Tag Archives: Risk

Problem-Owners Tolerate Risk

In theory, both risk aversion and value of life year relative to income are mainly set by utility function concavity. However, as both also show puzzlingly large empirical variation, there is apparently a lot more going on than a simple context independent utility function. But what?

While pondering this question, I saw some nature videos of cute young mammals, who start out as physically and emotionally fragile, and prone to crying, but who grow to be tough and emotionally steady like their parents. I wondered: might it be useful to frame this as a transition from risk-aversion to risk-tolerance?

Of course youngsters aren’t always risk averse. Sometimes they try to play with a scorpion, or rough-house too close to a cliff. In such cases, mom may restrain their risk-taking. But when kids suffer some mild depredation, such as wanting food or other help from mom, that’s when they act most emotionally needly and fragile:

Oh my, this problem I face is way too hard for my fragile emotions. I’m young, innocent, and nearly dying of fright to think about it. Won’t someone who loves me come to protect me from this terrible anxiety and suffering?

Plausibly parents are built to find it hard to resist wanting to help such puppy-dog eyes, to get them to take care of their kids. And when parents do tend to help when kids suffer, that actually makes kids more risk-tolerant regarding choices to take precautions to prevent future suffering.

Humans are famously neotenous, retaining more childlike features further into adulthood. And living in large social groups that share food and other resources, even human adults have incentives to show puppy-dog eyes, and to feel sympathetic to those eyes when they have something to share. So it seems plausible that human nature would have adapted the fragile-tough child-adult dynamic to apply to the helpee-helper relations in groups of adults.

That is, I’m suggesting that evolution has built we humans to strategically (if unconsciously) make a key attitude choice regarding each particular situation we face: do we project a cool tough self-reliant risk-tolerance, or a more emotionally-expressive risk-averse fragile vulnerability. In other words, do we choose to act vulnerable but sympathetic, or do we choose to act more “emotionally mature”? I’m not saying this is the only factor that influences risk-tolerance, but it may be one of the largest.

Yes, people often talk as if emotional maturity is the product of learning from experience, with perhaps some help from an admirable moral will. But if emotional maturity were always the better strategy, why would evolution have ever encoded in us any other tendency? Evolution could have made the very young take on emotionally mature attitudes if it had wanted. And in fact, it does sometimes want that, such as when kids “grow up too fast” due to getting little help in taking on adult-sized problems.

The key maturity choice here is of whether to “own” each problem that we face. It makes sense to own a problem, and act more risk-tolerant toward it (and more risk-averse re preventing it), when we seek to impress others with our confidence in handling the problem, when we bid for parent/leader roles, and when we want to avoid the embarrassment of others seeing that no one comes when we ask for help. However, when getting help with our problem seems more important, and likely enough, or when showing submission to and dependence on others seems important, it can make more sense to try to get others to own our problem by acting more risk-averse and hurt by the problem. We can suggest that others are to blame for causing our problem, and even if not they are responsible to help fix it.

So does my theory fit the data? In one key study, “Psychosocial maturity proved a better predictor of risk-taking behaviour than age.” Which is striking because age (at least below age 65) is one of our two usual best predictors of risk-tolerance, the other being male gender. Some data suggests that the following groups are also more risk-tolerant: the tall, the married, and those with more kids. People who work in finance, insurance, and real estate seem more risk-averse, and buying insurance on a risk is a strong sign that one is averse to it. Those with mid-level wages and extreme high or low wealth also seem more risk tolerant. The self-employed seem more risk averse.

Emotional maturity tends to increase on average with age, range of experiences, intelligence, self-esteem, and work performance, positive attitudes toward childcare, and not being an orphan. Here are some descriptions of the concept:

An emotionally mature individual gives off a sense of ‘calm amid the storm.’ They’re the ones we look to when going through a difficult time because they perform well under stress. … When you’re less mature, the world is full of minor annoyances, and you’re unaware of your own privileges. Think about how often a day you complain about others or different situations. (more)

Accept[s] the sorrows of life whole heartedly and … show[s] distress when there is occasion to be worried, without feeling a requisite to use a false facade of bravery. (more)

If my theory is right, and much of the variation in risk aversion (and value of life) we see results from strategic context-dependent choices to act risk-tolerant or risk-averse, this makes it harder to used measured risk-aversion (and value of life) to inform policy. Yes, if true risk aversion were higher, that would justify paying more to save lives, including via stricter regulation, and also justify more redistribution and social insurance. But if much of what we see people do is done for show, then we have to try to infer the real level of risk aversion behind all that show.

My guess is that on average in a social species with lots of sharing, free-riding is a bigger problem than excess autonomy. If so, we more often try to seem more needy to gain more help, than we try to seem less needy to gain respect. And thus typical behavior will exaggerate our real overall degree of risk aversion (and value of life). But I don’t yet know how to show this. It does seem worth further study; we may well figure out some way to see.

Added 3p: Related datum: “women are more sensitive to pain and less tolerant of pain than men.”

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Risk-Aversion Sets Life Value

Many pandemic cost-benefit analyses estimate larger containment benefits than did I, mainly due to larger costs for each life lost. Surprised to see this, I’ve been reviewing the value of life literature. The key question: how much money (or resources) should you, or we, be willing to pay to gain more life? Here are five increasingly sophisticated views:

  1. Infinite – Pay any price for any chance to save any human life.
  2. Value Per Life – $ value per human life saved.
  3. Quality Adjusted Life Year (QALY) – $ value per life year saved, adjusted for quality.
  4. Life Year To Income Ratio – Value ratio between a year of life and a year of income.
  5. Risk Aversion – Life to income ratio comes from elasticity of utility w.r.t. income.

The first view, of infinite value, is the simplest. If you imagine someone putting a gun to your head, you might imagine paying any dollar price to not be shot. There are popular sayings to this effect, and many even call this a fundamental moral norm, punishing those who visibly violate it. For example, a hospital administrator who could save a boy’s life, but at great expense, is seen as evil and deserving of punishment, if he doesn’t save the boy. But he is seen as almost as evil if he does save the boy, but thinks about his choice for a while.

Which shows just how hypocritical and selective our norm enforcement can be, as we all make frequent choices that express a finite values on life. Every time we don’t pay all possible costs to use the absolutely safest products and processes because they cost more in terms of time, money, or quality of output, we reveal that we do not put infinite value on life.

The second view, where we put a specific dollar value on each life, has long been shunned by officials, who deny they do any such thing, even though they in effect do. Juries have awarded big claims against firms that explicitly used value of life calculations to not to adopt safety features, even when they used high values of life. Yet it is easy to show that we can have both more money and save more lives if we are more consistent about the price we pay for lives in the many different death-risk-versus-cost choices that we make.

Studies that estimate the monetary price we are willing to pay to save a life have long shown puzzlingly great variation across individuals and contexts. Perhaps in part because the topic is politically charged. Those who seek to justify higher safety spending, stronger regulations, or larger court damages re medicine, food, environmental, or job accidents tend to want higher estimates, while those who seek to justify less and weaker of such things tend to want lower estimates.

The third view says that the main reason to not die is to gain more years of life. We thus care less about deaths of older and sicker folks, who have shorter remaining lives if they are saved now from death. Older people are often upset to be thus less valued, and Congress put terms into the US ACA (Obamacare) medicine bill forbidding agencies from using life years saved to judge medical treatments. Those disabled and in pain can also be upset to have their life years valued less, due to lower quality, though discounting low-quality years is exactly how the calculus says that it is good to prevent disability and pain, as well as death.

It can make sense to discount life years not only for disability, but also for distance in time. That is, saving you from dying now instead of a year from now can be worth more than saving you from dying 59 years from now, instead of 60 years from now. I haven’t seen studies which estimate how much we actually discount life years with time.

You can’t spend more to prevent death or disability than you have. There is thus a hard upper bound on how much you can be willing to pay for anything, even your life. So if you spend a substantial fraction of what you have for your life, your value of life must at least roughly scale with income, at least at the high or low end of the income spectrum. Which leads us to the fourth view listed above, that if you double your income, you double the monetary value you place on a QALY. Of course we aren’t talking about short-term income, which can vary a lot. More like a lifetime income, or the average long-term incomes of the many associates who may care about someone.

The fact that medical spending as a fraction of income tends to rise with income suggests that richer people place proportionally more value on their life. But in fact meta-analyses of the many studies on value of life seem to suggest that higher income people place proportionally less value on life. Often as low as value of life going as the square root of income.

Back in 1992, Lawrence Summers, then Chief Economist of the World Bank, got into trouble for approving a memo which suggested shipping pollution to poor nations, as lives lost there cost less. People were furious at this “moral premise”. So maybe studies done in poor nations are being slanted by the people there to get high values, to prove that their lives are worth just as much.

Empirical estimates of the value ratio of life relative to income still vary a lot. But a simple theoretical argument suggests that variation in this value is mostly due to variation in risk-aversion. Which is the fifth and last view listed above. Here’s a suggestive little formal model. (If you don’t like math, skip to the last two paragraphs.)

Assume life happens at discrete times t. Between each t and t+1, there is a probability p(et) of not dying, which is increasing in death prevention effort et. (To model time discounting, use δ*p here instead of p.) Thus from time t onward, expected lifespan is Lt = 1 + p(et)*Lt+1. Total value from time t onward is similarly given by Vt = u(ct) + p(et)*Vt+1, where utility u(ct) is increasing in that time’s consumption ct.

Consumption ct and effort et are constrained by budget B, so that ct + etB. If budget B and functions p(e) and u(c) are the same at all times t, then unique interior optimums of e and c are as well, and also L and V. Thus we have L = 1/(1-p), and V = u/(1-p) = u*L.

In this model, the life to income value ratio is the value of increasing Lt from L to L+x, divided by the value of increasing ct from c to c(1+x), for x small and some particular time t. That is:

(dL * dV/dL) / (dc * dV/dc) = xu / (x * c  * du/dc) = [ c * u’(c) / u(c) ]-1.

Which is just the inverse of the elasticity of with respect to c.

That non-linear (concave) shape of the utility function u(c) is also what produces risk-aversion. Note that (relative) risk aversion is usually defined as -c*u”(c)/u’(c), to be invariant under affine transformations of u and c. Here we don’t need such an invariance, as we have a clear zero level of c, the level at which u(c) = 0, so that one is indifferent between death and life with that consumption level.

So in this simple model, the life to income value ratio is just the inverse of the elasticity of the utility function. If elasticity is constant (as with power-law utility), then the life to income ratio is independent of income. A risk-neutral agent puts an equal value on a year of life and a year of income, while an agent with square root utility puts twice as much value on a year of life as a year of income. With no time discounting, the US EPA value of life of $10M corresponds to a life year worth over four times average US income, and thus to a power law utility function where the power is less than one quarter.

This reduction of the value of life to risk aversion (really concavity) helps us understand why the value of life varies so much over individuals and contexts, as we also see puzzlingly large variation and context dependence when we measure risk aversion. I’ll write more on that puzzle soon.

Added 23June: The above model applies directly to the case where, by being alive, one can earn budget B in each time period to spend in that period. This model can also apply to the case where one owns assets A, assets which when invested can grow from A to rA in one time period, and be gambled at fair odds on whether one dies. In this case the above model applies for B = A*(1-p/r).

Added 25June: I think the model gives the same result if we generalize it in the following way: Bt, and pt(et,ct) vary with time, but in a way so that optimal ct = c is constant in time, and dpt/ct = o at the actual values of ct,et.

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Covid19 Pizza-Risk Estimator

To keep us from catching and spreading Covid19, most of us are on “lockdown”, limiting non-home contact. But we aren’t completely isolated; we all do a few things that risk outside contact. And households really are the better unit of risk here; if one of you gets sick, the rest face a much higher risk.

To help households estimate and manage risk, I’ve made the following table, listing risks for 19 activities, all relative to the first: accepting delivery of and eating a pizza, paid for online. These risk estimates came from ~1000 respondents to each of 18 Twitter polls. (Technically, these estimates are medians of lognormal distributions fitted to poll response frequencies.) Yes, it would be better to get expert estimates, but I don’t have experts to poll. I hope experts who see these will publicly improve on them. Until they tell us more, we must act on what we know.

The above is actually part of a screenshot from this spreadsheet (copy it to edit it) that I’ve made to help you estimate household risk. (Anyone know how to embed it here, so each reader can edit their own version here?) On this sheet, you can combine these risk estimates with estimates of how often per week your household does each activity, and also any corrections for how you do it differently, to get your total household weekly “Pizza Risk”. That is, how many weekly risks you take equivalent to a pizza delivered & eaten.

In the spreadsheet, each row lists a risky activity, grouped into types. To use the sheet, consider the activity in each row and think of similar activities you do that risk outside contact. For each such activity, find the closest activity in the table, and for that row, enter how many times per week your household does that activity in the “Count” column. And if your activity seems to have a different risk from other households, such as because you do it for more or less time, or because it involves fewer or more outsiders, then enter a number other than 1 in the “Factor” column. For example, if doing it your way has twice the risk, enter the number 2.

If you mange to use this spreadsheet to get a Pizza Risk estimate, please complete the following two polls so we can learn about how Pizza Risk varies across households.

FYI, this post was up just 33 hours after I first tweeted the idea for this project.

Added 14Apr: Commentor Roman Kuksin did an explicit analysis of many of these risks, and finds a 0.74 correlation with the above risk estimates.

Added 18Apr: Here is another set of activity risk estimates: HT @diviacaroline.

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Small Change Good, Big Change Bad?

Recently I posted on how many seek spiritual insight via cutting the tendency of their minds to wander, yet some like Scott Alexandar fear ems with a reduced tendency to mind wandering because they’d have less moral value. On twitter Scott clarified that he doesn’t mind modest cuts in mind wandering; what he fears is extreme cuts. And on reflection it occurs to me that this is actually THE standard debate about change: some see small changes and either like them or aren’t bothered enough to advocate what it would take to reverse them, while others imagine such trends continuing long enough to result in very large and disturbing changes, and then suggest stronger responses.

For example, on increased immigration some point to the many concrete benefits immigrants now provide. Others imagine that large cumulative immigration eventually results in big changes in culture and political equilibria. On fertility, some wonder if civilization can survive in the long run with declining population, while others point out that population should rise for many decades, and few endorse the policies needed to greatly increase fertility. On genetic modification of humans, some ask why not let doctors correct obvious defects, while others imagine parents eventually editing kid genes mainly to max kid career potential. On oil some say that we should start preparing for the fact that we will eventually run out, while others say that we keep finding new reserves to replace the ones we use.

On nature preserves, some fear eventually losing all of wild nature, but when arguing for any particular development others say we need new things and we still have plenty of nature. On military spending, some say the world is peaceful and we have many things we’d rather spend money on, while others say that societies who do not remain militarily vigilant are eventually conquered. On increasing inequality some say that high enough inequality must eventually result in inadequate human capital investments and destructive revolutions, while others say there’s little prospect of revolution now and inequality has historically only fallen much in big disasters such as famine, war, and state collapse. On value drift, some say it seems right to let each new generation choose its values, while others say a random walk in values across generations must eventually drift very far from current values.

If we consider any parameter, such as typical degree of mind wandering, we are unlikely to see the current value as exactly optimal. So if we give people the benefit of the doubt to make local changes in their interest, we may accept that this may result in a recent net total change we don’t like. We may figure this is the price we pay to get other things we value more, and we we know that it can be very expensive to limit choices severely.

But even though we don’t see the current value as optimal, we also usually see the optimal value as not terribly far from the current value. So if we can imagine current changes as part of a long term trend that eventually produces very large changes, we can become more alarmed and willing to restrict current changes. The key question is: when is that a reasonable response?

First, big concerns about big long term changes only make sense if one actually cares a lot about the long run. Given the usual high rates of return on investment, it is cheap to buy influence on the long term, compared to influence on the short term. Yet few actually devote much of their income to long term investments. This raises doubts about the sincerity of expressed long term concerns.

Second, in our simplest models of the world good local choices also produce good long term choices. So if we presume good local choices, bad long term outcomes require non-simple elements, such as coordination, commitment, or myopia problems. Of course many such problems do exist. Even so, someone who claims to see a long term problem should be expected to identify specifically which such complexities they see at play. It shouldn’t be sufficient to just point to the possibility of such problems.

Third, our ability to foresee the future rapidly declines with time. The more other things that may happen between today and some future date, the harder it is to foresee what may happen at that future date. We should be increasingly careful about the inferences we draw about longer terms.

Fourth, many more processes and factors limit big changes, compared to small changes. For example, in software small changes are often trivial, while larger changes are nearly impossible, at least without starting again from scratch. Similarly, modest changes in mind wandering can be accomplished with minor attitude and habit changes, while extreme changes may require big brain restructuring, which is much harder because brains are complex and opaque. Recent changes in market structure may reduce the number of firms in each industry, but that doesn’t make it remotely plausible that one firm will eventually take over the entire economy. Projections of small changes into large changes need to consider the possibility of many such factors limiting large changes.

Fifth, while it can be reasonably safe to identify short term changes empirically, the longer term a forecast the more one needs to rely on theory, and the more different areas of expertise one must consider when constructing a relevant model of the situation. Beware a mere empirical projection into the long run, or a theory-based projection that relies on theories in only one area.

We should very much be open to the possibility of big bad long term changes, even in areas where we are okay with short term changes, or at least reluctant to sufficiently resist them. But we should also try to hold those who argue for the existence of such problems to relatively high standards. Their analysis should be about future times that we actually care about, and can at least roughly foresee. It should be based on our best theories of relevant subjects, and it should consider the possibility of factors that limit larger changes.

And instead of suggesting big ways to counter short term changes that might lead to long term problems, it is often better to identify markers to warn of larger problems. Then instead of acting in big ways now, we can make sure to track these warning markers, and ready ourselves to act more strongly if they appear.

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In the long run, we’re all still exposed to risk

Many of you will be familiar with the fact that past returns from notable stock indices, such as those in the US, are a biased indicator of the likely future returns to investing in equities. The problem is that due to war, government interference, and financial collapse, some stock markets disappeared altogether, wiping out investors. In some countries this has even happened multiple times. Historical stock indices that went to zero tend not to be remembered, and so are under-sampled. The result is ‘survivorship bias‘, a problem that shows up in many other research questions as well. When these defunct investments are put back in the sample, average returns are quite a bit lower than when you look at just, for example, the NY stock exchange.

A lesser known result is that a broader and representative sample of stock histories shows that investing over long time horizons doesn’t reduce the variability of your return. Contrary to convention wisdom, even young savers need to diversity across different assets types and countries in order to get that effect and be confident of retiring in comfort:

“One of the most enduring question in finance is the persistence of investment risk across time horizon. This issue of time diversification is crucial to long-term asset allocation decisions.

There is a widespread view that the longer the horizon, the more investors benefit from investing in equities. Young investors, for instance, are typically advised to allocate more to equities than those whose retirement is imminent, on the grounds that equities are less risky over long horizons. A common rule of thumb is that the percentage of stock allocation should equal 100 minus an investor’s age.

Some researchers claim to have found empirical evidence that equities are less risky over long horizons because of mean reversion. Mean reversion implies that the variance of stock retums does not grow linearly with time, contrary to a random walk. As a result, several authors have claimed that greater equity allocations are justified on the grormds that shortfall risk lessens as the horizon is extended.

This conclusion seems hardly justified. Previous findings of mean reversion have considered seventy years or so of U.S. data. For long-horizon retums, say ten years, this implies only seven truly independent observations, which seems insufficient to support robust conclusions about the risk of ten-year equity investments. The problem is that, with a fixed sample size, the number of efiective observations diminishes as the investment horizon lengthens. Another problem is that markets with long histories may not represent investment risk for reasons of survivorship bias.

One solution is to expand the sample by adding cross-sectional data. We describe the distribution of long-term returns for a sample of thirty countries for which we have long series of equity prices. The empirical evidence expands on the work of Jorion and Goetzmann (1999) and substantially extends results described by Dimson, Marsh, and Staunton (2002), who analyze a century of stock market returns in fifieen countries.

The results are not reassuring. We find no evidence of long-term mean reversion in the expanded data sample. Downside risk declines very little as the horizon lengthens. In addition, U.S. equities appear systematically less risky than equities of other markets.

Mean reversion is analyzed first in terms of variance ratio tests. There is no evidence of mean reversion from variance ratio tests across this sample, taking into account statistical properties of these tests. Furthermore, markets that sufiered interruption displayed mean aversion, or the opposite of mean reversion. Therefore, statistical properties such as high average retums and mean reversion may be an artifact of survival. Probabilities of losses on equities are reduced very slowly, if at all, with the horizon. In fact, shortfall measures such as value at risk (VAR) sharply increase with the horizon.

There is, however, some positive news. Diversification across assets pays. Over this century, a global stock market index would have displayed less downside risk than any single market. The conclusion is that across-country diversification is more effective than time diversification.” (HT Ben Hoskin)

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