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Problem 5. Bertrand Competition in discrete increments (64 points) (Note: This problem resembles one on last year's exam, but it's not the same, read carefully) Consider a variant of the Bertrand model of competition with two firms that we covered in class. The difference from the model in class is that prices are not a continuous variable, but rather a discrete variable. Prices vary in multiples of 1 cent. Firms can charge prices of 0, .01, .02, .03.... etc. The profits of firm i are (pi - c) D (p:) if pi Pj- Demand is extremely inelastic, that is, D (p) = @ for all p. Both firms have the same marginal cost o = c2 = c, and the marginal cost c is a multiple of 1 cent. (The firm can charge c-.01, c, c+ .01, c+ .02, etc.) Consider first the case in which the two firms move simultaneously (as we did in class), and apply the Nash Equilibrium concept. 1. Write down the definition of Nash Equilibrium as it applies to this game, that is, with p; as the strategy of player i and ;(pi, p; ) as the function that player 1 maximizes. Provide both the formal definition and the intuition. Do not substitute in the expression for a;- (8 points) 2. Show that pi = p; = c (that is, marginal cost pricing) is a first Nash Equilibrium of this game. (8 points) 3. Show that pi = p; = c + .01 is a second Nash Equilibrium of this game. (8 points) 4. (Harder) Can you find another Nash Equilibrium (you need to prove that it is a Nash Equilibrium) [Hint: The peculiar feature of this setup is that the firm can only charge prices that are multiples of 1 cent] Why does it matter that demand is inelastic? (10 points) 5. Now, we change the setup in just one way. The game is now played sequentially, that is, firm 1 moves first, and firm 2 follows after observing the price choice of firm 1. We apply Subgame Perfection to solve this game, and therefore start from the last period, from the choice of player 2. Player 2's strategy will be a function of Player I's price p1- Find the best response for player 2 as a function of pi, that is, find p; (pi) . (10 points) 6. Now let's continue with the backward induction and go back to player 1. Player 1 anticipates the best response of player 2 and chooses the price p, that will yiled the highest profit. What is this price pi? To simplify the solution, assume that player 2 responds to a price of c + .02 by also setting price c + .02 (8 points) 7. Write down the subgame perfect equilibrium. How does it differ from the set of Nash Equilibria of the symultaneous game? (8 points) 8. Can you conjecture how the solution of the dynamic Bertrand game will differ if firms can set price continuously? (4 points)3?. In a downward sloping yield curve environment, a) the liquidity premium cannot exist b} according to the expectations approach, long-term rates are no longer an average of current and expected lture rates c) expected future short term rates cannot be greater than the eta-rent short term rate d} a and b are correct 38. According to the Expectations Approach to the term structure a) the forward rate is not a good estimate of the expected future 1-year rate b} investors are risk averse c) when the term structure is in equilibrium. the forward rate is equal to the expected rate d} none of the above 39. According to the liquidity premium approach to the term structure a) The investors' subjective degree of risk aversion is embedded in the 2-year rate b} the equilibrium 2-year rate 2? an average of 1-year and expected tture 1-year rate c) the forward rate 23 the expected rate due to a risk premium d} all of the above 40. The buyer ofa put and seller ofa call a) both are potential sellers of the underlying asset b} both have rights and not obligations c) both prot if the price of the underlying asset falls d} a and c are correct 41. A protective put a) combines a long put with long stock b} creates a long call c) prots when the underlying asset's stock price increases d} has downside protection e) all of the above 42. Long a straddle a) is a bet on volatility b} prots when nothing happens c) prots with wide swings in either direction of the price of the underlying asset d} is the same as a protective put e) a and c are correct 43. The price volatility ofa bond during a year, in general1 depends upon {a} the duration of the bond (b) the volatility of interest rates {c} the volatility of expected inflation (d) all of the above 44. Which ofthe following statements is false: a) When current yield is greater than yield to maturity, the bond is selling at a premium b} The price of a semiqannual or an annual coupon paying bond will be the same if their coupon rate is the same as yield to maturity regardless of differences in maturity c) The concept of yield to maturity suffers born the reinvestment assumption for both semi-annual and annual coupon paying bonds d} Ifl invest $100 in a 10% coupon, 2-year bond at par. I will certainly get $121 at the end of the two years 45. An executive is given two choices, either receive nontransferable one-year European call options on 1000 shares with an exercise price oflt} or get an extra $1,000 in bonus at the end of the year for every point that the company's stock exceeds lIJCI dollars. Which bonus plan should she choose to provide her with the largest dollar payout?I a) Take the options b} Take the money c) They are the same 35. For a Treasury Bil], the EYE [{F-PJJ'T'Ft where t = x365] a} assumes compounding within a year b} assumes simple interest c} takes the periodic rate and multiplies by the number of periods d} b and c are true 36. Assuming you hold an annual pay coupon bearing bond tn maturity+ its annual rettu'n [r-ann = {thVoH lt't}-1] is equal to a} the "t'Tl'v'l if you can and do reinvest at a fixed rate b} the EAR c} the "t'Tl'v'l if you can and do reinvest at the YTlvi d} none of the above 3?. In a downward sloping yield curve environment. a} the liquidity premium cannot exist b} according to the expectations approach, long-term rates are no longer an average of current and expected future rates c} expected future short term rates cannot be greater than the current short term rate d} a and h are correct 38. According to the Expectations Approach to the term structure a} the forward rate is not a good estimate of the expected future 1-year rate b} investors are risk averse c} when tlte term structure is in equilibrium. the forward rate is equal to the expected rate d} none of the above 39. According to the liquidity premium approach to the term structure a} The investors' subjective degree of risk aversion is embedded in the 2-year rate b} the equilibritnn 2-year rate 3' an average of 1-year and expected future 1-year rate c} the forward rate :5 the expected rate due to a risk premium d} all of the above 28. The Liquidity Premium theory says {2 are right}: a. The equilibrium 2 year rate = forward rate b. The equilibrium 2 year rate is greater than the average of the current and expected future short term rates c. The expected future short term rate = the forward rate d. The expected future short-term rate is less titan the forward rate 29. The Liquidity Premium theory holds because investors are risk averse and because there are: a. More 2 year investors than one year investors b. More 2 year securities titan iTwo-year\" investors c. More one year securities than one year investors d. All of the above 3|}. Assume a zero coupon bond has duration = ill years and a 31} year 'bond has an 13% coupon and a duration =lIEI years. Assume urtber that the yields on both bonds are the same and then change by the identical intinitesimally small amount. Then, the price volatility of the 3|] year will be: a. Equal to the price volatility of the zero b. Less than the price volatility of the zero c. Greater than the price volatility of the zero d. Can't tell 3L The ability to replicate an option with a position in the underlying stock depends crucially on: {a} dynamically adjusting the hedge ratio on a continuous basis (b) correctly predicting tomorrowis stock price {c} properly estimating the stock's B {d} all of the above 32. Bonds with call provisions are more desirable than noncallable and generally higher priced less desirable than non callable and generally lower priced more desirable than non callable and generally higher priced are not worth buying sap-s 33. A party will enter a Swap agreement to: a} Reduce rislt exposure on its balance sheet b} Speculate c} Immunization against interest rate changes d} Not enough information given to determine 34. An upcoming event suggests that there will be signicant movement in the share price1 but you're not sure in which direction. Which position would you choose? a} Long a call b] Short a call c} Long a straddle d] Long a protective put 6. The efficient market hypothesis says (a) No one can ever beat the market over a ten year period (b) Insider trading should be illegal (c) Everyone should hold the same portfolio (d) None of the above 7. According to the liquidity premium theory, an upward sloping yield implies (a) Short-term rates are expected to rise (b) Long-term rates are expected to rise (c) Short-term rates are definitely not expected to decline (d) You cannot tell 8. A coupon bond that pays interest of $100 annually has a par value of $1,000, matures in 5 years, and is selling today at a $72 discount from par value. The yield to maturity on this bond is a) 6.00% b) 8.33% c) 10.39% d) 12.00% e) 60.00% 9. Comparing a long put position with a short call position reveals the following common element: (a) both positions have rights but no obligations (b) both positions benefit from an increase in interest rates (c) both positions will lose money if the price of the underlying remains unchanged (d) both positions are potential sellers of the underlying asset 10. If corporate insiders who buy stock in their companies earn the same risk adjusted return as other investors, then: (a) the market is not strong form efficient (b) the market is strong form efficient (c) they do not hold well diversified portfolios (d) then the beta of their portfolios must be one 1 1. To lend money starting at the beginning of next year for one year at a currently known rate you can: (a) sell short a two year security and buy a one year security (b) buy a two year security and sell short a one year security (c) buy a two year security (d) sell short a two-year security 12. One thousand dollars invested in a zero coupon bond with five years to maturity will produce the same amount of money after five years as $1000 invested in a 10 percent coupon bond with five years to maturity: (a) if they are both priced to yield 10 percent to maturity (b) if the investor can and does reinvest the coupons at the yield to maturity (c) if (a) and (b) hold (d) under no circumstances 13. A one basis point decrease in yield on a bond with a duration of 10 years and a yield to maturity of 11 percent produces a change in the price of a $100 face value bond from $90.00 to: (a) 90.05 (b) 89.92 (c) 90.11 (d) 90.08 14. The duration of a 5 year zero coupon bond is lower when the interest rate is: (a) higher (b) lower (c)unchanged (d) none of the above1. Assume next year's dividends are $3.00 per share and they are expected to grow at 4 percent per year. If the risk adjusted discount rate is 7 percent, then the constant rate of growth model implies a stock price of: (a) 80 (b) 90 (c) 100 (d) 110 2 Assuming the CAPM holds, the most appropriate discount rate to use in valuing a project to double the size of an all-equity firm's main factory is: (a) the risk free rate plus the company's beta times (RM-RF) . (b) the risk free rate plus the standard deviation of the project's returns (c) the internal rate of return on the project (d) the risk free rate plus the company's beta 3. A bond's duration is higher when (a) The coupon rate is higher (b) The coupon rate is lower ( c ) Yield to maturity is higher (d) None of the above 4. If the one year spot rate is 4% and the forward rates for years 2, 3, 4, 5 are 5%, 7%, 8%, and 6%, respectively, then today's interest rate on a five-year bond should be a. 4% 5% 6% 7% 8% 5. The Dividend Discount Model (DDM) a. is a valuation model only for companies that have paid dividends b is a dividend distribution model used by corporate managers for dividend decisions C. is a valuation model for new issues only accounts for risk by discounting with a risk adjusted discount rate e . a and d 6. The efficient market hypothesis says (a) No one can ever beat the market over a ten year period (b) Insider trading should be illegal (c) Everyone should hold the same portfolio (d) None of the above 7. According to the liquidity premium theory, an upward sloping yield implies (a) Short-term rates are expected to rise (b) Long-term rates are expected to rise (c) Short-term rates are definitely not expected to decline (d) You cannot tell 8. A coupon bond that pays interest of $100 annually has a par value of $1,000, matures in 5 years, and is selling today at a $72 discount from par value. The yield to maturity on this bond is a) 6.00% b) 8.33% c) 10.39% d) 12.00% e) 60.00%