story

Optimal financial investments over the life cycle

By Christopher Carroll and the Econ-ARK Team
Posted on April 27, 2020

Economists like to compare actual human behaviour to choices that would be made by a "rational" agent with perfect understanding of the most complex decisions. However, determining the mathematically optimal amount to save for retirement, and how best to invest those savings, turns out to be much harder than calculating how to land a spacecraft on the moon.

In fact, the computational tools that economists use to solve such problems descend directly from those originally developed to optimize Apollo trajectories - with 50 years of further mathematical and computational development. By 2005, those optimal choice tools were finally good enough to produce financial advice worthy of taking seriously enough that economists began recommending that real people follow the advice. A study by Cocco, Gomes and Maenhout (CGM for short) is the standard reference.

Computational tools today

But even today, these tools are not widely used, because it still takes years of study to master them. In 2015, the U.S. Consumer Financial Protection Bureau funded the creation of the Econ-ARK open source software project. Econ-ARK aims to make such tools much more accessible, both to researchers and to the wider public.

Thanks to subsequent funding by the Sloan Foundation and the Think Forward Initiative, we (the Econ-ARK team) are proud to be adding a new module to our toolkit. Our ConsPortfolioModel calculates the optimal solution to a lifetime optimal saving problem in which the consumer chooses how much to invest in risky (but expected high-return) versus safe (but low-return) assets (taking into account their risk tolerance).

Our hope is that such transparent and publicly available tools will eventually provide an alternative to the proprietary (and mysterious) advice that has recently become widely available, from so-called robo advisors, and the even more mysterious advice from human advisors.

The problem: uncertainties in life

Nobody saving for retirement knows what the future holds. You are likely to change jobs several times during your career, and each new job will come with a different profile of income growth and risk. You might die before retirement, in which case retirement savings would have been wasted, or you might live to be a century old (possibly outliving your savings).

Nor does anybody know what the payoffs will be for alternative investment choices. "Risky" assets like stocks have historically earned higher returns than "safe" assets like government bonds - but there is no guarantee that stocks will outperform bonds over any particular period (like until you retire).

Uncertainties like these are the reason that your personal financial decisions are mathematically harder than NASA's problems. The motion of a spacecraft is predictable: point it in a certain direction with a certain velocity, and Newton's equations tell you exactly where it will be, near or far into the future. In contrast, over a lifetime vulnerable to many risks, a wise financial decision must take into account all of the possible outcomes.

"Big data" now empowers us to quantify the most important risks. We can measure how often people change jobs (taking into account their education, occupation and so on), and what happens to income after job changes. Job-related income uncertainties can thus be represented as a statistical distribution over the many possible future outcomes. The same is true for other kinds of risks, like health.

When all the biggest risks have been quantified, we can calculate the joint probabilities of every conceivable draw, and weight each outcome by its probability and its desirability. ConsPortfolioModel calculates how the ultimate outcomes (such as retirement income) depend on the current choice of saving and portfolio choice, then computes the choices that are "optimal" for consumers with different preferences (toward risk, for example).

Replicating the standard model

Our first use of ConsPortfolioModel has been to replicate the results of the above-mentioned CGM study. A key input is the degree of consumers' "risk aversion". Researchers have found that many kinds of consumer behaviour are consistent with values of "relative risk aversion" in the range from 2 to 4.

The most striking conclusion of the CGM paper is captured in Figure 1, produced by our ConsPortfolioModel tool, under the assumption that consumers with risk aversion of 3 can choose between a "risky" asset with expected performance (for risk and return) like the stock market, versus a "safe" asset with lower returns historically typical of safe assets (like government bonds or a bank account). Figure 1 shows, by age, the optimal risky share - that is, the proportion of savings that it would be optimal to invest in the "risky" asset. The fact that the proportion is stuck at 1.0 at every age means that, according to our model, the optimal choice is always to invest 100% of your savings in stocks!

Figure 1: Portfolio choice for moderately risk averse consumers
Figure 1: Portfolio choice for moderately risk averse consumers

Parameters like "relative risk aversion" are hard to measure. Maybe the conventional value of 3, which works well to explain other choices, is inappropriate here - perhaps people just hate stock market risk more than other kinds of risk that would have similar financial consequences.

Figure 2 shows the profile of the mean risky share for a consumer with risk aversion of 6, twice the conventional value (think of your most risk-averse neighbour or relative). Even with such high risk aversion, the model says that until about age 35 it is still optimal to invest all of your savings in the stock market. After that, the risky share declines gradually until it stabilizes at around 65% at age 65.1

These results reflect two aspects of the model:

1. Young people start with little or no assets.

- Their income comes mostly from working in the labour market.

- If you have only a small amount of wealth, the absolute dollar size of the risk you are taking by investing that wealth in the stock market is small, so the risk is not large relative to the long-term reward.

2. For retirement, you plan to finance a lot of your future spending from your savings.

- So, investing everything in the stock market would put a large proportion of your retirement spending at risk.

- The "equity premium" is nevertheless large enough to make it worthwhile for most people to keep half or more of their assets in stocks.

Figure 2. Portfolio choice for highly risk averse consumers
Figure 2. Portfolio choice for highly risk averse consumers

What real people actually do

The pattern shown in Figure 2 is strikingly different from the actual choices that people make. Check out the data in Figure 3, from the Federal Reserve's triennial Survey of Consumer Finances, which measures the proportion of their assets that people at different ages actually have invested in stocks and other risky assets (from this article). The risky share that people choose in real life is much lower than the model says is optimal, even with extreme risk aversion of 6.

Below we examine two possible interpretations:

1. The model is basically the right framework for thinking about these questions.

- But some of its assumptions/calibrations are wrong.

2. People are behaving optimally, but the model is still missing some important features of reality.

Figure 3. Stockholdings as share of total financial Assets, by age of household head
Figure 3. Stockholdings as share of total financial Assets, by age of household head. Source: Survey of Consumer Finances. Federal Reserve Bank of St. Louis. Note: The share of stockholding is the average of all U.S. households within that age group.

Perhaps people are pessimistic about the equity premium. The calculations above assume that people expect an equity premium of 4%, which is a good estimate of what the average premium has been on stock market investments over the past century.

But nobody could have known beforehand that equity premia would turn out to be so high. Perhaps people chose lower risky shares because they mistakenly believed that risk premia would be lower. Figure 4 shows the consequences if people with a high risk aversion parameter (6) believe the equity premium will be only 2%.2

The shape of Figure 4 is much the same as before; in particular, the youngest people still hold 100% of their portfolios in risky assets. But the proportion of their portfolios that middle-aged and older people hold in stocks falls from over 50% to about 20%.

Figure 4. Portfolio choice for pessimistic and highly risk averse consumers
Figure 4. Portfolio choice for pessimistic and highly risk averse consumers

Figure 4 assumes that relative risk aversion is very high (6). When people are pessimistic about the equity premium, might their optimal risky shares be low even if they are not so risk averse? No. Figure 5 below shows that, even with the pessimistic beliefs that the equity premium will be only 2%, if relative risk aversion has a conventional value of 3, then the optimal risky share is still 100% for both young and old people. On average, the optimal risky share reaches a low point of about 90% for people nearing retirement.

Figure 5. Portfolio choice for pessimistic and moderately risk averse consumers
Figure 5. Portfolio choice for pessimistic and moderately risk averse consumers

Comparing to professional advice

Investment advisors sometimes advocate the "100 minus age" rule: one's portfolio share in risky assets should be 100 minus one's age. Therefore a 60-year-old would have 40% in stocks.

For highly risk averse people (risk aversion of 6), Figure 6 below shows that the rule's recommendation (orange dashing line) is not too different from what comes out of the model (in blue). While the rule would say that the 25-year-old should put 75% of their savings in the stock market, and the model says 100%, they agree that the young person's proportion should be high, and also agree that the proportion should decline during working life.

But the rule and the model disagree about what should happen after retirement. The rule recommends steadily reducing your exposure to risky assets as you get older, while the model says that after retirement your exposure should remain at about the same level as late in your working life, or even increase.

Figure 6. Portfolio choice for '100 minus age' rule vs optimizing highly risk averse consumers
Figure 6. Portfolio choice for '100 minus age' rule vs optimizing highly risk averse consumers

Financial advisors, who have daily contact with real human beings, may have an insight that the model does not incorporate. For example, perhaps the perceived riskiness of stock investments increases as you grow older.

The point of creating our ConsPortfolioModel tool was to make it easy for someone with moderate computer skills to explore possibilities like these. The README describes how you can use the tool to explore such model modifications; for example, if the perception of the riskiness of stock investments increases with age, the model can match the '100 minus age' rule's advice that retired people should reduce their exposure to stock market risk.

What might still be missing

Some experiments are not yet possible with our toolkit. Perhaps the most important is that we have no way to take into account the risks entailed in homeownership. Houses, like stocks, are assets whose price can go up or down. Since housing wealth constitutes the majority of the wealth of most consumers, the model's failure to consider the effects that homeownership should have on the optimal choice of risky financial investment is a failing serious enough to call into question the soundness of its conclusions.

The Think Forward Initiative grant that funded this work has a second component: calculating the optimal risky share taking homeownership risks into account. This question is at the frontier of what is possible using the kinds of tools we are developing. We are interested to see whether a proper treatment of homeownership will be enough to temper the recommendations of the model to invest heavily in risky financial assets. The answer is not clear - which is why we need a model.

Do it yourself! (if you know python, GitHub, and unix)
The computer code to reproduce all of the figures in this notebook, and a great many others, can be executed by installing the Econ-ARK toolkit and cloning the REMARK repository. The small unix program do_all_code.sh at the root level of the REMARKs/PortfolioChoiceBlogPost directory produces everything.
A replication of main results of the CGM paper is referenced in a link below.
The Econ-ARK toolkit is available at GitHub, and the ConsPortfolioModel is documented here.


Christopher D. Carroll is professor of Economics at Johns Hopkins University, and co-chair of the US National Bureau of Economic Research.

Find out more about the Econ-ARK project and team here.

Footnotes

  1. The dashing lines show the choices made by people at the 5th and 95th percentiles of the distribution of the risky share.
  2. This is around its historical average in the U.S. before 1941; some respected economists think such rates might prevail in the future.

References

  • Cocco, J. F., Gomes, F. J., & Maenhout, P. J. (2005). Consumption and portfolio choice over the life cycle. The Review of Financial Studies, 18(2), 491-533. doi.org/10.1093/rfs/hhi017
  • Velásquez-Giraldo, Mateo and Matthew Zahn. econ-ark/REMARK replication of [Cocco, Gomes, and Maenhout (2005)](doi.org/10.1093/rfs/hhi017](https://doi.org/10.1093/rfs/hhi017).