Here is the long-awaited conclusion to the wonky 4-part discussion of safe retirement spending. We went pretty far down the rabbit hole, and I think the conclusions are useful.

Attempting to finance a stable retirement from a risky portfolio is inherently fraught with risk and tradeoffs.

A desirable strategy achieves

• High initial and lifetime spending
• Low frequency and severity of shortfalls
• Low variability in spending

Increasing the level of spending today is desirable – but increases the risk of shortfall in the future. High variability in spending is bad – but upward adjustments let you increase spending to take advantage of better-than-expected investment performance, while downward adjustments reduce risk of even greater income shortfall when performance falls short. Finally, accepting spending variability in the future enables higher spending today, if you accept a thinner buffer against shortfall, and/or invest in higher-risk, higher-return portfolios.

The $64,000 question is, how risk-averse are you? How willing are you to trade current income for future income, variability and risk of shortfall?

• If you’re risk-neutral, you just want maximum expected lifetime spending. This leads to a highly equity-oriented, volatile portfolio, and variable spending something like a consistent percentage of the portfolio, starting low and allowing growth in the portfolio and spending over time (our test added smoothing and mortality updating, but spending is still highly variable).
• If you’re highly risk-averse, you want a highly diversified portfolio, and you’ll tend toward a traditional fixed 4% rule (which should probably be lower for conservative retirees in current market conditions).
• If you’re between those extremes, your portfolio, optimal lifetime spending, variability, and shortfall risk will tend to lie between those two extremes. You might consider combining a fixed amount and variable amount to achieve the best balance between variability, shortfall risk, and high initial and lifetime spending.
• What is more important than the specific initial rate is a strategy with an appropriate degree of spending flexibility, giving you the ability to take advantage of higher long-term equity returns without imposing intolerable variability, and a clear picture of the range of potential outcomes. To the extent possible, retirees should err on the side of moderate initial spending, embrace the volatility they can tolerate as the key to unlocking maximum lifetime spending, and accept that their retirement trajectory is ultimately dependent on how the timing of their retirement intersects long-term economic and market trends.

Here’s a slightly more technical discussion.

I got a little stuck trying to write a conclusion. First of all, I realized a constant spending parameter was needed for our model to make it more general, and felt compelled to run it again.

The updated model is



where:

• s_i is spending in period i
• P_i is the value of the portfolio for period i
• L_i is the retiree’s remaining life expectancy in period i
• K is the constant spending parameter
• n is the exponential moving average smoothing parameter
• h is the variable spending parameter
• b is the mortality insensitivity parameter

Other than the addition of K, this is the model in our last post. If n is 1, b is large, and K is 0, this reduces to something like a fixed percentage spending model; if h is 0, this is a constant spending model like Bengen’s 4% rule.

I tested about 20 values for each parameter, and 21 allocations from 0% to 100% equity in 5% steps, ultimately about 3.7m combinations.

Mapping the initial spending ‘severity frontier’, we see that the highest initial spending with no shortfall is basically the Bengen solution: 3.9% spending, 50% stocks, mostly fixed spending.


Shortfall: the worst observed decline below initial spending. The ‘frontier’ is the lowest achievable shortfall for an initial spending rate.



As an aside, below is a plot of probability of shortfall at various levels. Reading across, you can see that at 6%, the best you can do is about a 20% chance of some shortfall and about 4% probability of 50% shortfall (not with identical portfolios). Also, you can see that for each 1% you increase initial spending, best achievable probability of shortfall at any cutoff goes up about 10%.



Mapping the lifetime spending ‘severity frontier’, we see that highest lifetime expected spending with no shortfall is 118.06% of the initial portfolio. This is notably higher than maximizing initial spending with no shortfall. But initial spending is low – 2.33% of the initial portfolio, which is all equity, like all the other solutions for maximum lifetime spending.





This exposes the essential dilemma::

• If you want maximum initial spending with no shortfall, you have a low-risk portfolio, low lifetime spending. You leave a lot on the table.
• If you want maximum lifetime spending with no shortfall, you have low initial spending, high equity allocation in a portfolio that grows over time and gives you back-loaded spending.

This doesn’t really help us pick an optimal point. We need a ‘Sabermetric’ to combine the ‘holy trinity’ of high spending, low variability, and low risk of shortfall.

There is one in the literature. It’s ‘certainty equivalent’ (CE) spending, which takes an income stream and applies a discount according to 1) its level of variability and risk, and 2) a risk aversion parameter. If risk aversion is 0, you’re risk-neutral, no variability discount is applied. The higher your risk aversion the higher the discount. The form we use is constant relative risk aversion (CRRA), which means a stream that varies between $1 and $2 gets the same discount as one with similar variation between $100 and $200.

We calculate certainty-equivalent lifetime spending with a risk aversion parameter of 8 – the lowest that gives us reasonably diversified portfolios. Here is what the ‘severity frontier’ looks like:



This gives us a higher initial income than maximizing lifetime spending, and a higher lifetime spending than maximizing initial income. For instance, the solution in bold has initial spending of 3.83%, 1.7% probability of shortfall > 10%, lifetime spending > 110% of initial portfolio. Compared to maximizing initial spending for 0 shortfall in Table 1 (3.9% initial spending, 70.7 lifetime spending), we achieve a > 50% increase in lifetime spending at a price of a < 2% decrease in initial spending and a worst-case decline of 12.7% from initial spending.

What we find is:

• If we are completely risk neutral, our risk aversion parameter is 0 and we maximize lifetime spending…. we are all equities, low initial spending, high volatility.
• If we are highly risk averse, we will aim for the highest sustainable fixed spending, similar to the Bengen ‘4% rule’.
• If risk aversion is between these two extremes, we will seek a strategy with volatility and lifetime spending between these two extremes. As you increase risk aversion, the variable component of spending decreases and constant spending increases, the portfolio swings from all equities to a diversified mix, and initial spending generally decreases.

If you put full faith in this model and your estimate of risk aversion, you could just maximize certainty-equivalent spending. Or you might want to look up and down the ‘severity frontier’ and pick the best solution, one that combines high lifetime spending and CE spending with reasonable initial spending and shortfall risk.

In practice, maximizing certainty-equivalent spending depends on estimating risk aversion, which is not precisely knowable. We could query retirees on which potential outcomes and risk profiles they prefer, but that’s like describing two movies to them and asking which they would prefer. A portfolio outcome, like a movie, is an experience good, which can only be judged successful after experiencing the whole thing (although failures can often be discovered more quickly, or even from the description). Risk aversion may not even be consistent over time, but may increase abruptly at market extremes.

Nevertheless, certainty-equivalent spending has the advantage of being a consistent, objective measure that takes into account risk and variability, and it can identify a schedule of solutions at different levels of initial and lifetime spending and worst shortfall that are locally optimal, and you can pick the ones that look most desirable.

I wrote this up as a paper with all details, in case others want to delve into it and do more work on it, and will share code and data in this space shortly, as soon as I’ve cleaned it up a little.

I’ll throw out one other approach, which is to use a Kahneman-Tversky behavioral economics/prospect theory utility function, which is discontinuous and heavily penalizes losses from an initial frame of reference. That means it leads to inconsistent choices depending on the initial frame of reference and it’s path-dependent. Yet, it is highly predictive of human choices.

You could apply it to this problem by choosing the highest initial spending such that, if you set it as the initial frame of reference, future utility gains outweigh or equal losses. If you’re highly averse to losses, you’ll set initial spending low enough that the pain from small potential declines in income is offset by very large potential future gains. If you’re less loss averse, you’ll set initial spending a little higher, so moderate losses are offset by large future gains. We still have the thorny problem of estimating the degree of loss aversion. Maybe I’ll do a detailed exploration of that idea, but it might take a while, so I’ll just leave it out there, or if anyone wants to collaborate on a study like that, drop me a line.

In the last part of our look at dynamic rules for spending in retirement, we discussed how changing the allocation between stocks and bonds affects the maximum sustainable spending rate. We can summarize this relationship by plotting the highest feasible initial spending rate for any acceptable shortfall level.¹

Figure 15. Initial spending v. worst shortfall
equity=0-100%, s=0.05-1.2, smoothing=1.0-6.0)

Max shortfall	Init spend %	s	Equity %	Smoothing	Lifetime Spend Exp.
0.0%	1.7%	0.30	20%	5	58.0
4.3%	2.0%	0.35	40%	4	66.3
12.1%	2.3%	0.40	50%	3	77.4
16.9%	2.6%	0.45	60%	2.5	89.1
24.7%	2.9%	0.50	75%	2	105.9
32.2%	3.2%	0.55	80%	1.5	113.6
43.2%	3.5%	0.60	95%	1.5	132.5
50.4%	3.8%	0.65	95%	1	133.8
63.5%	4.1%	0.70	100%	1	141.0

Adjusting how fast spending rate increases with age

Suppose we are willing to accept the portfolio in bold and spending rate, which has never resulted in a decline greater than 24.7% below the initial spending amount. We use spending factor s=0.5, equity = 75%, smoothing factor = 2. We end up with a spending profile that looks like this:

Notice the ‘hump.’ In the average case, spending rises steadily, then drops. Ideally, we would like a flatter spending profile, preferring less variability in spending. If we could take the hump and spread part of it earlier, without increasing shortfall risk, it would increase the safe spending rate.

The spending rate at age 99 is s/life_expectancy=0.5/2.19 = 22.8%. Spending rates at advanced ages are too high, decimating the portfolio and causing the decline in spending. If we reduce them, maybe we don’t need to build as big a war chest in middle years to support such a high spending rate in late years.

There are several ways we could change the model to flatten the spending profile. The one we choose is to add a parameter we’ll call ‘life expectancy buffer’ to life expectancy, so we’ll spend at a rate of s/(life_expectancy+buffer). If the parameter is 2, then when our life expectancy is 3, instead of spending at a rate of s/3, we’ll spend at a rate of s/(3+2). If life expectancy buffer is 0 or low, the spending rate will rise relatively quickly as we age; if life expectancy buffer is large, the spending rate will change slowly as we age.

With a buffer of 2, we get a flatter spending profile later in life, without changing too much at the start of retirement.

Let’s compare efficient frontiers, with and without the buffer:

We find in all cases, the buffer gives us a higher level of initial spending for any level of shortfall risk. However, increasing the buffer does not always increase lifetime spending and sometimes worsens the tradeoff.

It’s a mixed picture. The buffer may in many cases allow us to spend safely at a higher initial rate, but doesn’t always improve lifetime spending. Sometimes it reduces portfolio growth and overall lifetime spending.

Third dynamic rule: Changing equity allocation over time.

Does it make sense to reduce the equity allocation as you get older? Let’s introduce a variable to gradually reduce the amount of equity. As long as your life expectancy is > 15 years, use the initial allocation. After that, we’ll reduce the equity percentage by stepdown percentage points per year.

What I thought would happen: stepping down equity over time would let you start with more equity, relying on smoothing to get you over early drawdowns and strong long-run real returns to catch up. Then with lower equity in later years you wouldn’t be exposed to big drawdowns late in life.

What actually happened: stepping down equities always worsens the risk/reward tradeoff.

Apparently my preconception was faulty, or this is the wrong model, stepping down too early and killing returns, or there’s a mistake somewhere (although I don’t think so). Watch this space for possible future elaboration or ninja-edit. I’ll consider this a first draft and move on.

Discarding the stepdown, we now have 4 parameters: spending factor s, starting equity, smoothing factor, and lifetime buffer. We have discussed how each one impacts the spending profile and shortfall risk. Let’s put it all together, try running a full range of reasonable values, and come up with the best efficient frontier over all values.

Let’s look at the rule in bold. (s=0.70, equity=75%, smooth=6.0, buffer=5)

• When you retire matters. A lot.
• The 4% rule doesn’t really work with a restrictive large cap/long term bond portfolio. The best solution is 95% equity and you still get a worst case 35% shortfall.
• < 3.5% has generally had a tolerable worst-case, and on average lets your spending grow over time to > 4% of the initial portfolio.
• Even at 3.2-3.5%, a dynamic rule gives you higher lifetime spending than a fixed 4% rule, and the failure scenarios are gradual, not sudden and catastrophic.
• A large equity allocation is desirable for real long-term growth, and stepping down the amount of equity as you get older doesn’t seem to improve risk-reward (at least, not a fixed annual stepdown when life expectancy is < 15 years).
• Taxes and investment expenses aren’t included and must be taken into account, reducing returns and the level of safe spending.
• On the other hand, including higher-risk, higher-return investments, like small caps, international equities, higher-yielding bonds should improve risk-reward and allow higher returns and spending.
• Of course, this analysis assumes the future will not be very different from the past.
• Finally, I would add that life annuities, in theory, could be a free lunch. In the analysis above, you put aside 100% of the money you may need in the 25% chance you live to be 90; alternatively, you could join an annuity pool which only needs to put aside money for the survivors, which also has tax advantages. However, fixed annuities are subject to inflation and credit risk, and the market for insurance products for the elderly is a jungle teeming with predators.

These dynamic rules are chosen for analytical simplicity, to be easy to understand and calculate. There’s no reason to believe that they are the best, ie spending a constant/(life expectancy+buffer) is the ideal model to get the highest spending rate at each point in time for the lowest shortfall risk.

In the last part of this series, we’ll think about what an ideal model would look, and how to maximize spending while keeping shortfall risk at an acceptable level.

¹On further review, the table in Part 2 isn’t the best way to look at it. No one thinks about the best spending factor they can achieve, they think about the initial spending %. And people don’t think about ‘lowest probability of 10% shortfall.’ Maximum acceptable shortfall is simpler and more informative.

The last post discussed a framework for evaluating simple dynamic spending rules.

• We defined a spending factor s as spending each year at a rate of s/(remaining life expectancy); and lifetime spending expectancy as the total amount you could expect to spend over your lifetime.
• We showed how, as you increase the spending factor, lifetime spend expectancy initially increases rapidly, but the curve flattens out as spending rate increases.
• We showed how, as you increase the spending factor, shortfall risk initially increases slowly, but the curve steepens as spending rate increases.

We found a simple dynamic spending rule could increase lifetime spending vs. a traditional fixed 4% rule, while keeping shortfall risk relatively low (arguably reducing risk by making the worst case more benign, at the cost of increased volatility, lowered starting spending, higher probability of modest shortfalls).

In this post we’ll look at how smoothing spending can improve outcomes, and how changing the equity/bond mix over time affects outcomes.
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How much can you safely spend out of a portfolio in retirement? Spend conservatively and you may be unnecessarily curbing the lifestyle and aspirations of you and your loved ones. Overspend and risk a shortfall and painful adjustment – in the extreme, the (hopefully apocryphal) “cat food” diet.

A traditional rule of thumb is a fixed 4% per year of your starting portfolio, adjusted each year for inflation. A previous post discussed why this rule may not be safe:

• Low bond yields – 1.8% for 10-year Treasurys and negative TIPS out to 10 years – mean historical bond returns are mathematically unobtainable.¹
• 2.2% real returns since 2000 on a 60/40 blended portfolio suggest that long-run return expectations need to be revisited. Low long-term interest rates are a forecast of low future returns, ie low growth and inflation expectations. To the extent equity risk premiums haven’t widened, they forecast lower than normal equity returns.
• Taxes and investment expenses must be included. Work supporting 4% tends to ignore them.
• US demographics are not very positive for growth, inflation, tax rates, and hence, real after-tax investment returns (which is reflected in the US fiscal position). The US dependency ratio is forecast to rise by 15 points over the next 20 years.

If the 4% rule hasn’t been decisively breached, forward-looking indicators are a bit worrisome. Could a more flexible rule not only be safer, but in favorable circumstances allow a higher level of spending? In this 3-part post, we test dynamic rules that vary withdrawal rates based on age and the size of the portfolio, and vary the composition of the portfolio over time.
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It’s not whether you get knocked down, it’s whether you get up. – Vince Lombardi

Playing around with DataNitro, an add-in that lets you run Python in Excel¹.

What is the worst that could happen to someone who owns stocks, bonds, bills, over a 1-, 5-, 10-year time-frame? Here are the worst rolling periods for each asset class for 1928-2010, adjusted for inflation.

Worst case real returns for rolling periods from 1 to 30 years, 1928-2010

Over short timeframes, stocks can do quite a bit worse. The worst 2-year period is -52.5% for stocks, v. -26% for bonds and -25% for bills. Around year 8, stocks cross over. The worst 20-year period for stocks sees you up 10.7%, and the worst 30-year period sees you up 243%! When bonds and bills get hurt by inflation, they stay down for very long periods.

Rerunning the analysis for the post-war era doesn’t change much. Most of the worst-case periods for stocks and bonds were after 1946, but bills did worst around the war and better afterwards.

Worst case real returns for rolling periods from 1 to 30 years, 1946-2010

Spreadsheet here.

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import numpy as np # not used in this example, but works!
 
def rolling_return(series, n):
    "given a series of m returns, compute m-n rolling n-period returns"
    m = len(series)
    retarray = []
    for i in range(m-n):
        rr = 1.0
        for j in range(n):
            rr = rr * (1+ series[i+j])
        retarray.append(rr-1)
    return retarray
 
active_sheet("Returns")
stocks = CellRange("Equities").value
bonds = CellRange("Bonds").value
bills = CellRange("Bills").value
cpi = CellRange("CPI").value
 
realbonds = [bonds[i]-cpi[i] for i in range(len(cpi))]
realbills = [bills[i]-cpi[i] for i in range(len(cpi))]
realstocks = [stocks[i]-cpi[i] for i in range(len(cpi))]
 
active_sheet("Data_1928")
for i in range(1,31):
    tempbonds = rolling_return(realbonds,i)
    tempbills = rolling_return(realbills,i)
    tempstocks = rolling_return(realstocks,i)
    Cell(i+1,2).value = min(tempstocks)
    Cell(i+1,3).value = min(tempbonds)
    Cell(i+1,4).value = min(tempbills)
 
active_sheet("Data_1946")
 
realbonds46=realbonds[18:]
realbills46=realbills[18:]
realstocks46=realstocks[18:]
 
for i in range(1,31):
    tempbonds = rolling_return(realbonds46,i)
    tempbills = rolling_return(realbills46,i)
    tempstocks = rolling_return(realstocks46,i)
    Cell(i+1,2).value = min(tempstocks)
    Cell(i+1,3).value = min(tempbonds)
    Cell(i+1,4).value = min(tempbills)

¹Why is Python a good thing? Lots of very powerful packages for data manipulation, optimization, statistical analysis, machine learning are available in Python. Also, Python is a powerful, expressive, readable language that makes it easy to manipulate complex data structures.

If ‘The Graduate’ were made today, Benjamin Braddock might hear a well-meaning uncle stage-whisper ‘Big Data’ instead of ‘Plastics.’ (Runners-up: ‘The Cloud’, ‘Social Discovery’, ‘Gamification’, the list goes on.) ‘Big data’ is a buzzword that people throw around a lot. What does it mean? Large data sets are not new. The IRS, the Census, Walmart, money center banks have always had big data sets.

What’s changed?
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I didn’t really post as much as I would have liked this year. I envy people whose thoughts come out in a more or less coherent, finished form. When I post something, I always think of what I really wanted to say after hitting ‘publish’.

Today, I’m going to just try to write for an hour and post what comes out, hopefully resisting the temptation to ninja-edit.

My buddy Josh does a post with quotes where a bunch of people say what they learned over the last year. So what did I learn?
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The first casualty when war comes is truth. – Hiram Johnson

Everybody talkin’ to their pockets
Everybody wants a box of chocolates
And a long-stemmed rose
- Leonard Cohen

Let’s talk a little about social capital.

According to studies, Greeks work the longest hours in Europe, and their retirement age is in the middle of the pack. Same goes for a lot of developing countries, and even some US inner cities. People work themselves to the bone, and they don’t get ahead.

Why are those countries in such a mess?
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So, some people are talking about Hurricane Sandy putting people back to work, and others are pointing out that this is the ‘broken windows fallacy.’ True, a massive superstorm is usually not a good thing. Nevertheless, three quick points.

English: Aerial view of roadbed collapse near the interface of the cantilevered truss sections of the San Francisco-Oakland Bay Bridge. View northwestward. Cropped from original version to better fit San Francisco–Oakland Bay Bridge article. (Photo credit: Wikipedia)

The Paul Ryan plan ‘Promotes saving by eliminating taxes on interest, capital gains, and dividends; also eliminates the death tax.’

So people like Mitt Romney, and all his heirs in perpetuity, would never pay another dime in income tax. How sweet is that? How fair is that?

It’s shocking that in this day and age, someone could make such a proposal, and be considered a serious person and politician.

The GOP has come an awful long way from its founder, who said, “Capital is only the fruit of labor, and could never have existed if labor had not first existed. Labor is the superior of capital, and deserves much the higher consideration.” – Abraham Lincoln