EC4024 Lecture 8. Arbitrage Pricing Theory
Stephen Kinsella
The Arbitrage Pricing theory, or APT, was developed to shore up some of the deficiences of CAPM we discussed in at the end of the last lecture. In
particular, CAPM only works when we make assumptions about preferences which don't make much sense: consumers only care about mean and
standard deviations in their wealth if their preferences are quadratic, as Markowitz and Sharpe specified them. Returns must also be normally (that is,
Gaussian) distributed. Finally and most importantly, people hold different beliefs, and these beliefs lead them to hold different portfolios. It is
therefore not quite clear what the market portfolio actually is. In practice we would use a large stock indx like the S&P 500, but this is not ideal.
CAPM shows us a world in which b is king, however, when investors hold different portfolios, the value of b changes. When you measure the market
portfolio differently (say, by taking a different broad index of stocks and shares), you get a different result for b.
APT was developed to shore up these deficiencies. Ross (1976), who developed APT, dropped the assumptions on preferences and strict maximisa-
tion. He kept the idea that firms and stocks are looking for profit maximising opportunities, and the market was hard to beat. Rather than evolving an e
quilibrium condition for the market from consumer preferences as Sharpe did, Ross snapped the market equilibrium onto the investors, merely
assuming that the search for arbitrage would keep investors at or near the CAPM-derived equilibrium.
The big idea of APT is to look at which combinations of assets one would hold to rule out any arbitrage. Arbitrage is possible when two assets with
the same risk have different returns. You can short the low return asset, go long on the other using the proceeds of the sale of the first, and in theory,
reap infinite rewards with no risk to yourself.
Single Factor APT
Begin with a single `factor', F, or driver, which generates the return Hrit L we see on every asset i, such that
E HRit L = ait + bi Ft + eit (1)
What fills the role of a factor?
The market rate of return, M, we talked about in the last lecture, or the rate of economic growth, or inflation, or some other macroeconomic factor.
The point is, the 'factor' is system wide, and there is only one.
As usual when modelling, we have to make some simplifying assumptions. Assume the following:
EHei L = 0,
e j M = EIei , e j M = 0,
(2)
covHei , FL = 0.
EHFL = 0
What do these conditions mean? Let's take them one by one. First, EHei L = 0. This means that the long run mean of the errors falls to zero, so we are
assuming the law of large numbers holds in this model. OK. Second, covIei , e j M = EIei , e j M = 0. This says the comovement of two assets, i and j, are
not related, so the returns on each stock (and the errors we get in measuring them) are independent of one another. Third, covHei , FL = 0.The errors in
measurement are not correlated with the factor. This is a bit of a stretch. Why? Can you think of an example where this might not hold? Well, we
have to assume it to get the model up off the ground, so let's do that. Fourth, the mean of the factor is zero.
Let's say there is no residual risk, so ei = 0.
Then returns would be calculated via
EHRi L = ai + bi EHFL
(3)
= ai .
Let's say we invest some fraction of our wealth l in asset i and (1-l), everything else, in asset j. What is the return on this portfolio, p?
EIR p M = lHai + bi FL + H1 - lL Ia j + b j FM
(4)
= lIai - a j M + IlIbi - b j M + b j M F.
Let's try to weight the porfolio to make some more money out of it. Say the weight, l* , looks like this:
bj
l* = . (5)
b j - bi
Don't forgt the portfolio has zero exposure to risk: the coefficient on F is zero, so we have
bj
EIR* p M = Iai - a j M + a j = EHRF L. (6)
b j - bi
Just rearrange the terms of the equation above, and we've got
a j - EHRF L ai - EHRF L
= (7)
bj bi
Let's call this ratio q.
Stephen Kinsella
The Arbitrage Pricing theory, or APT, was developed to shore up some of the deficiences of CAPM we discussed in at the end of the last lecture. In
particular, CAPM only works when we make assumptions about preferences which don't make much sense: consumers only care about mean and
standard deviations in their wealth if their preferences are quadratic, as Markowitz and Sharpe specified them. Returns must also be normally (that is,
Gaussian) distributed. Finally and most importantly, people hold different beliefs, and these beliefs lead them to hold different portfolios. It is
therefore not quite clear what the market portfolio actually is. In practice we would use a large stock indx like the S&P 500, but this is not ideal.
CAPM shows us a world in which b is king, however, when investors hold different portfolios, the value of b changes. When you measure the market
portfolio differently (say, by taking a different broad index of stocks and shares), you get a different result for b.
APT was developed to shore up these deficiencies. Ross (1976), who developed APT, dropped the assumptions on preferences and strict maximisa-
tion. He kept the idea that firms and stocks are looking for profit maximising opportunities, and the market was hard to beat. Rather than evolving an e
quilibrium condition for the market from consumer preferences as Sharpe did, Ross snapped the market equilibrium onto the investors, merely
assuming that the search for arbitrage would keep investors at or near the CAPM-derived equilibrium.
The big idea of APT is to look at which combinations of assets one would hold to rule out any arbitrage. Arbitrage is possible when two assets with
the same risk have different returns. You can short the low return asset, go long on the other using the proceeds of the sale of the first, and in theory,
reap infinite rewards with no risk to yourself.
Single Factor APT
Begin with a single `factor', F, or driver, which generates the return Hrit L we see on every asset i, such that
E HRit L = ait + bi Ft + eit (1)
What fills the role of a factor?
The market rate of return, M, we talked about in the last lecture, or the rate of economic growth, or inflation, or some other macroeconomic factor.
The point is, the 'factor' is system wide, and there is only one.
As usual when modelling, we have to make some simplifying assumptions. Assume the following:
EHei L = 0,
e j M = EIei , e j M = 0,
(2)
covHei , FL = 0.
EHFL = 0
What do these conditions mean? Let's take them one by one. First, EHei L = 0. This means that the long run mean of the errors falls to zero, so we are
assuming the law of large numbers holds in this model. OK. Second, covIei , e j M = EIei , e j M = 0. This says the comovement of two assets, i and j, are
not related, so the returns on each stock (and the errors we get in measuring them) are independent of one another. Third, covHei , FL = 0.The errors in
measurement are not correlated with the factor. This is a bit of a stretch. Why? Can you think of an example where this might not hold? Well, we
have to assume it to get the model up off the ground, so let's do that. Fourth, the mean of the factor is zero.
Let's say there is no residual risk, so ei = 0.
Then returns would be calculated via
EHRi L = ai + bi EHFL
(3)
= ai .
Let's say we invest some fraction of our wealth l in asset i and (1-l), everything else, in asset j. What is the return on this portfolio, p?
EIR p M = lHai + bi FL + H1 - lL Ia j + b j FM
(4)
= lIai - a j M + IlIbi - b j M + b j M F.
Let's try to weight the porfolio to make some more money out of it. Say the weight, l* , looks like this:
bj
l* = . (5)
b j - bi
Don't forgt the portfolio has zero exposure to risk: the coefficient on F is zero, so we have
bj
EIR* p M = Iai - a j M + a j = EHRF L. (6)
b j - bi
Just rearrange the terms of the equation above, and we've got
a j - EHRF L ai - EHRF L
= (7)
bj bi
Let's call this ratio q.
