Frequently Asked Questions In Quantitative Finance

Chapter 2: FAQs 59

We write the random return on theith asset as

Ri=αi+

∑n

j= 1

βjiRj+ (^) i,
where theRjare the factors, theαsandβs are con-
stants and (^) iis the stock-specific risk. A portfolio of
these assets has return
∑N
i= 1
aiRi=
∑N
i= 1
aiαi+
∑n
j= 1
(N
∑
i= 1
aiβji
)
Rj+···,
where the···can be ignored if the portfolio is well
diversified.
Suppose that we think that five factors are sufficient to
represent the economy. We can therefore decompose
any portfolio into a linear combination of these five
factors, plus some supposedly negligible stock-specific
risks. If we are shown six diversified portfolios we can
decompose each into the five random factors. Since
there are more portfolios than factors we can find a
relationship between (some of) these portfolios, effec-
tively relating their values, otherwise there would be
an arbitrage. Note that the arbitrage argument is an
approximate one, relating diversified portfolios, on the
assumption that the stock-specific risks are negligible
compared with the factor risks.
In practice we can choose the factors to be macro-
economic or statistical. Here are some possible macro-
economic variables.

• an index level


• GDP growth


• an interest rate (or two)