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Beyond Monte Carlo An Algorithmic Replacement for a Misunderstood Practice Shawn Brayman President, PlanPlus Inc. Shawn Brayman • PlanPlus Founder • 20+ years experience in financial services IT sector • B.Sc. (Applied & Computational Math), MES (Expert Systems) York University • Canadian Institute of Financial Planning (1994) prescribed course of study Shawn Brayman President PlanPlus Inc. • Software, training, speaking on financial planning in 9 countries Beyond Monte Carlo Our industry is filled with conflicting information. For both advisors and clients it is almost impossible to sort out “truth” from “fiction”. Beyond Monte Carlo Executive Summary SECTION ONE – MCS Myths MCS challenges on assumptions MCS tests higher & lower returns MCS tests bad years, client goals (sequence risk) Randomizing life expectancy tests longevity risk Beyond Monte Carlo Executive Summary SECTION TWO – Reliability Forecast Precision challenges with MCS Creating an algorithmic solution Validating the algorithmic solution Improved sensitivity analysis for a financial plan Beyond Monte Carlo Disclaimer! 1. MCS is simply a statistical procedure and as such cannot “be wrong”. It is how it is applied and interpreted that there may be issues. 2. My research specifically looked at MCS for longterm financial planning. I am sure many academics and researchers have valid applications of MCS in specific instances. Beyond Monte Carlo A Last Aside • • • • FPA Seattle had 7 of 30 sessions that deal directly with Monte Carlo or aspects of it (just over 23%) 3 are supportive/use it, 3 question it and 1 is “yes, but buyer beware” I went back and to FPA conference programs from Nashville, San Diego and Denver Did a randomization with 100,000 simulations and determined in 4 years the entire FPA conference will be about MCS with no “winner” Beyond Monte Carlo A Last Aside • • • • FPA Seattle had 7 of 30 sessions that deal directly with Monte Carlo or aspects of it (just over 23%) 3 are supportive/use it, 3 question it and 1 is “yes, but buyer beware” I went back and to FPA conference programs from Nashville, San Diego and Denver Did a randomization with 100,000 simulations and determined in 4 years the entire FPA conference will be about MCS with no “winner” Beyond Monte Carlo A Last Aside Beyond Monte Carlo Financial Planning is a Balancing Act Beyond Monte Carlo Financial Planning is a Balancing Act Return Risk Sequence Risk Beyond Monte Carlo Financial Planning is a Balancing Act Return Risk Longevity Risk Sequence Risk Beyond Monte Carlo Financial Planning is a Balancing Act Return Risk Longevity Risk Sequence Risk Beyond Monte Carlo Inflation & Return Assumptions Beyond Monte Carlo Case 1: Tom and Mary Both age 60 Assume mortality at age 85; Indexed pensions of $5,675 at age 60, $4,653 at 65 Both have $400,000 in retirement savings They have $5,000 in a joint savings Goal $60,000 after-tax during their lifetime; Assume a 15% average tax rate Beyond Monte Carlo Case 1: Tom and Mary Portfolio is Cash 15% Fixed income 40% Canadian equity 25% US equity 20% Beyond Monte Carlo What Have the Returns Looked Like? Beyond Monte Carlo Case 1: Tom and Mary Based on data from 1950 to 2005 (56 years) Inflation averaged 3.94% (geometric mean) Return on this mix averaged 8.96% (geometric mean) Real Return is (Nominal-Inflation)/(1+inflation) = 4.83% Beyond Monte Carlo Case 1: Tom and Mary Results Tom and Mary’s plan is perfect and they run out of money at the end of age 85, leaving an estate of $0 to their cat. Beyond Monte Carlo Monte Carlo Beyond Monte Carlo What is Monte Carlo? A statistical procedure where we pass an expected return and standard deviation and get random series of returns that “average” to our assumptions. If we passed a standard deviation of 0%, it would be identical to an algorithmic solution. Beyond Monte Carlo Monte Carlo and Returns – 3 Benefits It tests different returns – higher and lower? It tests for bad years up front or the impact of the order of returns? (Sequence risk) Gives an overall chance of success? Beyond Monte Carlo Run a Monte Carlo Enter 8.96% with standard deviation of 8.07% Results indicate 65% failure – timing? Beyond Monte Carlo Run a Monte Carlo Enter 8.96% with standard deviation of 8.07% Results indicate 65% failure – timing? NO • You just entered a different return than your plan (8.55%) Beyond Monte Carlo Geometric vs. Arithmetic Mean Planning uses geometric means Monte Carlo uses arithmetic mean G approx R – V/(2(1+R)) Many advisors attribute increased failures to timing when really different assumptions! Beyond Monte Carlo Test Higher and Lower Returns? For Tom and Mary, 8.96% return was “perfect”. The geometric mean is by definition the point of 50/50 probability. Beyond Monte Carlo Test Higher and Lower Returns? Enter proper 9.26% rate, 50.5% failures Beyond Monte Carlo Test Higher and Lower Returns? What if we increase arithmetic return and standard deviation with maintain the same 8.96% geometric mean? ? Beyond Monte Carlo Test Higher and Lower Returns? What if we increase arithmetic return and standard deviation with maintain the same 8.96% geometric mean? ! Beyond Monte Carlo Test Higher and Lower Returns? Conclusions: MCS results are strictly a function of the geometric mean and not sensitive to the level of risk. If MCS reports a single “probability of success”, it will average out results so no “test” of the range of returns is occurring. Recommendation: The success/failure result must be subtotaled to allow reporting based on higher or lower returns. Beyond Monte Carlo Does MCS Test Sequence Risk? 1. 2. 3. 4. 5. 6. 7. Even withdrawal of $52,159 indexed; $57,423 for the first 5 years (10% increase) then reduce the withdrawal to $49,932 for the duration of the plan. $65,000 for the first 5 years (25% more); $78,000 for the first 5 years (50% more); $104,300 for the first 5 years (100% more); Withdraw an additional lump sum of $100,000 in the 12th year of the plan; Withdraw an additional lump sum of $100,000 in the 20th year of the plan. All scenarios have the same PV for the withdrawals. Beyond Monte Carlo Test Sequence Risk? Beyond Monte Carlo Test Sequence Risk? So all withdrawal strategies with same PV of withdrawals resulted in the same 52% failure rate. Repeated the same 7 withdrawal strategies using 11% return +/- 20.63% (8.96% geometric mean). From 51.8% to 52.1% failures. No change. Why?? Beyond Monte Carlo Test Poor Initial Returns? Same PV of withdrawal and geometric mean results in same failures Beyond Monte Carlo Test Poor Initial Returns? Conclusion: If the PV of the withdrawals is constant, no difference in success/failure. Yes MCS can creates initial poor returns but no more so than great returns. If reports a single “probability of success”, it will average out results so no “test” regardless. Observation: Unless strategies exist to deal with initial poor returns without compromising initial great returns – so what? Beyond Monte Carlo Sidebar on Inflation Given consistent indexation of revenues and goals, planning is totally driven by the real rate of return. Changing inflation assumptions and return assumptions only matter to the extent they change the real rate of return. 10,000 simulations of inflation at 3.99% +/- 3.3% resulted in 49.1 success, assuming savings & withdrawals use the random variable for the indexation rate and constant real return. Beyond Monte Carlo Mortality Assumptions/Longevity Risk Beyond Monte Carlo Mortality Assumptions What mortality assumption would you use in financial plans? Mortality tables (77 for Tom) • Age 85 • Age 90 • Age 99 Beyond Monte Carlo Mortality Tom and Mary are 60 Tom’s life expectancy is age 77 Mary’s life expectancy is age 85 Beyond Monte Carlo What is the Impact of Mortality? Mortality Capital Required 85 $0 90 $69,000 95 $122,500 99 $158,200 Beyond Monte Carlo How do we Randomize? 1. Select a random number between 1 and the number of lives (i.e. 87,813) 2. Map to the tables to determine mortality Beyond Monte Carlo Results of Randomized Mortality When we randomize mortality, ½ will be less than expected mortality and ½ will be more. If we average the results, we have generated a plan based on the average life expectancy. If we sum Success/Failure, any scenario with mortality 85 or less succeeds, 86 or more fails Our results are 31% failure, 69% success – the same as the initial mortality assumption Beyond Monte Carlo Another Planning Risk In respect to life expectancy, aside from longevity risk, need to consider RAK-IT Risk Retire And Kick IT (IT = The Bucket!) When we obsess on Longevity Risk we may be planning for our compliance department, not the client I would say “Forget longevity risk if it means delaying my retirement and getting to my boat!” Beyond Monte Carlo Another Planning Risk Beyond Monte Carlo Longevity Risk Conclusion:By randomizing mortality to represent longevity, advisors are, on average, basing their planning on a lower mortality assumption than would be used in traditional plans. The success/failure is identical to initial mortality assumptions Observation: Results can be subtotaled based on the age parameter, which would allow reporting based on mortality bands. Beyond Monte Carlo Section One Summary Use of arithmetic means is subject to error and misinterpretation The practice of reporting a single result washes out all concept of testing extremes and results in a plan based on the mean. People talk as if randomization tested only a portion of the distribution – poor returns initially, living longer than planned, but it is a statistical process that equally looks at great initial returns or dieing sooner. MCS returns the same results as the geometric mean when consistent assumptions are maintained. Beyond Monte Carlo Section 2: Reliability Forecasts Pre-Decease Age 85 (69%) Outlive Age 85 (31%) Returns Above 8.96 (50%) A 34 ½ % chance the client will die before age 85 and have higher than projected returns (Surplus – Success) A 15 ½ % chance the client will outlive age 85 with higher than projected returns (Unclear outcome) Returns Below 8.96 (50%) A 34 ½ % chance the client will die before age 85 but have lower than projected returns (Unclear outcome) A 15 ½ % chance the client will outlive age 85 with lower than projected returns (Capital shortfall – Failure) MCS provides an interesting “partial probabilities” removing ambiguity around the “unclear outcomes” Beyond Monte Carlo MCS for Returns & Mortality 1. 2. 3. 4. 10,000 simulations 9.26% return +/- 8.07% or 8.96% geometric Randomized mortality 72.9% successful and 27.1 failed sims Is this good? • Dying at age 61 is success because $ remain • Living to 105 with funds exhausted at 103 is a failure Beyond Monte Carlo A Challenge with Precision… Simulated Returns/Age Planning Tool Beyond Monte Carlo A Challenge with Precision… Simulated Returns/Age Planning Tool Win/Loose Count Beyond Monte Carlo A Challenge with Precision… Simulated Returns/Age Planning Tool Win/Loose Count Beyond Monte Carlo A Challenge with Precision… Simulated Returns/Age Planning Tool Win/Loose Count The more comprehensive the planning algorithm, the longer per trial Beyond Monte Carlo A Challenge with Precision… Simulated Returns/Age Planning Tool Win/Loose Count The more comprehensive the planning algorithm, the longer per trial “While running the requisite 100,000 scenarios that would provide a minimal margin of error is computationally not feasible within most real-time engines, running as little as a few hundred scenarios can be woefully inadequate.” Beyond Monte Carlo 100 Sims Distribution Beyond Monte Carlo Goals for an Algorithmic Solution 1. 2. 3. 4. To generate partial probabilities like MCS To remove issues of speed and therefore precision To generate more usable results than a single likelihood of success To make the model easier to maintain Beyond Monte Carlo 2 Critical Observations Simulated Returns/Age Planning Tool • The randomized variables are independent of the plan! (This does not mean the plan is independent of these variables.) • MCS returns the same cumulative result as the geometric mean assumption, so… Beyond Monte Carlo So… Beyond Monte Carlo So… X Beyond Monte Carlo = Likelihood of Age x ROI 0.1% to 0.15% chance of dieing at 93 with a 10.46% to 10.96% return 0.35% to 0.4% chance of dieing at 87 with a 8.46% to 8.96% return Beyond Monte Carlo At each return, when are $ exhausted? Return 4.96 5.46 5.96 6.46 6.96 7.46 7.96 Age Funds Depleted 76.5 77 78 79 80 81 82.5 Remember age 85 and 8.96% return? Return 8.46 8.96 9.46 9.96 10.46 10.96 Age Funds Depleted 84 85 89 93 99 100+ So we run our planning model (in this case 13 times) to calculate the boundary for success and failure Beyond Monte Carlo Our Reliability Forecast Success Failure Beyond Monte Carlo Reliability Forecast Results 1. 2. Subtotal of cells in the “Success” region is 74.5% This compares to 72.9% +/- 1.57% with MCS and 10,000 simulations Beyond Monte Carlo Reliability Forecast Results (2) 1. Tested with withdrawal of $78,000 for 5 years and reduced $ thereafter. No change in RF solution (74.5%), 73.7% success with MCS. 2. Tested with $100,000 withdrawal in 12th year of plan. Same for RF (74.5%), 73.5% success for MCS. Beyond Monte Carlo Reliability Forecast Results (3) 1. 2. Tested accelerated withdrawal of $58,633 and funds gone at age 80. RF with new boundary gave 59.5% success. MCS resulted in 60.3% success. Beyond Monte Carlo Reliability Forecast Results (4) 1. Tested withdrawal of $65,000, 10% return and 11% standard deviation (9.5% geometric mean). 2. RF with new boundary gave 51.6% success. MCS resulted in 51.2% success with 10,000 sims. The Reliability Forecast provides the same partial probabilities as MCS with a dozen iterations of the planning model. Beyond Monte Carlo Subtotaling Results What if the client lives to age 90? Beyond Monte Carlo Subtotaling Results What if the client averages at least 6.96% return? Beyond Monte Carlo Improved Distribution Matrix Beyond Monte Carlo Sensitivity Analysis Sensitivity Analysis Age Probability Success 70 80.7% 100.0% 75 65.7% 97.8% 80 47.1% 83.4% 85 27.4% 68.3% 90 11.5% 58.9% 95 2.6% 54.6% Return Probability Success 11.5 22.4% 89.2% 10.5 37.2% 77.8% 9.5 54.4% 68.7% 8.5 70.9% 61.9% 7.5 84.0% 57.9% Beyond Monte Carlo Failure 0.0% 2.2% 16.6% 31.7% 41.1% 45.4% Failure 10.8% 22.2% 31.3% 38.1% 42.1% Sensitivity Analysis Probability of Success Age/Return 7.5 70 100.0% 75 100.0% 80 88.6% 85 73.6% 90 63.8% 95 59.2% 8.5 100.0% 100.0% 92.3% 78.0% 68.0% 63.2% 9.5 100.0% 100.0% 97.6% 85.2% 75.1% 70.1% Beyond Monte Carlo 10.5 100.0% 100.0% 100.0% 93.4% 84.2% 79.3% 11.5 100.0% 100.0% 100.0% 100.0% 94.7% 90.6% Sensitivity Analysis Beyond Monte Carlo Sensitivity Analysis Bad! Beyond Monte Carlo Sensitivity Analysis Annual Reviews Beyond Monte Carlo Summary of Findings Issue Return assumptions Monte Carlo Reliability Forecast Based on arithmetic mean. Estimation technique can convert. Advisors do not understand Not as utilized. Requires additional computation and subtotaling of results in return bands. Misunderstood. Misunderstanding. Does not occur. Based on geometric mean, same as financial plan Test longevity using randomized mortality Not as utilized. Could add additional layer of subtotaling. Easily subtotaled. Precision Poor due to computational load. Require 100,000 simulations. High. 12 calculations are necessary to generate the boundary line. Ability to generate overall probabilities of success Yes Yes Ability to provide improved granularity on sensitivity analysis Requires additional computation and subtotaling logic. Yes, easily subtotaled. Ability to maintain the planning tool Complex .May restrict certain capabilities. Simpler. The algorithmic solution is easier to maintain and audit. Test higher and lower returns Test impact of withdrawals and order of returns (sequence risk) Beyond Monte Carlo Easily subtotaled. Not calculated. Conclusion The Reliability Forecast approach outlined provides all the analytical results available using MCS in a fraction of the calculations, using more common assumptions, leading to higher precision. Beyond Monte Carlo Questions ? Beyond Monte Carlo