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Chapter 5

Harvests good and badInsurance and diversification

Harvests do not always arrive as expected; some years are rich and some poor. Pests may strike only one household's fields, while a drought makes the whole village's harvest fail at once. A household can store extra for bad years on its own, but the larger the reserve, the more it spoils and the more it ties up. When many households share the burden, each household's own bad luck can cancel out, and the more households there are, the smaller the swings each one bears. A disaster that strikes the whole village at once cannot be diversified away; it can only be borne by those willing to bear it, who must be compensated for doing so. Insurance is a state-contingent claim: give up a little in good years and take back a great deal in bad ones. It also brings two new problems: people stop taking care once insured, and those at higher risk are keener to buy cover.

1. Harvests rich and poor

For every household on the island, the harvest varies from year to year. In some years the weather is kind and the harvest is larger than usual; in others pests strike and little is gathered. By the logic of diminishing marginal utility, for the same average harvest, the larger the swings, the harder life is: the extra portions of good years cannot make up for the portions lost in bad years.

Today's counterpart. Fluctuating income, unexpected medical bills and property losses are risks that every household faces today.

Same average, larger swings, harder life

Two households both average 12 portions a year over twenty years. Drag the swings in one household's harvest and compare what each household's harvest is equivalent to as a steady amount each year.

Household with a steady harvestHousehold with a fluctuating harvest

Illustrative: the fluctuating household's harvest each year = 12 × (1 + size of swings × random number); the random numbers are fixed and adjusted so the average is exactly 12 portions. The steady equivalent is computed with logarithmic satisfaction.

2. Bearing it alone: precautionary saving

The most direct remedy is to store more in good years against bad ones. The pain of eating one portion less in a bad year is far greater than the satisfaction of eating one portion more in a good year; and the less one has, the faster the pain of one more portion lost grows. So the harder next year's harvest is to predict, the more valuable the portion kept for a bad year, and the more worthwhile it is to store an extra portion today. The extra saving kept against the unforeseen is called precautionary saving. Leland (1968) showed in a model that, as long as satisfaction has this property (satisfaction measured logarithmically does), the more uncertain future income is, the more people save today. But bearing it alone is costly. To withstand one large crop failure, a household must keep a large reserve year after year; it spoils in the store and cannot be kept as seed or lent to others. Store too little and the household still goes hungry in a bad year; store too much and most of it sits idle in most years.

Today's counterpart. A common piece of financial advice is to set aside several months of living expenses as an emergency reserve, which is precautionary saving.

Bearing it alone: how large a reserve is enough

A household wants to eat 12 portions a year, and its harvest fluctuates at random. Drag the reserve it keeps year-round to see the share of years of hunger over a thousand years, and how much grain the reserve wastes through spoilage each year.

Years of hunger
Spoiled each year

Illustrative: the harvest averages 12 portions with swings of about 40%; 10% of the reserve spoils each year, and it is topped up to the set level in good years; a year with less than 9 portions eaten counts as hunger. Computed on one fixed set of random harvests over a thousand years.

3. Bearing it together: mutual insurance

Suppose many households in the village agree that each will put a little grain into a common store every year, and whichever household is struck by pests will be compensated from it. Pest damage in different households is unrelated: this year it is one household, next year another, and what the common store takes in roughly equals what it pays out. The more households there are, the more fully this cancelling works: as long as each household's bad luck is unrelated to the others', the swings per household shrink roughly in proportion to the square root of the number of households.

This kind of mutual aid has a long history. Justinian's Digest of Roman law includes the so-called Rhodian sea law: when a ship in a storm jettisons part of its cargo to lighten its load, the loss is shared in proportion by all the owners of goods on board. Townsend (1994), studying household data from three villages in India, found that each household's consumption moved mainly with the village's average consumption and was little affected by its own income, illness or unemployment, which indicates a considerable degree of mutual insurance among villagers. That insurance was not complete, however: in one of the villages, the landless were less well protected.

Today's counterpart. An insurer pools the premiums of many policyholders and pays the few among them who suffer a loss. A portfolio holding many stocks lets each company's own bad luck cancel out on the same principle.

Bearing it together: the more households, the more bad luck cancels out

Pest damage in different households is unrelated. Drag the number of households in the common store and see how each household's share of the harvest clusters around 12 portions.

Illustrative: each household's harvest averages 12 portions with swings of about 40%, unrelated across households; the common store divides the total harvest equally among them. The bars show the distribution of each household's share over two thousand years; the shaded area is less than 9 portions.

4. Risk that cannot be diversified away

Mutual insurance has a limit. If a drought makes the whole village's harvest fail at once, the common store takes in little and must pay out much, and no number of households will help. Of each household's risk, the part that belongs to that household alone can be diversified away; the part shared by the whole village cannot be diversified away, however many households join together. From this one can work out what the best sharing arrangement looks like. As long as any household still bears swings that belong to it alone, handing those swings to the common store to be spread among others makes that household's life steadier while barely affecting anyone else, so every household can be better off. Under the best arrangement, therefore, each household's own bad luck is entirely shared away, and what each household finally receives rises and falls only with the village's total harvest.

Risk that cannot be diversified away does not disappear; it can only be borne by someone. Those with deeper reserves, better able to absorb losses, can accept a little grain from others and in a disaster year bear part of others' losses. Those who bear risk must be compensated for it, or no one will be willing to bear it.

Today's counterpart. A diversified portfolio still falls when the market as a whole falls. Insurers pass the catastrophe risk they cannot bear on to reinsurers.

Risk that cannot be diversified away

Each household's harvest swings have two parts: its own luck and the conditions shared by the whole village. Drag the weight of village-wide conditions and see how far the swings each household bears can fall as households are added.

Illustrative: the horizontal axis is the number of households. The weight is the share of each household's harvest variance due to village-wide conditions. After sharing, the swings each household bears = original swings × √(weight + (1 − weight) ÷ number of households); with infinitely many households they fall to original swings × √weight.

5. State-contingent claims

Insurance turns risk into a price. Each year the policyholder pays a fixed premium in exchange for a promise: if disaster strikes, he is compensated; if not, he receives nothing. This is a state-contingent claim. Take the idea to its end: drought years, flood years, pest years, bumper years. If for every state there were a claim, paying out only in that state, that could be bought and sold, the households most afraid of drought would buy more of the claim that pays in drought years, while households on high ground, unafraid of drought, would sell that claim in exchange for grain in other states. With each household buying and selling according to its own situation, risk would flow to those most willing and best able to bear it, without a separate common store for every kind of disaster.

A premium has two parts: the expected payout, plus compensation for bearing risk and the costs of running the business. For risk that can be diversified, the compensation can be very low; for risk that cannot, the compensation must be higher.

Today's counterpart. Premiums for property, health and life insurance can all be split into the expected payout and a loading for costs. An option is also a state-contingent claim.

Insurance: a state-contingent claim

In 90% of years the harvest is 13 portions; in 10% disaster strikes and it is only 3. Once insured, the household pays a premium every year and receives 10 portions in a disaster year. Drag the loading in the premium and see whether insurance is worth buying.

Good year (90%)
Uninsured
Insured
Disaster year (10%)
Uninsured
Insured

Illustrative: the expected payout is 1 portion a year; premium = expected payout × (1 + loading), where the loading covers compensation for bearing risk and running costs. Whether insurance is worthwhile is compared using logarithmic satisfaction.

6. New problems that insurance brings

Insurance solves the sharing of risk, but it brings two new problems. First, once insured, people may no longer be as careful: if the common store compensates for pest damage, why go to the trouble of controlling pests? This is called moral hazard, and Arrow (1963) and Pauly (1968) discussed how it shows up in health insurance. Second, those most eager to insure are often those at highest risk. Households on low-lying land, most afraid of floods, are the keenest to join, and each household knows its own land best, while the keeper of the common store cannot see it all. If everyone pays the same premium, low-risk households find it not worthwhile and leave; those who remain are riskier and riskier, and the premium has to keep rising.

The keeper of the common store cannot see each household's risk, but can let the households reveal it themselves by offering two agreements. One has a low premium, but in a disaster the household first bears part of the loss itself; the other has a high premium and compensates the loss in full. Households on high ground, rarely flooded, do not mind bearing that part themselves and choose the low premium; households on low-lying land, which need compensation almost every year, would rather pay more and choose full compensation. The households' choices sort them into different risk classes. The cost is that, to keep high-risk households from posing as low-risk ones, low-risk households can receive only partial cover.

The remedies follow from this. Policyholders bear part of the loss themselves, through deductibles and co-insurance, so that they still have an incentive to take precautions and people of different risk sort themselves; premiums are set according to risk, or everyone is required to join, so that only high-risk people do not end up insured.

Today's counterpart. Deductibles and co-payment shares in insurance contracts, premiums that vary with age and health, and compulsory social insurance are all responses to these two problems.

Deductibles: keeping the incentive to take precautions

A disaster causes a loss of 10 portions. Drag the share of the loss the policyholder bears himself and see how his pest-control effort, the chance of disaster and the premium change.

Pest-control effort
Chance of disaster
Annual premium
Self-borne in a disaster year

Illustrative: the more effort put into pest control, the lower the chance of disaster; the policyholder controls pests only for the part of the loss he bears himself, so his effort is proportional to that share. The premium is set at the expected payout.

7. A village of mutual aid: Civilization Simulator, version 5

The simulator unlocks the rule of "Insurance". The village has twenty households, and each harvest varies at random from year to year: part of the variation is each household's own luck, and part is the conditions shared by the whole village. You can choose whether each household keeps its own reserve, joins the mutual granary, or does both, and adjust the weight of village-wide conditions, then see over thirty years how many household-years of hunger there are and how much each household's food varies.

Individual reserves cannot cope with two disaster years in a row, and the mutual granary cannot cope with years when the whole village's harvest fails: relying on the granary alone without reserves, everyone goes hungry at once when the whole village's harvest fails. Diversification and reserves deal with two different kinds of risk, and using both together produces the least hunger.

Civilization Simulator · Chapter 5: a village of mutual aid

Twenty households, with harvests that vary at random each year. Choose individual reserves, the mutual granary or both, drag the weight of village-wide conditions, and see over thirty years how many household-years of hunger there are and how much each household's food varies.

SavingSeedLendingCommitmentMoneyInsuranceBank · Ch. 6

Illustrative model: each household's harvest averages 12 portions with swings of about 40%; the set weight of those swings comes from conditions shared by the whole village, and the rest from each household's own luck. Individual reserves: each household keeps a reserve of 6 portions year-round, 10% of which spoils each year. Mutual granary: the village's total harvest is divided equally among the households. Both: the harvest is divided equally first, then each household keeps a reserve. A year with less than 9 portions eaten counts as hunger.

8. What this step leaves open

With insurance, each household's own bad luck can be shared, but several problems remain unsolved:

  1. Savers want access at any time; borrowers need funds for the long term. The common store and stored grain must both be available at any time, but those clearing land or planting orchards need to borrow for many years.
  2. The risk of a large project is too much for one household. Building a ship or opening new land can succeed or fail on a large scale; one household can neither put up the resources nor bear the failure.
  3. Who bears the risk that cannot be diversified away, and how much they receive for it. This requires a way for those willing to bear risk to share in the gains in proportion to their stake.
  4. People take less care once insured, and those at higher risk are keener to insure. These problems are rooted in asymmetric information and become sharper when managing other people's property on trust.

What this step leaves open

Four problems still unsolved once there is insurance. Click a problem to see the assumption it hides.

A village of mutual aidReserves, sharing and insurance

9. Implications for products

Insurance products and diversified investment portfolios are both built on the mechanisms of this chapter. Three implications follow. First, diversification is an almost free improvement: risk that belongs only to individual holdings can be eliminated by diversifying, so there is no need to bear it, and no compensation should be demanded for it. Second, risk that cannot be diversified away does not disappear because of any product; it can only be transferred to those better able to bear it, and at a price. If a product claims to have eliminated such risk, one must ask where the risk has gone. Third, insurance and diversification both face asymmetric information. Their design should leave users with an appropriate share of responsibility and avoid attracting only those at highest risk.

Chapter summary: insurance and diversification

Four conclusions, each tied to one mechanism in this chapter.

Bearing it alonePrecautionary saving is costly

Withstanding a large crop failure means keeping a large reserve year after year, which spoils and cannot be put to other use.

Bearing it togetherEach household's own bad luck can cancel out

The more households, the fuller the cancelling; the swings per household shrink roughly with the square root of the number of households.

Cannot be diversifiedShared risk can only be transferred

When the whole village's harvest fails, no number of households helps; whoever bears this risk must be compensated.

New problemsInsurance changes behaviour

People take less care once insured, and those at higher risk are keener to insure; deductibles and risk-based premiums are the remedies.

References