# Inverted Withdrawals and risk aversion

On May 17, 2016, *Advisor Perspectives* published “Inverted Withdrawal Rates and the Sequence of Returns Bonus”. The author’s general idea is to start with a fixed-percentage withdrawal rate (4% of the portfolio’s current value; sometimes called “endowment withdrawals”) but then to add a *tilt* based on whether the portfolio is “up” or “down”. The idea is when the portfolio is up, take out more; when the portfolio is down, take out less. Of course, you’re *already* taking out less (or more) due to using a fixed-percentage withdrawal rate. So the author suggests taking out even less (or more).

Why try this?

With the normal inflation-adjusted 4% Safe Withdrawal Rate, you are subject to sequence of returns risk. With a fixed-percentage withdrawal, there is **no** sequence of returns risk. By *tilting* the author is trying to create a sequence of returns *bonus*.

My intuition is this will lead to wild income fluctuations that no one would be happy with in real life.

I picked a 1970 as an arbitrary date, just to give us something to look at. You can see that the Inverted Withdrawals are either below or above the Constant Percentage income; that’s by design. But what is more noticeable is that while Constant Percentage tends to have gradual changes in income, Inverted Withdrawals guarantees large jumps: in 1974 we hit a patch of bad returns and the portfolio value drops from $1.06 million to $838,000. That means we need to switch our tilt, instead of +1% we start using -1%.

But in practice that means *in 1973 we withdrew $53,000 and in 1974 we’re only allowed to withdraw $25,000*.

It is hard to imagine many retirees following through with a plan like that.

It gets worse, because the 1% is the least aggressive option Walton suggests. He also suggests a 3% tilt. This is what that same 1970-retiree would experience using a 3% tilt.

The swings are (by design) even more exaggerated. In 1974 income goes from $71,000 to $7,800. In 1986 incomes goes from $9,300 to $79,000.

It seems so far from what any retiree would accept that it is hard to understand what we can learn from modelling it.

# From charts to equations

There’s actually a way we can put some context around these wildly varying incomes: coefficient of relative risk aversion (CRRA) utility functions. We know that people place a value on certainty. Imagine I make you can offer:

- If you pick Option #1, I will give you $49.
- If you pick Option #2, I will flip a coin. Heads, I will give you $100. Tails, I will give you $0.

Option #2 has an expected return of $50, which is higher than the $49 from Option #1. Yet most people will pick Option #1, since it is certain.

Would you still pick Option #1 if it was $40? $30? $20? $10? CRRA utility functions are a way of calculating this. And empirical research on the behaviour of real people has consistently yielded estimates in the range of 1–4 for the coefficient of relative risk.

(I’ve also been told, but haven’t found any good references yet, that some researchers believe the CRRA should be *much* higher (i.e. much more risk averse) when talking about retirement portfolios based on the observed asset allocations of actual retirees.)

In any case…how do Inverted Withdrawals look when we put them through a CRRA utility function? Let’s start by looking at that 1970-retiree again:

`Constant 4% $32,499`

Tilt 1% $26,199

Tilt 2% $18,080

Tilt 3% $ 9,247

What this means is that most people consider the varying incomes of the 3% tilt and a constant $9,247 as being equivalent. They’d rather take $10,000 constant every year than the variations of a 3% tilt.

Instead of just looking at 1970, let’s look at every single year of retirement from 1871 onward:

The author says

If one is able to withstand large fluctuations in annual income, a high positive tilt preserves capital while increasing anticipated (average) income, keeping one more prepared for the vicissitudes of life.

But using a coefficient of risk aversion utility function and historical data we can investigate what “withstand large fluctuations” means in practice for a real retiree. And what we find matches our intuition: due to risk aversion, most people are going to treat the 3% tilt extremely negatively. In most cases it will be treated as equivalent to under $10,000 a year in income.

The author suggests that this kind of tilting makes sense when you have a large stable income and low liabilities: Social Security and pensions (or an annuity) cover most of your needs and you’re intent on preserving your initial portfolio (for spending shocks or legacy purposes).

Yet the CRRA utility function shows that, whatever the mathematical merits of this strategy, in the real world people will heavily discount such wildly varying income swings.

The costs of this approach — in terms of annual income through the lens of a CRRA utility function — are clear. What are the actual benefits to preservation of capital?

If we look at the “low volatility” portfolio (which is presumably what someone would choose if they were worried about preservation of capital), we can see that the standard deviation of capital goes from a bit over $200,000 with a 0% tilt to around $100,000 with a 3% tilt. That means that your capital hovers in the range of $900,000 to $1,100,000 instead of $800,000 to $1,200,000.

Is the tradeoff worth it?