Retirement plans that maximize certainty-equivalent spending (conclusion)

In part 1 and part 2, we developed a framework for evaluating and identifying a good plan for retirement spending and asset allocation.

  • We discussed how a CRRA (constant relative risk aversion) utility function and the related concept of certainty-equivalent (CE) spending can discount a stream of future cash flows based on their risk and variability, and the retiree’s risk aversion.
  • We solved a toy problem: how the retiree maximizes certainty-equivalent spending if he or she can invest at a guaranteed fixed risk-free real return rate.
  • We generated CE-optimal spending schedules using that fixed risk-free real return rate for retirees with different levels of risk aversion (in this case, no investment risk, just longevity risk, trading off current income for future risk of outliving the portfolio)
  • We moved from a fixed rate assumption to using historical US real returns on stocks and bonds. We generated a spending schedule that maximized CE spending based on historical real returns for a 50% equity portfolio.
  • Using that spending schedule, we solved the other side of the problem, and generated an equity allocation that would have maximized CE spending for that spending schedule.
  • We looked at that solution, and found it seemed pretty good.

So, where we left off, we had independently solved the spending schedule and then the equity allocation schedule. Of course, that does not mean that when you put those two solutions together, they are the best we can do. It just means the equity allocation is the best available given that spending schedule. So today, we’ll try to solve them simultaneously.

The framework in which we try to solve the retirement spending problem is:

Maximize expected CE spending for a 25-year retirement…
Which is modeled as a function of a 2×25 matrix
• 25-years of retirement
• Spending % each year
• Equity % each year (the balance to be allocated to bonds)

And we find CE spending as:

Starting with the 2×25 vector of portfolio allocations and spending %
-> generate cash flows using historical returns for each retirement cohort
-> compute CE spending using CRRA function and gamma
-> compute expected CE spending for each cohort based on life table
-> compute expected CE spending across all cohorts and across all survival scenarios

That gives us a value: the CE spending a random retiree at any year 1926-1987, with the given life table, and given risk aversion, could have expected from that 2×25 spending/allocation schedule.

Now, the problem is to maximize that value: find the 2×25 spending/allocation schedule that maximizes the CE spending function.

So we fire up our optimizer, using this function, gamma=4, and the starting solution we previously found solving the two schedules independently. We try a few different optimization methods. Some of them fail, but the Powell method comes up with a pretty good solution after about six hours on our PC. We use that as our starting solution and run the optimization again using several different methods, and with a very slight improvement it holds up as the best we can find.

Age Equity % Spending %
65 63.8% 6.1%
66 63.7% 6.4%
67 72.1% 6.7%
68 73.7% 6.9%
69 74.4% 7.2%
70 75.2% 7.5%
71 85.3% 7.9%
72 86.1% 8.3%
73 75.3% 8.7%
74 76.8% 9.2%
75 80.9% 9.8%
76 86.3% 10.5%
77 91.6% 11.3%
78 100.0% 12.2%
79 98.9% 13.0%
80 100.0% 14.1%
81 100.0% 15.3%
82 100.0% 16.6%
83 100.0% 18.5%
84 100.0% 20.9%
85 100.0% 24.1%
86 100.0% 28.8%
87 100.0% 36.8%
88 100.0% 52.8%
89 50.0% 100.0%

We see that our initial spending is higher (6.1% vs. 5.9% when we optimized spending and equity independently). We see that in our median case, spending is flatter. We see that the worst-case outcome is a bit worse. Nevertheless it seems credible that the tradeoff is preferable for a moderately risk averse retiree.

Actual spending using computed schedule, % of initial portfolio, 25-year retirement cohorts 1926-1987

Actual Spending using computed schedule, 25-year retirement cohorts 1926-1987

It’s quite interesting that the equity % starts at 63.8% and rises throughout retirement. Conventional wisdom, as implemented in many target date funds would be to reduce the equity allocation as you get older, since you have less time to recover any shortfall from a major market decline. So that result bears investigation to see if there is an error, or if it’s inherent in the unconventional aspects of this approach.

Otherwise, this seems like an analytically sound approach that yields a good practical result.

Comments are invited.

Retirement plans that maximize certainty-equivalent spending, part 2

Last time we solved the problem of the perfect retirement spending plan, assuming a fixed known real return, and a CRRA utility function.

This time, we’ll try to look at the problem from the other angle:

  • Let’s assume a fixed spending schedule
  • Then, let’s solve the problem of the perfect portfolio allocation schedule between US stocks and bonds (of course, the approach could be generalized to other/more assets)

First, a brief digression to make the case that a CRRA utility function is a good thing to use.

CRRA utility

Here is a visualization of what a CRRA utility function looks like for different levels of gamma (use the slider to see how it changes as you adjust risk aversion parameter gamma.

Think of 1 as the ideal income or base case, where utility=0. With risk neutrality or gamma=0, gains and losses generate the same change in utility. As risk aversion increases and gamma goes up, small losses generate bigger and bigger drops in utility, while big gains generate smaller and smaller increases in utility. A ‘no-loss’ utility function with gamma=∞ would be utility=0 for consumption >=1 and a straight line down to -∞ for consumption<1.

One possible objection is, OMG, CRRA utility is such a strange complicated abstraction! No one actually thinks that way.

But we’re all used to thinking about mean-variance (or some of us, anyway). Clearly there is a tradeoff between the volatility of a portfolio, the distribution of potential outcomes, and the return we are willing to accept. So at some level a lot of our thinking about finance involves something very similar to applying a discount to future income streams based on how risky and volatile they are. That’s what the CRRA utility function does – apply a discount based on distribution of outcomes or volatility.

Another possible objection is, ‘utility’ is unobservable in the real world.

But if the utility function correctly ranks the outcomes consistently with the way a human would, at some level that’s all that matters1. The actual value is arbitrary. And as far as I know, any consistent ordinal ranking can be mapped to a cardinal utility function. And we can ask people which outcomes they prefer, either a priori asking them to rank risky outcomes to estimate their risk aversion, or simply generating CRRA-consistent retirement profiles with varying levels of risk aversion, and asking them to choose one.

Finally, why CRRA utility? The important property of CRRA utility is that it’s scale-invariant. A distribution of cashflows between 10 and 15 gets the same discount as a similar distribution between 100 and 150. So if you use a non-CRRA function, you’re going to get different answers depending on the size of the income streams that get generated.

So, if we think that, to reasonable approximations, humans are risk averse, they make consistent choices about risky outcomes, and their risk aversion is scale invariant over the range of outcomes we are studying, a CRRA utility function seems like a reasonable thing to use, as an approximation that leads to a problem we can solve.

Optimal allocation schedules

Last time we looked at the problem of finding the optimal spending schedule, given a known future return.

How do we find an optimal portfolio allocation schedule, given a known e.g. 25-year spending plan?

What we want: the expected CE spending for a given spending schedule and equity allocation schedule.

We write a function that for a given cohort, eg people who retired in 1987:

  • Takes as input the equity % in each year (a 1×25 vector – the bond% is implied as 1-equity%)
  • Uses the known returns for the 25 years 1987-2012, and the known spending schedule, to compute the retirement cash flows
  • Returns the certainty-equivalent cash flow for the 25-year period.

Now that we can calculate the CE spending over 25 years, we can write a second function that

  • Takes as input the cash flows for a period
  • Uses our life table with how many people survived in each year, and computes the expected value of the CE spending that 1987 retiree would have expected across all survival scenarios. (Since some people died in each year, it’s the CE cash flow through each year i of retirement, weighted by the percentage of retirees who lived to year i)

We can further write a third function that calls the second function on each cohort from 1926-1987, and computes the CE spending that each cohort retiring from 1926-1990 would have expected. That gives a distribution of outcomes which we also discount using the CRRA utility function, giving us what we want: the expected CE spending for someone who retired in a random year 1926-1990.

So the sequence is:

2×25 vector of portfolio allocations and spending %
-> cash flows
-> CE spending
-> expected CE spending for a cohort based on life table
-> expected CE spending across all cohorts and across all survival scenarios, e.g, for a random 25-year retiree at any year 1926-1987.

Finally, we can systematically try a universe of allocation schedules and spending schemes and find the one that maximized the expected CE spending for someone who retired in a random year in our history.

The optimal retirement plan is the one that would have maximized expected utility over all historical cohorts and survival timelines.

This does not seem completely computationally intractable in this day and age, so let’s try to compute it.

We add to our code from last time to

  1. 1) find an optimal spending schedule for a fixed return (what we did last time)
  2. 2) find an optimal spending schedule for a 50% equity portfolio that would have maximized CE spending using historical returns 1926-2012)
  3. 3) Use gamma=4, and the fixed spending schedule we found in 2), we find the equity allocation schedule that would have maximized CE spending using historical returns 1926-2012

In other words, to make the problem a little more tractable, first we find an optimum spending schedule for a 50% equity portfolio. Then, using that spending schedule, we find the optimal equity allocation schedule.

We end up with this schedule:

Age Equity % Spending %
65 61.5% 5.9%
66 61.9% 6.1%
67 70.7% 6.3%
68 72.4% 6.6%
69 73.2% 6.8%
70 73.6% 7.1%
71 83.7% 7.4%
72 84.9% 7.8%
73 75.1% 8.2%
74 76.3% 8.6%
75 80.2% 9.1%
76 85.6% 9.6%
77 89.7% 10.2%
78 99.8% 10.9%
79 96.4% 11.7%
80 100.0% 12.7%
81 100.0% 13.8%
82 100.0% 15.3%
83 100.0% 17.1%
84 100.0% 19.4%
85 100.0% 22.7%
86 100.0% 27.6%
87 100.0% 35.7%
88 100.0% 51.8%
89 50.0% 100.0%

This is somewhat counterintuitive insofar as conventional wisdom would be to reduce equity as your time horizon shrinks.

Let’s look at what that expected spending profile would have looked like.

Actual Spending using computed schedule, 25-year retirement cohorts 1926-1987

Actual Spending using computed schedule w/gamma=4, 25-year retirement cohorts 1926-1987

This schedule does in fact seem to perform pretty well. Pretty high initial spending, pretty smooth outcomes, pretty good median case, not very catastrophic worst case. Assuming I had a way to annuitize my longevity risk after 90 (which only about 20% of 65-year-olds will outlive), I would be pretty OK with this range of outcomes.

This seems like a sound approach, and the outcome looks like the kind of solution I would be hoping to find.

In a future post, we’ll see if we can determine both simultaneously – an optimal spending plan and portfolio allocation for a given level of risk aversion.

1If, on the other hand, humans care about things like the order in which income streams are experienced, ie you prefer an income of 10 followed by an income of 15 to the other way around, then CRRA utility is not going to capture that. Then maybe we need to move to a Kahneman-Tversky prospect theory utility function. And if people’s risk aversion changes over time, for instance at market peaks at troughs, then that’s also a problem.

Optimal certainty-equivalent spending retirements with DataNitro

Let’s see if we can come up with an ideal spending plan for a retirement, if you have a guaranteed annual return, for different levels of risk aversion.

It’s probably been done before, but seems like a fun illustration of the power of numerical optimization with Excel, Python and DataNitro.

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Bitcoin is the Linux of payments. And its killer apps will be for US dollars.

bernanke-ronpaulI was scanning the news the other day, and someone on Hacker News mentioned that half the items above the fold on StreetEYE were about Bitcoin. And I said to myself, I haven’t seen the neckbeards this excited since the early days of Linux.

And it hit me, Bitcoin is the new Linux.

Go back to 1998, the days of The Cathedral and the Bazaar and the ‘Halloween Document’, and open source zealots were gleefully foreseeing the day when freedom-loving hackers would take down the evil Microsoft empire.

Linux was how virtuous hackers were going to end the hegemony of robber barons who stifled freedom and innovation and extracted monopoly rents.
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Why Bitcoin is here to stay

Bitcoin Magazine

Bitcoin Magazine (Photo credit: zcopley)

In 2011 I blogged about why Bitcoin is a Ponzi scheme doomed to fail.

In the unlikely event these mad scribblings dissuaded anyone from hopping aboard the Bitcoin train, I humbly apologize. (Although it did subsequently fall 90% from its June 2011 high.)

I don’t think any of the analysis was off base, but nevertheless I have moderated my views a bit, and now suspect Bitcoin, or something like it, is here to stay.
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Amazon is making money

Profits, like sausages… are esteemed most by those who know least about what goes into them. – Alvin Toffler

Amazon founder Jeff Bezos starts his High Orde...

Amazon founder Jeff Bezos starts his High Order Bit presentation. (Photo credit: Wikipedia)

The punditocracy is blabbering on again about Amazon’s supposedly profitless business, see The Daily Beast and Slate, more discussion here and here.

If you are growing an ever more massive business without ever having to go back to markets for more capital…you are making money.

Profits are an opinion. Cash is a fact.

Amazon is generating a ton of cash ($4b in annual operating cash flow).

If the cash flow keeps growing and net income stays zero… at some point one has to conclude the net income is not really economically accurate or relevant.
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The StreetEYE manifesto

Rogue Trader (film)

Rogue Trader (film) (Photo credit: Wikipedia)

“Being good…is not good enough! Everyone must be connected to our strategy, or we will find you, and weed you out!

Information arbitrage is our business. If you don’t know what an information curve is, then find out!

Position yourself in an information curve. Dominate the curve!

Nick Leeson, who most of you know and all of you have heard of, runs our operation in Singapore, which l want all of you to try to emulate.”

— Ron Baker, in Rogue Trader (1999)

“It is this semi-sucker rather than the 100 percent article who is the real all-the-year-round support of the commission houses. He lasts about three and a half years on an average, as compared with a single season of from three to thirty weeks, which is the usual Wall Street life of a first offender. He knows all the don’ts that ever fell from the oracular lips of the old stagers-excepting the principal one, which is: Don’t be a sucker!” – Jesse Livermore

In 2005-2006, briefly, inexplicably, I had an information edge over most of Wall Street. I was reading the top financial blogs of the era, Calculated Risk and The Housing Bubble, marveling at flippers, NINJA borrowers (no income, no job or assets), negative-amortization. I asked friends in Wall Street mortgage departments how come, after Greenspan started raising rates, they weren’t shrinking profits and laying people off, what with a flat yield curve, shrinking net interest margin and all that. They said they were pushing ARMs, originating to sell, making it up on fees. I asked if they weren’t worried about defaults, was told the deals were overcollateralized to resist high defaults, and anyway it was the bondholders’ problem. I downloaded some applications for option ARMs and thought, ‘these guys are out of their cotton-picking minds.’ This was at a time when certain banks’ net profits consisted entirely of negative amortization, interest that was being tacked onto loan balances without any cash changing hands.

That was my formative experience in the power of crowd-sourced research. This is not a Monday morning quarterback, 20/20 hindsight claim. Those blogs saved my ass. And, in truth, a long list of people called the crisis1. The thing they had in common was, they thought for themselves, they did their homework, they were willing to bet against the crowd.

ANYTHING on talking head TV or put out by a bank, is a) selling something and b) conventional wisdom. There is just no money in anything else. At its best, it’s listening to awesome guys like Icahn or Druckenmiller, but who are primarily talking their book.2

OK, I didn’t really have an information edge. That was pretty much BS.

In the old days, you could say that the floor traders at the exchange had an information edge from being at the nexus of the flow, the upstairs trader had the customer flow.

I have the speculator’s edge, which is that I don’t have to do a thing. The market-maker has to provide a bid-ask, the institutional investor has to put his the clients’ money to work using the strategy he pitched. I just wait until the market does something that looks stupid and I try to apply calculated aggression, invest in meaningful size in an attractive risk-reward opportunity. Most of the time, I just try not to do something stupid. It’s not information arbitrage, it’s stupidity arbitrage.

If you’re like most people almost everybody, there is no information arbitrage, just old-fashioned hard work. I’m sure there were a bunch of suckers who thought they were geniuses being in Mike Milken’s orbit, or Bernie Madoff’s, or even just as Steve Jobs fans. Most of the time, there is exactly one guy in that information curve who is getting rich, and it’s not you. You’re their bitch, their greater fool, their sucker, their mark.

Even the floor traders and the upstairs traders don’t have the edge any more. They got information-arbitraged out of the loop by direct-access trading and HFT bots.

The real edge is, doing your homework, listening to a diversity of opinion, thinking out of the box. And not doing something dumb just because everyone else is doing it. And that’s all I had, a divergent opinion with an attractive risk-reward, and a healthy level of fear over what I was reading.

So, where does StreetEYE fit into this story? In the mid-2000s, I was fortunate to fall in love with blogs and find smart people with divergent opinions doing their homework. But it was a hard road for the early adopter, building a blogroll of dozens of blogs in FeedDemon, Google Reader. And (somewhat tragically) ultimately blogs never really achieved their full potential because there were too many, it was too hard for a lot of people to navigate, there was no central front page where great stuff filtered up, it was all ad-hoc blogrolls and hat-tips.

Then Twitter came along and it was a similar process… find some awesome people to follow, like these. Then say, hmmh… who do these guys follow, and find some more. Then say… I could probably write a script to do that.

So, at the first level StreetEYE is, let’s find the best people to follow, so you can leverage the twittersphere and blogosphere without being a total nerd.

And then at the next level it’s …. what are the stories everyone is talking about right now? I don’t want to come into the office and have everyone say, “did you see Soros’s Op-Ed in the FT?” I want to see everything as soon as it gets popular. Great news aggregation sites like Memeorandum and Hacker News and Reddit and Buzzfeed and Digg have cracked the code and become the ‘front page of the Internet,’ finding the most popular stuff in their respective fields. At some point I looked at what my filters were popping up and I said, to some extent I’ve cracked the code, somebody’s got to do the same thing for financial markets, and I might as well put this out and give it a shot.

But at the ultimate level, it’s about you, dear reader, and it’s about us. My hunch is, if we get a smart bunch of investors to come to the site every day, read, share, and especially upvote the stuff everyone needs to know, together, using the wisdom of crowds, the power of technology, and a light touch from a humble editor/curator, we can find the best journalism, blogging, research, and analysis, and make the best goddamn front page for investors on the Internet.

And that, for now, is the StreetEYE manifesto. Together, we can find the best content on the Internet, be better informed, elevate the quality of the conversation, and if we don’t all get rich, maybe we can at least avoid the next Really Stupid Thing.

I welcome any and all suggestions, issues, complaints. Please email me at druce@streeteye.com. The most important thing I need now is feedback on how to make it better.

And keep reading, doing the hard work of staying informed, and upvoting/retweeting to pass it along to your fellow investors.

Your humble fellow investor and curator -
Druce

1Even excluding talking heads like Roubini, Zelman, … Paulson, Ackman, Einhorn, Falcone, Eisman… I could go on for a while. (As an aside, for anyone jumping on the anti-Ackman bandwagon and mocking his HLF and JCP plays… go back and look at what he said about MBIA before the crisis.)

2It’s not just that there’s no skepticism and thinking outside the box. My pet peeve is the way every other word is invested with heavy emotional weight, fraught with meaning, an incantation to soothe the faithful. It’s all so tribal and inflammatory, when the essence of good investing is to be independent, dispassionate and unemotional. And then, anyone who shows up and says the emperor has no clothes is mocked mercilessly. Anybody willing to stake their reputation and go against the crowd is worth a respectful listen… and even better to find the ones too crazy to even be mentioned.

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Obama goes all-in on Inspector Clouseau for the Fed

Peter Sellers as Chief Inspector Clouseau in t...

Peter Sellers as Chief Inspector Clouseau in the The Pink Panther (Photo credit: Wikipedia)

So, the Obama Administration is ‘all-in’ on Summers, despite nearly everyone who hasn’t worked for him (and a number who have) thinking he’s not the best candidate.

The argument: ‘crisis experience,’ and the need for a ‘steady hand.’

Summers’s crisis experience is like Inspector Clouseau’s, the master detective who always seems to be at the scene of the crime.
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Risk arbitrage – Investing and poker

When I was young people called me a gambler. As the scale of my operations grew, I became known as a speculator. Now I am called a banker. But I have been doing the same thing all the time. – Ernest Cassel

To win, you must understand the game, you must understand the players, and above all you must understand yourself.

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“Cat Food” revisited – final thoughts – part 4

Here is the long-awaited conclusion to the wonky 4-part discussion of safe retirement spending. We went pretty far down the rabbit hole, and I think the conclusions are useful.
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The work an unknown good man has done is like a vein of water flowing hidden underground, secretly making the ground green. - Thomas Carlyle

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