Terms and Definitions
Firm’s / Client’s Portfolio
Benchmark Portfolio
Symbol: B B B
Weights: b \pmb b b
Inner-Product: b ′ 1 = 1 \pmb b' \pmb 1= 1 b ′ 1 = 1
Active Holdings
Symbol: h \pmb h h
Weights: h \pmb h h
Inner-Product: h ′ 1 = 0 \pmb h' \pmb 1= 0 h ′ 1 = 0
Active Portfolio
Symbol: A A A
Weights: w = b + h \pmb w = \pmb b + \pmb h w = b + h
Inner-Product: w ′ 1 = 1 \pmb w' \pmb 1= 1 w ′ 1 = 1
I R = E [ r W ] − E [ r B ] T E \begin{align}
IR = \frac{E[r_W] - E[r_B]}{TE}
\end{align} I R = T E E [ r W ] − E [ r B ]
Tracking error / active risk :
T E = V a r ( r A − r B ) = V a r [ ( w − b ) ′ r ] = V a r ( h ′ r ) = h ′ V h = σ e \begin{align}
TE &= \sqrt{Var(r_A - r_B)} \\
&= \sqrt{Var[(\pmb w - \pmb b)' \pmb r]} \\
&= \sqrt{Var(\pmb h'\pmb r)} \\
&= \sqrt{\pmb h'V\pmb h} \\
&= \sigma_e
\end{align} T E = V a r ( r A − r B ) = V a r [( w − b ) ′ r ] = V a r ( h ′ r ) = h ′ V h = σ e The Problem
Our client holds a portfolio that looks quite a bit like some benchmark. We assume they’ve optimized their portfolio to have no alpha with respect to to the benchmark (α B F = 0 \alpha_{BF} =0 α B F = 0 β A B = 1 \beta_{AB}=1 β A B = 1
We want to construct an active portfolio which will be constructed as a combination of the benchmark portfolio with active positions .
To provide our client with a higher alpha portfolio, the we can employ one of two distinct methods.
Maximimizing the Information Ratio.
Benchmark Hedging.
One method to supply alpha to A A A information ratio .
Numerator : It can be shown that when β A B = 1 \beta_{AB} =1 β A B = 1 h ′ α = E [ r A ] − E [ r B ] \pmb h' \pmb \alpha = E[r_A] - E[r_B] h ′ α = E [ r A ] − E [ r B ]
Denominator : We enforce that h ′ V h ≤ σ ˉ \sqrt{\pmb h' V \pmb h} \leq \bar \sigma h ′ V h ≤ σ ˉ h ~ \pmb{ \tilde h } h ~
Beta Constraint : Turns out that β A B = 1 ⟺ β n B ′ h = 0 \beta_{AB} = 1 \iff \pmb \beta_{\pmb nB}'\pmb h =0 β A B = 1 ⟺ β n B ′ h = 0
Active Weights Constraint : By definition of h \pmb h h h ′ 1 = 0 \pmb h'\pmb 1=0 h ′ 1 = 0
We can maximize the information ratio and satisfy the constraints by solving:
L = h ′ α − λ h ′ V h − γ ′ ( C h − c ) \begin{align}
L = \pmb h' \pmb \alpha - \lambda \pmb h' V \pmb h - \pmb \gamma'(C\pmb h - \pmb c)
\end{align} L = h ′ α − λ h ′ V h − γ ′ ( C h − c ) Where
C = [ 1 ′ β n B ′ ] , c = [ 0 0 ] \begin{align}
C= \begin{bmatrix}\pmb 1' \\\pmb \beta_{\pmb nB}'\end{bmatrix}, \ \ \pmb c=\begin{bmatrix}0 \\ 0\end{bmatrix}
\end{align} C = [ 1 ′ β n B ′ ] , c = [ 0 0 ] After solving we find that:
h ~ = ( 2 λ ) − 1 V − 1 [ I − C ′ ( C V − 1 C ′ ) − 1 C V − 1 ] α + V − 1 C ′ ( C V − 1 C ′ ) − 1 c = ( 2 λ ) − 1 V − 1 [ I − C ′ ( C V − 1 C ′ ) − 1 C V − 1 ] α \begin{align}
\pmb{\tilde h} &= (2\lambda)^{-1}V^{-1}\big[I - C'(CV^{-1}C')^{-1}CV^{-1}\big]\pmb \alpha + V^{-1}C'(CV^{-1}C')^{-1}\pmb c \\
&= (2\lambda)^{-1}V^{-1}\big[I - C'(CV^{-1}C')^{-1}CV^{-1}\big]\pmb \alpha
\end{align} h ~ = ( 2 λ ) − 1 V − 1 [ I − C ′ ( C V − 1 C ′ ) − 1 C V − 1 ] α + V − 1 C ′ ( C V − 1 C ′ ) − 1 c = ( 2 λ ) − 1 V − 1 [ I − C ′ ( C V − 1 C ′ ) − 1 C V − 1 ] α Now, we scale our answer to the appropriate tracking error, σ ˉ \bar \sigma σ ˉ k k k
( k h ~ ) ′ V ( k h ~ ) = ∣ k ∣ h ~ ′ V h ~ = σ ˉ ⟹ k = σ ˉ h ~ ′ V h ~ = σ ˉ σ e \begin{align}
\sqrt{(k\pmb{\tilde h})'V(k\pmb{\tilde h})}= |k|\sqrt{\pmb{\tilde h}'V\pmb{\tilde h}}&= \bar \sigma \\
\implies k &= \frac{\bar \sigma}{\sqrt{\pmb{\tilde h}'V\pmb{\tilde h}}} \\[12pt]
&=\frac{\bar \sigma}{\sigma_{e}}
\end{align} ( k h ~ ) ′ V ( k h ~ ) = ∣ k ∣ h ~ ′ V h ~ ⟹ k = σ ˉ = h ~ ′ V h ~ σ ˉ = σ e σ ˉ Finally, h ∗ = k ⋅ h ~ \pmb h^* = k \cdot \pmb {\tilde h} h ∗ = k ⋅ h ~ active portfolio will have tracking error equal to σ ˉ \bar \sigma σ ˉ
We supply w ∗ = h ∗ + b \pmb w^* = \pmb h^* + \pmb b w ∗ = h ∗ + b
Benchmark Hedging
A difficulty with maximizing the information ratio is that it requires views on every stock in the benchmark. Instead of having N N N P P P P P P
By definition:
V a r ( r E T F ) = b ′ V b ∈ R C o v ( r p , r E T F ) = ( V b ) p ∈ R P × 1 \begin{align}
Var(r_{ETF}) &= \pmb b' V \pmb b \in R\\[5pt]
Cov(r_p, r_{ETF}) &= (V\pmb b)_p \in R^{P\times 1}
\end{align} V a r ( r E T F ) C o v ( r p , r E T F ) = b ′ V b ∈ R = ( V b ) p ∈ R P × 1 We construct:
α A = [ α p 0 ] ∈ R ( P + 1 ) × 1 V A = [ V p ( V b ) p ( V b ) p ′ b ′ V b ] ∈ R ( P + 1 ) × ( P + 1 ) \begin{align}
\pmb \alpha_A &=\begin{bmatrix}\pmb \alpha_p \\ 0 \end{bmatrix} \in \mathbb R^{(P+1)\times 1}\\[12pt]
V_A &= \begin{bmatrix}V_p & (V\pmb b)_p \\ (V\pmb b)_p' & \pmb b' V \pmb b\end{bmatrix} \in \mathbb R^{(P+1)\times(P+1)}
\end{align} α A V A = [ α p 0 ] ∈ R ( P + 1 ) × 1 = [ V p ( V b ) p ′ ( V b ) p b ′ V b ] ∈ R ( P + 1 ) × ( P + 1 ) We want to find:
h A = [ h p h E T F ] ∈ R ( P + 1 ) × 1 \begin{align}
\pmb h_A = \begin{bmatrix}\pmb h_p \\ h_{ETF}\end{bmatrix} \in \mathbb R^{(P+1) \times 1}
\end{align} h A = [ h p h E T F ] ∈ R ( P + 1 ) × 1 The Lagrangian for our problem is:
L = h A ′ α A − λ h A ′ V A h A − γ ′ ( C h A − c ) \begin{align}
L = \pmb h_A'\pmb \alpha_A - \lambda \pmb h_A' V_A \pmb h_A - \pmb \gamma'(C\pmb h_A - \pmb c)
\end{align} L = h A ′ α A − λ h A ′ V A h A − γ ′ ( C h A − c ) Where
C = [ 1 p ′ 1 β p ′ 1 ] , c = [ 0 0 ] \begin{align}
C = \begin{bmatrix} \pmb 1_p' & 1 \\ \pmb \beta_p' & 1 \end{bmatrix}, \ \ \pmb c=\begin{bmatrix} 0 \\ 0\end{bmatrix}
\end{align} C = [ 1 p ′ β p ′ 1 1 ] , c = [ 0 0 ] After solving we find that:
h ~ A = ( 2 λ ) − 1 V A − 1 [ I − C ′ ( C V A − 1 C ′ ) − 1 C V A − 1 ] α + V A − 1 C ′ ( C V A − 1 C ′ ) − 1 c = ( 2 λ ) − 1 V A − 1 [ I − C ′ ( C V A − 1 C ′ ) − 1 C V A − 1 ] α \begin{align}
\pmb{\tilde h}_A &= (2\lambda)^{-1}V_A^{-1}\big[I - C'(CV_A^{-1}C')^{-1}CV_A^{-1}\big]\pmb \alpha + V_A^{-1}C'(CV_A^{-1}C')^{-1}\pmb c \\
&= (2\lambda)^{-1}V_A^{-1}\big[I - C'(CV_A^{-1}C')^{-1}CV_A^{-1}\big]\pmb \alpha
\end{align} h ~ A = ( 2 λ ) − 1 V A − 1 [ I − C ′ ( C V A − 1 C ′ ) − 1 C V A − 1 ] α + V A − 1 C ′ ( C V A − 1 C ′ ) − 1 c = ( 2 λ ) − 1 V A − 1 [ I − C ′ ( C V A − 1 C ′ ) − 1 C V A − 1 ] α Now, we scale our answer to the appropriate tracking error, σ ˉ \bar \sigma σ ˉ k k k
( k h ~ ) ′ V A ( k h ~ ) = ∣ k ∣ h ~ ′ V A h ~ = σ ˉ ⟹ k = σ ˉ h ~ ′ V A h ~ = σ ˉ σ e \begin{align}
\sqrt{(k\pmb{\tilde h})'V_A(k\pmb{\tilde h})}= |k|\sqrt{\pmb{\tilde h}'V_A\pmb{\tilde h}}&= \bar \sigma \\
\implies k &= \frac{\bar \sigma}{\sqrt{\pmb{\tilde h}'V_A\pmb{\tilde h}}} \\[12pt]
&=\frac{\bar \sigma}{\sigma_{e}}
\end{align} ( k h ~ ) ′ V A ( k h ~ ) = ∣ k ∣ h ~ ′ V A h ~ ⟹ k = σ ˉ = h ~ ′ V A h ~ σ ˉ = σ e σ ˉ Scaling, h A ∗ = k ⋅ h ~ A \pmb h^*_A = k \cdot \pmb {\tilde h}_A h A ∗ = k ⋅ h ~ A active portfolio will have tracking error equal to σ ˉ \bar \sigma σ ˉ
We need to unpack h A ∗ = [ h p ∗ ′ h E T F ∗ ] ′ \pmb h^*_A= \begin{bmatrix} \pmb h_p^{*\prime} & h_{ETF}^* \end{bmatrix}' h A ∗ = [ h p ∗ ′ h E T F ∗ ] ′
w A ∗ = ( h A ∗ + b ) + h E T F ∗ ⋅ b = h A ∗ + ( 1 + h E T F ∗ ) ⋅ b \begin{align}
\pmb w^*_A &= (\pmb h^*_A + \pmb b) + h_{ETF}^* \cdot \pmb b \\[5pt]
&= \pmb h^*_A + (1+h_{ETF}^*)\cdot \pmb b
\end{align} w A ∗ = ( h A ∗ + b ) + h E T F ∗ ⋅ b = h A ∗ + ( 1 + h E T F ∗ ) ⋅ b 