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Jan 9, 2024

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1. Suppose that satisfies the independence axiom and there are three lotteries available: p, q and r. Write down one implication of p>q. 1.Indeed, the independence axiom in Expected Utility Theory implies that the preference ordering between lotteries is preserved even when they are mixed or compounded with a third lottery. Given p ≻ q and then we get αp + (1-α)r ≻ αq + (1-α)x. This result means that if an individual prefers lottery p over q, they will still prefer a mixture of lottery p and r over a mixture of lottery q and the risk-free payoff x, regardless of the value of α. 2. Consider the following four assets whose payoffs are as follows, with 0<X<Y, Py<Px, YPy>XPx and a (0,1). Asset A: X with probability Px, with probability 1-Px; Asset B: Y with probability Py, 0 with probability 1-Py; Asset C: X with probability aPx, 0 with probability 1- aPx; Asset D: Y with probability aPy, 0 with probability 1-aPy. An agent chooses A over B. When given the choice between C and D, the same agent chooses D over C. Could this individual’s preference be consistent with vNM expected utility theory? 2.Such preferences are not consistent with vNM expected utility theory. They violate the independence axiom. Consider the set of payoffs {0, X, Y}. Asset C can be a compound lottery that with probability a returns asset A and with probability (1-a) returns 0. 3. You are a basketball player. There are two time periods t ∈ {1, 2}. When you take a shot at time t one of two events can be realized: hitt and misst. Let P(hitt) be the unconditional probability that you make a basket in period t and let P(hitt , hits) be the joint probability if making baskets in periods t and s. Suppose P(hit1)=0.5 and P(hit1, hit2)=0.3: Based on the stated probabilities above, do you have a hot hand? Why or why not? 3.P{Hit2∩Hit1} =P{Hit2|Hit1} *P{Hit1}; Since P {Hit2, Hit1} =0.3, P{Hit1} =0.5, get P{Hit2|Hit1} =0.6. By using Bayes’ rule, P{Hit2∩Miss1} =P{Hit2| Miss1} *P{Miss1}; P {Hit2, Miss1} =P{Hit2}-P {Hit2, Hit1} =0.5-0.3=0.2; P{Hit2|Miss1} =P {Hit2, Miss1}/P{Miss1} =0.2/0.5=0.4. Since P{Hit2| Hit1}>P{Hit2|Miss1}, it is likely to have a hot hand. 4.Risk aversion. Gamble A: A certain payoff of 500; Gamble B: win 1000(50%) or 0(50%). Jaime prefers gamble A over B. Show that Jaime is risk-averse regarding the attitude towards risk. 4. U[E(W)]>E[U(W)]. Jaime prefers A over B, therefore, U(500) =0.5*U(1000)+0.5*U(0) 5.Describe a self-control problem that an economic agent might face.

Sophisticated agents are aware of future self-control problems. How does this awareness result in behaviors that are different from those of naïve agent who are ignorant of future self-control problems? 5. Gym attendance. Individuals typically believe that going to the gym is a good thing but may fail to go regularly due to failures of self-control. Sophisticated agents can only have long-term gym contracts when they are aware of an increased likelihood of going. 6. Iyengar and Kamenica (2010) study simplicity-seeking using evidence from allocation to funds in 401(k) plans. What is their null hypothesis for the relationship between the proportion that individuals invest in equities and the number of funds offered in their 401(k) plan? What do they find? Is this evidence of simplicity-seeking by 401(k) investors? 6.Their null hypothesis is not relationship between equity share and the number of funds offered in a 401(k) plan. They found a negative relationship between equity share and the number of funds offered. This is not simplicity- seeking. They find that allocations and the number of funds is related: more funds lead to lower allocation to equity and higher allocations to bond and money market funds. However, they don’t provide any reasons why funds are simpler than equity funds, they cannot conclude their evidence supports simplicity-seeking. 7. Benartzi and Thaler (2001) study naïve diversification in retirement plans. Under their null hypothesis, what should be the relationship between the number of equity funds in a retirement plan and the average proportion of assets allocated to equity? What do they find? Are their findings consistent with naïve diversification? 7.Under their null hypothesis, there should be no relationship between the number of equity funds offered in a plan and the average proportion of assets allocated to equity in the same plan. They reject the null and their finding suggests that investors may practice naïve diversification. 8. A mutual fund manager claims that the superior performance of her fund is due to in-depth research of firms and economic fundamentals. Does she believe that markets are efficient (in which form of market efficiency)? Does she believe that markets are strong- form efficient? Why or why not? 8. Market efficiency can be classified into three forms: Weak-form efficiency, Semi-strong form efficiency, Strong-form efficiency. She may believe that markets are weak-form efficient, so that historical price information in impounded in price. She doesn’t believe that markets are semi-strong form efficient. If there’s value to research, then prices cannot contain all public information. She certainly doesn’t believe in strong-form efficiency. If she did,

there’s no point to research since insiders will have beat her to profit opportunities. 9. What is Disposition effect? Provide theoretical support for disposition effect . 9.The disposition effect is an anomaly discovered in behavioral finance. It relates to the tendency of investors to sell assets that have increased in value, while keeping assets that have dropped in value. Hersh Shefrin and Meir Statman identified and named the effect in their 1985 paper, which found that people dislike losing significantly more than they enjoy winning. 10. Bernard and Thomas (1989) present evidence of market inefficiency around the time of quarterly earnings announcement. If you were an arbitrageur with deep pockets, how would you exploit their research to make money? Would this strategy involve risk? 10.To exploit their strategy would require investing in a portfolio of stocks with high standardized unexpected earnings (SUE) and shorting a portfolio of stocks with low SUE. If you could implement such a strategy, would be exposed to risk. The alleged mispricing between the high and low SUE portfolios could persist, forcing you to incur borrowing costs. In the short-term the mispricing could increase, causing you to lose money in your position. 11. a) In event studies, how do we calculate abnormal returns? Why do we use abnormal returns instead of raw returns? Does rejecting the null hypothesis of no abnormal returns indicate markets are (semi strong) efficient? 11.(R_it-E[R_t]); Because abnormal returns provide a more accurate expected return in following reasons: Adjust for market movements, Control for risk factors, Evaluate event impact, Comparability; Rejecting the null hypothesis of no abnormal returns suggests that there is evidence of abnormal returns, which could be interpreted as a sign of market inefficiency. In the context of the semi-strong form of market efficiency, if abnormal returns can be consistently achieved by using publicly available information, it would indicate that the market is not fully incorporating that information into asset prices, and therefore, the market is not semi-strong efficient. 12. a) Describe how mistrust of stock market might affect an individual’s decision to participate in the stock market. b) Describe a way to increase stock market participation by reducing mistrust in the market. c) What effects, if any, could reducing investor mistrust have on stock price? 12.a) Mistrust in the market could result in lower expected returns to stocks. While in models that include no other frictions, this could simply result in smaller stock positions. If investors must pay transactions costs to buy and sell shares, low expected returns relative to

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these costs could stop them from buying stock at all. b) Disclosure, law enforcement, insurance schemes. c) Increased participation induces increase in demand, leading to increase in stock prices. 13.Describe one reason why investors may prefer dividend to capital gains? Can you suggest a reason why they might prefer capital gains to dividends? 13.For behavioral explanations for dividend relevance, Self-control; Prospect theory and Avoiding regret will cause you prefer dividend. For someone who prefer capital gains, they may consider Taxes (capital gains are normally taxed at a lower rate than dividends) and Transactions cost(incurring fewer transaction costs compared to regular dividend payouts). 14. In Stein’s (1996) model, are managers with short horizons or managers with long horizons more likely to respond to mispricing in the stock market? 14.Short horizons. 15. Bernard and Thomas (1989) study post-earnings announcement drift. 1.In a market that is semi-strong form efficient. What level of post-earnings announcement drift should you expect? No drift 2. In a market that is strong form efficient, when should information about earnings become impounded in price? Price will change before the earnings announcement if any private information is known to the market . 3. If the authors reject the null hypothesis that abnormal returns (defined as realized returns, less expected returns,) are equal to zero, is this equivalent to rejecting semistrong form efficiency? Why or why not? Not necessarily. Any test of market efficiency is a joint test of market efficiency and asset pricing models.

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