The “Checklist’’ > 9b. Construction: cross-sectional strategies> Portfolio construction: “factors”
Portfolio construction: “factors”
Policy
Signal(function of current
information)
Characteristics(“Expected return”from the signal)
Portfolio(vector ofholdings)
h (·) : st ≡
s1,t
·sn̄,t
∈ it → βsignalt ≡
βsignal1,t
·βsignaln̄,t
→ hsignalt ≡
hsignal1,t
·hsignaln̄,t
(9b.4)
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The “Checklist’’ > 9b. Construction: cross-sectional strategies> Portfolio construction: “factors”
Exposures
Exposures
Policy
Signal(function of current
information)
Characteristics(“Expected return”from the signal)
Portfolio(vector ofholdings)
h (·) : st ≡
s1,t
·sn̄,t
∈ it → βsignalt ≡
βsignal1,t
·βsignaln̄,t
→ hsignalt ≡
hsignal1,t
·hsignaln̄,t
(9b.4)
• Exposure of a portfolio ht
βcharh,t ≡ h′tβchar
t =
n̄∑n=1
hn,tβcharn,t (9b.36)
Normalized value characteristic
Market capitalization characteristic
characteristics
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The “Checklist’’ > 9b. Construction: cross-sectional strategies> Portfolio construction: “factors”
Characteristic portfolio
Characteristic portfolio
Policy
Signal(function of current
information)
Characteristics(“Expected return”from the signal)
Portfolio(vector ofholdings)
h (·) : st ≡
s1,t
·sn̄,t
∈ it → βsignalt ≡
βsignal1,t
·βsignaln̄,t
→ hsignalt ≡
hsignal1,t
·hsignaln̄,t
(9b.4)
• Signal exposure of a portfolio ht
βsignalh,t =
1
ictEt{Πh,t→t+1 − rrft→t+1|st} (9b.40)
information coefficient (9b.35)
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The “Checklist’’ > 9b. Construction: cross-sectional strategies> Portfolio construction: “factors”
Characteristic portfolio
Characteristic portfolio
Policy
Signal(function of current
information)
Characteristics(“Expected return”from the signal)
Portfolio(vector ofholdings)
h (·) : st ≡
s1,t
·sn̄,t
∈ it → βsignalt ≡
βsignal1,t
·βsignaln̄,t
→ hsignalt ≡
hsignal1,t
·hsignaln̄,t
(9b.4)
• Characteristic portfolio
hchart ≡ argmin
h′βchart =1
V{Πh,t→t+1|st} = argminh′βchar
t =1
h′σ2Π;th (9b.41)
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-28-2017 - Last update
The “Checklist’’ > 9b. Construction: cross-sectional strategies> Portfolio construction: “factors”
Characteristic portfolio
Characteristic portfolio
Policy
Signal(function of current
information)
Characteristics(“Expected return”from the signal)
Portfolio(vector ofholdings)
h (·) : st ≡
s1,t
·sn̄,t
∈ it → βsignalt ≡
βsignal1,t
·βsignaln̄,t
→ hsignalt ≡
hsignal1,t
·hsignaln̄,t
(9b.4)
• Characteristic portfolio
hchart ≡ argmin
h′βchart =1
V{Πh,t→t+1|st} = argminh′βchar
t =1
h′σ2Π;th (9b.41)
⇓Analytical solution
hchart ≡
(σ2Π;t)
−1βchart
βchar′t (σ2
Π;t)−1βchar
t
(9b.42)
Characteristic portfolio: Normalized value
Characteristic portfolio: Market capitalization
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-28-2017 - Last update
The “Checklist’’ > 9b. Construction: cross-sectional strategies> Portfolio construction: “factors”
Characteristic portfolio
Characteristic portfolio
Policy
Signal(function of current
information)
Characteristics(“Expected return”from the signal)
Portfolio(vector ofholdings)
h (·) : st ≡
s1,t
·sn̄,t
∈ it → βsignalt ≡
βsignal1,t
·βsignaln̄,t
→ hsignalt ≡
hsignal1,t
·hsignaln̄,t
(9b.4)
• Characteristic signal portfolio
hsignalt ≡ argmin
h′βsignalt =1
V{Πh,t→t+1|st} = argminh′βsignal
t =1
h′σ2Π;th. (9b.46)
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-28-2017 - Last update
The “Checklist’’ > 9b. Construction: cross-sectional strategies> Portfolio construction: “factors”
Characteristic portfolio
Characteristic portfolio
Policy
Signal(function of current
information)
Characteristics(“Expected return”from the signal)
Portfolio(vector ofholdings)
h (·) : st ≡
s1,t
·sn̄,t
∈ it → βsignalt ≡
βsignal1,t
·βsignaln̄,t
→ hsignalt ≡
hsignal1,t
·hsignaln̄,t
(9b.4)
• Characteristic signal portfolio
hsignalt ≡ argmin
h′βsignalt =1
V{Πh,t→t+1|st} = argminh′βsignal
t =1
h′σ2Π;th. (9b.46)
⇓Analytical solution
hsignalt =
(σ2Π;t)
−1βsignalt
βsignal′t (σ2
Π;t)−1βsignal
t
(9b.41)
Signal characteristic portfolio stock market
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The “Checklist’’ > 9b. Construction: cross-sectional strategies> Portfolio construction: “factors”
Characteristic portfolio of reversal strategy
• Risk drivers: log-values of 392 stocks of the S&P500 index• Signal : negative momentum (reversal)
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The “Checklist’’ > 9b. Construction: cross-sectional strategies> Portfolio construction: “factors”Flexible portfolio construction
Flexible portfolio construction
Policy
Signal(function of current
information)
Characteristics(“Expected return”from the signal)
Portfolio(vector ofholdings)
h (·) : st ≡
s1,t
·sn̄,t
∈ it → βsignalt ≡
βsignal1,t
·βsignaln̄,t
→ hsignalt ≡
hsignal1,t
·hsignaln̄,t
(9b.4)
hsignalt = argmin
h′βchart =1
h′σ2Π;th (9b.46)
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-28-2017 - Last update
The “Checklist’’ > 9b. Construction: cross-sectional strategies> Portfolio construction: “factors”Flexible portfolio construction
Flexible portfolio construction
Policy
Signal(function of current
information)
Characteristics(“Expected return”from the signal)
Portfolio(vector ofholdings)
h (·) : st ≡
s1,t
·sn̄,t
∈ it → βsignalt ≡
βsignal1,t
·βsignaln̄,t
→ hsignalt ≡
hsignal1,t
·hsignaln̄,t
(9b.4)
hsignalt = argmin
h′βchart =1
h′σ2Π;th (9b.46)
hsignalt ≡ argmaxh∈Ct{h
′βsignalt }
Dual Problem:Ct ≡ {h′σ2
Π;th ≤ 1
βchar′t (σ2
Π;t)−1βchar
t}
m
(9b.54)
(9b.55)
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The “Checklist’’ > 9b. Construction: cross-sectional strategies> Portfolio construction: “factors”Flexible portfolio construction
Flexible characteristic portfolio of reversal strategy
• Risk drivers: log-values of 392 stocks of the S&P500 index• Signal : negative momentum (reversal)• Constraints: bound of variance, market neutrality, with stock
index, zero investment long/short
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The “Checklist’’ > 9b. Construction: cross-sectional strategies> Portfolio construction: “factors”Flexible portfolio construction
Flexible portfolio construction
multiple d̄-dimensional signals βsignalt ≡ (β
signal1t | · · · |βsignalk̄
t ) (9b.59)
n̄× k̄number of instrumentsin the strategy
number of signal typesn̄× 1 n̄× 1
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-28-2017 - Last update
The “Checklist’’ > 9b. Construction: cross-sectional strategies> Portfolio construction: “factors”Flexible portfolio construction
Flexible portfolio construction
multiple d̄-dimensional signals βsignalt ≡ (β
signal1t | · · · |βsignalk̄
t ) (9b.59)
n̄× k̄number of instrumentsin the strategy
number of signal typesn̄× 1 n̄× 1
⇓
hsignalkt ≡ argmax
h∈Ct{h′βsignalk
t } (9b.60)
hk′σ2
Π;thk ≤ 1
βsignalk′t (σ2
Π;t)−1β
signalkt
hk′βsignaljt = 0, for j 6= k
⊆ Ct(9b.61)
(9b.62)
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-28-2017 - Last update
The “Checklist’’ > 9b. Construction: cross-sectional strategies> Portfolio construction: “factors”Flexible portfolio construction
Flexible portfolio construction
multiple d̄-dimensional signals βsignalt ≡ (β
signal1t | · · · |βsignalk̄
t ) (9b.59)
n̄× k̄number of instrumentsin the strategy
number of signal typesn̄× 1 n̄× 1
⇓
hsignalkt ≡ argmax
h∈Ct{h′βsignalk
t } (9b.60)
hk′σ2
Π;thk ≤ 1
βsignalk′t (σ2
Π;t)−1β
signalkt
hk′βsignaljt = 0, for j 6= k
⊆ Ct(9b.61)
(9b.62)
⇓
multi-signal char. portfolios (hsignal1t | · · · |hsignalk̄
t ) = βsignal†t (9b.63)
n̄× 1 n̄× 1
optimal pseudo-inverse (9b.64)
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The “Checklist’’ > 9b. Construction: cross-sectional strategies> Portfolio construction: “factors”Flexible portfolio construction
Flexible portfolio construction
Relevant constraints
• Uncorrelation constraint
{hk′σ2Π;th̃t = 0} ⊆ Ct (9b.64)
exogeneous portfolios
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-28-2017 - Last update
The “Checklist’’ > 9b. Construction: cross-sectional strategies> Portfolio construction: “factors”Flexible portfolio construction
Flexible portfolio construction
Relevant constraints
• Uncorrelation constraint
{hk′σ2Π;th̃t = 0} ⊆ Ct (9b.64)
exogeneous portfolios
• market impact/trans. costs penality
hsignalkt ≡ argmax
hk∈Ct{hk′β
signalkt − λc(hk,hk
t−1)} (9b.65)
c(h,k) ≡ (h− k)′q2(h− k) (9b.66)
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