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These items below are great, (sometimes) simple little formulas for retirement planning -- all of which are not, you'll notice pretty quickly, the 4% rule. Retirement planning can often be complex if not a completely unanswerable problem domain or as Wade Pfau once opined: "A truly safe withdrawal rate is unknown and unknowable". On the other hand, by using some simple, deterministic formulas to wrap your head around what might or might not work before you get into the deep weeds with a planner and his or her complex, proprietary models I think you can, if not necessarily DIY, at least save a little time and money and make the conversations-to-come more efficient. Moshe Milevsky, in a great article on de-emphasizing complex simulator-based retirement ruin calculations (It’s Time to Retire Ruin (Probabilities) ), points out the pros and big cons of stochastic modeling (e.g., simulators). He tells us, helpfully, that "the highly technical and subtle stochastic 2.0 lecture makes no sense until the deterministic 1.0 lecture is crystal clear." This just means walk with simple formulas before you run with a heavy duty model. So this post is part of the 1.0-walk "lecture." In my mind, some of the benefits of starting with the deterministic 1.0 level include:
These items below are great, (sometimes) simple little formulas for retirement planning -- all of which are not, you'll notice pretty quickly, the 4% rule. Retirement planning can often be complex if not a completely unanswerable problem domain or as Wade Pfau once opined: "A truly safe withdrawal rate is unknown and unknowable". On the other hand, by using some simple, deterministic formulas to wrap your head around what might or might not work before you get into the deep weeds with a planner and his or her complex, proprietary models I think you can, if not necessarily DIY, at least save a little time and money and make the conversations-to-come more efficient. Moshe Milevsky, in a great article on de-emphasizing complex simulator-based retirement ruin calculations (It’s Time to Retire Ruin (Probabilities) ), points out the pros and big cons of stochastic modeling (e.g., simulators). He tells us, helpfully, that "the highly technical and subtle stochastic 2.0 lecture makes no sense until the deterministic 1.0 lecture is crystal clear." This just means walk with simple formulas before you run with a heavy duty model. So this post is part of the 1.0-walk "lecture." In my mind, some of the benefits of starting with the deterministic 1.0 level include:
- It's free and not all that hard
- It provides a degree or two of (current and future) freedom from advisors and planners and throws a healthy dose of control into your own hands
- It provides, if you are willing, a good self-directed tutorial on some of the most gnarly core assumptions you need to make, especially when you really spend some time thinking about them. For example, think about the "expected real rate of return" for a minute; there is a lot that goes into that number and it is always a good conversation to have, especially in 2016
- With simple formulas one can also, for a brief blissful moment, steer widely clear of the large number of buried, invisible assumptions and biases of modelers and programmers and advisors that are implicit in advisor provided software (or even the free ones on the internet for that matter)
- Used in concert, multiple formulas and techniques provide informative context and multiple points of view in the planning process. It can set the stage for more complicated planning and discussions later. Note that many of these formulas will converge on roughly similar answers.
- Since planning, in its heart of hearts, is or should be a continuous process, you now have an entry level tool-set at your disposal to that will allow you to adapt to changing circumstances at your leisure. My mental model here is skiing: try to ski without being aware of changing terrain and without adjusting your speed, direction, and where you are looking based on everything that is going on around you…and you will get hurt. Stay aware, in other words.
The Formulas...
This is the formula that Moshe Milevsky -- using a formula that originated from Leonardo Fibonacci in the 1100s and that was created to figure out how long money would last given spend and growth assumptions -- suggests using to frame a discussion between advisor & client. His reasoning is that it can start a more constructive conversation around retirement than a simulator can do...that is, before someone understands the pros and cons and subtleties and assumptions of simulators. It also focuses the conversation on a small number of important variables like the growth rate.
Output is the number years the portfolio will last given the inputs. Then compare that result to what you might expect for your average, long, and then longest terminal retirement ages in your plan. Note that the formula doesn't work that well if you happen to spend less than you earn. I'm thinking that is closer to a perpetuity in which case you are in pretty good shape and probably don't need a formula (or blog post) like this.
Here is how Michael Zwecher does it in Retirement Portfolios: Theory, Construction, and Management 2010 (notes to chapter 3, page 252). Similar math and result. I usually ask my daughter (Stanford) to help me reconcile formulas like these two but last time I asked she chided me on my lack of what she calls middle school math so I haven't asked yet. I didn't promote this to a new rule so it could stay here with Milevsky's.
ln(x) natural logarithm of x
r discount rate
i(e) expected inflation
lifestyle spend rate
wealth portfolio or endowment
Excel is Excel of course but the PMT function, an ever useful thing, was recently mentioned in an article by M. Barton Waring and Laurence B. Siegel (The Only Spending Rule Article You’ll Ever Need) who used it to create a retirement spending rule they called ARVA (Annually Recalculated Virtual Annuity). Enter the simple assumptions into the formula and then repeat that process each year. If you can spend what it says, you'll likely never run out of money. Like the Milevsky formula this has the benefit of incredible simplicity (ignore the spending volatility it implies for now). It also tends to focus the discussion on the growth rate assumption. It is also free. Depending on the person you are talking to, it is sometimes called the mortgage formula or an annuity formula. It also happens to be more or less the same math as the previous formula but turned inside out; it's solving for P rather than n.
Where:
Ifa = annual future expected income
ef = a life expectancy scaling factor to account for stochastic life span (1.5 was found to be a good value)
nu = 0.00942 for the model when gamma is 4 and no subjective time discounting is being performed
A better visualization might be this...
When in doubt about what I've written here, go back to earlyretirementnow for the source...
Postscript. For me, I use RH40 like this especially since it is rather conservative: when I have no better tools at hand, I can wing a withdrawal rate by using RH40 thus:
where c is a constant for "do I feel lucky today." It feels like c could be something like .005 (half percent) or .01. But at that point, if I am getting into serious analysis, I am not using RH40, I am using a simulator or a differential equation or something.
Formula 15 - Annuity calc
The left side represents either a portfolio value (end of year after returns and spending) or a similar net wealth "process" when it is animated through simulation by randomizing mu and where the 20th percentile of the resulting distribution in some year is the value of interest. The right side is either the value of an annuity with a load factor times a buffer zone percentage or it is the same thing projected out over the same number of years as one would simulate the left side. mu is a return variable that is randomized, w is wealth units in terms of (1/spend). iPx is a cash flow weighted by a conditional survival probability, R is an annuity discounting rate. l is the annuity load and z is buffer zone percent, say 10%. Someday I'll get someone to tell me how to do the notation the right way.
Formula 17 - Custom Spend Liability Calculations
A. Deterministic Version
where CSL is a custom spend liability or estimate; c is the spend path and ct is the spend at time t; tPx is a conditional survival probability for someone age x at time t which is extracted from a lifetime process that can be described by ln[p] = (1-e^t/b)e^((x-m)/b) where x is age, t is time in years, m is the mode, and b is a dispersion factor (see Milevsky's 7 Equations book, Ch2) or alternatively this can be pulled from an actuarial life table (e.g., SOA or SS tables); and d is the discount rate (with some superfluous hyperbolic discounting parameters in there to capture some time subjectivity should one want to do that kind of thing). If you ditch the hyperbolic terms and change ct to 1 this is basically an annuity formula. Also, the infinity term could be changed to 120-minus-age without much effect because the conditional survival goes towards zero around that point.
The purpose of the difference between this and #15 is to allow one to design a custom spending expectation "c" and to also design a custom discount (using the alpha and Beta in front of "r") if one were to have some odd, asymmetrical time preference. I can't really imagine the latter in any normal person's daily grind but there it is anyway.
B. Stochastic PV Version
This is the animated version with simulation where the result is a stochastic present value (spv) spending distribution.
There are other ways to do this sim, by the way. This is just the way I recently did it and the equation was my amateur attempt at describing it. Note that I had an earlier version of the simulation below where the discount was neither stochastic nor chained in a series which means it might be a closer (discrete non-utility) proxy to Yaari's eq13 [1965].
Formula 18. Bond Total Returns
bonds(i)=100*(yields(i)./yields(i+1) + (1-yields(i)./yields(i+1))./((1+yields(i+1)/100).^10) - 1 + yields(i)/100);
I've calculated bond total returns in several models I've done based on a variation of the formula above. I confirmed this against feedback from Aswath Damodaran at the Stern Shool, Wade Pfau, and the developer behind cFireSim.com. The formula was adapted depending on whether I was using 10 year or 5 year data. Data was typically from St Louis Fed data series.
Formula 19. Wealth Depletion Time Simulation math
This is more or less an adaptation of stochastic consumption formulas in #17 to now include utility as well as to then animate it with simulation. This formula cannot exist independently from the simulation code that generates things like random returns and inflation. The code also does things like "snap" spending to (a) available income when wealth depletes, or (b) to available income if spending, in spend rule mode, drops below available income. The math framework for the sim proper is this:
where
E[V(c)] is the expected discounted utility of lifetime consumption
c - is a consumption plan, c(t) is consumption in period t
S - is the number of iterations
k - is a subjective discount on utiles
g(c) - is the CRRA utility function but using a constant (-1) in the numerator
(income - in the sim; that and the snap can't be seen in the above formula)
I will add more formulas here as I run into them or as they are sent to me…
[1] It took me a bit to figure out what Ken meant by what I am calling "X" so this is what I came up with that matches his math...I think...for a thirty year planning horizon where infl is .014 and discount is .04 and a planning horizon is 30 years. Those are his assumptions not mine. My assumptions are more conservative and hence my spending budget would b e quite a bit lower [postscript: Ken has a spreadsheet on his site with how to exactly do this. I should have gone there first] :
Formula 1: Portfolio Longevity Using Milevsky's Fibonacci Formula
EL estimated portfolio longevity
M nest egg (money)
w withdrawal rate in dollars per year
g annual real growth rate of portfolio as a %. It should also be
net of fees, taxes, and maybe market risk. Much of the constructive
discussion, in an advisory conversation, will revolve around g
discussion, in an advisory conversation, will revolve around g
Here is how Michael Zwecher does it in Retirement Portfolios: Theory, Construction, and Management 2010 (notes to chapter 3, page 252). Similar math and result. I usually ask my daughter (Stanford) to help me reconcile formulas like these two but last time I asked she chided me on my lack of what she calls middle school math so I haven't asked yet. I didn't promote this to a new rule so it could stay here with Milevsky's.
 |
| Formula "1a" -- Annuity number of periods |
r discount rate
i(e) expected inflation
lifestyle spend rate
wealth portfolio or endowment
Formula 2: Spend Rate Using the Excel PMT Function
PMT(r,n,Pv) in Excel
or
P = ( Pv * r ) / [ 1 - (1 + r)-n ]
P the "payment" or spend rate estimate
r annual real growth rate; g in the previous formula
n number of retirement years expected (I use 95 minus my age but 30 is often used)
Pv money or nest egg or endowment
Output is the estimated spend rate for this year. W&S recommend doing all again next year because, well because everything changes. Technically things change continuously but a continuous recalc might be a little OCD. Playing around with it doesn't hurt anyone, though.
Blanchett's Two Simple Spending Formulas
Based on a whole pile of research and a bunch of Monte Carlo simulations, David Blanchett derived a rule of thumb formula (or formulas) that allows one to get results pretty close to what a simulator might generate for successful retirement spending rates…that is if one does not happen to have a simulator [article here]. It seems to work pretty well from my own look-see. He found that one formula does not work for all ages, though. The first formula (Dynamic) apparently works better for periods that are 15 years or more and the second (RMD Approach) is better for periods 15 years or less.
Formula 3: Dynamic Formula for Retirements >= 15 Years
w withdrawal rate
Years distribution period
PoS target probability of success
Alpha fees as a negative percent like -00.50%
Equity% equity allocation
Output is a proxy for a simulator-generated suggested retirement withdrawal rate for the given input variables and a target success rate. Designed more for advisors than retail investors, this is not as hard as it looks for self-managing retirees.
Formula 4: RMD Approach for Retirements <= 15 years
Blanchett says that the equation is based on the IRS’ required minimum distribution (RMD) and works better for short retirements (with respect to the underlying simulator results) than the previous formula. See the article for his rationale
w% [<=15 years] = 1 / Years
w withdrawal rate
Years distribution period
Output is withdrawal rate percent for a short retirement.
Formula 5: Divide by 20 Rule of Thumb
I've already written more than I probably need to on this slick little rule (here). This rule was created by Evan Inglis and is a rule of thumb designed to calculate a simple "safe" retirement spend rate on an age-adjusted basis. The purpose of the rule is, I gather, to be: a) simple, b) conservative and relatively safe but not foolproof, c) age adjusted, d) get spending pretty close to a reasonable level of expected real returns, and e) more than likely leave a little bequest if you last long enough. Seems to work pretty well as far as I can see.
w = Age / 20
w withdrawal rate percentage for a given year/age
Age age
20 this rule of thumb number is a range boundary. That means that below 20 is more than likely pretty safe; above 10 is pretty dangerous. After about age 70 "20" might get a little conservative and will either tend to leave a legacy or perhaps it can be adjusted to something else like divide by 18 or 19 or 17…
Output of this formula is a withdrawal rate percent for spending in a given year for a given age.
Formula 6: Ken Steiner's Actuarial Equation.
Ken Steiner at howmuchcanIaffordtospendinretirement.com is one of my favorite commentators. He, along with a very short list of retirement finance writers, is able to combine correct math with common sense and good communication to get their message out. The article that got me to add this equation is here but his site (a place I've been perusing for at least three years) is a good repository of actuarial correct retirement math that is probably worth a look. Not least of Ken's virtues is his insistence on an actuarial balance sheet for retirees, an idea to which I adhere personally. With apologies to Ken, his basic equation -- which presumes some prior steps to get to the ability to complete this -- is this. The equation determines the "actuarial correct" amount for this year's spending budget. Go to his site for more detail on why he likes this equation especially as it relates to making sure spending is tuned to not only current assets but also expectations about future sources if income:
TYSB = C/X +Y/X
TYSB This year's Spending Budget
C Current endowment - PV of future non-recurring expenses
X divisor = PV of $1 for (nbr of planning years) growing at i [1]
Y PV of income from other sources including SS
i inflation or growth factor
For example, if the endowment is $1M, inflation is 1.5%, PV of future non-recurring is 200k, income from other sources is 2k at age 70, and the discount rate is 4% and the planning horizon is 30 years (if age 58 that's 88 or 87 if you include this year): the spend budget would be $42,251 or a 4.23% spend rate. If one were to have a planning horizon of 95 that would be a spend budget of $37,699 or a 3.77% spend rate. As you can see this is not out of bounds with respect to the 4% rule for this set of assumuptions and it varies with longevity expectation. I think it might be a little optimistic but it is what it is and you can always set your longevity assumption farther out to be more conservative. This is a useful addition to the set of simple equations especially if one were inclined to triangulate amongst many tools to make a self-judgement on retirement risk.
Gordon Irlam's Simple Rules of Thumb for investing and consumption.
Formula 7: A mostly simplified version of the annual portfolio consumption equation
Source is this article here which I would recommend perusing before applying the formulas. The first form of his annual retirement portfolio consumption looks like this:
annual retirement portfolio consumption =
Formula 7: A mostly simplified version of the annual portfolio consumption equation
Source is this article here which I would recommend perusing before applying the formulas. The first form of his annual retirement portfolio consumption looks like this:
annual retirement portfolio consumption =
nu * (W + Ifa * e * ef) / (1 - exp(- nu * e * ef)) - Ifa
Where:
Ifa = annual future expected income
ef = a life expectancy scaling factor to account for stochastic life span (1.5 was found to be a good value)
nu = 0.00942 for the model when gamma is 4 and no subjective time discounting is being performed
e = remaining life expectancy
W = portfolio size
Irlam: "Merton provides a formula for optimal consumption but it depends upon life expectancy being fixed. I attempted to modify it to handle a stochastic lifespan [to get the above]. I am being lazy in not discounting estimated future income. I can get away with this because the discount rate is small. If nu is small enough, and life expectancy is fixed, the optimal consumption is W / e. In reality we should consume more because the portfolio experiences growth over time (ignoring withdrawals), but we should consume less to set aside funds for the possibility that we live longer than our life expectancy."
Formula 8: Irlam's super-simple version of portfolio consumption.
annual retirement portfolio consumption =
W / e
Where:
W= portfolio size
e = remaining life expectancy
Looks like the RMD math to me -- and in fact saying this is the 8th equation is a little bit of a cheat because this is the same exact math as #4 above -- but let's let Gordon riff on this. Irlam on W/e : "...the simple rule, W / e, seems to work quite well. A more sophisticated approach of using the complex formula was tried, but while it performed 3% better for large portfolios it performed 4% worse for small portfolios (where there was a shorting constraint). Lacking a consistent advantage for the complex formula, I elected to use the simple formula, W / e. (Using VPW is another possibility. If you get the parameters right, it performed 1% better than the simple formula for affluent and constrained investors alike. W / max(e, 8) was also found to be a contender: 3% better for the affluent, 1% better for the constrained.) These two rules are very simple, but they have strong theoretical underpinnings, and they work well empirically. Like all rules of thumb there are limits. These limits are likely to be hit when gamma is not a constant but drops as consumption becomes satiated, if real interest rates ever rise significantly and we are no longer able to ignore the need for time discounting of future income, or there is interest in leaving a bequest. In these cases it may be better to perform stochastic dynamic programming (SDP) to compute the optimal asset allocation and consumption."
Formula 9: Milevsky and Robinson's Sustainable Spending Rate without Simulation.
While the underlying formula in the link behind in the last sentence will look hairy if you click through, the Excel implementation is a little easier to navigate. Here are two forms of the equation in Excel format, one for risk of ruin and the other for spending given a portfolio success probability:
Prob(stochastic spending > W) = gammadist(x, (2r+4l)/(v^2+l)-1, (v^2 + l)/2, TRUE)
Sustainable Spend Rate = gammainv(P, (2r+4l)/(v^2+l)-1, (v^2 + l)/2)
x = spend rate = spend / nest egg
r = expected mean return
v = volatility
l = mortality rate = ln(2)/(median age at death - curr age)
P = Probability of success equivalent to output of first equation
I recommend reading the underlying article. Also there is a version of this discussion in a related essay by Milevsky in "Retirement Income Redesigned" by Evensky and Katz. The intent was to create a formula that can be done without simulation so that the variables are more transparent and some education can be done on the trade-offs of retirement risk and return.
Formula 10: Estrada's Maximum Withdrawal Rate
This is from a paper by Javier Estrada at IESE. In "Maximum Withdrawal Rates:An Empirical and Global Perspective" 2017 he asks
himself the question "Given a desired bequest [say zero, he says later],
what are the maximum inflation-adjusted withdrawals an individual could make
during his retirement?" to which he replies "Analytically, this problem has a closed-form solution,
referred to here as the maximum withdrawal rate.
This variable, which is at the center of this article, has at least two critical applications: It provides a way to
assess how likely is a retiree to sustain a target level of inflation-adjusted
withdrawals during his retirement, and it provides a comprehensive way to evaluate retirement strategies. The focus
here is largely, though not exclusively, on the first application." I wrote about it here and it looks like this
in his article:
I've never been particularly good at notation (or reading notation for that matter) but I
think this can be represented in more concise form like the following (any math whizzes
can correct me on this and full acknowledgement will be given):
MWR =
MWR - Max withdrawal rate
T - end of retirement or (longevity expectation - current
age)
R - real return of portfolio during a given period*
PT - The end portfolio or bequest is zero and by
derivation falls out of equation entirely
W - withdrawal is constant and by derivation drops out
* In the paper R is only ex-post. I'm cheating here and looking forward using constant rate. Estrada says it can be swagged from history or simulated
Since I am not sure I got that notation above right, here is another version from earlyretirementnow.com. His technical appendix 8 is the source for the code I built and is probably worth the read if you are interested:
A better visualization might be this...
When in doubt about what I've written here, go back to earlyretirementnow for the source...
Formula 11: Zwecher's Percent of Assets Committed to a Floor
In Michael Zwecher's highly recommended book (Retirement
Portfolios: Theory, Construction and Management (2010)) he provides a nifty
formula (p. 82) for determining, in the absence of a precise "pinpoint"
bottom-up allocation number based on current prices, the "centering point"
estimate of the percent of assets allocated to a retirement income floor.
To quote: "The general rule for finding flooring
allocations by age, length of window, and expected inflation can be found by
valuing an annuity that begins in M periods, and lasts for N periods. Assuming
a constant with a payout ratio of (L/W)%, constant rates r, and constant
expected inflation ie , the allocation to flooring can be found by using the following
formula:"
where
A% = flooring allocation
L = lifestyle
W = wealth
r = constant rate
i = expected inflation
N = number of years payments are meant to cover
M = lag before payments start
He goes on to say: "One useful adjunct of the previous
formula is that it can be used to help provide a lifestyle feasibility test."
It's described in more detail on p83 but to summarize: if, from above, A=(L/W)K
where K is everything to the right of L/W and K is a present value factor, then
(to paraphrase) W < KL/A is bad or infeasible for planning a floor, W>KL/A
is good, and "if the entire portfolio is to be used for a floor" then
W = KL.
Formula 12. RiversHedge Age-based Retirement Spending Rule of Thumb.
This is a home brew formula I came up with by riding the coat tails of Evan Inglis' "divide by 20" rule. It combines the simplicity of a rule of thumb with an age-based formula that trounces the 4% rule with the luck that it happens to match the results and "curve" of more sophisticated simulation and regression based approaches.
w - withdrawal rate
A - age
I profiled this in a series of posts in May 2017+. I think it stands up to scrutiny:
w = RH40 + c
where c is a constant for "do I feel lucky today." It feels like c could be something like .005 (half percent) or .01. But at that point, if I am getting into serious analysis, I am not using RH40, I am using a simulator or a differential equation or something.
Formula 13 - Milevsky's "All or or Nothing" optimal age for annuitization (Cotton)
No explanation is provided since it is covered better in this link by Dirk Cotton than I could do even in summary. But let's briefly quote: "The authors develop models for two annuity markets. The first, referred to as “All or Nothing”, calculates an optimal age for purchasing a life annuity once in retirement. This single-purchase limitation might be the result of a retiree preferring to make a single annuity purchase, a pension plan that limits the participant to a single purchase, or a country's annuity market." The underlying formulas use Gompertz math for "force of mortality." The original Milevsky article, which I have seen before (I might have covered this before) is here: Annuitization and Asset Allocation, Moshe Milevsky and Virginia Young, 2007. In brief:
Formula 13 - This is the Kolmogorov Ruin Equation from Milevsky's book.
This is not directly usable in this form but I do have some solver software, one in VBA/Excel and another that I re-wrote in R. The formula, a partial differential equation, produces output that is scarily a lot like my simulator. I wrote up the first draft of what I know here.
Formula 14 - FRET (flexible ruin estimation tool)
This estimates the lifetime risk of ruin. I instantiated this in a tool I created that produces the same results as formula 13 above but it is easier, more tractable and more transparent than the PDE. The explanation for the terms is in a post here. This is not directly usable in this form. The R-script necessary for this is pretty easy and runs to about 1 page of code ex-comments.
Also, I just (11/15/17) ran into the following in a 2001 German paper on self-annuitization by Albrecht and Maurer. It is the same (swap in a sum for the integral; same idea) as above but it is probably more technically correct for my purposes because I am doing discrete math in the software rather than continuous.
Also, I just (11/15/17) ran into the following in a 2001 German paper on self-annuitization by Albrecht and Maurer. It is the same (swap in a sum for the integral; same idea) as above but it is probably more technically correct for my purposes because I am doing discrete math in the software rather than continuous.
This is an adjusted version of the Edmond Halley formula in chapter 3 of Milevsky's "7 Equations" book. I just added a factor for "load" since I know the insurance companies are expensive. I've heard various ranges as an estimation aid like 10-15% or 13-18%. No idea what it should be. This formula is not usable in this form. While the denominator is the discounting factor, the numerator is a cash flow weighted by conditional survival probability. For that you'd have to have another formula like this below which is also from his book, chapter two. x is age, t is number of years out, M is mode of the longevity expectation with b as the dispersion.
Formula 16 - Free Boundary Process Approximation
This was not really "collected." I made this up and the notation is not kosher because I am not a mathematician and I don't really know how to do formal notation. On the other hand it represents what I am trying to do if one can make sense of it
Formula 17 - Custom Spend Liability Calculations
A. Deterministic Version
where CSL is a custom spend liability or estimate; c is the spend path and ct is the spend at time t; tPx is a conditional survival probability for someone age x at time t which is extracted from a lifetime process that can be described by ln[p] = (1-e^t/b)e^((x-m)/b) where x is age, t is time in years, m is the mode, and b is a dispersion factor (see Milevsky's 7 Equations book, Ch2) or alternatively this can be pulled from an actuarial life table (e.g., SOA or SS tables); and d is the discount rate (with some superfluous hyperbolic discounting parameters in there to capture some time subjectivity should one want to do that kind of thing). If you ditch the hyperbolic terms and change ct to 1 this is basically an annuity formula. Also, the infinity term could be changed to 120-minus-age without much effect because the conditional survival goes towards zero around that point.
The purpose of the difference between this and #15 is to allow one to design a custom spending expectation "c" and to also design a custom discount (using the alpha and Beta in front of "r") if one were to have some odd, asymmetrical time preference. I can't really imagine the latter in any normal person's daily grind but there it is anyway.
B. Stochastic PV Version
This is the animated version with simulation where the result is a stochastic present value (spv) spending distribution.
- x is the number of iterations; N is a random lifetime process.
- npv(c) is random because the terms (especially D [the discount], N [a random lifetime process in years; see above] and parts of Ct [the spend plan, similar to "c" above and Yarri's c(t); think inflation, random spending var, and maybe spend shocks]) are random.
- EV is the expected value but I think that extracting the Pth percentile is as or more interesting.
There are other ways to do this sim, by the way. This is just the way I recently did it and the equation was my amateur attempt at describing it. Note that I had an earlier version of the simulation below where the discount was neither stochastic nor chained in a series which means it might be a closer (discrete non-utility) proxy to Yaari's eq13 [1965].
Formula 18. Bond Total Returns
bonds(i)=100*(yields(i)./yields(i+1) + (1-yields(i)./yields(i+1))./((1+yields(i+1)/100).^10) - 1 + yields(i)/100);
I've calculated bond total returns in several models I've done based on a variation of the formula above. I confirmed this against feedback from Aswath Damodaran at the Stern Shool, Wade Pfau, and the developer behind cFireSim.com. The formula was adapted depending on whether I was using 10 year or 5 year data. Data was typically from St Louis Fed data series.
Formula 19. Wealth Depletion Time Simulation math
This is more or less an adaptation of stochastic consumption formulas in #17 to now include utility as well as to then animate it with simulation. This formula cannot exist independently from the simulation code that generates things like random returns and inflation. The code also does things like "snap" spending to (a) available income when wealth depletes, or (b) to available income if spending, in spend rule mode, drops below available income. The math framework for the sim proper is this:
where
E[V(c)] is the expected discounted utility of lifetime consumption
c - is a consumption plan, c(t) is consumption in period t
S - is the number of iterations
k - is a subjective discount on utiles
g(c) - is the CRRA utility function but using a constant (-1) in the numerator
| - gamma ≠ 1: CRRA utility with a constant: (c(t)^(1-g) - 1)/(1-g) - gamma = 1: log utility of c(t) T* - is the random lifetime |
(income - in the sim; that and the snap can't be seen in the above formula)
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I will add more formulas here as I run into them or as they are sent to me…
Just for fun also see: Simple vs Complex by Josh Brown
also on complexity at Farnam Street.
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also on complexity at Farnam Street.
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[1] It took me a bit to figure out what Ken meant by what I am calling "X" so this is what I came up with that matches his math...I think...for a thirty year planning horizon where infl is .014 and discount is .04 and a planning horizon is 30 years. Those are his assumptions not mine. My assumptions are more conservative and hence my spending budget would b e quite a bit lower [postscript: Ken has a spreadsheet on his site with how to exactly do this. I should have gone there first] :
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