FINA6372 Behavioural Finance MIDTERM EXAM (Sample Exam Paper with Solutions) Last Name: _____________________ First Name:___________________________ Student # _______________________ Instructions: Important: Students are allowed to bring calculators and a crib sheet (A4-size). The crib sheet can be double-sided. Only hand written crib sheets are allowed. Invigilators will take your crib sheet away if printed, photocopied or scanned materials are found. – Total number of questions: 3 ; – Full Mark: – Write down your answers in the space provided below each question; if you need more space, you can write at the back of the examination papers with proper indications. – Always clearly show all steps in your calculations. Your grades will NOT be based on final solutions only. – Always leave 4 decimals in the ($) numbers in your calculations (e.g. PMT = $10.8924) and, particularly, 6 decimals for interest rates (e.g. r = 0.078643 or 7.8643%). – If you use a pencil while answering the exam questions, you cannot request for a re-grading after all grades are issued. – Good luck! For Instructors/TAs Use ONLY Q1 Q2 Q3 Total 2 Q1: The utility functions of three individuals, X, Y and Z, are provided as follows X: U(w)=ln(w) Y: U(w)=2w Z: U(w)=w2 where w is the wealth of states. All of them have zero initial wealth and faces the following lottery Lottery Probability Outcome Good State 0.6 10 Bad State 0.4 1 1. Compute the expected utility for X, Y, and Z in this lottery. X: E(U)=0.6×ln(10) + 0.4×ln(1) = 1.3816 Y: E(U)=0.6×2×10+0.4×2×1=12.8 Z: E(U) = 0.6×(10)2+0.4×(1)2=60.4 2. Compute the certainty equivalent wealth (CE) for X, Y, and Z for this lottery X: ln(CE)= 1.3816, therefore, CEX=3.9811 Y: 2×CE=12.8, therefore, CEY=6.4 Z: CE2=60.4, therefore, CEZ=7.7717 3. Describe the risk attitudes of X, Y, and Z; briefly state the reason of your argument. Expected payoff = 0.6×10 + 0.4×1=6.4 X: CEX< Expected payoff, therefore, X is risk averse 3 Y: CEX= Expected payoff, therefore, Y is risk neutral. Z: CEZ>Expected payoff, therefore, Z is risk seeking. (One can also prove this by the convexity of the utility curve.) 4. “The expected utility of Z is higher than the expected utility of X; therefore, Z likes the lottery better than X”. Is this statement true or false Please briefly explain. False. The utility function represents an ordinal preference. The number from the utility function is not meaningful and is not comparable across different individuals. The number is only meaningful when we compare the order of two states for the same individual. 4 Q2: Answer following questions based on the story below. Your answers should not exceed the space provided. Nancy bought a goose that lays golden eggs at below fair market value (the expected value of owning the goose for the rest of its life). If the goose lays a golden egg it is worth a lot of money, otherwise the normal egg is worth nothing. The goose lays an egg every month, and the egg is golden a completely random 1% of the time (in other words, the draws are from an independent and identically distributed random variable). Several years pass, and the goose has yet to lay a golden egg for Nancy, although the previous owner told Nancy that last time the goose laid a golden egg it took several years. Nancy gets an offer to sell the goose at above its fair market value, which would nonetheless be less than the price Nancy originally paid for the goose when it was younger because the goose is getting old. Nancy rejects the offer. 1. “Nancy’s behavior can be explained by expected utility theory, assuming Nancy is risk averse.” Is this statement true or false Please briefly explain. False. Nancy is being offered more than the fair market value for the goose. Just like Andrei, if Nancy is being offered the expected value of the goose as a sure thing and is risk averse, she should sell the goose. Since she is being offered more than the fair market value, she should certainly sell the goose. (b) “Nancy’s behavior can be explained by prospect theory.” Is this statement true or false Please briefly explain. True. If Nancy’s reference point is the price at which she bought the goose, prospect theory implies that she should be risk seeking in the loss domain. This is captured by a convex value function in the loss domain. She would also be loss averse, which might drive her behavior, but the curvature of the value function is the right thing to focus on here. (c) If we do not consider utility functions, which behavioral bias can explain Nancy’s behavior best Nancy’s behavior can be explained by the representativeness heuristic. Specifically, even though the goose’s egg laying is totally random, the fact that it took several years to lay the golden egg last time might lead Nancy to over-estimate the probability that the goose will lay a golden egg soon since the goose has not laid a golden egg in past few years. This is an example of ignoring base rates and Gambler’s Fallacy due to representativeness heuristics. We give full marks if you use the phrases “Base-rate Neglect” or “Gambler’s Fallacy” even if you did not mention representativeness heuristics Disposition effect can explain this observation as well, however, disposition effect is related to the prospect theory, which is utility-function-related. So it should not be included in the answer. 5 Q3: Please read the following tables carefully and answer the following questions. 1. The diagram below suggests that, when a firm is recently added into S&P500 index, the stock of this firm is likely to outperform its peers. How would you explain this pattern What is the role of limits of arbitrage in this observation The figure suggests that firms, after being included in the major indices (such as S&P 500), outperform other firms with similar firm characteristics. The main driver is the purchase from index funds. Index funds have to track indices really closely. When the constituents of indices are changing, they have to update their portfolios, i.e., selling the stocks that are dropped from the indices and buying the stocks that are added into the indices. When all index funds and quasi-index funds (funds that follow indices closely) are buying the added stocks, prices of these stocks go up. For noise traders, a stock, after its addition into major index, is more visible. With the interactive effect between stock price increase and investor’s attention, noise traders will start purchasing this stock. 6 Obviously, the upward price movement is not driven by cash flow or discount news of these stocks. It is purely driven by the excessive demand of financial institutions to replicate the performance of indices and the attention of noise traders. Therefore, the added stocks are temporarily overvalued. Limits of arbitrage plays an important role in the persistence of overvaluation, i.e., the arbitrageurs are unwilling to trade against the overvaluation because of certain risks and costs. This explains why such price effect does not disappear in the short run. 2. If this empirical pattern indeed reflects the mispricing driven by the limits of arbitrage, describe the type of risks/costs that can deter arbitrageurs to eliminate mispricing. Risks/Costs Yes/No (circle one) Brief Explanations Fundamental risks Yes It is not always easy to find a similar stock as the stock added into the index. If it is difficult to find such a stock, it is very difficult to construct a hedging portfolio to “cancel out” fundamental risk factors (i.e., such as industry-level risks) in the hedging position. Noise trader risks Yes When index funds are buying, noise traders can follow. If noise traders continue to buy, the price of this stock will increase for a while before going back to its fundamental. Arbitrageurs are not willing to trade against the overvaluation before most noise traders jump into the market. Horizon risks Yes Same as above (one can combine these two risks together in the solution). Arbitrageurs don’t know when noise traders would stop buying these stocks. In this case, arbitrageurs will avoid trading against the mispricing too early. Model risks No For this setting, there is no model risks Implementation costs Yes, but very minor For stocks added into major indices, they are usually of large companies. Therefore, the transactional costs are very minor. Also, for these stocks, it is not very 7 costly to borrow them from brokers and short them. So overall, the implementation costs are quite minor. 3. Can you think of an argument that this empirical pattern might be driven by rational economic reasons instead of the mispricing under limits of arbitrage If so, how would you distinguish these two arguments (i.e., rational explanations vs. mispricing under limits of arbitrage) The rational explanation: Firms are added into indices for a reason. Arguably, one reason can be that Standard & Poor expects the added firm represents the industry better than the dropped firm given its profitability and industry leadership is currently underestimated. If this is true, then the addition into S&P500 index itself signals a new information to the market about S&P’s judgment about the added firm’s future cash flows and discount rate. In this case, the adjustment of price should be rational and permanent where increased prices just reflect a high cash flow and a low discount rate in the future.
