A Theory of Intermediated Investment with Hyperbolic Discounting Investors

1. A Theory of Intermediated Investment with Hyperbolic Discounting Investors GAO Feng (Tsinghua SEM) HE Ping (Tsinghua SEM) HE Xi (MITEconomics)

2. Preface • A long time ago, a visitor from out of town came to a tour in Manhattan. At the end of the tour they took him to the financial district. When they arrived to Battery Park the guide showed him some nice yachts anchoring there, and said, "Here are the yachts of our bankers and stockbrokers." "And where are the yachts of the investors?" asked the naive visitor.

3. Investment • Investment decisions are intertemporal choices involving tradeoffs among costs and benefits occurring at different times, which not only affect one's health, wealth, and happiness, but may also determine the economic prosperity of nations • Fisher (1930): investment is not an end in itself but rather a process for distributing consumption over time • Major concerns: return, risk, information acquisition, life-cycle, liquidity constraint, risk preference, background risk, etc.

4. Related Studies • Samuelson (1969, REStat), Merton (1969, REStat): dynamic programming with uncertainty • Ehrlich and Hamlen (1995, JEDC): precommitment strategy with intermittent revision • Campbell and Viceira (1999, QJE): time varying investment opportunities • Viceira (2001, JF): background risk, life-cycle • Gollier (2002, JME): Liquidity constraint, decreasing aversion to risk on wealth • Chacko and Viceira (2005, RFS): incomplete market with stochastic volatility

5. Empirical Facts • Investment behaviors are more complex than what most standard theories could explain • Barber and Odean (2002, RFS): online trading make investors trade more actively but less profitable • Barnea, Cronqvist and Siegel (2010, JFE): genetic factor is critical for investor behavior • He and Hu (2010, RBF): horizon effect • Mastrobuoni and Weinberg (2009, AEJ-EP): consumptions are not smoothed • Meier and Sprenger (2010, AEJ-AE): individuals with present-biased preference over-borrow on their credit cards

6. Behavioral Theories and Investment • Barberis & Huang (2001, JF): mental accounting and loss aversion • Angeletos, Laibson, Repetto, Tobacman and Weinberg (2001, JEP); Harris & Laibson (2001, Econometrica); Salanie and Treich (2006, EER): hyperbolic discounting • Grenadier & Wang (2007, JFE): real options investment model with hyperbolic discounting entrepreneurs • Munk (2008, JEDC): habit formation

7. Hyperbolic Discounting • Frederick, Loewenstein and O’Donoghue (2002, JEL): given two similar rewards, humans show a preference for one that arrives sooner rather than later, but valuations fall very rapidly for small delay periods, but then fall slowly for longer delay periods

8. Facts Related to Hyperbolic Discounting • Time inconsistent preferences, implying a motive for consumers to constrain their own future choices (Laibson, QJE 1997) • Under-saving (Laibson, EER 1998; Diamond and Koszegi, JPubE 2003; Salanie and Treich, EER 2006) • Over-borrowing (Heidhues and Kőszegi, AER 2010) • Use of commitment device (Basu, AEJ-Micro 2011)

9. The Role of Intermediaries for Investors • Information production: He (2007, RFS); Gorton and He (2008, RES) • Monitoring: Diamond (1984, RES) • Screening: Bernanke and Blinder (1988, AER) • Liquidity provider: Diamond and Dybvig (1983, JPE) • Risk transformation: Diamond (1984, RES) • Maturity transformation: Diamond and Dybvig (1983, JPE) • Payment methods: He, Huang and Wright (2005, IER)

10. Goal of This Paper • Time inconsistent preference generates a liquidity shortage for the investor who invests on his own • Financial intermediaries make investments on behalf of the investors and provide liquidity for unsophisticated investors • The financial intermediaries in our model can be interpreted as banks, pension funds, mutual funds, etc.

11. Related Works • DellaVigna and Malmendier (2004, QJE): contract design with time inconsistency (monopoly firm) • Heidhues and Kőszegi (2010, AER): credit contract with time inconsistency (competitive firm) • Basu (2011, AEJ-Micro): individuals join rotational savings and credit associations (roscas) to fund repeated purchases of nondivisible goods without defect even when there is no punishment, roscas serves a commitment device

12. Agenda • Basic model • Competitive equilibrium • Linear contract and term premium • Discussions • Conclusion

13. Basic Model

14. Investment Technology • Three dates (t=0,1,2) • Each agent is endowed with 1 unit of good at date 0, and consumes at date 1 and 2 • The good can be stored with 0 return, or invested in a project at date 0 with a return R > 1 at date 2, if it is liquidated at date 1, one can get 1

15. Time Inconsistent Preferences • Self 0’s utility is u(c1) + u(c2) • Self 1’s utility is u(c1) + βu(c2) • Self 0 believes that self 1’s utility

16. First Best • We measure welfare using long-run self-0 utility • The first best solution does not depend on degree of time-inconsistency

17. Diagram of Proof c1 1 R c2

18. Autarky • Investors cannot commit, and liquidation has no cost, so they will liquidate some of the investment for consumption at date 1 based on their date 1 preference regardless what they believe at date 0

19. Diagram of Proof c1 1 R c2

20. Ineffective Market • In the autarky case, if we allow for trading at date 1, that is, an investor can trade his date 2 consumption from his investment for date 1 consumption, investors will have the same consumptions as in autarky case • Proof: The price of date 2 consumption, p, must be 1/R, otherwise either (1,0) or (0,R) will dominate all other points on the budget line and it cannot be equilibrium

21. Diagram of Proof c1 pR p > 1/R 1 p = 1/R p < 1/R R 1/p c2

22. Role of Intermediary • At its own best interest, an intermediary can improve the welfare of an investor with time inconsistent preference by offering a contract that punishes early withdraw

23. Competitive Equilibrium

24. Incentive Compatible Contract • Assume there are finite β’s among people, with β1 < β2 < … < βN, and , and financial intermediaries offer a finite menu of repayment options C = {(c1s, c2s)} . • An incentive compatible map (c1(.), c2(.)): {β1, β2, … ,βN} R+ satisfies the following condition:

25. Equilibrium Definition • We define a competitive equilibrium as a contract C offered by the financial intermediaries with an incentive compatible map (c1(.), c2(.)) that satisfies the following properties: • Zero-profit • No profitable deviation, there exists no contact C’ with incentive-compatible map (c1‘(.), c2‘(.)) such that for some β, u(c1‘(β)) + βu(c2‘(β)) > u(c1(β)) + βu(c2(β)), and C’ yields positive profits • Non-redundancy

26. Observable Naïve Investors • The financial intermediary solves • u is the perceived utility from the perspective of date 0 if she accepts the contract

27. Equilibrium Outcome • PC must be binding • IC must be binding • PCC is equivalent to • Perceived date 1 consumption is zero: • Competitiveness will drive the financial intermediary’s profits to zero • The problem is equivalent to setting the profit to be zero with PC binding through lifting u

28. Equilibrium Contract • For a naïve investor, the competitive-equilibrium contract has two repayment options, with the investor expecting to choose , and actually choosing c1 and c2 satisfying • Equivalent to the autarky case

29. Diagram of the Result c1 1 R c2

30. Intuition • At date 0, the intermediary will offer a perceived-choice contract with very high date 2 consumption but zero date 1 consumption while expecting the investor with a need of immediate gratification at date 1 will switch to a contract with early withdraw of date 2 consumption despite of a high penalty • The more date 2 perceived-consumption, the greater drop in utility when date 1 comes, the more desperate the investor is, and the less the intermediary needs to offer in an alternative contract

31. Observable Sophisticated Investors • The financial intermediary solves

32. Equilibrium Outcome • PC must be binding • Competitiveness will drive the financial intermediary’s profits to zero • The problem is equivalent to setting the profit to be zero with PC binding through lifting u

33. Equilibrium Contract • For a sophisticated investor, the competitive-equilibrium contract has a single repayment option satisfying • Equivalent to the first best case

34. Diagram of the Result c1 1 Equilibrium consumption for naïve investors Equilibrium consumption for sophisticated investors R c2

35. Intuition • A sophisticated investor rationally expect his own preference change and his choice at date 1, which is the only relevant choice for his utility at date 0

36. Summary for Observable Preference • For a naïve investor, the financial intermediary offers a contract with a punishment for early withdraw, the welfare of a naïve investor is NOT improved • However, if liquidation is costly, then the financial intermediary can improve welfare as it avoids costly liquidation • For a sophisticated investor, the first best is achieved, and his welfare is strictly improved • If everyone else is as naïve as you are, or everyone knows that you are naïve, making investment through a zero-profit intermediary does not help nor hurt

37. Unobservable Preference • Again we study the most simple case: all investors has the same at date 0, and investors are naïve ( ) with probability π, and investors are sophisticated ( ) with probability 1 – π • All investors choose the same contract (c1s, c2s) at date 0, but naïve investors will switch to (c1n, c2n) at date 1

38. Equilibrium Outcome • PC must be binding • ICn must be binding • ICs implies c1s < c1nand c2s > c2n • Competitiveness will drive the financial intermediary’s profits to zero • The problem is equivalent to setting the profit to be zero with PC binding through lifting u

39. Equilibrium Contract • Suppose all investors has the same at date 0, and investors are naïve ( ) with probability π, and investors are sophisticated ( ) with probability 1 – π, the competitive-equilibrium contract has two repayment options. All investors choosing the same contract (c1s, c2s) at date 0, but naïve investors will switch to (c1n, c2n) at date 1. We have

40. Interpretation of the Results • “Efficiency-at-the-top”: the repayment schedule of naïve investors is similar to the case with known preference, but this is not the case for the sophisticated investors, who get a more back-loaded repayment schedule • There is a discontinuity at full sophistication

41. Cross-Subsidy Effect • Suppose all investors has the same at date 0, and investors are naïve with probability π, and investors are sophisticated with probability 1 – π. In a competitive equilibrium, the intermediary makes money on the naïve investors but loses money on the sophisticated investors. Moreover, the sophisticated investors’ welfare in the competitive equilibrium is strictly increasing in π

42. Diagram of the Result c1 (c1n,c2n) (c1s,c2s) c2

43. Intuition • The intermediary offers a contract with very high long-term return and large penalty upon early withdraw, and it makes a profit from the naïve investors, who suffer from the need for immediate gratification, while losing money to the sophisticated investors, who enjoy the high long-term return

44. Summary for Unobservable Preference • The welfare of a naïve investor is LOWER than the case of autarky • For a sophisticated investor, the first best is NOTachieved, and his welfare is strictly improvedupon the autarky case • If you are naïve, do not pretend to be sophisticated, because that will hurt you • The sophisticated investors are happier if there are more naïve investors, but they always think their repayment structure is distorted with the existence of naïve investors

45. Linear Contract and Term Premium

46. Restricted Linear Contracting • Our earlier analyses focus on the case in which investors can only liquidate a predetermined fixed portion of his investment contract • In practice, restricted linear contract corresponds to the case in which investors can liquidate any portion of his investment contract • But do more options bring welfare improvement to the investors? in particular, the naïve investors?

47. Observable Sophisticated Investors • The financial intermediary solves

48. Diagram of the Result c1 1 c2 R

49. Intuition • A perfectly sophisticated depositor is fully aware of her time inconsistency, so it would be profit maximizing to offer her a contract with an interest rate of which aligns self 1's interest with the self 0’s welfare • The first best is still achieved

50. Observable Naïve Investors • The financial intermediary solves