EXERCISE A3 Due: June 16
PART 1 - BASIC QUESTION 1 Suppose that there are two independent economic factors, F 1 and F 2. The risk-free rate is 6%, and all stocks have independent firm-specific components with a standard deviation of 45%. The following are well-diversified portfolios:
Portfolio A B
Beta on F 1 1.5 2.2
Beta on F 2 2.0 -0.2
Expected Return 31% 27%
What is the expected return–beta relationship (APT prediction) in this economy?
The APT predicts: E(rp ) = rf + βP1 [E(r1 ) rf ] + βP2 [E(r2 ) – rf ] We need to find the risk premium (RP) for each of the two factors: RP1 = [E(r1 ) rf ] and RP2 = [E(r2 ) rf ] In order to do so, we solve the following system of two equations with two unknowns: .31 = .06 + (1.5 × RP1 ) + (2.0 × RP2 ) .27 = .06 + (2.2 × RP1 ) + [(–0.2) × RP2 ]
The solution to this set of equations is RP1 = 10% and RP2 = 5% Thus, the expected return-beta relationship is E(rP ) = 6% + (βP1 × 10%) + (βP2 × 5%)
QUESTION 2 Consider the following data for a one-factor economy. All portfolios are well diversified.
Portfolio A F
E(r) 12% 6%
Beta 1.2 0.0
Suppose that another portfolio, portfolio E, is well diversified with a beta of .6 and expected return of 8%. Would an arbitrage opportunity exist? If so, what would be the arbitrage strategy?
The expected return for portfolio F equals the risk-free rate since its beta equals 0. For portfolio A, the ratio of risk premium to beta is (12 − 6)/1.2 = 5 For portfolio E, the ratio is lower at (8 – 6)/0.6 = 3.33 This implies that an arbitrage opportunity exists. For instance, you can create a portfolio G with beta equal to 0.6 (the same as E’s) by combining portfolio A and portfolio F in equal weights. The expected return and beta for portfolio G are then: E(rG ) = (0.5 × 12%) + (0.5 × 6%) = 9% βG = (0.5 × 1.2) + (0.5 × 0%) = 0.6
Comparing portfolio G to portfolio E, G has the same beta and higher return. Therefore, an arbitrage opportunity exists by buying portfolio G and selling an equal amount of portfolio E. The profit for this arbitrage will be rG – rE =[9% + (0.6 × F)] [8% + (0.6 × F)] = 1%
That is, 1% of the funds (long or short) in each portfolio.
QUESTION 3 Consider the following multifactor (APT) model of security returns for a particular stock. Factor Inflation Industrial production Oil prices
Factor Beta 1.2 0.5 0.3
Factor Risk Premium 6% 8% 3%
a. If T-bills currently offer a 6% yield, find the expected rate of return on this stock if the market views the stock as fairly priced. b. Suppose that the market expected the values for the three macro factors given in column 1 below, but that the actual values turn out as given in column 2. Calculate the revised expectations for the rate of return on the stock once the “surprises” become known. Factor Inflation Industrial production Oil prices
Expected Rate of Change 5% 3% 2%
Actual Rate of Change 4% 6% 0%
Question a. E(r) = 6% + (1.2 × 6%) + (0.5 × 8%) + (0.3 × 3%) = 18.1%
Question b.
Surprises in the macroeconomic factors will result in surprises in the return of the stock: Unexpected return from macro factors = [1.2 × (4% – 5%)] + [0.5 × (6% – 3%)] + [0.3 × (0% – 2%)] = –0.3%
E(r) =18.1% − 0.3% = 17.8%
QUESTION 4 Assume that both X and Y are well-diversified portfolios and the risk-free rate is 8%. Portfolio X Y
Expected Return 16% 12%
Beta 1.00 0.25
In this situation you would conclude that portfolios X and Y: a. Are in equilibrium. b. Offer an arbitrage opportunity. c. Are both underpriced. d. Are both fairly priced.
The correct answer is b. Since portfolio X has = 1.0, then X is the portfolio associated to the risk factor. Using E(RX ) = 16% and rf = 8%, the expected return for portfolio Y is not consistent (the model predicts 0.25(16% - 8%) + 8% = 10%).
PART 2 - INTERMEDIATE QUESTION 5 Assume that it has been shown empirically that there are only two risk factors that determine expected returns, F1 and F2. Assume, as well, that it is possible to model the future according to five possible scenarios or states of nature, and that the returns on assets X, Y and Z are as given in the following table, where you can also find the returns on the respective portfolios associated to every risk factor, and
δ1
δ2 :
State
Probability
Horrible Bad Average Good Excellent
20% 20% 20% 20% 20%
Returns on asset (%) X Y Z -55.23 70.70 -9.00 -12.47 61.00
623.99 10.00 25.00 -3771.42 3237.44
53.00 413.37 -1493.12 1058.75 83.00
Returns on portfolios associated to F1 & F2
δ1
δ2
-10.00 -5.00 25.00 40.00 50.00
-5.00 38.48 8.00 -1.44 0.00
If the risk-free rate equals 10%,
a. Identify some arbitrage opportunity. b. Build the corresponding arbitrage portfolio, assuming that initially you are equally invested in each of the three assets. Do not consider idiosyncratic risk.
Calculate expected returns for each asset (11%, 25%, 23%). Calculate expected returns for each portfolio associated to each risk factor (20%, 8%). Calculate the covariances between each asset returns and the returns on the portfolios associated with the risk factors. Calculate the variances of the returns on the portfolios associated to each risk factor. Find coefficients b1 and b2 for assets X, Y and Z dividing the corresponding covariance by the variance of the portfolio associated with each the risk factor (X => 0.5 and 2.0; Y => 1.0 and 1.5; Z => 1.5 and 1.0).
Use APT forecast model (keep in mind that this model uses the excess returns). Verify that expected returns on assets X and Z coincide with the prediction and that expected return on asset Y is greater than what the APT model predicts (17%). Build an arbitrage portfolio so as to invest more of the undervalued asset Y, by selling some of asset X and asset Z. To do that, it is necessary to write a three-equation system (zero investment and zero systemic risk) that yields wX = -1/3; wY = 2/3; wZ = -1/3, namely, positions on assets X and Z are closed and all the proceeds are invested in asset Y. This is the example in CWS book:
QUESTION 6 What is the arbitrage equilibrium price of asset C in the example below (statecontingent payoffs and prices)?
State 1 State 2 Price
4/3x5 + 1/12x4 = 7
Asset A 9 4 5
Asset B 0 8 4
Asset C 12 6 ?
PART 3 - ADVANCED This assignment does not include advanced questions.
