EXERCISE A3Due: June 16

PART 1 - BASIC

QUESTION 1Suppose 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:

PortfolioBeta on F 1Beta on F 2Expected Return

A1.52.031%

B2.2-0.227%

What is the expected returnbeta relationship (APT prediction) in this economy?

The APT predicts:

E(rp ) = rf + P1 [E(r1 ) rf ] + P2 [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% + (P1 10%) + (P2 5%)

QUESTION 2Consider the following data for a one-factor economy. All portfolios are well diversified.

PortfolioE(r)Beta

A12%1.2

F6%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 Es) 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%

G = (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 3Consider the following multifactor (APT) model of security returns for a particular stock.

FactorFactor BetaFactor Risk Premium

Inflation1.26%

Industrial production0.58%

Oil prices0.33%

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.

FactorExpected Rate of ChangeActual Rate of Change

Inflation5%4%

Industrial production3%6%

Oil prices2%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 4Assume that both X and Y are well-diversified portfolios and the risk-free rate is 8%.

PortfolioExpected ReturnBeta

X16%1.00

Y12%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 5Assume 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 :

Returns on asset (%) Returns on portfolios associated to F1 & F2

StateProbability XYZ

Horrible20%-55.23623.9953.00-10.00-5.00

Bad20%70.7010.00413.37-5.0038.48

Average20%-9.0025.00-1493.1225.008.00

Good20%-12.47-3771.421058.7540.00-1.44

Excellent20%61.003237.4483.0050.000.00

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 6What is the arbitrage equilibrium price of asset C in the example below (state-contingent payoffs and prices)?

Asset AAsset BAsset C

State 1 9012

State 2 486

Price 54?

4/3x5 + 1/12x4 = 7

PART 3 - ADVANCED

This assignment does not include advanced questions.