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The return covariance has the factor form Σ = F QF T + D, where F ∈ Rn×k (with rank K ) is
the factor loading matrix, Q ∈ Sk is the factor covariance matrix, and D is a diagonal matrix
with positive entries, called the idiosyncratic risk (since it describes the risk of each asset that is
independent of the factors). This form for Σ is referred to as a ‘k -factor risk model’. Some typical
dimensions are n = 2500 (assets) and k = 30 (factors).
(a) What is the ﬂop count for computing the optimal portfolio, if the low-rank plus diagonal
structure of Σ is not exploited? You can assume that λ = 1 (which can be arranged by
absorbing it into Σ).
(b) Explain how to compute the optimal portfolio more eﬃciently, and give the ﬂop count for your
method. You can assume that k ≪ n. You do not have to give the best method; any method
that has linear complexity in n is ﬁne. You can assume that λ = 1.
Hints. You may want to introduce a new variable y = F T w (which is called the vector of
factor exposures). You may want to work with the matrix
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- Fall '13
- The Aeneid