If a sovereign country were to retire...

If a sovereign country were to retire and forgo income, they's have the same issues, except for the immortality thing, as any run of the mill retiree. This was recently covered in

The Optimal Extraction Rate versus the Expected Real Return OfA Sovereign Wealth Fund,  By Knut K. Aase and Petter Bjerksund [2019], a fun but very dense read that has a ton of affinities with retirement finance.  

Here are some selected extracts. Not all of these will make sense out of context and I'm not going to correct copy paste errors in all of the characters. 

• the optimal extraction rate from the fund is significantly smaller than the expected real rate of return on the underlying fund.
• Optimal portfolio choice and spending are then inconsistent.
• It follows from optimal consumption and portfolio choice theory that the optimal consumption per time unit, c[∗,t] , and the optimal wealth at time t, W[∗, t] , are connected. [presents a general formula for optimal W(t) being the same as a stochastic present value (instantaneous)]
• With a very long horizon T, it is optimal for the agent to consume approximately a fraction of the remaining wealth at any time t. In reality this fraction is a stochastic process. [emphasis added]
• If the horizon is unbounded at the outset, the fraction k is consumed forever. We may consider the factor k[-sub]T (t) as an estimate as of time t.
• when half the expected excess return on the fund over the risk-free rate is larger than the right-hand side of (19), then the extraction rate is lower than the expected rate of return on the wealth portfolio. [requires awareness of the math before and after…]
• We notice that for plausible values of the parameters, the optimal extraction rate is strictly smaller than the expected real rate of return on the wealth portfolio.
• Seen from time 0,the end wealth of the agent corresponds to the random variable Wt , not the sure amount W0.
• To use the expected return on the endowment fund as the extraction rate, is on the other hand consistent with investing everything in the single risky asset, or group of assets, with the largest expected return(s) one can find, and completely ignore risk. [emphasis added]
• Proposition 1 Assuming that the optimal extraction rate k is a constant, it depends on the return from the fund only via the certainty equivalent rate of return [italics in original]
• Proposition 2 When (i) the objective is to maximize utility and, (ii) we consider a particular fund in isolation, the optimal drawdown rate will be lower than the expected real rate of return on the fund, for any reasonable levels of the impatience rate and the relative risk aversion. [italics in original]’
• Proposition 3 With recursive utility, assuming a deterministic investment opportunity set, the optimal extraction rate k is a constant and depends on the return from the fund only via the certainty equivalent rate of return [italics in original]
• it is an empirical fact that the volatility of aggregate consumption in society is different from the volatility of the securities market
• There are several important lessons we can draw from this example. Firstly, for reasonable parameter values, it is optimal to consume considerably less than the expected rate of return of the fund. Secondly, if the utility impatience rate and the certainty equivalent fund return are equal, the optimal consumption rate equals the two regardless of EIS. Thirdly, if the utility impatience rate is less than the certainty equivalent fund return, the latter is an upper bound for the optimal consumption rate.
• In the model of the present section where the fund is part of society as a whole, and the objective is to maximize recursive utility of total consumption, the optimal extraction rate X is randomly fluctuation with time having dynamics given in (60). 
• Suppose the parameters satisfy α > θ² . Then the expected value m(t) of the optimal extraction rate X is approximately stationary in the short run, with level X0 strictly smaller than the corresponding constant optimal extraction rate k. [italics in original, bold added]
  The intuition is again that the added uncertainty caused by having θ =/= 0 [i.e., consumption and wealth correlation are not 1] implies that a more cautious, optimal extraction policy is called for, provided we interpret m(t) as the optimal extraction rate at time t. [emphasis added]
• We have derived concrete formulas for optimal extraction from an endowment fund consistent with risk aversion, and demonstrated that the optimal extraction rate is strictly smaller than the expected rate of return. The difference is far from negligible, and amounts to several percentage points in most real situations.

Discussion

Ok, this paper was dense but not without some common sense intuition.  If we render this as a retiree and not a sovereign and make him or her, the retiree, immortal...or very early retiree or a long duration trust or an endowment, we can perhaps say something similar. And note that a lot of this leans on the long-horizon aspect:

• -          If the horizon is infinity then we’d want to consume at or below the expected rate of return… except…
• -          If we realize less than we expect, and we will due to volatility and the nature of the geometric processes involved, then we'd have to consume less, somewhere around an adjusted Geo mean return expectation at infinity or close, except…
• -          If we think the return processes are not stable and will vary randomly, we might want to consume even less, except…
• -          If the variability is itself not very stable, that’d just make things worse so we’d want to consume less, except…
• -          If the consumption process is not constant, we might want to consume less. If it covaries directly with wealth it might be ok maybe but not necessarily, except…
• -          If the consumption process and wealth process do not covary exactly then we might want to consume even less due to the uncertainty (this is sorta like sequence risk) and the fact that it could take the perpetuity to not-perpetuity. This is not precisely sequence risk and I have very rarely seen this discussion in the literature so it’s a good catch. Rivershedge has pounded on this a couple times before.

So, in a seriously non-scientific mind-game, take some expected arithmetic nominal return (for sake of argument, let’s use .10) with some variance (again, just for argument’s sake, let’s say .20 std dev),

• -          knock off whatever for inflation expectations, say 3 or 4%, maybe more for random i,
• -          penalize that for volatility at ~vol/2, call it a half percent at a 10% std dev and 2% for 20%
• -          and then subtract a percent or two more for the other uncertainty in the points above

Then pray there is something left -- starting at a 10% AR, I myself reduce this game to around 2%, which is about the rate for a perpetuity. And then, maybe now that we are a little stressed out, we can perhaps breathe easier with the following givebacks:

• -          If your horizon T is << ∞ add some back due to living less long, how much idk in this post.
• -          If your risk aversion is strong and you are a “do ya feel lucky kid” kind of person, add some more

Probably still under 4%. Unless you are 90 and not immortal. Then it’s higher. But the paper is right. If you are immortal you need to spend quite a bit less than the expected return to fund all of your immortality.