- Wealth looks like it matters in determining the spend rate optima in the analysis but as far as I can tell only indirectly in the sense that lower levels of wealth support lower absolute dollar spend rates and those spend rates present less of a "cliff" over available income when wealth is depleted. There is a lower height to fall from when consumption depletes wealth and then drops to available income. That means higher spend rates in a low wealth situation are not penalized as severely (in the utility math) as they would be for larger wealth and bigger falls.
- Optimal spend rates tend to go up with age, though this is already known. But in this model, it appears as if the impact of the previous point is accelerated or accentuated at later ages as longevity probabilities come in a bit.
- Risk aversion, in this model, has a significant impact on optimal spend rates as well as the rate at which spend-rates change for changes in wealth and age in the presence of pensionized income like social security or annuities. The means and methods of measuring risk aversion in real life are beyond me. Contemplating the idea of the stability of risk aversion over time makes me woozy.
The goal of this post is to use a lifecycle utility model [1] -- one that anticipates a "wealth depletion time" i.e., a time interval in the late lifecycle when non-pensionized wealth runs out and consumption snaps to available income -- to evaluate "optimal spend rates" by optimizing a value function (expected discounted utility of lifetime consumption) that is calculated across different spend rates, wealth levels, a range of ages, and different coefficients of risk aversion.
This exploration is neither sufficiently rigorous nor exhaustive enough to make hard conclusions or recommendations about spend rates. There are too many dials to turn...and one has to accept the model as meaningful in the first place. The exploration, rather, is intended to help me see the general shape of lifetime consumption utility in terms of spend rates and in graphic form. It also helps me shake out the software one more time.
Just So You Know
- Longevity is treated as random and conforms to the SOA IAM table with extension to 2018.
- The intervals of interest (age, wealth) are narrow due to the large number of iterations required but probably conform to most early stage retirement analysis. For taking an informal look at the shape of the output this is enough for now.
- Since the model is highly sensitive to input parameters, and since the output also depends heavily on a coefficient of risk aversion, which is subjective, this post is not a recommendation on spend rates. It is just me playing with the model as it's designed to see what it tells us and to see what it looks like for one set of parameters.
The model is obviously not "real life." It represents no more and no less than the shape of a few lines of code I built. That means that the model is like a bottle that holds water. The shape of the water depends on the shape of the bottle. The model in this post is merely a bottle of arbitrary design; there are other bottles. The question then is whether the model has anything useful to say about reality and maybe also whether you think I'm any good at modeling and/or whether I have gone far enough. TBD.
- Annuities, though part of the model, are not considered here (yet) but would have a significant impact. Social security is set to a low value and then discounted even more. This is not particularly realistic but helps me think through some issues in modeling.
- For modeling convenience and efficiency and to support the more generalized goals of my exploration -- as opposed to me trying to give specific, realistic retirement advice -- I have assumed a constant inflation adjusted spend rate. This is not always received as a happy assumption; I attempt to defend my choice here (link). Also note that the model is still naive on things like taxes and fees not to mention autocorrelation or return regimes.
- In the output, nothing is chained. The results at age 65 do not depend on what happened at 60. Each age-wealth-spend-risk "set" is run independently as if it is the first and only time it has ever been run. The way I was thinking about it was that if next year my wealth halves or doubles and if I were inclined to recalculate EDULC, this might be a map of what to expect (if my assumptions were to be similar).
The Model
The "wealth depletion time" model is explained in detail here. The schematic looks like this:
The Model
The "wealth depletion time" model is explained in detail here. The schematic looks like this:
tPx - Conditional survival probability at time t for a person age x.
k - a subjective discount on utiles
c(t) - consumption at time t; c is the consumption plan
S - number of iterations
g[c(t)] - CRRA utility function of form (c(t)^(1-gamma)-1)/(1-gamma) for gamma <> 1 else ln(c(t))
Assumptions and Parameters
Age: 60 - 75 in 5 year steps
Endowment: $.4M to 2M in steps of .2M
Spend Rates: 2.75% to 12% in .25% steps
Risk aversion: Coefficients of 1 and 3 (gamma) in a CRRA (and log) utility function
Asset Allocation: 50/50
Assumptions and Parameters
Age: 60 - 75 in 5 year steps
Endowment: $.4M to 2M in steps of .2M
Spend Rates: 2.75% to 12% in .25% steps
Risk aversion: Coefficients of 1 and 3 (gamma) in a CRRA (and log) utility function
Asset Allocation: 50/50
Asset class returns and correlation*:
- Risk-high = 8% arithmetic return and 18% standard deviation
- Risk-low = 3.5% arithmetic return and 4% standard deviation
- correlation -.10 (in our real world this is highly variable)
- Risk-high = 8% arithmetic return and 18% standard deviation
- Risk-low = 3.5% arithmetic return and 4% standard deviation
- correlation -.10 (in our real world this is highly variable)
Utility is calculated on real spending which snaps to available income at wealth depletion
Inflation, where it matters, is 3%
Sim Iterations: 10,000
Subjective discount on utiles: .005
Longevity: from SOA IAM table with longevity extension to 2018; for age x = 60, done two ways:
a) T* is a random variable in each sim iteration and the lifetime is simulated for T*- x years
b) a vector of conditional survival properties is calculated for age x over age -> infinity (121)
Convention b is chosen here for reasons described in the model.
Social Security: 11,000 annually starting at 70 and inflated thereafter. This is low vs reality, but:
(a) it creates a stub of minimum income for the WDT module, and/or
(b) it represents a heavy risk-discounting of SS income for subjective reasons of my own
The Process
1. For age 60, a wealth level of .4M, and a risk aversion coefficient of 3 the value function is estimated via simulation for each spend rate between .0275 and .1200. The spend rate with the maximum lifetime consumption utility in that series is selected and stored.
2. Step 1 is repeated for the next wealth level (.6M) and then repeated again at each wealth level up to 2M in steps of 200k.
3. Steps 1 and 2 are repeated for ages 65, 70 and 75.
4. Steps 1-3 are repeated for a risk aversion coefficient of 1.
This process can perhaps be visualized by the figure presented in this post if one were to select the spend rates at the critical points of the curves. I used common sense to not run all of the spend rates at each step, otherwise the total iteration count would have been something like 27 million if I got the math right. I mean I have kids and cooking and laundry to do when I am not over-simulating stuff...
The Output
For given assumptions and a risk aversion coefficient (gamma) = 3, spend rate optima look like:
For given assumptions and a risk aversion coefficient (gamma) = 1, spend rate optima look like:
Conclusion
The main conclusions today, which are preliminary, based on a very limited exploration of just a few parameters, and which will be revisited over time are these:
-----------------------------------
[1] Saying "lifecycle" is probably not completely fair since this only looks at the decumulation stage but that seems to be the convention for this type of analysis. Send me a better name.
Inflation, where it matters, is 3%
Sim Iterations: 10,000
Subjective discount on utiles: .005
Longevity: from SOA IAM table with longevity extension to 2018; for age x = 60, done two ways:
a) T* is a random variable in each sim iteration and the lifetime is simulated for T*- x years
b) a vector of conditional survival properties is calculated for age x over age -> infinity (121)
Convention b is chosen here for reasons described in the model.
Social Security: 11,000 annually starting at 70 and inflated thereafter. This is low vs reality, but:
(a) it creates a stub of minimum income for the WDT module, and/or
(b) it represents a heavy risk-discounting of SS income for subjective reasons of my own
The Process
1. For age 60, a wealth level of .4M, and a risk aversion coefficient of 3 the value function is estimated via simulation for each spend rate between .0275 and .1200. The spend rate with the maximum lifetime consumption utility in that series is selected and stored.
2. Step 1 is repeated for the next wealth level (.6M) and then repeated again at each wealth level up to 2M in steps of 200k.
3. Steps 1 and 2 are repeated for ages 65, 70 and 75.
4. Steps 1-3 are repeated for a risk aversion coefficient of 1.
This process can perhaps be visualized by the figure presented in this post if one were to select the spend rates at the critical points of the curves. I used common sense to not run all of the spend rates at each step, otherwise the total iteration count would have been something like 27 million if I got the math right. I mean I have kids and cooking and laundry to do when I am not over-simulating stuff...
The Output
For given assumptions and a risk aversion coefficient (gamma) = 3, spend rate optima look like:
For given assumptions and a risk aversion coefficient (gamma) = 1, spend rate optima look like:
Conclusion
The main conclusions today, which are preliminary, based on a very limited exploration of just a few parameters, and which will be revisited over time are these:
- Wealth looks like it matters in determining the spend rate optima in the analysis but as far as I can tell only indirectly in the sense that lower levels of wealth support lower absolute dollar spend rates and those spend rates present less of a "cliff" over available income when wealth is depleted. There is a lower height to fall from when consumption depletes wealth and then drops to available income. That means higher spend rates in a low wealth situation are not penalized as severely (in the utility math) as they would be for larger wealth and bigger falls.
- Optimal spend rates tend to go up with age, though this is already known. But in this model, it appears as if the impact of the previous point is accelerated or accentuated at later ages as longevity probabilities come in a bit.
- Risk aversion, in this model, has a significant impact on optimal spend rates as well as the rate at which spend-rates change for changes in wealth and age in the presence of pensionized income like social security or annuities. The means and methods of measuring risk aversion in real life are beyond me. Contemplating the idea of the stability of risk aversion over time makes me woozy.
-----------------------------------
[1] Saying "lifecycle" is probably not completely fair since this only looks at the decumulation stage but that seems to be the convention for this type of analysis. Send me a better name.
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