Here are five [now nine, maybe 10 depending on how I count] great, simple little formulas for retirement planning -- all of which are not, you'll notice, 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.
FIVE [9] SIMPLE FORMULAS
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. 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
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
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
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 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 =
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 =
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.
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."
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:
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.
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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
[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:
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.
---------------------------------------------------------
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
[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:
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