4. Martha S. has a choice of two assets: The first is a risk-free asset that offers a rate of return of r

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1 Spring / IA 350, Interediate Microeconoics / Proble Set 3 1. Suppose that a stock has a beta of 1.5, the return of the arket is 10%, and the risk-free rate of return is 5%. What is the expected rate of return this stock according to the CAPM?. If the risk-free rate of return is 6%, and if a risky asset is available with an expected return of 9% and a standard deviation of 3%, what is the axiu rate of return you can achieve if you are willing to accept a standard deviation of %? What percentage of your wealth would have to be invested in the risky asset? 3. If a stock has a b of 1., the return on the arket is 8%, and the risk-free rate of return is 3%, what expected rate of return should this stock offer according to the CAPM? If the expected value of the stock one year in the future is $100, what price should the stock be selling for today? 4. Martha S. has a choice of two assets: The first is a risk-free asset that offers a rate of return of r f, and the second is a risky asset that has an expected rate of return of r and a standard deviation of. a. If x is the percent of wealth that Martha invests in the risky asset, what is the equation for the expected rate of return on the portfolio? What is the equation for the standard deviation of the portfolio? b. By solving the second equation above for x and substituting the result into the first equation, derive an expression for the rate of return on the portfolio in ters of the portfolio's riskiness. c. Suppose that Martha can borrow oney at an interest rate of rf and invest it in the risky asset. If r 0%, r 10%, and 10%, what will be Martha's expected return if she borrows an f aount equal to 100% of her initial wealth and invests it in the risky asset? d. Suppose that Martha can borrow or lend at the risk-free rate. If r 0%, r 10%, and 10%, what is the forula for the "budget line" that Martha faces?. Plot this budget line below. e. Which of the following risky assets would Martha prefer to her present risky asst, assuing she can only invest in one risky asset at a tie and that she can invest a fraction of her wealth in whichever risky asset she chooses. Write the word "better", "worse", or "sae" after each of these assets: Asset A, with 17%,, 5% r A Asset B, with 30%,, 5% A r B B f

2 Spring / IA 350, Interediate Microeconoics / Proble Set 3 Asset C, with 11%,, 1% r C Asset, with 5%,, 14% r U r x r,. How uch of f. Now suppose that Martha's utility function has the for x x x her portfolio will she invest in the original risky asset? Expected Return Assuing that the Capital Asset Pricing Model is valid, coplete the following table. In this table, p 0 is the current price of asset i, and E[p 1 ] is the expected price of asset i in the next period. r f r r i i p 0 E[p 1 ] 10% 0% 10% % 0% % 0% % 30% % % 0 80

3 Spring / IA 350, Interediate Microeconoics / Proble Set 3 6. The precise forula for the variance of a portfolio of two securities is: 1, 1 w 1 w w w 1 1, and the covariance of securities1 and, where 1, 1, Variance of the portfolio wi Weight of security i in theporfolio, where i Variance of security i 1, Covariance of securities1and, is thecorrelation coefficient of the twosecurities. 1, 1, 1 Using these forulas, calculate the expected returns for portfolios A, B, and C as directed below, where the underlying securities have the following characteristics: Security I Security II Expected Return 16% 8% Risk (Standard eviation) 30% 15% a. In portfolio A, the weight of security I is 0.50 and the weight of security II is 0.50, and the correlation coefficient of the two securities is 0.5. b. In portfolio B, the weight of security I is 0.50 and the weight of security II is 0.50, and the correlation coefficient of the two securities is c. In portfolio A, the weight of security I is 0.5 and the weight of security II is 0.75, and the correlation coefficient of the two securities is 0.5. d. Record these results in the table below, and briefly coent on the, focusing on the iportance of the correlation between the securities. Expected Return Portfolio A Portfolio B Portfolio C Risk (Standard eviation)

4 Spring / IA 350, Interediate Microeconoics / Proble Set 3 7. Katie has $60,000 to invest. Of this aount, she reserves $15,000 for risk-free assets, leaving $45,000 to allocate aong the risky securities. There are only three firs in the econoy, each of which issues coon stock. The following table provides inforation of the price per share and total nuber of shares outstanding for each of the three firs: Fir Share Price Shares Outstanding 1 $150 70,000,000 $80 93,750,000 3 $5 1,000,000,000 a. Calculate the arket value of each fir and the aggregate arket value of the firs. b. According to the capital asset pricing odel, in what proportions should Katie allocate the $45,000 across the risky securities in the econoy if she is to construct the arket portfolio? c. How uch oney will Katie allocate to each security? 8. Annualizing returns and volatility: a. If average daily returns are (0.04%) and the standard deviation of daily returns is 0.01 (1%), calculate the annualized return and annual volatility. b. If average weekly returns are (0.3%) and the standard deviation of weekly returns is 0.05 (5%), calculate the annualized return and annual volatility. c. If average onthly returns are 0.01 (1%) and the standard deviation of onthly returns is 0.06 (6%), calculate the annualized return and annual volatility. 9. For each of the following deand functions, derive an expression for the price elasticity of deand: a. p 60 p b. p a bp p 40 p c. p Ap b d. p p 3 e. p p a f. b : : : : : :

5 Spring / IA 350, Interediate Microeconoics / Proble Set Suppose that the deand for kitty litter, in pounds, is ln p, 000 p ln price and is incoe. 1, where p is the a. What is the price elasticity of deand for kitty litter when p = and = 500? When p = 3 and = 500? When p = 4 and = 1,500? b. What is the incoe elasticity of deand for kitty litter when p = and = 500? When p = and = 1,000? When p = 3 and = 1,500? c. Write out a general expression for the price elasticity of deand when price is p and incoe is : d. Write out a general expression for the incoe elasticity of deand when price is p and incoe is : 11. The deand function for football tickets for a typical gae at a large Midwestern university is qp 00, , 000p. The university has a clever athletic director who sets ticket prices for football gaes so as to axiize revenue. The seating capacity of the university's football stadiu is 100,000. a. Write down the inverse deand function: b. Write an expression for total revenue: Write an expression for arginal revenue: c. On the graph below, use blue ink to draw the inverse deand function and use red ink to draw the arginal revenue function. Also draw a vertical blue line representing the capacity of the stadiu. d. What price will generate the axiu revenue? How any tickets will be sold at this price? e. At this quantity, what is arginal revenue? At this quantity, what is the price elasticity of deand? Will the stadiu be full? f. Suppose that a series of winning seasons has caused the deand curve for football tickets to shift upward. The new deand function is qp 300, , 000p. Write out the new inverse deand function: g. Write an expression for arginal revenue as a function of output. MR q

6 Spring / IA 350, Interediate Microeconoics / Proble Set 3 Add this new deand function (red ink) and new arginal revenue function (black ink) to the diagra. Ticket Price Quantity (in thousands) h. Ignoring stadiu capacity, what price would generate axiu revenue? What quantity of tickets would be sold at this price? i. As you should have noticed, the quantity that would axiize total revenue given the new higher deand curve is greater than the capacity of the stadiu. As clever as he is, the athletic director cannot sell seats that he doesn't have. He notices that his arginal revenue is positive for any nuber of seats that he sells up to the capacity of the stadiu. Therefore, in order to axiize his revenue, he should sell tickets at a price of. j. When he does, this, his arginal revenue fro selling an extra seat is. The price elasticity of deand for tickets at this price is. k. How uch could the athletic director increase the total revenue per gae fro ticket sales if he added 1,000 new seats to the stadiu and adjusted the ticket price to axiize revenue?

7 Spring / IA 350, Interediate Microeconoics / Proble Set 3 l. How uch could he increase the revenue per gae by adding 50,000 new seats? 60,000 new seats? (All the tie aintaining a revenue axiization goal.). A local corporation offers to build as large a stadiu as the athletic director would like, as long as it has naing rights to the stadiu and the athletic director pledges to price his tickets so as to keep the stadiu full. If the athletic director wants to axiize his revenue fro ticket sales, how large a stadiu should he choose? 1. The deand curve for a particular product is p 00 5p and the supply curve is Sp 5p. a. On the graph below, use blue ink to draw the deand curve and the supply curve. The equilibriu arket price is and the equilibriu quantity sold is. b. A quantity tax of $ per unit sold is place on the product. Use red ink to draw the new supply curve, where the price on the vertical axis reains the price paid by deanders. The new equilibriu price paid by the deanders will be and the new price received by the suppliers will be. The equilibriu quantity sold will be. c. What is the deadweight loss associated with this tax? Shade in this area on the diagra. Price Quantity

8 Spring / IA 350, Interediate Microeconoics / Proble Set 3 P 18 3Q 13. Suppose that the inverse deand function for bananas is d d and the inverse supply function is P s 6 Q s, where quantity is easured in pounds and price is easured in cents. a. If there are no taxes or subsidies, what is the equilibriu quantity and equilibriu arket price? b. If a subsidy of cents per pound is paid to banana growers, then in equilibriu it still ust be that the quantity deanded equals the quantity supplied, but now the price received by sellers is cents higher than the price paid by consuers. What is the new equilibriu quantity? What is the new equilibriu price? c. If the cross-elasticity of deand between bananas and apples is +.5, what will happen to the quantity of apples deanded as a consequence of the banana subsidy (assuing the price of apples stays constant)? 14. According to a study by econoists on Fullerton and Thoas Kinnaan, the arc price elasticity of trash disposal over a broad range of prices is Suppose the town of Lincolnwood is currently charging $1.00 per bag for trash reoval. The town currently collects 10,000 bags of trash each week. To reduce trash collection to 9,000 bags weekly, what price ust Lincolnwood charge per bag?