There are some new players here just to mix things up and so that we get a variety of approaches. In addition I have added a dynamic process. By dynamic I mean several things: 1) for a prescriptive rule like RH40 or even a dynamic rule like PMT, I am calculating at each step -- given the info available at the time (portfolio value, age, longevity using the SSA life table 95th percentile for each age[1], spend rate[2]) -- the probability of success in order to gauge a subjective sense of the risk taken along the path, 2) for the PMT function the spend rate depends on portfolio value and longevity (distribution period or DP = SSA 95th percentile for that age - current age) so it is dynamic by definition, and 3) for the dynamic simulated constant risk (MC sim and kolmogorov) approaches I will solve for a spend rate in each year -- given all ambient conditions in that year as in "#1" -- by a trial and error method in order to yield a 90% POS. Note that for "dynamic" I do not mean decision rules as they are normally understood the retirement finance lit nor do I mean "adaptive" in the way I define that term to myself. Me? I consider adaptive to be something like this: I get a bad medical diagnosis and I change my medical liability/reserve on my balance sheet from 0 to $1M and I change my longevity expectation from SSA 95th percentile to maybe 3 years and then I start from zero again with a new plan.
The Players:
1. The 4 % rule. You know him. Spend 4% calculated at time zero and the inflation adjust it from there. This is the rule for those that fetishize smooth income. Remember we have static growth of ~5.7% and inflation of 3% rather than random/simulated.
2. RH40. This should be familiar by now. This is a prescriptive variable-percentage rule of thumb in the form of: withdrawal = age / (40-age/3)/100.
3. Excel PMT function. This shows up in the lit more often these days. Waring and Seigel had a good demonstration in their Annually Recalculated Virtual Annuity (ARVA) article. In this case I take it a step farther and I make longevity vary with the 95th percentile expectation in the SSA life table for any given age.
4. Dynamic forward MC sim. This is my personal MC simulator tuned to several conservative assumptions like 50/50 allocation, a small degree of return "regime" suppression for a few years, fees, etc. What I do here is in any time t I take what we know from t such as age and (dynamic) longevity expectation, and also what we know from t-1 such as the value of the portfolio and I solve for the spend rate that results in a 90% POS [2][3] so that we have a constant risk spend. In real life, constant risk might create extremely volatile spending. While I think there are other "utilities" than smooth consumption ( fail risk, status and social stuff, etc) extreme consumption vol is probably not that great.
5. Kolmogorov equations. These are from Milevsky's 7-Equations book. This is analytic simulation by any other name. I added it here just as a reality check on my simulator. Note that I am not doing the math. I couldn't integrate or differentiate my way out of a paper bag. I am using a spreadsheet someone else created from Milevsky's book. I am always amazed how elegant and effective these things are without all the apparatus of modern simulation methods which are like riding a bike in first gear: lots of motion for small forward progress. The mode is set to 90 and dispersion to 9, returns 6%, inflation 3%, vol = 10%.
The Rules:
The game is more or less the same as the last post.
- start with $1M
- 40k is defined as "preferred lifestyle" at the begin age
- each year (starting at age 58) consume part of portfolio using each method
- assume some rate of return to grow the portfolio
- [new] for dynamic methods, take state info and solve for spend rate that yields 90% POS
- [new] for prescriptive rules calculate forward risk by simulation
- update the rule/method each year and repeat
- render each year's consumption and residual wealth in PV terms (.03 discount)
1. Stay ahead of a fair price for annuitizing[4], at a given age at time t, the value of the t-0 preferred lifestyle inflated to time t. Try to stay ahead all the way to age 95. The purpose here, if I got the numbers right, is to reserve the option to annuitize wealth as and if I choose and to keep any market upside in the interim. A more dynamic and randomized model makes this (a lot) more complex than I can handle. I recommend G. Irlam on spending and annuitization. Papers searchable at ssrn.com. For the game at hand I am just keeping it simple while acknowledging the annuitization option should probably be part of any game.
2. Provide the highest utility (certainty equivalent[5]) income over the age range 65-95 for gamma = four.
3. Keep the forward-estimated risk under a 30% probability of failure each step of the way. The purpose of this is to dampen the possibility of any behavioral risk such as panic selling and stupid allocations when under stress.
The Game in Charts:
A. Spend Rates. First, here are the prescriptive (RH40) and descriptive (effective rates for the others) spend rates that came out of the game:
Figure 1. Effective Spend Rates
B. Present Value of Lifestyle. These lines are the period t spend rates in present value terms for each of the methods/players over time.
Figure 2. PV of lifestyle outcomes
C. PV of Residual Wealth. Figure 3 is the wealth along the paths of the spend methods over time in PV terms. The dotted line is not a spend method. This is the cost of an annuity at a given age in time t that would annuitize the value in time t the 4% lifestyle calculated at t-0 and inflated to t. This is amateur hack territory. Not sure I got this right or whether I can legitimately do it this way[2]. Kolmogorov isn't shown partly because I forgot to include it and partly because it would have been overlapped with the MC sim almost all of the way. Next time.
Figure 3. PV of residual wealth
D. Path-wise recalculated risk. Figure 4. is the simulated risk of failure calculated at each step using only state-info with the main adaptive/dynamic updates being to: portfolio value, spend rate, age and actuarial/longevity-adjusted distribution period. Note that the MC sim and Kolmogorov are forced to a 10% risk by design.
Figure 4. Recalculated Fail Risk
The Outcome:
Test 1: Stay ahead of the annuity curve to age 95.
The 4% rule and PMT are off the table because they fail test 1: The other three all made it or were very close but RH40 was slightly ahead at the end.
Test 2: Deliver high utility income over range x.
The 4% rule and PMT are off the table again because they fail test 3.
Conclusion:
Since test 1 was a photo finish and there was some subjectivity to "beating the annuity curve" and because the age range in test 2 was defined at the outset as 65-95 for the range to evaluate utility of income, I give the game to Kolmogorov. That plus "constant risk," just like "smooth income," has high utility to me. RH40 acquitted itself quite well I have to say in a predictably self-serving way.
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[1] The 95th percentile expectation looks like this for each age from 60+. I have my version of the SS data but this is from aacalc.com because it was faster.
[2] This is a bit of a mismatched process. The sim will sim based on a constant inflated spend for the then current spend rate but we know that in the next step we will not be constant we will, rather, be dynamic. This is just one way of doing it.
[3] This process is quite imprecise for several reasons such as: my sim is an amateur device, the sim runs are small because it is a tedious and long process to do this type of game, and the "solve for" is trial and error so I am not hitting 90% on the nose because it is hard to do that. Also note that I am rounding the longevity expectation to the nearest whole age which creates jumps in the solutions especially at later ages. But I'm thinking this is like real life where we are all winging this on the fly with imprecise knowledge and tools.
[4] Fair price for annuity is not done by me. I am again leaning on aacalc.com. Assumptions/inputs for given age are: inflated value of the 4% lifestyle, TIPS for bonds, male, single, annual distribution, 90% Moneys Worth ratio. Unfortunately this way of doing it will project the current world's assumptions into the future, which is unfortunate, but that's all I've got right now. Note that in real life, random movements in wealth at various ages will push people into "it's optimal to annuitize now" channel based on Irlam's rigorous economic application of stochastic dynamic programming to the annuity question. So I guess real life is more complex than my charts; how surprising.
[5] CRRA utility: C^(1-gamma)/(1-gamma). As in the last post. As an amateur hack, I'm not sure I got this right but I'll roll with it the way I've done it for now.
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