Providing an Active Portfolio

Supplying alpha to a benchmark portfolio.

Terms and Definitions

  • Firm's / Client's Portfolio
    • Symbol: FF
  • Benchmark Portfolio
    • Symbol: BB
    • Weights: b\pmb b
    • Inner-Product: b′1=1\pmb b' \pmb 1= 1
  • Active Holdings
    • Symbol: h\pmb h
    • Weights: h\pmb h
    • Inner-Product: h′1=0\pmb h' \pmb 1= 0
  • Active Portfolio
    • Symbol: AA
    • Weights: w=b+h\pmb w = \pmb b + \pmb h
    • Inner-Product: w′1=1\pmb w' \pmb 1= 1
  • Information ratio:
IR=E[rW]−E[rB]TE\begin{align} IR = \frac{E[r_W] - E[r_B]}{TE} \end{align}
  • Tracking error / active risk:
TE=Var(rA−rB)=Var[(w−b)′r]=Var(h′r)=h′Vh=σe\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}

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 (αBF=0\alpha_{BF} =0). We also assume, (and will impose by constraint), that the the beta of the active fund with respect to the benchmark is one (βAB=1\beta_{AB}=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.

  1. Maximimizing the Information Ratio.
  2. Benchmark Hedging.

Maximizing the Information Ratio

One method to supply alpha to AA is to maximize the information ratio.

Numerator: It can be shown that when βAB=1\beta_{AB} =1 then h′α=E[rA]−E[rB]\pmb h' \pmb \alpha = E[r_A] - E[r_B].

Denominator: We enforce that h′Vh≤σˉ\sqrt{\pmb h' V \pmb h} \leq \bar \sigma. Though we'll do this via a scaling after solving for an unscaled h~\pmb{ \tilde h } .

Beta Constraint: Turns out that βAB=1  ⟺  βnB′h=0\beta_{AB} = 1 \iff \pmb \beta_{\pmb nB}'\pmb h =0. We enforce the latter.

Active Weights Constraint: By definition of h\pmb h we want h′1=0\pmb h'\pmb 1=0

We can maximize the information ratio and satisfy the constraints by solving:

L=h′α−λh′Vh−γ′(Ch−c)\begin{align} L = \pmb h' \pmb \alpha - \lambda \pmb h' V \pmb h - \pmb \gamma'(C\pmb h - \pmb c) \end{align}

Where

C=[1′βnB′],  c=[00]\begin{align} C= \begin{bmatrix}\pmb 1' \\\pmb \beta_{\pmb nB}'\end{bmatrix}, \ \ \pmb c=\begin{bmatrix}0 \\ 0\end{bmatrix} \end{align}

After solving we find that:

h~=(2λ)−1V−1[I−C′(CV−1C′)−1CV−1]α+V−1C′(CV−1C′)−1c=(2λ)−1V−1[I−C′(CV−1C′)−1CV−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}

Now, we scale our answer to the appropriate tracking error, σˉ\bar \sigma. We want to find kk such that:

(kh~)′V(kh~)=∣k∣h~′Vh~=σˉ  ⟹  k=σˉh~′Vh~=σˉσe\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}

Finally, h∗=k⋅h~\pmb h^* = k \cdot \pmb {\tilde h}. And the active portfolio will have tracking error equal to σˉ\bar \sigma.

We supply w∗=h∗+b\pmb w^* = \pmb h^* + \pmb b to the client.

Benchmark Hedging

A difficulty with maximizing the information ratio is that it requires views on every stock in the benchmark. Instead of having NN predicted alphas, suppose we only have PP views for constructing the active portfolio. We only invest in the PP assets for which we have views and a benchmark ETF.

By definition:

Var(rETF)=b′Vb∈RCov(rp,rETF)=(Vb)p∈RP×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}

We construct:

αA=[αp0]∈R(P+1)×1VA=[Vp(Vb)p(Vb)p′b′Vb]∈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}

We want to find:

hA=[hphETF]∈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}

The Lagrangian for our problem is:

L=hA′αA−λhA′VAhA−γ′(ChA−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}

Where

C=[1p′1βp′1],  c=[00]\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}

After solving we find that:

h~A=(2λ)−1VA−1[I−C′(CVA−1C′)−1CVA−1]α+VA−1C′(CVA−1C′)−1c=(2λ)−1VA−1[I−C′(CVA−1C′)−1CVA−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}

Now, we scale our answer to the appropriate tracking error, σˉ\bar \sigma. We want to find kk such that:

(kh~)′VA(kh~)=∣k∣h~′VAh~=σˉ  ⟹  k=σˉh~′VAh~=σˉσe\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}

Scaling, hA∗=k⋅h~A\pmb h^*_A = k \cdot \pmb {\tilde h}_A. And the active portfolio will have tracking error equal to σˉ\bar \sigma.

We need to unpack hA∗=[hp∗′hETF∗]′\pmb h^*_A= \begin{bmatrix} \pmb h_p^{*\prime} & h_{ETF}^* \end{bmatrix}' to recover the weights we offer the client:

wA∗=(hA∗+b)+hETF∗⋅b=hA∗+(1+hETF∗)⋅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}