Playing a feasibility game against the 1970s

Why Play a "Feasibility v 1970s" game?

1) I was re-reading Patrick Collins "Monitoring and Managing a Retirement Income Portfolio"(2015) which, unsurprisingly, focuses on managing and monitoring a retirement portfolio (a worthy read, even on the third go-through). Collins has a strong emphasis on feasibility, annuity boundaries and balance sheets. These are all good things and will be important parts of one of my next "process" posts. I wanted to run some feasibility analytics just for fun and to shake off some cobwebs from what I did in this area last year.

2) I was partly inspired by a post by EREVN (The myopia of failure rates) along with some correspondence on the topic (can see this in my post that happened to have echoes of his point) of what gets missed when saying that a spend rule (like the 4% rule) "worked." What gets missed is the dynamic technical evaluation as one flows through the time between the beginning and the "end." The psychological states that might fall out of that evaluation get missed as well.

3) I lived most of my childhood supported by someone running a SWR program through the 1970s and I saw first hand what that kind of environment can do to portfolio longevity.  Maybe next time we'll play against Venezuela but I'm not sure there would be any winners. 1990s Japanese equities might be fun, too.

The combination of these three points got me interested to see what would happen to a run-of-the-mill 4%-rule run deterministically through the 1970s along with some additional variations on that rule-theme while using some elements of modern retirement practice, such as feasibility analysis, on all of them. Hopefully this sets me up for doing a better job on my next post. Maybe it will also add some insights to my own planning efforts.

The Players

A) Constant inflation adjusted spend as a percent (e.g., 4%) of the initial portfolio,
B) Constant inflation adjusted spend as a percent of the initial portfolio but based on initial feasibility
C) Dynamically reevaluated spend based on feasibility in each period
D) Dynamically reevaluated spend based on feasibility in each period but "capped" by player A


Determining The Winner

The player with the longest portfolio longevity and also with the highest discounted utility of lifetime consumption wins. A little money left at around age 100 is worth looking at too. This will be subjective in the end since these are all different goals.

The Rules or maybe "A Series of Unfortunate Assumptions"

Several of these assumptions range between bad and insupportable but I have kids and laundry and cooking to do and I can't afford more sophisticated rules right now just to make you happy...unless maybe you want to come over and help me fold some laundry...

• Age and sex: 60 year old male. Not trying to be toxic, that's just me. 
• Endowment: $1M
• Start year: 1966. I hear that is a bad year to start (i.e., good for me) but I have not validated it personally.
• Data: Stern School annual market data for S&P and treasuries. Inflationdata.com for inflation.
• Allocation: 50/50 (arbitrarily selected) allocation in an annually rebalanced portfolio. Note that the low risk asset is the 10 year; Bengen used the five year. I'd love to do this with a third alt-risk asset some day.
• Spending is at the end of the year for reasons I can't really defend. I could do this again with spend at the beginning but I won't. Spending is inflated each year at the current inflation rate. 
• Social income and players spend rates are independent. In a better game there would likely be optimality gains for depleting wealth with a higher spend in the presence of a sufficiently high lifetime income. 
• Process is deterministic based on historical data from start year to the latest year for data, 2018. This limits me to 52 years which in most cases is enough but sometimes isn't. In cases where I say a spend rate lasts "forever" what I mean is that it lasts for at least 52 years for a 60 year old and after that I'm faking it and asserting it lasts forever. 
• In calculating the present value of spending or it's annuity proxy I assume that the discount rate and inflation net out for a constant real spend or cash flow. This is highly convenient and suspect but makes life easier. I saw Laurence Seigel and Andrew Rudd do it in a 2018 paper so I feel sorta justified for now.  
• To calculate the real value of spend in year "t" I use the chained inflation rates annualized or what we'd call a compound annual growth rate. 
• Spend rates when adjusted to get to feasibility are ratcheted in 1k granular increments.
• I assume in cases where the player is not doing 4% constant real spend that they know what that lifestyle would be in any given year relative to where they are at that point.
• Annuity pricing is asserted to be the same as the PV of a spend plan and that the market is more or less complete and, if we are thinking in annuity terms, that that completeness extends past age 85. 
• When wealth depletes spend rates snap to constant real social income.
• At wealth depletion I assume there is a subsistence income of 5k real. This is not social security or pension which I have not modeled. It is maybe better understood as help from society in the absence of SS sought only when nothing is left. It is maybe even better understood as a really bad assumption that prevents the infinite disutility that comes from zero consumption. The addition of real social income at age 70 would change all the results and be worthy of examination in another post. The presence of lifetime income in a utility model is a big deal.  This assumption may be the biggest flaw in the whole game. 
• There is no bequest utility or plan.
• Just to make it really unrealistic (but easy for me) we have no taxes or fees but you can assume that would make things worse in this game for everyone but the IRS and advisory firms.
• If spending ever gets below 50% of the preferred 4% lifestyle after age 90 spending is held constant in nominal terms thereafter.

Key Game Elements and Measurements

1. The feasibility barrier or boundary -- in any time (t) satisfies the following inequality

Where W(t) is starting wealth at some time t after game start and F(t,x) is the valuation of the spend liability at time t for player aged x or alternatively a proxy for an annuity price for income similar to the player's projected spend path. F is calculated as the present value of the then current spend rate projected into the "then future" where the inflator and discount are (a little too) conveniently the same. The conditional survival probability vector is re-calculated for the age attained at (t) and used to probability-weight the projected cash flows.  This boundary valuation can be viewed either as an annuity price and/or a stochastic present value of spending (except I guess here it is not stochastic). Either way it is a "solvency benchmark." If life were perfect, spending and income would always match and there would be a fairly priced annuity to match the required income pattern, but alas...  There is a load of 10% added here which in the case of annuity-style-thinking it is the fully loaded insurance overhead and risk costs added on top of the fair annuity. If we are in spend-PV mode, the load is maybe better described as a margin of error.  Either way the calc can be conceived like this where I'm pretty sure I've botched the notation:

F(x,t) is the boundary valuation for the game player aged x at future time t. cf is the cash flow at time t that is then projected out to infinity and inflated at a rate R which is the then-available discount rate determined by the compounded annualized inflation rate starting at t=0 to t-whatever. This is a little sketchy but I had to come up with something. iP(x',t) is the conditional survival probability for the person age x at game start but now advanced to age x' at time t. l is the load or margin of error. My confidence in this nasty little construct wanes a little bit every time I try to write it down but it works for now.

Note that the goal here is to evaluate W/F to be greater than or equal to 1.  For what it's worth, an infeasible plan can still work for a pretty long time, it is just technically insolvent in an asset/liability sense.  Note that F is always a moving target because things change: age, survival probabilities, wealth, inflation and discount expectations, etc. Here's Collins on the boundary and its significance:

"In terms of portfolio management, [the feasibility "free boundary"] is perhaps the single most critical piece of information that the investor should know....The investor needs to know if they are in trouble, not weather their equity position has outperformed the S&P500...Reference to the free boundary may be the clearest way to depict the portfolio's current economic condition... As wealth depletion pushes farther and farther toward the region of infeasibility, the consequences of investor actions or inactions are magnified in the sense that an ill-considered asset management election may not only generate losses, but losses from which the portfolio can never recover." Collins 2015 [emphasis added]

2. The utility calc -- is what is used to evaluate the consumption paths that are, in the end, shaped by the possibility of failure or at least the sudden move from normal consumption off of the portfolio to consumption that is constrained by available income when we run out of money.  We are conveniently ignoring the idea that the presence of life income creates the possibility that high spending and early depletion can be more optimal than a lower spend with later depletion. That's another post.  The consumption path over the remaining life of the players is evaluated via a value function that looks like this:

where V is the value function of consumption path c, tPx is the conditional survival probability for a player aged x at time t, theta is a subjective discount and g is the utility function for spend at time t which is also additive and separable, an assumption which I gather can be debated. g can be framed as a CRRA utility function like g(c) = [c^(1-gmma) -1] / (1-gmma) for gamma <> 1 and g(c) = ln(c) for gamma = 1. I am using the ln(c) form because it is easy and represents a baseline low risk aversion.


The Game Process 

Player A

- Start with $1M endowment and spend 4% of the initial value adjusted for inflation.
- Returns are a 50 50 mix of S&P and 10 year treasury from 1966+
- Spending is effectively at the end of the year for reasons all my own
- When wealth depletes spending snaps to spending of 5k of "social" income
- in each year, F is valued and a ratio of W(t,begin)/F(x,t) is calculated
- Portfolio longevity is measured in years
- The discounted utility of lifetime consumption is calculated on the entire path to infinity

Player B

- same as player 1 except that spending is adjusted until the feasibility condition is satisfied in year one and constant inflation adjusted thereafter.

Player C

- same as player 2 except that the adjustment is done every year

    o if infeasible spending is reduced until W/F >= 1

    o if feasible with a surplus, spending is increased until W/F just >= 1 

Player D

- same as player 3 except that when spending gets to or above player 1, spending stays at 4% level

- if later on it becomes infeasible again, player 3 rules apply subject to player 4 constraint

End of game

- compare portfolio longevities

- compare the discounted utility of lifetime consumption

- look at remaining wealth at some age like 100

- pick a winner or winners

The Game in Action

Move 1 - Player A

Player A is basically the Bengen study except that we are not using five year Treasuries and I have no idea if he did it monthly or annually and I also don't know how he sequenced spending and returns withing each time period. I'll have to look sometime.

The interesting thing here, and this is the point that EREVN made and that I also made, is that while the 4% rule "worked" for 30 years: a) that's all it worked for, and b) in this feasibility game, it was infeasible not only in the first year but in every year over the whole time-frame when looking at the dynamic evaluation of feasibility. How do you think that would have felt!?  Here is a chart of W/F for player A

So, it "worked" but it was formally infeasible and increasingly so for 30 years and then failed at age 90. That is a plan that worked but it also looks pretty unhappy. It didn't work and had lower lifetime utility because at 30 years there is still a pretty high chance of survival. This is what is usually missed in Bengen-style analyses of retirement choice.

Portfolio Longevity in years   - 30 years
Lifetime Utility, gamma=1    - 251.97

Wealth at age 100                  - 0 

Move 2 - Player B

Player B's main move is to adjust spending, before the plan starts, to something that is feasible under the rules and constraints above. In this case an adjustment of spending to 34k in year 1 creates a feasibility condition in year 1 (but not all subsequent years). Player B then spends 34k in constant inflation adjusted terms in all subsequent years. The W/F chart looks like this. Note different X and Y scales:

Generally very feasible with a few dips in the middle and a soaring change at the end.

Portfolio Longevity in years  - infinite*
Lifetime Utility, gamma=1   - 250.70

Wealth at age 100                  - 2.8M nom

* ok, it's not infinite. Since I started with 1966 the data only goes to 2018 which is to about age 112. At that point the survival probabilities are pretty low anyway. I forced it to infinite and will go back and validate how long it lasts later.

Move 3 - Player C

Player C's main move is to take advantage of the fact that spending and feasibility can be evaluated dynamically and that spending can be increased or decreased to reflect the feasibility calc.  After the first year of spending at 34k it is adjusted in all subsequent years to make W/F = ~1.  Chart of W/F:

Frankly I thought it would last forever. But it didn't. That makes me doubt my spreadsheet and method. I'll have to look closer to figure out how the end-game killed it. I'm pretty sure it was inadequate growth in the portfolio combined with skyrocketing spending (we're not looking at sustainability!), the spending rule/policy in the last bullet above, and the collapsing-longevity-expectation's influence on the boundary valuation. But it could also be an error.  Either way it looks like some cascade of bad luck, bad choices, and aging conspired to ruin the game for player C. TBD because this is important in order for anyone to take dynamic feasibility processes seriously. 

Portfolio Longevity in Years  - 39 years

Lifetime Utility, gamma=1   - 253.00
Wealth at age 100                  - 0 

Move 4 - Player D

Player D is really player C who here attempts to keep her spending under or at the 4% lifestyle assuming that she is still keeping track of that benchmark.  The goal here is to keep spending at feasibility levels when things are bad but to raise them when things are good but to respect the preferred lifestyle (4%) that was in our head at the beginning in order to not over-spend and succumb to lifestyle creep and so to enhance portfolio longevity and bequest capacity over the long haul.   

Portfolio Longevity in years   - 47 years
Lifetime Utility, gamma=1    - 251.68

Wealth at age 100                  - 1.6M nom

An Attempt at a Synthesis 

Here are the paths in real spending contextualized with conditional survival probability for the 60 year old players:

And here are the results in tabular form:  

An Attempt at Picking a Winner

I'm going to give this to Player D (feasibility tuning early with a limit on spending to what the preferred lifestyle would have been) for now.  Lifetime utility was not optimal here but it was at least close to the constant 4% spend and higher than the static-feasibility of player B. Plus the portfolio longevity was really high with residual dollars for bequest or margin of error still in the bank at a late age. Lifetime income would change everything, though. It would have put the constant spend back in play for second place or higher but that is another post.  

Some additional Thoughts 

• It looks like my subjective risk assessment on the winner diverged here from classical utility analysis but I guess I have other fish to fry. 
• It appears as if technically "infeasible" plans can run pretty far in time and technically feasible plans can crash.
• A focus on feasibility early in the retirement cycle is probably helpful.
• I'm still confused by this but a late-age obsession with feasibility seems to not be as helpful as I thought it was. There is also interaction with small portfolio effects I think. I still need to work through this.
• Conservatism in lifestyle assumptions over the arc of a life seems to be helpful.
• "Set and forget" approaches look like they can be sub-optimal in utility terms, especially in the context of random lifetime.
• Constant 4% spend failed because there was a ton of right tail on the longevity distribution that "30 year fixed" analyses always miss. But we knew that.
• This post is a type of spend rule game but the rules here at least have some basis in economic rationality rather than being ad-hoc, something Collins called "single bright-line rules-for-all-occasions" which he also calls an "elusive oxymoron...a set of rules-to-follow-for-a-random-process" by which I think he means it doesn't make sense analytically except maybe in the context of some individual and particular backfit data series, i.e., it's not enough.
• My guess is that triangulating an answer with the assistance of sustainability (and other) analysis (simulation related to fail rates or shortfall or lifetime ruin) would go a long way to getting a pretty good process going over time. Note that at one point (recently) it looked to me that in the calculus of ret-fin, feasibility and sustainability are mathematically equivalent, which might still be true, but maybe they aren't really when playing it out like this in this kind of game.  

See  Supplement to my "playing a feasibility game against the 1970s"