volatile wages tend not to hold equities, or, if they hold equities, they do so as a smaller share of their portfolio. Betermier et al. (2012) employ a data set that tracks 3% of the Swedish population and examine people changing jobs. As individuals move from jobs with less to more volatile wages, they reduce their exposure to risky assets. In their sample, the least volatile wages are earned by individuals working in the field of recycled metal waste and the most volatile wages are earned by those working in asset management (!). An individual moving from the former to the latter slashes her holdings of risky assets by 35%. Dimmock (2012) reports that universities with more volatile non-financial income (tuitions, fees, grants, donations, etc.) hold more fixed income and other low-risk securities. Very volatile nonfinancial income markedly reduces allocations to alternative assets, except for the most selective universities.

3. Life Cycle

Before we describe optimal asset allocation over the life cycle, let’s review. . . .

3.1. Financial Planners’ Advice

Ubiquitous advice from financial professionals is to reduce your stock allocation as you age. A popular rule of thumb is one hundred minus your age, which says that the proportion of your portfolio invested in equities should equal the difference between your age and one hundred. There are many variants, but I will refer to any plan that reduces risky asset holdings as you approach retirement as the “one hundred minus your age rule.”

Panel A of Table 5.1 lists the asset allocation for the Vanguard Target Retirement 2050 Fund. Such target date funds are designed to modify an investor’s asset allocation as he approaches retirement—in this case, a date that falls between (p.158) 2048 and 2052. These are actually funds-of-funds and change the mix of underlying funds over time. Investors pay fees both on the underlying funds and for the privilege of having their asset allocation automatically adjusted as they age. (These funds can be expensive!) According to Vanguard, an investor with twenty-five years to retirement should hold 90% equities and 10% bonds. This evolves over time until, a year before retirement, the mix is half and half (how the equity and bond mix changes as the individual ages is called the glide path). At retirement the investor further cuts equity holdings to 30%.

Table 5.1

Asset Allocations over the Life Cycle

Panel A: Vanguard Target Retirement 2050 Fund

Stocks

Bonds

Cash

25 Years before Retirement

90%

10%

0%

1 Year before Retirement

50%

50%

0%

In Retirement

30%

65%

5%

Panel B: Life-Cycle Allocations to Equities

Age

100−Age Rule

Malkiel (1990)

Shiller (2005)

Federal Thrift

25

75%

70%

85%

85%

35

65%

60%

71%

75%

55

45%

50%

26%

50%

Panel B of Table 5.1 reports allocation to equities recommended by several sources: from the finance professor Burton Malkiel (1990), the economics professor Robert Shiller (2005), and from the “L Fund” of the federal Thrift Savings Plan, the retirement program for federal employees or members of the uniformed services (this is Uncle Sam’s version of the 401(k), see chapter 1).16 While the individual equity proportions slightly differ, the overall pattern is that they all hold less equity with increasing age. The most aggressive decline is recommended by Shiller, who recommends moving from 85% equities at age twenty-five to 26% equities at age fifty-five.

3.2. Equities Are Not Less Risky over the Long Run

A common reason given for the one hundred minus your age rule is that stocks are less risky over the long run. This was made popular by adherents of (p.159) Siegel (1994). Saying that stocks are less risky over the long run is equivalent to saying that stock returns are mean-reverting. But the evidence for this is weak: chapter 8 shows that theory predicts and empirical evidence confirms that the amount of mean reversion in stock returns is very small.

Furthermore, if stock returns are predictable and mean-reverting, optimal asset allocation would not follow the one hundred minus your age rule. If stocks are mean-reverting, you want to load up on stocks when stock prices are low and expected returns are high (buy cheap), and you would reduce equity holdings when stock prices are high and expected returns are low (sell dear). Your portfolio composition would depend on the relative valuation of stocks and bonds; that is, your asset allocation would be sensitive to market conditions and your age. Luis Viceira, a finance professor at Harvard Business School, says: “It is logically inconsistent to count on reduced long-term risk while ignoring the variation in returns that produces it.”17

Pástor and Stambaugh (2012a) argue that for an investor needing to forecast the variability of stock returns (predictive variance, for a statistician), the risk of stocks actually increases in the long run. They show that although there is weak mean reversion in equity returns, it is swamped by the uncertainty of current and future expected returns and the large estimation risk of parameters in an econometric model used to forecast returns. Mean-reverting returns do cause stocks’ long run risk to decrease, but the other two components outweigh this effect, causing the riskiness of stocks to increase with horizon.

Protecting against Stock Underperformance in the Long Run

Zvi Bodie (1995), a pension finance expert at Boston University, demonstrates that stocks are not less risky over the long run by computing the cost of insuring against stocks underperforming bonds.

In Panel A of Figure 5.2, I graph the probability that stocks will outperform bonds as a function of horizon. All the panels in Figure 5.2 use U.S. stock and bond data from 1926 to 2012 from Ibbotson Associates.18 The probability that stocks outperform bonds rises from 58% at a one-year horizon to 86% at a thirty-year horizon. This makes sense: stock returns have a mean of 11% versus 5.7% for bonds, so stocks will eventually dominate on average. But this is not a correct statement about risk.

Investing over the Life CycleInvesting over the Life CycleInvesting over the Life Cycle

Figure 5.2

Panel B of Figure 5.2 plots the cost of shortfall risk. Bodie shows that this insurance cost is a put option: it pays out when the cumulated amount in stocks is less (p.160) (p.161) than the cumulated value in bonds at the end of the period, and zero otherwise. If stocks are less risky in the long run, then the cost of this insurance should decrease over time. Panel B shows exactly the opposite: the cost of insuring against stocks underperforming bonds increases with horizon. For a thirty-year horizon, the insurance cost is 40 cents per dollar of initial investment in equities.

Bodie’s analysis is clever because it doesn’t matter whether stock returns are mean-reverting. One of the key insights of the Black–Scholes (1973) option pricing framework is that the expected return of stocks does not affect the value of the option.19 Whatever the expected return of the stock, Bodie’s cost of insurance in Panel B of Figure 5.2 shows that stocks are not less risky in the long run.

How can we reconcile Panel A of Figure 5.2, where the probability of stocks outperforming bonds increases over time, and Panel B, where the cost of insuring against shortfall risk also increases over time? The insurance cost in Panel B has two parts: the probability of stocks underperforming bonds and the loss contingent upon underperforming. When stocks underperform, they can really, really underperform. Investors who experienced the Great Depression or even more recently the financial crisis of 2007–2008 know that stock losses can be huge. In extreme cases you lose everything, as investors in Chinese or Russian equities learned in the early twentieth century.

(p.162) Theoretically, it’s possible that mean reversion of equity volatilities, rather than the conditional means of equity returns, could produce a downward-sloping cost of insurance over time. But in the real world, it doesn’t happen. In Panel C of Figure 5.3, I compute shortfall costs using a model of time-varying volatility developed by Heston (1993). Heston’s model specifies stock volatility as mean-reverting, and there is much more predictability in stock volatilities than in conditional means of stock returns. Panel C shows that the cost of insurance with time-varying volatility also increases over time and is quite similar to the cost in Panel B with constant volatility. I plot the cost of insurance using two volatility levels: the long-run steady state volatility of 19% through the 1926–2012 sample and a starting level of volatility of 60%, which we experienced during the Great Depression and the more recent financial crisis. In both cases, stocks are still risky over the long run.

Nevertheless, even though the conventional wisdom that stocks are less risky over the long run is incorrect, the advice that financial advisors give can be appropriate. To state this rigorously requires using . . .

3.3. Life-Cycle Models

Individuals earn wages during their working lives, but this flow of income ceases when they retire. Thus, people have to save during their working lives so that they can support themselves during retirement. Economists use life-cycle models to study the consumption, savings, and investment decisions of people as they age. The first such model was developed by Franco Modigliani and his student Richard Brumberg. They wrote two papers, one published in 1954 and the other in 1980. Brumberg died early from heart disease, and Modigliani worked on the second paper again only when it was published in a volume of his collected papers.20

Life-cycle models contain at least two periods: an accumulation period in which individuals work and save and a decumulation period during which individuals draw down their savings and so dis-save. There are sometimes other periods, like early versus late retirement, being a young worker and then a middle-aged worker, and so. Individuals tailor their consumption, savings, and investment decisions to their needs at different ages, and pensions enable agents to smooth consumption over their life cycles.

Modigliani and Brumberg also created overlapping generations (OLG) economies where individuals each follow life-cycle models, and there are different cohorts or generations. In the setting of our simple two-period life-cycle model, there are two generations alive at each time. One generation is young and working, and their parents in the other generation are old and retired. In the next period, a new young generation is born and the previous young generation ages and becomes old. The previous old generation leaves the economy through death. In OLG (p.163) models, pensions paid to one generation can be financed by the generations following them.21 OLG models allow economists to measure intergenerational inequality: one of Raimondo’s desires in reforming Rhode Island’s pension plan was to reduce the subsidy paid by young workers to old workers and retirees. Modigliani won the Nobel Prize in 1985.

3.4. Life-Cycle Asset Allocation

The key determinant of whether your stock allocation should shrink as you age is the correlation of human capital with equities. At a point in time, human capital is an asset, and if it is stock-like, we wish to hold a smaller equity allocation (and vice versa). The life-cycle approach takes into account how human capital changes through time.

Bond-Like Human Capital

A young person, at the start of his career, starts with relatively little financial wealth. Most of what he has is pluck, grit, and talent: human capital. If his human capital is bond-like, then his financial portfolio should offset that human capital bond position and be heavily weighted toward equities. We’ll first take the extreme case and assume human capital is equivalent to risk-free bonds. Suppose that we can represent a young person’s total wealth, H+W, comprising human capital and financial wealth by the following balance sheet:

Young Person

Human Capital (Bonds)

$800,000

Financial Wealth: Stocks

$200,000

Financial Wealth: Bonds

$ -

Total

$1,000,000

Equity Fraction of Total Wealth

20%

Equity Fraction of Financial Wealth

100%

Suppose the young person would like a target 20% equity holding over his whole portfolio, including human capital and financial wealth. Since his human capital is equivalent to bonds, this means that he holds only stocks—and no bonds—in his financial portfolio. We can represent this balance sheet by the pie (p.164) chart in Panel A of Figure 5.3. Although the young person holds 20% in equities as a fraction of total wealth, his equity allocation as a fraction of financial wealth is 100%. If human capital is bond-like, this young person should hold a large fraction of equities.

Investing over the Life Cycle

Figure 5.3

Now take this person when he is old and nearing retirement. We’ll assume that his total wealth is the same, at $1 million, but most of it is now financial wealth; the investor has transferred his human capital to financial capital over his working career. His balance sheet is now:

Old Person

Human Capital (Bonds)

$100,000

Financial Wealth: Stocks

$200,000

Financial Wealth: Bonds

$700,000

Total

$1,000,000

Equity Fraction of Total Wealth

20%

Equity Fraction of Financial Wealth

22%

(p.165) The old person still targets an overall 20% equity holding. To maintain this risk profile, he has replaced his bond-like human capital with actual bonds. His equity holding as a fraction of financial wealth is now 2/(2+7)=22%. Thus, the old person has reduced his equity allocation in his financial portfolio compared to that of the young person. Panel B of Figure 5.3 is the pictorial representation.

So, if you’re a Supreme Court Justice with risk-free labor income, you should reduce your equity holdings as you grow older. The standard financial planner advice is tailor made for you, as are the slew of target date funds offered by many asset management companies.

Stock-Like Human Capital

Let’s consider the case of an investor whose human capital is stock-like. This investor targets an overall 80% proportion of equities in her portfolio (yes, she is more risk seeking than our first investor in this example). Suppose that her human capital and financial wealth sum to $1 million just as in our other example. When young, she will allocate heavily to bonds because her human capital is already equivalent to owning stocks. Here’s a balance sheet:

Young Person

Human Capital (Stocks)

$800,000

Financial Wealth: Stocks

$ -

Financial Wealth: Bonds

$200,000

Total

$1,000,000

Equity Fraction of Total Wealth

80%

Equity Fraction of Financial Wealth

0%

(p.166) Figure 5.4, Panel A shows this balance sheet in a pie chart. Although the fraction of equity in her total wealth portfolio is 80%, her financial portfolio contains only bonds. Her stock-like human capital crowds out stocks in her financial portfolio. A more detailed analysis would derive these figures using the techniques of section 2. We’re going to take these as given and focus on what happens when this person ages.

Investing over the Life CycleInvesting over the Life Cycle

Figure 5.4

When this person is old, she needs to substitute stocks in her financial portfolio for the lost stock-like human capital. The balance sheet:

Old Person

Human Capital (Stocks)

$100,000

Financial Wealth: Stocks

$800,000

Financial Wealth: Bonds

$100,000

Total

$1,000,000

Equity Fraction of Total Wealth

80%

Equity Fraction of Financial Wealth

89%

The corresponding pie chart is in Panel B of Figure 5.4. The investor holds a large fraction of her financial portfolio, 8/(1+8)=89%, in equities when old compared to the 0% position in equities when she was young. In both cases, her overall equity allocation is 8/10=80% of combined human capital and financial wealth. Thus, a person with stock-like human capital tends to hold more equity as she ages.

So, a person with stock-like labor income, like a stockbroker, should do the exact opposite of the one hundred minus your age rule. It seems that for the average reader of this book working in asset management, with labor income (p.167) highly correlated with the stock market, the traditional financial planner advice is inappropriate!

You are now going to object because you know a sixty-year-old stockbroker who lost his job in the Great Recession in 2008 and he held all his wealth in equities. He lost both his job and his wealth at one fell swoop and seemed to (unintentionally) follow the life-cycle investing advice. The life-cycle theory is a relative statement on how allocations change with time: for an individual whose labor income is stock-like, he should increase the allocation to equities as he ages to maintain a constant overall exposure for risk—the theory does not say that there should be a large allocation to equity when he is old. As section 2 shows, an individual would generally prefer to hold more bonds, all else equal, if her labor income is correlated with equities. Thus, according to life-cycle theory, the stockbroker might have optimally held mostly bonds, but his (hopefully very small) equity portion would have gradually increased as he approached retirement.

While these examples illustrate the main intuition of labor income across the life cycle—that human capital is an asset and our holdings of financial assets should adjust to counter-balance the bond-ness or stock-ness of human capital—actual life-cycle models used by economists add several features. We have abstracted from leverage constraints, how a person consumes, and changing liabilities. Many students today graduate with large debt—so they begin with negative net wealth. What is most lacking, however, is that we treated the young person’s problem independently of the old person’s problem. In a true life-cycle model, the young person preemptively takes into account the reduced labor income when she is old. Our next (stylized) example rectifies this but shows that the same intuition holds.

Sovereign Wealth Fund Example

Sovereign wealth funds (SWFs) are tantamount to a retirement plan for a nation. Any SWF set up to diversify from (and outlast) commodity wealth is like a retirement fund. Just as the correlation of human capital returns with asset returns drives asset allocation over an investor’s life cycle, the correlation of commodity returns with asset returns should determine a SWF’s asset allocation.

Consider the following simple model of a SWF.22 There will be two periods: the first is when oil wealth is plentiful and the second when the resource has been mostly extracted. To simplify, we’ll abstract from a spending rule (consumption) and just maximize the expected utility of terminal wealth. (Recall this is not the same as maximizing the amount of money because the utility function trades off risk and return; see chapter 2.) There are two financial assets: risk-free bonds, (p.168) which we assume have an interest rate of zero, and equities. The SWF has CRRA utility (which is the academic version of mean-variance utility; see chapters 2 and 4) and decides on the amount of equities to hold in its portfolio in each period. These are given by α1 and α2, respectively:

(5.4)
maxα1,α2EW21γ1γ,

where γ is the nation’s degree of risk aversion, which I set to 10. Equation (5.4) states that we want to maximize the country’s expected utility, where the expression in the brackets is CRRA utility over terminal wealth, and we do that by finding the optimal period 1 and 2 portfolio weights. I constrain the portfolio weights to lie between 0 and 1 (so there can be no shorting).23

Oil is like labor income. The SWF can harvest oil of Lt at the beginning of each period, and fluctuating oil prices generating a return at the end of the period. Oil income reduces by 90% from period 1 to period 2, so L1=0.1×L0. Calibrating this model using data from 1983 to 2012, the correlation of oil returns with equities (S&P 500) is a low 4%. (This might seem surprising, and it turns out that commodities are also not very good inflation hedges, as chapter 11 discusses.)

Figure 5.5 graphs the asset allocations to equity as a function of L/W, the ratio of oil to financial wealth at the beginning of each period. Panel A shows the allocations to equities at the empirical correlation of 4%. In each period, the allocations to equity slope upward. As there is more oil, the SWF holds more equities because of diversification. Equities are lowly correlated with oil, so there is large scope for equities to pay off when oil does not, making equities increasingly attractive when oil reserves are large. Notice that the first period equity holdings are larger than the second period for a given L/W; the SWF reduces equities with time because oil is “bond-like,” which in this case means oil is minimally correlated with equities. The SWF is endowed with a lot of bond-like securities at the first period, so holds more equities. In period 2, oil is mostly extracted, and so the SWF lowers the equity weight to counter the disappearing bond-like oil income.

Investing over the Life CycleInvesting over the Life Cycle

Figure 5.5

In Panel B of Figure 5.5, I contrast the case where oil returns and S&P 500 returns are positively correlated at 0.3. In this case, the equity shares are decreasing in L/W and the period 0 and 1 weights flip. The SWF holds fewer equities for a given level of L/W in the second period because oil is now stock-like. In the first period, the SWF is endowed with a stock-like oil resource. After most of the oil is extracted, the SWF wishes to replace that stock-like exposure to maintain the same overall risk profile, so it increases equity holdings in the second period. (p.169)

(p.170) This example also illustrates that looking only at the SWF’s financial portfolio can result in the mistaken impression that risk aversion is changing. If the oil (or labor) income is bond-like, then reduced oil inflows imply that the SWF should reduce equities. An observer only looking at the financial portfolio might think the country has grown more risk averse. If oil income is stock-like, then the SWF increases equity holdings when oil inflows shrink. It may look as if the SWF has become more risk seeking, but all the SWF is doing is balancing oil risk across its life cycle.

Are You a Stock or a Bond?

A number of studies try to estimate the correlation between human capital returns and equities. At the overall level—a representative agent summing up the income of all workers in the United States or large subsets of workers—the correlation of human capital with stocks is low. Cocco, Gomes, and Maenhout (2005), for example, report that the correlation of labor income with stock returns is –2% for those with a college degree, 1% for those with only a high school education, and –1% for those not finishing high school. Thus for the average investor, human capital tends to be bond-like.24 (Perhaps the one hundred minus your age rule is right after all, at least for the average person.)

The early life-cycle models relied on similar estimates.25 Since they assumed the average worker had labor income that was bond-like with low correlations with equities, they predicted that young people should hold more equity than old people. They also predicted that young people should always hold some equities and poorer households would hold more equity than rich households. To understand the latter, bear in mind that, assuming they have the same risk profiles as rich households, the total wealth of poor families is dominated by human capital. Thus they would offset this bond-like capital by preferring equities in their financial portfolios.

It turns out all of these predictions are counterfactual.

3.5. What People Actually Do over Their Life Cycles

There are three important facts of households’ asset allocation over their life cycles:26

  1. (p.171) 1. About half of households do not hold any equity

    This is called the stock non-participation puzzle, and I discuss it further in chapter 3.

  2. 2. Older households hold bigger equity allocations than do young, working households.

    Another way to put this is that household equity shares are constant or even increase with age. Even at retirement, the equity fraction is very high. In retirement, households hang onto their equity holdings for as long as possible instead of moving as soon as they can to bonds and other low-risk assets.

  3. 3. The rich have larger equity weights than the poor.

There are also three stylized facts of households’ consumption over their life cycles:27

  1. 1. Consumption is very smooth.

    Life-cycle models predict that when an individual’s permanent income increases suddenly, the individual should consume more—after all, you’ve just enjoyed a windfall. But spending does not respond very fast to changes in income. This is called the excess smoothness puzzle in the literature.

  2. 2. At the same time, consumption is excessively sensitive.

    Although consumption doesn’t move when projected income changes, consumption does react—and it reacts too much—to past changes in income.

  3. 3. Consumption is hump-shaped over the life cycle.

    The peak consumption years are in middle age. At retirement, consumption declines in line with lower labor income. Figure 5.6 shows consumption expenditures in different age brackets from 1984 to 2011 from the Consumption Expenditure Survey conducted by the Bureau of Labor Statistics. I have normalized the forty-five to fifty-four age bracket, where consumption peaks, at 100%. Note that the more recent retirees have started to consume more compared to those who retired in the 1980s.

Investing over the Life Cycle

Figure 5.6

It is an exciting time to be working in the life-cycle area. The newest models can explain these patterns by incorporating many realistic features: they take into account the inability to borrow (especially by the young), use utility functions that capture salient consumer behaviors, and model complex dynamics of labor income and asset returns.28 These models can match the stylized facts of what people do (p.172) over their life cycles. In all of them, whether your labor income is stock-like or bond-like is a major determinant of your asset allocation over the life cycle.

Benzoni, Collin-Dufresne, and Goldstein (2007) and Lynch and Tan (2011) specify labor income processes that match the low correlation of labor income and stock returns for the average investor but allow more realistic dynamics where labor income and stock returns can co-move over business cycle and longer frequencies. This is intuitive: we expect labor income and stock returns to be correlated over long frequencies as both human capital and stock capital should benefit from growing economies. Anyone losing a job in the Great Recession also knows that labor income and stock returns can be highly correlated over certain short periods.

Benzoni, Collin-Dufresne, and Goldstein allow labor and stocks to move with each other in the long run (labor and stock returns are cointegrated). Because of this long-run comovement, the young person with a long horizon has stock-like human capital. The old person with a short horizon has bond-like human capital. (p.173) Thus, the young optimally hold no equities (they actually want to short equities if they can), and equity portfolio shares increase with age.29

Lynch and Tan make the point that although the correlation of labor income and equities is close to zero unconditionally, it can mask pronounced conditional correlation over the business cycle. Put simply, while there isn’t much correlation between labor income and equities overall, there is significant correlation when it really matters, like when stocks crash. In their estimates, conditional shocks to permanent components of labor income are 48% correlated with conditional stocks to stock returns even though the unconditional correlation between labor income and stock returns is just 2% in their sample. Since labor income is conditionally stock-like, the young optimally hold low equity portfolio weights.

3.6. Other Considerations

The surprising preference for more equities in old age is predicted by the life-cycle model if labor income is risky, like a stock. (I reiterate: this is the opposite of standard financial planner advice.) Other circumstances, however, may warrant reducing equities as retirement approaches. Perhaps the most important is . . .

Health-Care Risk

Health-care shocks can be truly debilitating. Health care is expensive and spending is concentrated in late life. Marshall, McGarry, and Skinner (2010) report that the median medical spending in the last year of an American’s life exceeds $5,000, but for the top 5% it exceeds $50,000. Both the mean and variance of medical expenditure rise with age, especially in old age. Health-care shocks do not mean just a one-off medical expenditure. One serious medical problem is too often followed by others. De Nardi, French, and Jones (2010) unpack health-care spending into persistent and transitory components and show that the persistent components have autocorrelations of more than 90% per year.

Health-care risk is not just about being able to pay medical bills. Serious ill health means the loss of current income for a working family and the loss of future income as well. Substantial declines in wealth and future labor income follow bad health-care shocks.30 Sickness often produces a downward spiral: ill health affects earnings, and lower earnings lead to poorer health.

Some health-care costs are borne by insurance and some by the government (Medicare and Medicaid). But a lot of health-care expenses in the United States are paid by individuals. Munnell et al. (2008) estimate the present value of out-of-pocket health-care costs from age sixty-five at $100,000 per person, excluding long-term care. Webb and Zhivan’s (2010) estimates are even higher, (p.174) at $260,000 for a typical couple aged sixty-five. But there is large skew, and a 5% chance the couple will spend more than $570,000. Many individuals also lack insurance—according to the Census Bureau, 21% of Americans earning between $25,000 and $49,000 per year do not have health insurance. Close to 20% of Americans between ages twenty-five and thirty-four do not have health insurance.31

Individuals can meet out-of-pocket health-care shocks and insure against the loss of income produced by poor health by holding safe assets. Health shocks are a form of background risk, which is the term given by economists to any risk on which individuals can’t buy insurance.32 Background risk raises precautionary savings and the demand for safe assets.33 Empirical evidence is consistent with this: Rosen and Wu (2004) show that people in poorer health hold fewer stocks and other risky assets. Life-cycle models with health-care risk, like De Nardi, French, and Jones (2010) and Pang and Warshawsky (2010), demonstrate that it is optimal to increase holdings of risk-free assets as retirement approaches. In retirement, individuals should optimally continue to save to meet out-of-pocket health-care expenses, whereas standard life-cycle models predict that individuals should dis-save.

Changing Utility

It’s well established that people tend to become more risk averse as they get older.34 (Perhaps this explains why your father used to attend anti-establishment rallies when young and then later switched political affiliations.) By itself, this would favor the 100 minus your age rule because, as the investor ages, he becomes more risk averse and wishes to hold fewer equities.

Endogenous Labor Income

Investment in labor income is a choice variable; that is, labor income is (partly) endogenous. Earning an MBA, for example, improves one’s future earnings, but you trade off the cost of being out of the work force and paying tuition in exchange for the benefit of higher pay when you graduate. Labor income can also be increased by working longer hours or switching occupations. Shocks to wealth can be partially offset by working harder.35 A young person has great flexibility, but that declines with age.

(p.175) On the one hand, the flexibility of a young person’s human capital could argue for decreasing stocks with age. Youth has greater capacity to absorb risk—so when young you can hold more equities because you can afford the losses by absorbing them through the human capital dimension. The old person has less flexible human capital (you can’t teach an old dog new tricks, supposedly), which limits the old person’s ability to hold risky financial assets. On the other hand, if the flexibility of a young person’s human capital is just a substitute for equities, then she will want to replace that dwindling capacity to take risk with equities as she grows older.

There are other complex and subtle interactions between labor income and capital markets. Higher wages can spur people to work harder (goofing off has a high opportunity cost). An offsetting effect is that earning a higher wage causes a person to become wealthier, and when you’re really wealthy, you may really prefer to goof off.36 Chai et al. (2011) show that the first effect dominates, so those with positive wage shocks early in life tend to work harder. These households consume more and build up relatively little financial wealth, but the savings that they do hold has a portfolio share in equities that decreases with age. The reason is that these households save through Social Security (which I comment on below) and because of high wage profiles, their Social Security benefits are higher at retirement.

3.7. Summary

All else equal, the life-cycle theory predicts:

Bond-like labor income => Reduce equities as you age.

Stock-like labor income => Increase equities as you age.

Financial planner advice that you should reduce your equity holdings as you age (or the one hundred minus your age rule) is not appropriate for people whose labor income is positively correlated with equities. Life-cycle models predict that individuals with stock-like labor income should increase their equity holdings with age, as they wish to replace their decreasing stock-like human capital with the financial equivalent. This is what most households actually do. The need to meet health-care expenses, however, entails setting aside risk-free assets, and these risk-free positions should increase as the health-care risk rises over the life cycle.

Most financial planners and target date funds, which blindly follow versions of the one hundred minus your age rule, do not take into account how an individual’s (p.176) labor income co-moves with equities. Whether you should hold more or fewer equities as you approach retirement crucially depends on whether you are a stock or a bond.

4. Retirement

4.1. Replacement Rates

In reforming Rhode Island’s pension plan, Raimondo started at the end and asked what would be optimal for an individual to do at retirement. Then, with a secure retirement in mind, she asked how a retirement plan should be designed to get there. Raimondo is not an economist (she graduated from Yale Law School), but she employed the same solution that an economist would have used. Economists solve life-cycle models by starting at the end of the problem, finding its solution, and then stepping back one period to solve the younger person’s problem given the optimal solution of the older retiree. This procedure is called dynamic programming (see chapter 4).

In examining Rhode Island’s pension plan, Raimondo’s team found some cases where some individuals had replacement rates well above one. However, while most working Americans can rely on Social Security to establish a (minimal) base for their replacement income, over 50% of teachers and many public safety employees in Rhode Island’s plan did not participate in Social Security.

Economists believe that the replacement rate, the amount of income in retirement as a fraction of pre-retirement income, should be less than 100% for most people. There are a number of justifications for this advice.37 At retirement, households usually face lower tax rates. Social Security benefits, for example, are more lightly taxed than regular wages, or not taxed at all, in some states. Retirees sometimes downsize their housing (but many do not, see below), leading to lower property taxes. But most important, households do not need to save for retirement—they are now in retirement!

The World Bank, the Employee Benefit Research Institute, and other academic and policy institutions advocate a target replacement rate of 75% to 80%.38 Scholz and Seshadri (2009) compute optimal replacement rates using a life-cycle model and find median optimal target rates of 75% for married couples and 55% for singles. There is, however, a large amount of dispersion as many other factors, like the number of dependent children, the level of income, and other household-specific considerations enter the computation.

Some people prefer other patterns of retirement income. Some prefer their income never to fall. In fact, for many institutions with their own life cycles, this is (p.177) exactly the preferred “retirement” income pattern for a country with a SWF whose oil wells have run dry or a foundation whose benefactor has died. Surveys also indicate that most people prefer steady or rising real incomes, even in retirement: the declining consumption shares that we observe in data (see Figure 5.6) could be because people are forced to cut back consumption in retirement by a lack of money coming in and poor planning, not because they like replacement rates less than one.39 Life-cycle models allow us to compute optimal pre-retirement savings and investment patterns for such investors by using special utility functions.

With habit formation utility, investors intensely feel the pain of being unable to live on in the style to which they’ve become accustomed. Countries and universities, for example, find it tough to cut spending below certain levels, and these levels (or habits) rise over time. Using habit formation in life-cycle models generates very smooth consumption patterns in late life.40 The investor has to save a very large amount during her working life to finance this and continues to save after retirement (there is large wealth accumulation at all ages across the life cycle).

Ratchet Consumption

Dybvig (1995, 1999) develops a model of ratcheting spending, which should be attractive to investors wishing to maintain spending; it even allows spending to rise when investments have sufficiently appreciated.41 To sustain their spending pattern, investors would set aside some of their portfolios in a risk-free asset. This risk-free portion can be thought of as an insurance policy guaranteeing future minimum spending, which is called the liability-hedging portfolio. The remainder of the funds is invested in a (levered) stock portfolio. When the risky portfolio has sufficiently appreciated, the investor ratchets up spending and can protect it by transferring some of the appreciated monies into the risk-free portfolio.

While Dybvig’s framework is ideal for endowments, individuals, and even countries that do not wish to experience declines in their standards of living, current circumstances present two challenges in following ratcheting spending rules. First, interest rates now are very low (real rates are negative at the time of writing, see chapter 11), and the necessary minimum spending rates from Dybvig’s model are very, very low. Second, a large liability hedging portfolio is held in bonds—and expected returns on bonds are very small at the moment and potentially negative. Nothing, unfortunately, comes for free.

(p.178) By starting at the retirement problem and working backward, life-cycle models make clear that the true risk-free asset for retirees is not T-bills. Risk-free assets are securities that can lock-in increasing retiree spending in real terms.42 The only way to do this is to hold a portfolio of real bonds (TIPS, see chapter 11) that completely immunizes a retiree’s cash flow needs at retirement. Real bonds, however, have prices that are quite volatile and funding these retirement needs with real bonds can open up the investor to substantial investment risk as real rates change over time. Today’s negative real yields mean that an individual, instead of counting on bond earnings to help achieve the savings goal, has to set aside even more money than will be needed later in retirement.

4.2. Are We Saving Enough?

Using life-cycle models, Skinner (2007) estimates how much we should save for a comfortable retirement. The numbers are startlingly high. Consider a forty-year old planning to retire at sixty-five, living to ninety-five, saving 7.5% of pre-tax earnings, and hoping income in retirement is just 30% of pre-retirement earnings. At age forty, this individual needs to have a wealth-to-income ratio of approximately 2, rising to 3 at age fifty and peaking at 5 when retirement occurs. This wealth does not include housing, and for non-homeowners the numbers are even larger at 4, 5, and 8 for ages forty, fifty, and sixty-five, respectively.

Most people are nowhere near these levels of savings.

Most households fail to save the amounts required by life-cycle models to ensure a comfortable retirement. Lusardi and Mitchell (2006) report that the early baby boomers had a median wealth-to-income ratio of just 2, far less than the optimal savings numbers calculated by Skinner (2007). Poterba, Venti, and Wise (2011a) report that half of households headed by someone sixty-five to sixty-nine had less than $52,000 in total net wealth in 2008, and 43% of such households could not scrounge up $25,000—which is often the minimum investment in annuities (see below). According to the Employee Benefit Research Institute in 2013, 57% of U.S. workers reported less than $25,000 in total savings excluding their homes, and 28%—a clear-eyed minority—were not confident they had enough money to retire comfortably.43 An amazing 34% report that they had not saved anything for retirement!

The first-order problem is getting people to save. Consumption today is always more appealing than saving for tomorrow. Economists have modeled this type of behavior using a hyperbolic discounting function, which is the St. Augustine version (p.179) of utility (“Lord make me chaste, but not yet”).44 Fortunately, some economists have also exploited behavioral biases to entice people to save.

Thaler and Benartzi (2004) created the Save More Tomorrow program, described in Thaler and Sunstein’s (2009) bestseller, Nudge. Thaler and Benartzi exploit human nature to get people to commit to saving future wage increases. People won’t save out of today’s funds, but they will save out of funds that they haven’t yet received. When they receive those funds, they can’t be bothered to change their original plans and inadvertently end up saving. Inertia is thus enlisted on behalf of thrift at no cost to current consumption. Thaler and Benartzi found savings rates increased from 3% to over 13% when employees committed to a program where they saved more tomorrow.

Conversely, Scholz, Seshadri and Khitatrakun (2006) provocatively argue that most households have adequately saved for retirement. There is, however, no mortality risk in their model for working people, no out-of-pocket health-care risk prior to retirement, and no risky assets. It’s likely that taking into account these risks, households are not saving enough. Also, their study was completed before the Great Recession of 2008–2009, which decimated many households’ balance sheets.

The lack of savings is also reflected in the underfunding of defined benefit pension plans. This is behind Rhode Island’s travails and Raimondo’s desire to reform its pension system. In these plans, employers save on behalf of workers, but employers have not been contributing enough. Or the benefits promised are too high. Or both. In the long run, there is an immutable adding up constraint: the money that we spend during retirement cannot exceed the savings (plus investment earnings) that we have set aside to fund retirement.

If we don’t save enough, then we’ll have to consume less during retirement. Skinner (2007) argues that we can consume less during retirement and still be happy. There is a lot more leisure during retirement. We may place great value on this unexpected richness of time, which can offset our lower consumption. Skinner had better be right, because most people are going to need to consume a lot less after they retire.

4.3. The Four Percent Rule

The 4% rule was created by William Bengen (1994), a financial planner.45 Bengen found that an investor holding 50% stocks and 50% bonds could make her portfolio last at least thirty years if she spent approximately 4% per year. In Bengen’s original formulation, the investor started retirement by consuming 4% of her wealth at that time—so she starts with a nest egg of twenty-five times her desired (p.180) retirement spending. Each year, she increases her spending in line with inflation. Although she started spending 4% of her wealth at her start of retirement, her spending rate fluctuates up and down depending on asset prices and inflation rates. For individuals retiring over 1926–1976, Bengen found that their wealth was sufficient to last for at least thirty-three years. Increasing the initial withdrawal rate to 6%, wealth would be often exhausted before twenty years, especially for those retiring during the 1960s and early 1970s.

The 4% rule is now generally known as a fixed proportional spending rule—spend a constant 4% of your wealth each year. Milevsky (2012) observes that “this 4% rule has taken on the status of a mythical beast in the retirement planning world. (You slay it, or it slays you.)” But the 4% rule is simple, easy to implement, and can be followed regardless of how much wealth an individual has accumulated. Consequently, the 4% rule is very popular. Many financial planners recommend rules similar to the 4% rule—if not 4%, then some other fixed fraction of wealth. It also helps that 4% is very close to the spending rates of endowments and foundations (see chapter 1) and is often used as a long-run average real interest rate (see chapter 11).

The main problem of the fixed 4% rule is that it forces retirees’ consumption to go up and down with the market. That is, individuals are forced to spend more when markets are performing well and cut drastically back when markets crash. Most individuals do not like such volatility in their consumption plans—many of their outflows are fixed in dollar terms, not percentage of wealth terms.46 Thus retirees tend to prefer regular fixed payments that remain approximately constant in real terms. There is also severe longevity risk with the 4% rule. The 4% rule is proportional to wealth, so if you live for a long time your wealth might dwindle to nearly nothing. And 4% of nothing is, well. . . .

Bengen never originally advocated a fixed rule of 4%. His was an initial spending rule, and he held spending fixed in real terms and adjusted the spending rate over time. The sure-fire way to ensure that your spending never falls, in nominal or real terms, is to follow Dybvig’s ratchet rule. In Bengen’s formulation, there is always risk that an investor will outlive his savings: even if in Bengen’s original sample the retirement nest egg lasted for at least thirty years, pity the poor seniors who live past one hundred who will eventually have nothing.

Finance theory also says that the spending rate (consumption) should be determined simultaneously with the portfolio choice. If you’re holding a risky portfolio, you probably want to spend less, and vice versa. This applies to both retirement and to how we save and consume getting to retirement. The 4% rule says nothing about optimal portfolio choice. The best financial planners do not use the 4% rule and instead craft a retirement plan that takes into account longevity risk, health (p.181) status, risk aversion, and all the other important considerations of a client—all the factors in an agent’s life cycle.

4.4. Annuities

Annuities should be a retiree’s best friend. They provide steady payments, there is no risk of a retiree outliving his savings, and some versions index payments to inflation.

The beauty of annuities is that you receive other people’s money. The best part is that they don’t miss it because they are dead, so everyone is better off. This intuition was first shown in a seminal paper by Menahem Yaari (1965), an economist at Hebrew University. Yaari showed that an annuity is like an insurance policy with a negative price (in fact, annuities are reverse death insurance): those who die early subsidize the consumption of the lucky Methuselahs who die late. But everyone benefits because at the start you don’t know if you’re going to die early or late, and you can’t consume once you die. Yaari demonstrated there were gigantic benefits of investing in annuities.

The original setting by Yaari was restrictive: he assumed the annuities were fairly priced, there were no fees or frictions, individuals didn’t want to save anything to pass onto their heirs, and there were no uninsurable risks. Taking these into account reduces the attractiveness of annuities, but there are still large benefits. Yogo (2011) includes health shocks, bequest motives, and other considerations and shows that there are gains of more than 15% of household financial wealth if households were to purchase additional annuities (over those already embedded in Social Security).47

You never outlive your money with an annuity. The risk of an individual outliving her savings has been steadily increasing because lifespans have been getting longer. The Society of Actuaries reports that life spans have improved by around two years each decade for males, from 66.6 years in 1960 to 75.7 years in 2010, and around 1.5 years per decade for females, from 73.1 years in 1960 to 80.8 years in 2010.48 “There is a general misunderstanding of what “average life expectancy” means, and when people are told that they will live to an age such as eighty or eighty-five, they don’t realize that this means there is a 50% chance that they could live past that age,” says an actuary. Many people don’t even get the average right, with 41% underestimating their life expectancies by at least five years. The longevity risk is substantial: by age sixty-five, U.S. males in average health have a (p.182) 40% chance of living beyond age eighty-five, and females have more than a 50% chance. For healthy people, these probabilities increase to 50% for males and 62% for females.

The beneficiaries of Rhode Island’s pension plan do not face longevity risk because the state bears it for them. Increasing longevity substantially raises the cost of providing for pensions, and increasing lifespans are one reason behind the underfunding of Rhode Island’s pension plan. Longevity risk also increases during recessions—people tend to die later during recessions—probably because the unemployed and underemployed have more time, exercise more, and spend more time with their families during recessions.49 Defined benefit plan sponsors then face a double whammy during recessions as asset values fall and liabilities swell as a result of increased longevity and lower interest rates.

While annuities are in theory a great deal, they are not very popular, with fewer than 6% of households buying them.50 This is termed the annuitization puzzle by Modigliani (1986). Why are people reluctant to buy annuities?51

Credit Risk

Once you buy an annuity, you are exposed to the credit risk of the insurance company that sold you the annuity. Unfortunately for consumers, many insurance companies are no longer staid and boring. In 2011, Liberty Life Insurance Co. was acquired by Athene Holding Ltd, a company funded by the private equity firm Apollo Global Management. Before its acquisition by Athene, Liberty Life held a “squeaky clean” portfolio of state government (muni) bonds and bonds issued by highly rated corporations. After its acquisition, Liberty Life’s products were backed by securitized subprime mortgage products, time-share vacation homes, and a railroad in Kazakhstan. “All the upside would go to Athene if it worked out. And the downside would go to the annuity holders if it didn’t,” says a former bond (p.183) portfolio manager at Liberty Life.52 This behavior has been repeated in insurance companies acquired by various private equity firms and investment banks, including Apollo, Goldman Sachs, Harbinger, and Guggenheim.

If an insurance company goes bust, the annuities are protected, but the protection is capped and the maximum protection differs across states. In my home state of New York, the cap is $500,000, which would be significantly below the value of annuities purchased by the average reader of this book. In California, the limit is $250,000. Consumers can minimize this risk by choosing highly rated insurance companies, but this doesn’t guarantee a risk-free annuity. AIG was considered a safe name until it was downgraded in September 2008 and then would have spiraled into bankruptcy had it not been rescued by the federal government in what was at the time the largest bailout of a private company in U.S. history. Individuals can diversify this risk by buying annuities from several companies, but it’s hard enough just to get a retiree to purchase one annuity. Partly, it’s because . . .

They’re Complicated

There are certain annuities, immediate annuities, deferred annuities, variable and indexed annuities, investment-linked annuities, joint and survivor annuities, fixed guarantees, contingent guarantees, annuities with floors or caps (or both), benefits that increase periodically at various high-water marks, sliding surrender scales, contingent protection, bundled life insurance, various degrees of inflation protection, annuities with and without (contingent) withdrawal penalties, and so on.53 Annuities are really complicated.

“There is almost no investor capable of making an informed choice about this,” says a director for the Consumer Federation of America.54 The people selling annuities have terrible reputations, and some deserve them. “If you need to act in the customers’ best interest,” says a former SEC official, “you can’t sell this crap.”

The closest annuities to those in academic life-cycle models are “single premium immediate annuities,” which investors buy at a single point in time with their retirement savings.55 These investors then receive payments for the duration of their lives.

Bad Deals

Not only are these complicated products very far from the nice, clean products that academics treat in their life-cycle models, they can be much more expensive (p.184) than their actuarially fair values. Milevsky and Posner (2001) value the insurance guarantees in annuities and find these are worth between 0.01% and 0.10%, but the insurance industry charges 1.15%.56 Mitchell et al. (1999) estimate the value of annuities offered by life insurance companies and find that it’s 80 to 90 cents for each dollar. However, Mitchell et al. show that the difference between fair value and purchase price has shrunk over time.

Even if annuities are fairly priced for an average policyholder, they can be expensive for certain segments of the population. Brown, Mitchell, and Poterba (2002) report a male buying an annuity at age sixty-five would get 91 cents of value for every dollar put in, but a more general member of the public would be only getting 81 cents of value. This is because insurance companies selling annuities face . . .

Adverse Selection

Early in Jane Austen’s Sense and Sensibility, Mrs. Dashwood convinces her husband not to help the family of our heroines (Elinor, Marianne, and Margaret), explaining: “People always live forever when there is an annuity to be paid them.” Insurance companies take into account adverse selection when pricing annuities to avoid bankrupting themselves by paying people who “always live forever.”

Perhaps there is a spiraling negative interaction of adverse selection with demand: adverse selection drives up the price of annuities, this leads to lower demand, which in turn leads to greater adverse selection, and so on. Chalmers and Reuter (2012b) rule out this explanation in a study of the Oregon Public Employees Retirement System. They do not find that participants in Oregon’s pension plan buy more annuities when annuity prices fall, which must point to other investor-specific characteristics driving (lack of) annuity demand. Getting value for money is also not an issue in Oregon’s system, as the lucky participants receive more than $1 in benefits in annuities for each $1 that they buy.

Framing

Consumers often view annuities as gambles. Benartzi, Previtero, and Thaler (2011) argue that consumers view annuities in an “investment” mindset, or frame, rather than in a “consumption” mindset. They come to an annuity decision with the question, “Will I live long enough for this investment to pay off?” The risk is that all the (usually considerable) money placed in the annuity, earned with blood, sweat, and tears, vanishes with early death. The investor benefits from guaranteeing his consumption each year (subject to credit risk) but does not view the annuity in terms of consumption benefits. Changing the frame can enhance demand.

(p.185) Health-Care Risk, Again

Dylan Thomas doesn’t take account of exponentially increasing health-care expenses in senescence when he says, “Old age should burn and rave at close of day; Rage, rage against the dying of the light.” The upside of annuities is that they lock away a big chunk of capital to ensure regular payments. The downside is that this capital is not available to meet unexpected health-care shocks or the needs of your children (who have come back home to live with you).57

Annuities are initially priced by valuing the expected payments given the health of an individual at the time the annuity is purchased. But health varies over time and after a severe health shock, the value of annuities can decrease substantially. This loss happens exactly when an individual most needs resources to cushion the blow from reduced income. This makes annuities unattractive to hold, especially for investors facing large uncertainty about their health.58

Some annuity products, like variable annuities with guaranteed income benefits, allow policyholders to withdraw more than the regular amount (with penalties, of course) in emergencies. But understanding whether these are good deals is hard for most consumers (see They’re Complicated, above).

They Already Have Some

Most Americans are entitled to receive Social Security—which is an inflation-indexed annuity. Inflation protection is extremely valuable for recipients but very expensive for the government. According to MetLife, an annuity paying out the maximum Social Security benefit for a couple at age sixty-six would cost almost $1.2 million.59

It is possible that Social Security meets the annuity needs of many Americans. Authors like Yogo (2011), however, report that it is still in the average consumer’s best interests to purchase additional annuities over those in Social Security. I also bet that the desired retirement lifestyle of the average reader of this book would not be sustained by Social Security.

Social Security is itself underfunded; the Social Security Administration estimates that the fund will be empty in 2033. The additional money required to pay all scheduled benefits is equivalent to 80% of the amount of outstanding U.S. Treasury debt.60 So it is not only private annuities that have credit risk.

(p.186) Bequests

Perhaps the dead do have utility functions. The very rich use trusts to ensure that their heirs follow certain rules as a condition of inheriting money, and donors tie up endowment money to universities and foundations with restricted use provisions (see chapter 1). Many individuals want to leave some kind of legacy to their children through bequests. Inkmann, Lopes, and Michaelides (2010) show the bequest motive is a particularly powerful incentive not to purchase annuities; many individuals would rather manage money themselves and bear the longevity risk so that they have something to pass onto their children.61

Encouraging Annuitization

Although few individuals purchase annuities, there is evidence that annuity participation rates increase with wealth, education, and life expectancy—the kind of factors that seem to determine whether investors are comfortable holding equities in the first place (see chapter 2).62 These people are able to understand the benefits conferred by annuities, so there is scope for better financial education to expand the annuity market. Framing certainly can help. So too can the push by regulators to allow partial annuitization when retiring from defined benefit plans (allowing retirees to take combinations of annuities and lump sums).63

Some defined benefit plan sponsors are actively encouraging retirees to adopt annuities. United Technologies is automatically enrolling beneficiaries into a “secure income fund” that gives workers some upside, if stocks and bonds do well, but gives workers a minimum amount each year that is guaranteed for life.64 The protection is bought periodically by United Technologies from insurance companies. (In finance terms, United Technologies’ workers are receiving a straight annuity plus call options, or a deferred version of Dybvig’s ratchet spending.) In Rhode Island’s pension reform, Raimondo tried to keep cost-of-living adjustments (COLAs) in pension payments rather than eliminate them as other states have. Under her proposal, COLAs would be paid, but the amount depended on the funding status of the pension plan and was based on investment returns. In this way, the state would not bear all the inflation risk of the annuity payments.65

(p.187) But these efforts can only go so far if the products themselves are deficient. Before the arrival of the market index fund in asset management, investors had no cheap way into a broad, diversified stock portfolio. They were stuck with expensive active management products, and active management on average underperforms the market (see chapter 16). The market index fund changed everything. Its introduction took immense effort by dedicated individuals, a large dollop of investor education, and the firms that pioneered it were relative outsiders to the asset management industry. And it almost didn’t happen—the first attempt failed. Regime shifts are rarely engineered by companies profiting from the status quo.66

The annuity market is similarly ripe for innovation. There is currently no annuity product similar to the market index fund: an annuity that is dirt cheap, simple, can be sold in scale, avoids costly intermediaries, and is actually bought by investors rather than sold to them. There are more challenges here compared to the market index fund because insurance regulation varies by state, and adverse selection makes pricing tricky. But what a benefit this would be to investors!

4.5. Are We Dis-Saving Enough in Retirement?

For those lucky, or prescient enough, to enjoy a comfortable retirement, there is a final retirement savings puzzle: the elderly tend to dis-save much more slowly than predicted by simple life-cycle models. In fact, for those with relatively large retirement savings, Poterba, Venti, and Wise (2011b) show that retirement account balances continue to increase after individuals leave the workforce! Fewer than 20% of households withdraw prior to age 70.5, the age when households are forced by law to take minimum distributions, and when they do withdraw, the withdrawal rate is low. Even after age 70.5, the withdrawal rate tends to be below the rate of return earned on retirement assets. Not surprisingly, many people die with piles of savings.

The need to pay for medical care is one big reason. Another is the attachment to one’s house, which is the largest investment for most individuals.67 (Houses are highly illiquid assets with huge idiosyncratic risk, funded by a short position in bonds.) While the basic life-cycle model predicts that agents should reduce their housing when they retire, homeowners choose to stay in their homes for as long as possible and cannot easily borrow against them. This can be alleviated by reverse mortgages, but this market is still small and suffers from many of the same drawbacks as the annuity market. Housing wealth is not typically used to support (p.188) consumption during retirement and is liquidated for the most part only following the death of a spouse or a forced move to a nursing home. Even in these cases, liquidated housing wealth is not used to support nonhousing consumption. The other reason saving continues in retirement is that individuals are looking beyond their own life cycle, to grant inheritances to their children and other relatives.

4.6. Summary

The most important problem is to save now so that something will be available for retirement. Save, save, and save! While annuities meet most of retirees’ needs—they remove longevity risk and provide guaranteed smooth payments during retirement—most people do not hold them. Current annuity markets are highly inefficient and hard for consumers to navigate; the flipside is that there is still large scope for innovation in annuity markets by new players.

5. Employees’ Retirement System of Rhode Island Redux

On November 17, 2011, the pension regulation championed by Raimondo passed in the Rhode Island House and Senate, and the next day the Rhode Island Retirement Security Act of 2011 was signed into law by the governor. As a result, ERSRI’s unfunded pension liability fell from $7.3 billion to $4.3 billion and the taxpayer contribution for 2012–2013 was cut to $300 million, from over $600 million. ERSRI’s funding ratio was projected to rise steadily over the next few years. A senior director at Fitch Ratings said, “Fitch believes that Rhode Island’s pension reform is the most comprehensive measure undertaken by any of the states in recent years.”68

There were some reductions in benefits for state workers. The plan increased the retirement age to sixty-seven for employees with less than five years of service and to fifty-nine for employees with more than five years of service. The reform created a defined contribution plan that operated alongside the defined benefit plan, and contributions would be split between the defined benefit and defined contribution plans. The calculation of the COLA was subject to a 4% cap and a 0% floor and only applied to a retiree’s first $25,000 of retirement income. The COLA would not be paid until the pension system had reached an 80% funding ratio and was calculated as the five-year average investment return less 5.5%. But as Raimondo promised, all benefits accrued prior to June 30, 2012 remained unchanged.

The changes didn't please everyone. Unhappy pensioners and unions sued. After federal mediation, a settlement rolled back some of the original reforms: COLAs, for example, were no longer contingent just on investment returns but were based on half on returns and half on the inflation rate. COLAs could also be paid when the plan was less than 80% funded. The retirement age was scaled back to 65 from 67. Nevertheless, the settlement preserved 95% of the original savings.69

(p.189) The full effects of the pension reform will take decades to manifest, just as it took decades for the pension fund to land in so much trouble. Raimondo tackled public pension reform by starting at the end, describing an optimal retirement income, and then seeking a path to that goal—in other words, by using the life-cycle approach to investing. ERSRI is not in the clear, but at least Rhode Island is on the right track.

Notes:

(1) This is based on the case “Saving Public Pensions: Rhode Island Pension Reform,” Columbia CaseWorks #120309.

(2) Reported by Barkley, R., “The State of State Pension Plans: A Deep Dive into Shortfalls and Surpluses,” Morningstar, Nov. 26, 2012.

(3) Raimondo, G., Truth in Numbers: The Security and Sustainability of Rhode Island’s Retirement System, Rhode Island Office of the General Treasurer, June 2011.

(4) Editorial, “Stand Up for Rhode Island,” Providence Journal, Nov. 17, 2011.

(5) See Novy-Marx and Rauh (2009, 2011a, 2011b) and Mitchell (2012).

(6) Glaeser and Ponzetto (2013) argue that public pension promises are high partly because they are “shrouded,” meaning that they are hard for the taxpayer to understand. The study is worth reading if only for the detailed history of California, Pennsylvania, Massachusetts, and Ohio pensions. Public pensions are decentralized in the first two states and centralized in the latter two. Centralization appears to have reduced pension benefits by drawing state-level media attention to help taxpayers understand pension costs. Glaeser and Ponzetto also include a useful table describing the public pension plans in all fifty states.

(7) Quoted in Pew Center on the States, The Trillion Dollar Gap: Underfunded State Retirement Systems and the Road to Reform, 2012.

(8) Quoted in Burton, P., “Pension Reformer Honored,” The Bond Buyer, Jan. 9, 2012.

(9) Quoted by Walsh, M. W., “The Little State with a Big Mess,” New York Times, Oct. 22, 2011.

(10) This approach follows Merton (1990) and Bodie, Merton, and Samuelson (1992).

(11) In this example, we take risk aversion and the amount of human capital wealth (and total wealth) as given. In reality, risk aversion affects both the choice of financial assets and the choice and type of human capital—a highly risk-averse person is unlikely to pursue a career as a mountaineer, for example. See Ranish (2012) and section 3.6 on endogenous labor income.

(12) See Baxter and Jermann (1997).

(13) This is assumed by Shiller (1995), Jagannathan and Wang (1996), and Eiling (2013), among others. Denoting wages as Lt and the growth rate of wages as gH, suppose wages follow the process:

Lt=(1+g)Lt1+εt,

where wages start at time 0 from the level L0 and εt is an independent and identically distributed (i.i.d.) shock. If we discount the wage stream at the expected rate of return on human capital, rH, then we can value the present value of wages as

Ht=LtrHg,

and the growth in wages, g, is equal to the return on human capital, Ht/Ht11. Earlier work by Friedman (1957) and Hall (1978) discounted labor income at risk-free rates to compute human capital.

(14) The major risk is that when you lose your job in a recession, your wage permanently falls. See Meghir and Pistaferri (2011) for a summary of permanent and temporary components in labor income.

(15) As shown by Heaton and Lucas (2000).

(16) These are taken from Social Security, Portfolio Theory, Life-Cycle Investing, and Retirement Income, Social Security Policy Brief 2007-02.

(17) From Viceira (2008). Viceira’s papers and Campbell and Viceira (1999, 2001) are seminal references in dynamic portfolio allocation for individual investors. Viceira (2001) looks specifically at the case with labor income.

(18) The analysis assumes that stock returns are log-normally distributed and that the bond return is held constant and fixed at the mean log return of long-term government bonds. This is the Black–Scholes (1973) world.

(19) For a readable explanation sans differential equations, see Kritzman (2000).

(20) As recounted by Deaton (2005).

(21) See Kotlikoff (1988) for a survey. A deep insight generated by OLG models is that population and economic growth should drive savings. If there are more young than old people, the saving of the young outweighs the dissaving of the old, and there is positive net savings. If incomes are growing, the saving of the young will outweigh the dis-saving of the old.

(22) Some more formal models of SWF asset allocation are Scherer (2011) and Bodie and Brière (2013).

(23) For those who are technically adept: the case of power utility, i.i.d. log-normal returns for asset returns and labor income is not analytical. See Duffie et al. (1997) and Koo (1998). All interesting life-cycle models are solved numerically.

(24) See also Campbell et al. (2001). There is some variation, of course, in the estimates. Lustig and Van Nieuwerburgh (2008) are the exception in presenting a significantly negative correlation between labor income and stock returns. Bansal et al. (2011) estimate a large 35% correlation of labor income and equity returns. Neither paper uses direct estimates of labor income; they rely instead on asset prices to infer its behavior.

(25) These papers include Zeldes (1989), Heaton and Lucas (1997), Viceira (2001), and Cocco, Gomes, and Maenhout (2005).

(26) The first paper in this literature is Friend and Blume (1975). See also Heaton and Lucas (2000), Vissing-Jørgensen (2002), Ameriks and Zeldes (2004), and Calvet, Campbell, and Sodini (2006). Curcuru et al. (2004) is a literature summary.

(27) This is a much larger literature than the life-cycle portfolio choice literature. The canonical references are Deaton (1991), Carroll (2001), and Gourinchas and Parker (2002). A literature summary is Attanasio and Weber (2010). Most of these papers, however, do not have portfolio choice decisions involving risky assets.

(28) For life-cycle models with financial constraints, see Rampini and Viswanathan (2013). Laibson, Repetto, and Tobacman (2012) and Pagel (2012) develop behavioral life-cycle models with hyperbolic discounting utility and reference-dependent preferences, respectively. Peijnenburg (2011) uses ambiguity aversion in her life-cycle model. In addition to the papers in the main text, see also Gomes and Michaelides (2005).

(29) They are actually slightly hump-shaped, with the old slightly decreasing their equity portfolio weights compared to the middle-aged.

(30) The large literature is summarized by Poterba, Venti, and Wise (2011).

(31) As reported by Rampini and Viswanathan (2013).

(32) See Kimball (1990).

(33) See Hubbard, Skinner, and Zeldes (1995) and Palumbo (1999). Yogo (2011) shows that spending on medical care can reduce some health-care background risk.

(34) A recent review is Paulsen et al. (2012).

(35) When there is a stock market crash, the investor could consume less, work harder, or both. This labor margin makes a person effectively less risk averse, as Swanson (2012) notes. Other authors developing life-cycle models with endogenous labor income choices include Farhi and Panageas (2007) and Gomes, Kotlikoff, and Viceira (2008).

(36) Economists label these income and substitution effects, and there is a positive relation between wages and labor supply. See Bodie, Detemple, and Rindisbacher (2009).

(37) See Scholz and Seshadri (2009).

(38) The OECD estimates the average replacement rate is 60% across all OECD countries. See OECD, Pensions at a Glance 2011: Retirement-Income Systems in OECD and G20 Countries.

(39) See Beshears et al. (2012).

(40) Fuhrer (2000) and Michaelides (2002) were the first to put habit utility in life-cycle models. See also Polkovnichenko (2007).

(41) Dybvig’s (1995, 1999) ratchet consumption model is a dynamic generalization of constant proportion portfolio insurance. This was invented by UC Berkeley finance professors Hayne Leland and Mark Rubinstein, which they marketed through their firm LOR (Leland–O’Brien–Rubinstein). It was blamed for a large part of the 1987 stock market crash. See Bernstein (1992) for a history. Riedel (2009) generalizes Dybvig’s original formulation, allowing for jumps and fat tailed processes.

(42) See Bodie (1990b) and Campbell and Viceira (2001).

(43) The 2013 Retirement Confidence Survey: Perceived Saving Needs Outpace Reality for Many, Employee Benefit Research Institute, Issue Brief, March 2013, No. 384.

(44) See Laibson (1997).

(45) The rule has many variants, several due to follow-up work by Bengen. Milevksy and Huang (2011) develop versions with mortality risk, which Bengen (1994) omits.

(46) Harvard University experienced some problems in meeting liabilities with its fixed spending rule from its endowment. See “Liquidating Harvard,” Columbia CaseWorks ID #100312.

(47) Davidoff, Brown, and Diamond (2005) also showed large benefits of annuitization even with health-care risk or when available annuity products could not exactly meet agents’ desired consumption.

(48) Numbers and quotation are from McKeown, K., and E. Michalak, “Retirees Underestimate Life Expectancy, Risk Underfunding Retirement,” Society of Actuaries press release, July 30, 2012. The full report is Society of Actuaries, 2011 Risks and Process of Retirement Survey.

(49) Stevens et al. (2012) identify one more reason: in expansions it is easier for low-skilled health-care workers (especially those working at nursing homes and outpatient centers) to find other employment. During recessions, these workers return to their low-paid jobs in health care, and thus the standard of care rises during recessions.

(50) See Inkmann, Lopes, and Michaelides (2010). And those that buy annuities often do so non-optimally. Many purchase “period certain” annuities, which are the same as bonds as they pay a certain amount every year for a fixed number of years. Period certain annuities are not the life annuities that we consider in this section. Investors would be better off buying a regular bond and simultaneously buying a deferred annuity that starts making payments when the bond matures. See Scott, Watson, and Hu (2011).

(51) All of these considerations are shared by long-term care insurance products, except for some time many insurers actually underpriced long-term care insurance. See Brown and Finkelstein (2011). One important difference between annuity markets and long-term care insurance is that the presence of Social Security’s means tests serves as an impediment (it is an implicit tax) on the private market providing long-term care insurance. See Pauly (1990) and Brown and Finkelstein (2007).

(52) Quotation and details are from Mider, Z. R., “Apollo-to-Goldman Embracing Insurers Spurs State Concerns,” Bloomberg, April 22, 2013.

(53) Obfuscation is in the interest of the general asset management industry, as Carlin and Manso (2011) show formally.

(54) Quotations from Olen (2012), who provides a blistering critique of this industry.

(55) The new generation of literature on annuities explores deferred and index annuities, like Milevsky (2005), Gong and Webb (2010), and Maurer et al. (2013).

(56) The basic benefit is that at least the original investment will be returned upon death. This is a put option with a strike price equal to the purchase price of the annuity. There are many variants, see They’re Complicated.

(57) See Sinclar and Smetters (2004), De Nardi, French, and Jones (2010), and Pang and Warshawsky (2010).

(58) Reichling and Smetters (2013) show that the best hedge for most young individuals against time-varying mortality risk is actually life insurance, rather than annuities.

(59) As reported in “Falling Short,” Economist, April 9, 2011.

(60) Numbers from The 2012 Annual Report of the Board of Trustees of the Federal Old-Age and Survivors Insurance and Federal Disability Insurance Trust Funds. Social Security liabilities do not have the “full faith and credit” backing of the U.S. government and are not counted as a government liability. The liabilities of Fannie Mae and Freddie Mac also didn’t have an explicit guarantee, and now the federal government is the sole owner of both entities. Hamilton (2013) estimates the total of all off-balance-sheet commitments by the federal government to be $70 trillion, which is six times the reported on-balance sheet liabilities.

(61) See also De Nardi, French, and Jones (2010) and Yogo (2011). Conversely, Hurd (1989) estimates bequests that are indistinguishable from zero.

(62) See Brown (2009), Inkmann, Lopes, and Michaelides (2010), and Beshears et al. (2012).

(64) Bernard, T. S., “A 401(k) That Promises Never to Run Dry,” New York Times, Nov. 14, 2012. Top–down proposals from regulators to mandate partial or full annuitization have not been greeted with favor by some parts of industry or the public; see Schieber (2012).

(65) The Wisconsin Retirement System is the only other public pension system that has a contingent COLA explicitly depending on investment returns. See Novy-Marx and Rauh (2012).

(66) See Christensen (1997).

(67) See Venti and Wise (2002). Nakajima and Telyukova (2012) show this behavior results from including housing in a life cycle model. Fischer and Stamos (2013) also develop a life-cycle model with housing.

(68) Quote from Burton, P., “Reform in Rhode Island Could Start Trend,” The Bond Buyer, Nov. 21, 2011.

(69) See Economist, “Little Rhody, Big Debts,” Feb. 22, 2014. (p.190)