The “Checklist” > 9b. Construction: cross-sectional strategies > From signal to instruments P&L

Signal characteristic

• Topic: use the fundamental law of active management to build signalcharacteristics

• Generalize the standard construction discussed e.g. in [Grinold andEaston, 1998b] to the case of general instruments

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The “Checklist” > 9b. Construction: cross-sectional strategies > From signal to instruments P&L

Signal characteristic

To build signal characteristics βsignalt

1 explore the effect of the signa onto the risk drivers2 map the risk drivers into P&L’s3 compute the conditional expected excess P&L

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The “Checklist” > 9b. Construction: cross-sectional strategies > From signal to instruments P&L

Signal-conditioned moments of the risk drivers

Step 1 Step 2 Step 3Xt+1|st → Πt→t+1|st → βsignal

t

(9b.8)

Postulate homogenous cross-correlation

pX,S;t ≡ Crt{X̃t+1, S̃t} = λt × Id̄ (9b.25)

⇓

Conditional expectation

Et{Xt+1|st} = µX;t + σvolX;t ◦ st × λt (9b.26)

See Example 9b.9

d̄× d̄

d̄× 1 d̄× 1 d̄× 1 d̄× 1

X̃t+1 ≡ σ−1X;t(Xt+1 − µX;t)

S̃t ≡ σ−1S;t(St − µS;t)

Information coefficient

µX;t ≡ Et{Xt+1} (9b.24)σ2X;t ≡ Cvt{Xt+1} (9b.24)

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The “Checklist” > 9b. Construction: cross-sectional strategies > From signal to instruments P&L

Signal-conditioned moments of the P&L

Step 1 Step 2 Step 3

Xt+1|st → Πt→t+1|st → βsignalt

(9b.8)

Πt→t+1|it ≈ θt + δt (Xt+1|it − xt) (9b.29)

⇓ affine equivariance (31.8)

Conditional expectation of the P&L

µΠ;t ≡ Et{Πt→t+1|st} = rrft→t+1vt +αt + βsignalt λt (9b.30)

See Example 9b.10

First order pricing approximation

No-signal alpha (9b.32)

αt ≡ θt − rrft→t+1vt︸ ︷︷ ︸time

+ δt (µX;t − xt)︸ ︷︷ ︸risk

Signal beta (9b.33)

βsignalt ≡ δt (σvol

X;t ◦ st)

(??)

n̄× 1 n̄× 1 n̄× d̄ d̄× 1

n̄× 1 n̄× 1 n̄× 1 n̄× 1

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The “Checklist” > 9b. Construction: cross-sectional strategies > From signal to instruments P&L

“Smart beta”Step 1 Step 2 Step 3

Xt+1|st → Πt→t+1|st → βsignalt

(9b.8)

Assume APT-like LFM (19.29) on the P&L Πt→t+1 ⇒

αt = Et{Πt→t+1|st} − rrft→t+1vt − βsignalt λt ≈ 0 (9b.35)

⇓

Conditional expectation of the P&L

Et{Πt→t+1 − rrft→t+1vt|st} ≈ λt × βsignalt (9b.36)

See Example 9b.11

λt = ict =tr(Cvt{Πt→t+1,B

signalt })

tr(Cvt{Bsignalt })(9b.37)

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