Chapter 2: FAQs 55

return on the market as a whole (or some representative
index),RMby

Ri=αi+βiRM+ (^) i.
The (^) iis random with zero mean and standard deviation
ei, and uncorrelated with the market returnRMand the
other (^) j. There are three parameters associated with
each asset,αi,βiandei. In this representation we can
see that the return on an asset can be decomposed into
three parts: a constant drift; a random part common
with the index; a random part uncorrelated with the
index, (^) i. The random part (^) iis unique to theith asset.
Notice how all the assets are related to the index but
are otherwise completely uncorrelated.
Let us denote the expected return on the index byμM
and its standard deviation byσM. The expected return
on theith asset is then
μi=αi+βiμM
and the standard deviation
σi=
√
βi^2 σM^2 +e^2 i.
If we have a portfolio of such assets then the return is
given by
δ


∑N
i= 1
WiRi=
(N
∑
i= 1
Wiαi
)
+RM
(N
∑
i= 1
Wiβi
)

• ∑N
  i= 1
  Wi (^) i.
  From this it follows that
  μ=
  (N
  ∑
  i= 1
  Wiαi
  )
  +E[RM]
  (N
  ∑
  i= 1
  Wiβi
  )
  .
  Writing
  α=
  ∑N
  i= 1
  Wiαi and β=
  ∑N
  i= 1
  Wiβi,