Final Exam Solutions - Investment, Capital, and Finance | ECON 422, Exams of Economics

Download Final Exam Solutions - Investment, Capital, and Finance | ECON 422 and more Exams Economics in PDF only on Docsity! Econ 422 Summer 2008 Final Exam Suggested Solutions I Miscellaneous Questions (15 points) 1. Tool Makers, Inc. uses tool and die machines to produce equipment for other firms. The initial cost of one customized tool and die machine is $850,000. This machine costs $10,000 a year to operate. Each machine has a life of 3 years before it is replaced. What is the equivalent annual cost of this machine if the required return is 9%? (5 points) To compute the equivalent annual cost, we first compute the present value of the costs: 2 3 $10,000 $10,000 $10,000$850,000 $875,312.95 1.09 1.09 1.09 PV = + + + = Next, we compute the present value of annuity that pays $1 for 3 years at 9% 31 1(0.09,3) 1 2.53 0.09 1.09 PVA ⎛ ⎞⎛ ⎞= − =⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ The equivalent annual cost is then 875,312.95 $345,796.54 2.53 EAC = = 2. Consider an individual whose utility of wealth function is U(W)=W1/2 . If the person chooses occupation A, his or her wealth is given by the following wealth distribution: Wealth: 1,000,000 2,000,000 Probability: 0.8 0.2 If the person chooses occupation B, his or her wealth is given by the following wealth distribution: Wealth: 1,100,000 1,300,000 Probability: 0.5 0.5 a. Which occupation will the person choose and why? Or will the person be indifferent to the alternatives? Explain. (2.5 points) The individual will choose the occupation that maximizes the expected utility of wealth. The expected utilities for the two occupations are 1/ 2 1/ 2 1/ 2 1/ 2 [ ( )] 0.8(1,000,000) 0.2(2,000,000) 1082.84 [ ( )] 0.5(1,100,000) 0.2(1,300,000) 1094.49 A B E U W E U W = + = = + = Since occupation B gives the higher expected utility, choose B. b. Compute the certainty equivalent wealth and risk premium for occupation A. (2.5 points) The certainty equivalent wealth is the amount of wealth with certainty that gives the same expected utility as the gamble: ( ) [ ( )]c aU W E U W= . Therefore, we solve 1/ 2 2( ) 1082.84 (1082.84) $1,172,548.34c cW W= ⇒ = = The risk premium is [ ] cAE W W− so we compute [ ] 0.8($1,000,000) 0.2($2,000,000) $1,200,000 [ ] $1,200,000 $1,172,548.34 $27,451.66 A c A E W E W W = + = − = − = 3. The probability distribution for next year’s price of Google stock is given by Price P1 $300 $400 $500 $600 Probability 0.1 0.2 0.5 0.2 a. Compute the expected value and variance of next year’s price. (2.5 points) The expected stock price next year is 1[ ] 300(0.1) 400(0.2) 500(0.5) 600(0.2) 480E P = + + + = To compute the variance use 2 2 1 1 1 2 2 2 2 2 1 2 2 2 1 1 1 ( ) [ ] [ ] [ ] (300) (0.1) (400) (0.2) (500) (0.5) (600) (0.2) 238,000 ( ) [ ] [ ] 238,000 (480) 7,600 V P E P E P E P V P E P E P = − = + + + = = − = − = Normal pdf 0 0.5 1 1.5 2 2.5 -0.5 -0.25 0 0.25 0.5 0.75 return pd f μ - 2σ μ + 2σ μ b. What is the approximate probability that the return in any given year is less than - 7.9%? Notice that 0.124 0.203 0.079μ σ− = − = − . Therefore, Pr( 0.079) Pr( )R R μ σ< − = < − . Since the normal curve is symmetric and roughly 67% of the area is between μ σ± , it follows that Pr( 0.079) 0.33/ 2 0.165R < − = = III. Portfolio Theory and CAPM (30 points) 1. Transfer the diagram below to your blue book and use it to answer the following questions (2.5 points each) a. The curved line in the diagram represents expected return and standard deviation values for portfolios of Amazon and Boeing stock. Using this curved line, indicate the location of the efficient portfolios of Amazon and Boeing stock. Efficient portfolios are portfolios with the highest expected return for a given level of risk. Therefore, the efficient portfolios are those portfolios that start at the edge of the bullet (minimum variance portfolio) and are above it. b. On this graph, indicate a portfolio that would be chosen by a very risk averse individual and a portfolio that would be chosen by a very risk tolerant individual. Very risk averse investors will choose a portfolio close to the minimum variance portfolio. Risk tolerant investors will choose a portfolio close to Amazon. 2. Consider creating a portfolio of three stocks A, B and C (Amazon, Boeing and Costco). (2.5 points each) a. Suppose [ ] 0.10, [ ] 0.06, [ ] 0.15A B CE R E R E R= = = . What is the expected return on an equally weighted portfolio of the three assets? [ ] (0.10 0.06 0.15) / 3 0.1034pE R = + + = b. Suppose ( ) 0.20, ( ) 0.15, ( ) 0.25A B CSD R SD R SD R= = = and that ( , ) ( , ) ( , ) 0.2A B A C B CCORR R R CORR R R CORR R R= = = . What is the standard deviation of an equally weighted portfolio of the three assets? For an a portfolio of three assets 2 2 2 2 2 2 1 1 2 2 3 3 1 2 12 1 3 13 2 3 23( ) 2 2 2pV R x x x x x x x x xσ σ σ σ σ σ= + + + + + With equal weights and equal correlations, the above formula simplifies considerably 2 2 2 2 1 1 1 1 2 1 3 2 3( ) ( ) 2 ( ) 0.01911 pV R x σ σ σ ρ σ σ σ σ σ σ⎡ ⎤= + + + + +⎣ ⎦ = The standard deviation is then ( ) 0.01911 0.1382pSD R = = 3. Assume that the T-Bill rate is 5 percent, and the expected return on the market portfolio (e.g., S&P 500) is 13 percent. Using the CAPM, answer the following (2.5 points each): a. What is the market risk premium in this example? What is the beta of the market and what is the beta of T-Bills? The market risk premium is defined as [ ]M fE R r− . Hence, for this example [ ] 0.13 0.05 0.08M fE R r− = − = . By definition, the beta of the market is 1 and the beta of T-Bills is zero (since the T- Bill rate is constant). b. Draw a graph showing how the expected return on any asset varies with its beta (β). That is, draw the security market line (SML). c. What are the expected returns on assets with betas of 0.5 and 1.5? Using the CAPM pricing equation we get [ ] ( [ ] ) [ ] 0.05 0.5(0.13 0.05) 0.9 [ ] 0.05 1.5(0.13 0.05) 1.7 f M fE R r E R r E R E R β= + − = + − = = + − = d. If an asset has a beta of 1.5, what fractions of wealth must be invested in the market and T-bills to have the same expected return as the asset? Invest xM = 1.5 in the market portfolio and invest xf = -0.5 in T-bills (borrow 50% of wealth). IV. Market Efficiency and Miscellaneous (15 points, 5 points each) 1. State the efficient markets hypothesis, and describe the information sets associated with weak-form, semi-strong form and strong form market efficiency. The efficient markets hypothesis says that asset prices fully reflect all current relevant information. The information sets associated with the three forms of market efficiency are Weak form: past prices Semi-strong form: public information Strong form: all information – public and private 2. It is sometimes said that if the securities market is efficient, an investor need only throw darts at the stock pages to pick securities and be just as well off. Do you agree or disagree with this statement? Briefly explain. I would disagree with this statement. The efficient markets hypothesis says that assets are correctly priced and there is no right time to purchase assets. Throwing darts at the stock page will create a portfolio of stocks but you have no control over the risk of the portfolio. Rational investors will choose portfolios commensurate with the risk that they are willing to tolerate. This will not entail a random selection of stocks. 3. What is the random walk model of stock prices? Which form of market efficiency implies that stock prices follow a random walk? The random walk model of stock prices says that price changes are serially random. Mathematically, the random walk model is 1ln ln [ ] random error term t t t t P P E gain ε ε −= + + = V. Option pricing and hedging (20 points) 1. Suppose you think the price of Yahoo! stock will go down significantly over the next three months. a. What type of option investment would you use to profit on this belief? Justify your answer with a payoff diagram. (2.5 points) One option strategy would entail purchasing a put option on Yahoo!, which entitles the holder to sell Yahoo! stock at strike price X over some specified length of time. The put has a payoff equal max(0,X - S(T)). So if the stock price goes down beyond the sum of the price of the put and the exercise price, a profit is made.