The Insurance Is the Lemon: Failing to Index Contracts

• Date: 01 February 2020
• Author: BENJAMIN HÉBERT,BARNEY HARTMAN‐GLASER
• DOI: http://doi.org/10.1111/jofi.12856
• Published date: 01 February 2020

THE JOURNAL OF FINANCE •VOL. LXXV, NO. 1 •FEBRUARY 2020

The Insurance Is the Lemon: Failing to

Index Contracts

BARNEY HARTMAN-GLASER and BENJAMIN H ´

EBERT∗

ABSTRACT

We model the widespread failure of contracts to share risk using available indices.

A borrower and lender can share risk by conditioning repayments on an index. The

lender has private information about the ability of this index to measure the true state

that the borrower would like to hedge. The lender is risk-averse and thus requires a

premium to insure the borrower. The borrower,however, might be paying something

for nothing if the index is a poor measure of the true state. We provide sufficient

conditions for this effect to cause the borrower to choose a nonindexed contract instead.

ACENTRAL IMPLICATION OF THE LITERATURE on financial contracting is that agents

should structure contracts to share risk as efficiently as possible. In many fi-

nancial markets, standard contracts are simple and do not include risk-sharing

arrangements that condition payments on publicly available indices. A leading

example of this phenomenon is the mortgage market. In this market, home-

owners are exposed to the risk that their homes will decline in value. Lenders

are arguably better equipped to bear this risk and could insulate homeowners

against declines in house prices by making mortgage repayment terms con-

tingent on a house-price index. These types of mortgage contracts have been

widely proposed as a solution to problems facing the mortgage market, such as

the subprime default crises of 2007,1but have failed to supplant the standard

∗Barney Hartman-Glaser is at the Anderson School of Management, University of California,

Los Angeles. Benjamin H´

ebert is with the Graduate School of Business, Stanford University and

the National Bureau of Economic Research. The authors would like to thank Vladimir Asriyan;

Francesca Carapella; Eduardo Davila; Peter DeMarzo; Emmanuel Farhi; Valentin Haddad; David

Hirschleifer; Christopher Hrdlicka; Roger Myerson; Batchimeg Sambalaibat; Jesse Shapiro; Alp

Simsek; Amir Sufi; David Sraer; Sebastian Di Tella; Victoria Vanasco; Jeff Zwiebel; and seminar

and conference participants at Stanford University, UC Berkeley Haas, the University of Wash-

ington Foster, the Federal Reserve Board, the Finance Theory Group Meeting at the University of

Minnesota, Columbia Law School, the Adam Smith Conference, and the 2018 WFA Meetings. We

would particularly like to thank John Kuong and Giorgia Piacentino (discussants), Philip Bond

(Editor), an anonymous associate editor, and two anonymous referees for comments that helped

improved the paper. All remaining errors are our own. We have read The Journal of Finance

disclosure policy and have no conflicts of interest to disclose.

1See, for example, the “Shared Responsibility Mortgage” proposed by Mian and Sufi (2015),

in which interest and principal payments are contingent upon local house price indices, or the

“Shared-Equity Mortgage” proposed by Caplin et al. (2007), in which a borrower receives a second

mortgage where payment is due only upon the sale of the house and is contingent on house value.

DOI: 10.1111/jofi.12856

C2019 the American Finance Association

463

464 The Journal of Finance R

mortgage. Two common explanations for this type of market failure are that

the space of feasible contracts is incomplete (Hart and Moore (1988)) or the

transaction costs associated with implementing risk-sharing contracts entails

are high transaction. Neither of these explanations applies, however, when in-

dices are available that would allow agents to share risk efficiently and that

are almost costless to contract upon.

In this paper, we develop a model in which the failure to condition on indices

and thus efficiently share risk is an equilibrium outcome resulting from asym-

metric information. In our model, an agent, who we call the borrower, seeks

financing from a set of lenders. This financial contract must be written in view

of potential conflicts of interest between the lender and the borrower, which are

related to an “internal,” or idiosyncratic, state. For example, this internal state

could represent the hidden ability of a mortgage borrower to make payments

to the lender. At the same time, there may be some benefits of risk-sharing be-

tween the lender and the borrower over some imperfectly measured state (e.g.,

local area house prices). We refer to this state as “external” to indicate that

it is unaffected by the actions of the lenders and the borrower. The external

state is not directly observable. To realize any risk-sharing benefits, the con-

tracts must condition on some potentially imperfect measurement of the state,

which we call an index (e.g., a house price index). Lenders know the true joint

distribution of the index and the external state (i.e., the quality of the index),

while the borrower does not. In effect, the borrower faces an adverse selection

problem over basis risk when lenders offer an indexed contract.

At least two equilibria can arise in the model. In the first type of equilib-

rium, which we refer to as the full-information optimal contracts equilibrium,

all lenders offer a contract featuring the optimal amount of insurance condi-

tional on the true quality of the index. The full-information optimal contracts

equilibrium exists when there is competition between lenders and features no

loss in efficiency due to asymmetric information about the index. In the second

type of equilibrium, which we refer to as the noncontingent contracts equilib-

rium, all lenders offer a contract that does not condition on the index. To see

why such an equilibrium can arise, consider the borrower’s response when a

single lender deviates and offers a contingent contract. To at least break even

on such a contract, the lender must charge the borrower an insurance premium.

At the same time, the borrower may be concerned that the index is in fact un-

correlated with the risk that she is aiming to insure, that is, that the basis risk

for the contract is too high to justify the premium. Lenders who know the basis

risk is high are happy to offer insurance and charge a high premium because

the insurance is cheap for them to provide (precisely because the basis risk

is high). As a result, the borrower will reject the indexed contract in favor of

a standard noncontingent contract. We note that the noncontingent contracts

equilibrium exists even though the contracting space allows for the use of an

index, there are no transactions costs, and lenders make competing offers.

To illustrate the intuition behind these two equilibria, suppose that there

are two equally likely external states, “good” and “bad.” Now suppose that

a borrower receives offers of one dollar of financing from several competing

The Insurance is the Lemon 465

lenders. The borrower is risk-averse with respect to the external state, meaning

that her expected marginal value of a dollar is 1/2 in the good state and 3/2in

the bad state. The lenders are also risk-averse, but less so than the borrower.

Their expected marginal value of a dollar is 3/4 in the good state and 5/4in

the bad state. Lenders can offer contracts that are contingent upon some index

but not upon the true external state directly. The index can be “high quality,”

in which case it is perfectly correlated with the true underlying state, or “low

quality,” in which case it is independent of the true state and hence unrelated

to the either the borrower’s or the lender’s preferences. The borrower believes

that these two cases are equally likely but lenders observe the quality of the

index before making their offers. Finally, the lenders cannot offer contracts

that specify positive transfers from the lender to the borrower.

Suppose that the lenders make the following offers, depending on the quality

of the index. If the index is of high quality,they offer a contract that calls for the

borrower to repay 8/3 dollars if the realization of the index indicates the good

state and nothing otherwise. If the index is of low quality, they offer a contract

that calls for the borrower to repay one dollar regardless of the realization

of the index. These offers constitute what we call a full-information optimal

contracts equilibrium. To see why they can arise in equilibrium, note that all

lenders earn weakly positive profits and could not possibly earn more by making

different offers. Moreover, given that all lenders have common information, the

borrower can perfectly infer the quality of the index by observing the contracts

that the lenders offer. In other words, it is not possible for a single lender

to convince the borrower that the index is of high quality if all of the other

lenders offer a noncontingent contract. This same intuition carries over to the

second type of equilibrium we describe, that is, the noncontingent contracts

equilibrium.

Now suppose that all lenders offer a contract that calls for the borrower to

repay one dollar regardless of whether the index is of high or low quality.These

offers constitute what we call the noncontingent contracts equilibrium. Can

a single lender gain by deviating and offering the best contingent contract?

Again, the answer is no. If a single lender deviates by offering a contingent

contract, then she will have to charge a premium for it to at least break even,

where by premium, we mean that the contract calls for the borrower to repay

an amount that in expectation exceeds the amount financed. In the case of the

best contingent contract, the expected repayment of the borrower is 4/3, that

is, 8/3 (the repayment if the index is in a good state) times 1/2 (the probability

that the index is in a good state), while the amount financed is one, so that

the premium is 1/3. If the index is of low quality, lenders are risk-neutral with

respect to the index and the premium is pure profit. If the index is of high

quality, the premium is compensation for risk and leaves the lenders with zero

net present value. Consequently, a lender would be at least as willing to make

this offer given a low-quality index as when given a high-quality index. As such,

standard belief refinements imply that the borrower can believe that the index

is low quality after observing this deviation. Given these beliefs, the borrower is

strictly better off when accepting one of the offers of a noncontingent contract.