At Last! Portfolio Guidance Based On Expected Utility Maximization

by: Laurence Kotlikoff
Summary

Economic Theory has a single method for providing portfolio guidance. It's called Expected Utility Maximization (EUM).

The financial industry's investment advice bears no relationship to EUM.

No wonder. Until now, economists have not had a method to calculate EUM in a reliable, let alone quick manner.

My financial planning software company just made a huge computational breakthrough.

Our tool, MaxiFi Premium at www.maxifi.com, now does EUM in seconds with perfect precision. Consequently, we've found Economics' Holy Grail of portfolio guidance.

Every week, it seems, there’s a new financial website with just the right solution for managing your money. “We’ll bucket you.” “We’ll goal you.” “We’ll de-tax you.” “We’ll balance you.” “We’ll life-cycle you.” “We’ll advise you.” “We’ll robo you.” “We’ll ….”

These sites have one thing in common -- a complete disconnect with economic theory. None of these sites references economics’ investment mantra – invest to maximize your expected utility from consumption. Expected utility references the lifetime happiness you’ll experience on average, not from looking at your money, not from weighing your money, not from fondling your money, but from spending (consuming) your money.

The safest way to spend your assets is to do so immediately. Then you can stop worrying about investing. But none of us wants to splurge today and starve tomorrow or do the opposite. Doing so would violate our physiology. Our guts, mental and gastrointestinal, understand satiation.

That first hot dog, with the mustard, relish, and, ok, ketchup (but God forbid mayo) is heaven. The 11th hot dog, not so much. And after you’ve chomped down 74 Nathans (made by Nathans Famous, Inc.’s hotdog chain) inside 10 minutes – the current world record, the 75th may not even stay down.

Economists have a special expression for getting full (increasing satiation). It’s called diminishing marginal utility. It means the extra happiness from the first hot dog exceeds that from the 2nd, which far exceeds that from the 11th, which kills compared to the 75th.

Diminishing marginal utility leads us to smooth (spread out) our consumption over time. That’s why we save for retirement and why we arrange to eat hot dogs at a steady pace. Diminishing marginal utility also leads us to smooth our consumption over time, good times and bad times. This keeps us from putting all our money on Maximum Security at the Kentucky Derby.

Economists capture our happiness and its diminishment due to diminishing marginal utility via a simple mathematical expression called a utility function. The standard formula for your happiness at any point in time involves just two things – what you consume at that point in time and your degree (coefficient) of risk aversion. The higher your degree of risk aversion, the faster diminishing marginal utility (satiation) sets in.

Since we live for many years, our lifetime utility consists of the sum of annual year-specific utility functions all of the form just described (with an added factor that accounts for how much we prefer to consume when young compared to when old). The tricky thing is deciding how much to spend and how much to invest knowing that the future is uncertain, including the returns you’ll earn on whatever assets you decide to hold over time.

Enter expected utility maximization (EUM), which figures out how aggressively or cautiously you should spend and invest your money to achieve the highest lifetime utility averaged over all the different future returns you may earn. If you are more risk averse, you’ll care very little for the 11th hot dog, but be hyper-concerned about being able to consume the first few no matter what. Hence, cautious spending and investing will maximize your expected utility. If you are more risk tolerant, more aggressive spending and investing will deliver the greatest expected utility.

EUM is easy to say, but hard to do. The earliest EUM frameworks, developed in the Fifties and Sixties, were primitive. They assumed households lived for just one period, i.e., they ignored the need to adjust, each year, our spending and, typically, our portfolio in light of what returns we’ve earned in the prior year.

Many a finance theory paper and many a simulation study have been produce by many an economics or finance professor, myself included. This research has taught us general lessons. But we could never actually calculate precisely what households with particular degrees of risk aversion and particular investment options should do, let alone come up with the answer in a matter of seconds.

I know this all too well. I’ve been working to crack this nut for the past 26 years through my financial planning software company, Economic Security Planning. The main technical problem has been the need to incorporate household cash flow (borrowing) constraints. Those constraints make the standard method -- dynamic programming – used to achieve EUM very imprecise. Fortunately, over the past year, my company made a major computational breakthrough that can perfectly calculate EUM in seconds. The following, reproduced from Advisors Perspectives, describes and illustrates the new methodology.

Super Quick Expected Utility Maximization Is Finally Possible!

The new method, included in our tool, MaxiFiPRO, is called certainty equivalence. It entails generating, within seconds, 500 living standard trajectories given a client’s spending/investment strategy.

The trajectories are produced simultaneously via parallel processing. The client’s living standard in each year on each trajectory is calculated on an as if (for-sure) basis using our iterative dynamic programming technique for which we received a patent. Then asset returns are drawn to advance the client to the next year on the trajectory.

Once the 500 trajectories are computed, they are plugged into a lifetime utility function, whose degree of risk aversion is specified by the client via a simple question about their risk tolerance. We then repeat the exercise for alternative spending/investment strategies and compare levels of expected (average) lifetime utility.

To restate, along each trajectory we solve, for each future time period, a certainty problem. Thus, if a household is 30 and could live to 100, we solve 500 x 70 or 35,000 dynamic programs, which we then use to calculate expected utility.

Let me clarify what I mean by a spending/investment strategy. An investment strategy is familiar. It involves specifying the assets you’ll hold in each future year and their portfolio shares. To set their spending strategy, our users set an “as if” safe real rate of return for their clients. If the safe rate is set at, say, 1 percent real, it means that their clients will, each year, spend as if they will earn 1 percent for sure on their investments.[1]

MaxiFiPRO compares expected lifetime utility for the client’s Base strategy as well as for two alternative strategies. The strategy with the highest expected lifetime utility is the one the client will likely prefer.[2] We present various comparisons of the sets of living standard trajectories under the base strategy and the alternatives. But summarizing 500 living standard trajectories in a single number – the client’s index of expected lifetime utility -- makes deciding what’s best for the client far easier.

Illustrating Expected Utility Maximization in MaxiFiPRO

To illustrate portfolio guidance based on expected utility maximization, consider a hypothetical 54-year-old couple, Martha and Sam. Martha earns $200,000, Sam $50,000. They will both retire at 62. Their combined retirement accounts total $1 million and they have $400,000 in regular assets. They also have a $1 million house with a $500,000 mortgage plus standard annual housing expenses.

Martha and Sam’s Base strategy is to hold half of all their assets in stock and half in bonds. But they also want to consider both “Safe” (20 percent stock and 80 percent bonds) and “Risky” (80 percent stock and 20 percent bonds) strategies. In all three cases, they specify, for their spending strategy, a 1 percent “As If” safe real rate of return.

The chart below shows the couple’s expected utilities for their specified degree of risk aversion. Note that expected utility from the safe strategy is highest when the client is very risk averse and can tolerate almost no risk. The risky strategy produces the highest expected utility when the client can tolerate a lot of risk. And the Base strategy is best if the client is moderately risk averse.

EUM -- The Holy Grail of Portfolio Guidance

MaxiFiPRO doesn’t recommend particular spending or investment strategies. It simply lets planners explore options with their clients based on the modern finance of consumption smoothing. Expected utility maximization has been the Holy Grail when it comes to portfolio guidance. It’s good to know that we’ve finally found it!

Index of Expected Lifetime Utility

How Much Risk Can You Tolerate?

Almost None

Very Little

Moderate

Some

A Lot

Base Strategy

50-50 Stocks/Bonds

100

100

100

100

100

Safe Strategy

20-80 Stocks/Bonds

102

101

99

96

89

Risky Strategy

80-20 Stocks/Bonds

92

94

97

101

111

Index of Expected Lifetime Utility: This table compares your expected lifetime utility (average welfare) from adopting your Base strategy and alternative spending/investment strategies. MaxiFi calculates your expected lifetime utility for a given strategy by averaging your lifetime welfare arising under each of the 500 living standard trajectories it simulates. Whether your Base strategy generates a higher or lower level of expected utility depends on your tolerance for living standard risk. If you're very concerned about experiencing years of low living standards, i.e., if you are highly risk averse, your expected lifetime utility will be lower for strategies that have trajectories with downside living standard risk. If you're less concerned about the downside and more focused on the upside to your living standard, i.e., you are less risk averse, your expected lifetime utility will be higher for strategies that feature considerable upside living standard risk. To read the chart, look at the different columns and pick the one that best describes your willingness to tolerate the risk of a lower living standard. Once you've picked a column, look down that column at each row. Notice that Base Strategy values are always set to 100. If the value of an alternative is, for example, 112, it means that strategy is 12 percent better than the base case. If the value is 80, it means that strategy is 20 percent worse. The highest scoring strategy in each column is dark green and worse strategies are lighter green. Note that you can't compare the numbers across the different columns. A 100 in the "Almost None" column does not equal a 100 in the "Very Little" column because you can't compare the happiness of people with different risk tolerances. Note: The percentage difference in expected utility is calculated relative to the Base Strategy. If an alternative is, say, 10 percent higher, it means that it delivers the same expected utility were you to stick with the Base Strategy but somehow be able to enjoy a 10 percent higher living standard under all Risk Analysis Overview circumstances. Economists call this a consumption equivalent calculation. Disclaimer: MaxiFi does not provide investment advice. It uses a standard form of the utility function, which may not properly represent your own or your clients' preferences. Moreover, its statistical assessment of the likelihood of users receiving particular returns on particular assets may be wrong.


[1] Note that the spending and investment strategies chosen can be set up to change over time and also change based on the situation the clients find themselves in.

[2] We assume the standard constant relative risk aversion form for annual utility. If the client values consumption differently, our expected utility calculations and strategy comparisons will, obviously, be biased.

Disclosure: I/we have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it. I have no business relationship with any company whose stock is mentioned in this article.