Question

Assume that you are in the two-factor exact APT world. There are two portfolios (portfolio 1 and portfolio 2) which have loadings on the two factors as follows:

Loadings	factor 1	factor 2
portfolio 1	1.5	0.55
portfolio 2	1.41	-1.1

The expected return on portfolio 1 is 8.04% and the expected return on portfolio 2 is 14.09%. The risk-free rate is 2.1%.

There is a new portfolio just formed (portfolio 3). It has loadings of 3 and 1.5 on factor 1 and 2, respectively. The expected return on this portfolio is 10%. Is this consistent with APT? If it is consistent with the APT fill the boxes below with zeros. If not, construct an arbitrage strategy that generates $100 today for sure and costs nothing in the future.

Answer 1

ER = Rf + b1 * Market Risk Premium + b2 * Market Risk Premium

x = Reurn on factor 1 and, y = return on factor 2

Portfolio 1

.0804 = 0.021 + (x - 0.021)*1.5 + (y - 0.021)*0.55

0.0594 = 1.5x +0.55y - 0.0315 - 0.01155

1.5x +0.55y = 0.10245 (1)

Portfolio 2: 0.1409 = 0.021 + (x - 0.0201)*1.41 + (y - 0.0201)*(-1.1)

0.1199 = 1.41x - 1.1y - 0.02961 + 0.0231

1.41x - 1.1y = 0.12641 (2)

x= 0.0751 and y= -0.0186

Return of Portfolio 3 based factors and returns x and y:

= 0.021 + 3* (0.0751- 0.021) +1.5 * (-0.0186 - 0.021)

= 12.39%

Risk Premium of First Factor =  3* (0.0751- 0.021 = 16.23%

Risk Premium of Second Factor = -5.94%

For a given factor model (Fi), if the expected return calculated by the model (12.39%) does not equal the expected return quoted in the market (10%), an arbitrage opportunity exists.