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.

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We have two economic factors F1 and F2 in
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Stock
Expected return
bi1
bi2
A
7%
2
-1
B
17%
1
2
C
12%
1
?
If the risk free rate is 2%, what is stock C's bi2 so
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-1
2
1

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Expected Excess Return
Beta 1
Beta 2
A
4.9%
1
0
B
4.0%
0
1
C
4.9%
0.5
0.5
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Portfolio
ββ on F1
ββ on F2
Expected Return
A
1.0
2.0
19
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B
2.0
0.0
12
%
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Beta (2nd factor)
A
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1.8
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1.1
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