<p>If you want to describe excess returns in terms of exposure to common risk factors, you want the risk factors to be orthogonal. However, if you have $k$ factors with no perfect collinearity, you can always orthogonalize them and use those. You then call the orthogonalized factors the risk factors. </p>

CC BY-SA 3.0
2015-01-26T13:54:25.333Z
what would happen if i use factors with no perfect collinearity?
jtomasrl
CC BY-SA 3.0
2015-01-26T15:37:42.793Z
That was addressed here. "However, if you have k factors with no perfect collinearity, you can always orthogonalize them and use those." Did you mean something else?
jmbejara
CC BY-SA 3.0
2015-01-26T16:18:21.367Z
I know i can always orthogonalize them, but what would happend if i dont orthogonalize them. If i use them just as they are.
jtomasrl
CC BY-SA 3.0
2015-01-26T20:56:58.743Z
Then the linear equation (excess returns as a linear function of exposure to risk factors) doesn't really make sense the way you want it to. You want the coefficients in the factor model to represent the impact of marginally more exposure to one of the risk factors controlling for all the others. If they aren't orthogonal, then the coefficients don't have this interpretation.
jmbejara
CC BY-SA 3.0
2015-01-26T04:11:31.430Z
2015-01-26T04:11:31.430Z
jmbejara
<p>Why in the Arbitrage Pricing Theory (APT), one of the assumptions is that the factors has to be orthonogal? what if not?</p>

CC BY-SA 3.0
2015-01-26T15:37:21.260Z
Do you have a citation for this? Campbell, Lo, and Mackinlay (on p. 220) doesn't have an orthogonality restriction on f in their introduction to APT. Nor, when they talk about portfolios of risk factors (on p. 223) do they require the portfolios to be orthogonal.
BKay
<asset-pricing>
Arbitrage Pricing Theory (APT), orthogonal factors
jtomasrl
# Arbitrage Pricing Theory (APT), orthogonal factors

(Asked by: username_0 on 2015-01-26)

Why in the Arbitrage Pricing Theory (APT), one of the assumptions is that the factors has to be orthonogal? what if not?

**Tags:** asset-pricing

> **Comments:**
> **username_1**: Do you have a citation for this? Campbell, Lo, and Mackinlay (on p. 220) doesn't have an orthogonality restriction on f in their introduction to APT. Nor, when they talk about portfolios of risk factors (on p. 223) do they require the portfolios to be orthogonal.  

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## 1 answers

### Answer by username_2 on 2015-01-26. Score: 3 | Accepted answer

If you want to describe excess returns in terms of exposure to common risk factors, you want the risk factors to be orthogonal. However, if you have $k$ factors with no perfect collinearity, you can always orthogonalize them and use those. You then call the orthogonalized factors the risk factors.

> **Comments:**
> **username_0**: what would happen if i use factors with no perfect collinearity?  
> **username_2**: That was addressed here. "However, if you have k factors with no perfect collinearity, you can always orthogonalize them and use those." Did you mean something else?  
> **username_0**: I know i can always orthogonalize them, but what would happend if i dont orthogonalize them. If i use them just as they are.  
> **username_2**: Then the linear equation (excess returns as a linear function of exposure to risk factors) doesn't really make sense the way you want it to. You want the coefficients in the factor model to represent the impact of marginally more exposure to one of the risk factors controlling for all the others. If they aren't orthogonal, then the coefficients don't have this interpretation.  

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