Implied Risk Premia: Theoretical Foundation

BRISMA Project Documentation Version: 1.0 Last Updated: 2025-12-25 Status: Issue #48 - Asset-Level Implied Returns (Complete)

Table of Contents

1. Introduction
2. Asset-Level Implied Returns - Issue #48
3. Factor-Based Implied Risk Premia - Issue #49
4. Bantleon Calibration Methods - Existing
5. Worked Example: BRISMA Portfolio - Issue #51
6. Validation and Quality Checks
7. References - Issue #50
8. Appendices

1. Introduction

1.1 The Problem: What Returns Does the Market Expect?

Portfolio managers observe their current asset allocation $\mathbf{w}$ and can estimate the covariance matrix $\mathbf{Q}$ from historical data. The fundamental question is:

What expected returns $\boldsymbol{\mu}$ would make this allocation optimal?

This is the inverse optimization problem: instead of finding optimal weights given expected returns, we find the implied returns that rationalize observed weights.

1.2 Why Implied Returns?

Key Insight: Implied returns extract the market's collective wisdom embedded in current prices and allocations.

1.3 Literature Review

2. Asset-Level Implied Returns

This section addresses Issue #48

2.1 Mean-Variance Framework

2.1.1 The Utility Function

An investor with mean-variance preferences maximizes:

$$U_{MV}(\mathbf{w}) = \mathbf{w}'\boldsymbol{\mu} - \frac{\lambda}{2}\mathbf{w}'\mathbf{Q}\mathbf{w}$$

where: - $\mathbf{w} \in \mathbb{R}^n$: Portfolio weights (column vector) - $\boldsymbol{\mu} \in \mathbb{R}^n$: Expected returns (column vector) - $\mathbf{Q} \in \mathbb{R}^{n \times n}$: Covariance matrix (symmetric, positive semi-definite) - $\lambda > 0$: Risk aversion parameter

Interpretation: - First term $\mathbf{w}'\boldsymbol{\mu}$: Expected portfolio return - Second term $\frac{\lambda}{2}\mathbf{w}'\mathbf{Q}\mathbf{w}$: Penalty for portfolio variance - $\lambda$ controls the trade-off: higher $\lambda$ = more risk-averse

2.1.2 First-Order Condition Derivation

Theorem 2.1 (Unconstrained Inverse Solution)

For the unconstrained mean-variance problem, the implied expected returns are:

$$\boxed{\boldsymbol{\mu}^* = \lambda \mathbf{Q} \mathbf{w}}$$

Proof:

Starting from the utility function: $$U(\mathbf{w}) = \mathbf{w}'\boldsymbol{\mu} - \frac{\lambda}{2}\mathbf{w}'\mathbf{Q}\mathbf{w}$$

Take the gradient with respect to $\mathbf{w}$: $$\nabla_{\mathbf{w}} U = \boldsymbol{\mu} - \lambda\mathbf{Q}\mathbf{w}$$

At the optimum, the first-order condition requires $\nabla_{\mathbf{w}} U = \mathbf{0}$: $$\boldsymbol{\mu} - \lambda\mathbf{Q}\mathbf{w} = \mathbf{0}$$

Solving for $\boldsymbol{\mu}$: $$\boldsymbol{\mu}^* = \lambda\mathbf{Q}\mathbf{w} \quad \blacksquare$$

Second-Order Condition:

The Hessian is: $$\nabla^2_{\mathbf{w}} U = -\lambda\mathbf{Q}$$

For a maximum, we need $\nabla^2_{\mathbf{w}} U \prec 0$ (negative definite), which holds when $\mathbf{Q} \succ 0$ (positive definite) and $\lambda > 0$.

2.1.3 Constrained Case (Budget Constraint)

Theorem 2.2 (Constrained Inverse Solution)

With the budget constraint $\mathbf{1}'\mathbf{w} = 1$, the implied returns become:

$$\boxed{\boldsymbol{\mu}^ = \lambda\mathbf{Q}\mathbf{w} + \gamma^\mathbf{1}}$$

where $\gamma^*$ is the Lagrange multiplier (shadow price of the budget constraint).

Proof:

The Lagrangian is: $$\mathcal{L} = \mathbf{w}'\boldsymbol{\mu} - \frac{\lambda}{2}\mathbf{w}'\mathbf{Q}\mathbf{w} - \gamma(\mathbf{1}'\mathbf{w} - 1)$$

KKT conditions: 1. Stationarity: $\boldsymbol{\mu} - \lambda\mathbf{Q}\mathbf{w} - \gamma\mathbf{1} = \mathbf{0}$ 2. Primal feasibility: $\mathbf{1}'\mathbf{w} = 1$

From stationarity: $$\boldsymbol{\mu}^ = \lambda\mathbf{Q}\mathbf{w} + \gamma^\mathbf{1} \quad \blacksquare$$

Interpretation of $\gamma^*$: - $\gamma^*$ is the marginal utility of relaxing the budget constraint - Can be set to $r_f$ (risk-free rate) for zero-beta interpretation - Can be determined by normalization (e.g., target average return)

2.1.4 Geometric Interpretation

The formula $\boldsymbol{\mu}^* = \lambda\mathbf{Q}\mathbf{w}$ has a beautiful geometric interpretation:

1. $\mathbf{Q}\mathbf{w}$ represents the marginal contribution to risk of each asset
2. $\lambda$ scales this to expected returns
3. Assets with higher risk contribution must have higher expected returns to justify their weight

Visualization: - High-variance assets: Higher implied $\mu_i$ - Highly correlated assets: Share implied returns through off-diagonal $Q_{ij}$ - Diversifying assets: Lower implied returns due to negative correlations

2.2 Risk Aversion Parameter $\lambda$

2.2.1 Economic Interpretation

$\lambda$ represents the investor's marginal rate of substitution between expected return and variance:

$$\lambda = \frac{\partial \mu_p / \partial w_i}{\partial \sigma_p^2 / \partial w_i}$$

Typical Values:

2.2.2 Estimation Methods

Method 1: From Observed Returns

Given external estimates of expected returns $\tilde{\boldsymbol{\mu}}$:

$$\lambda^* = \frac{\mathbf{w}'\tilde{\boldsymbol{\mu}}}{\mathbf{w}'\mathbf{Q}\mathbf{w}} = \frac{\text{Expected portfolio return}}{\text{Portfolio variance}}$$

Derivation: Pre-multiply $\boldsymbol{\mu}^ = \lambda\mathbf{Q}\mathbf{w}$ by $\mathbf{w}'$: $$\mathbf{w}'\boldsymbol{\mu}^ = \lambda\mathbf{w}'\mathbf{Q}\mathbf{w}$$ $$\lambda = \frac{\mathbf{w}'\boldsymbol{\mu}^*}{\mathbf{w}'\mathbf{Q}\mathbf{w}}$$

Method 2: Black-Litterman Equilibrium

Assuming the market portfolio is mean-variance efficient:

$$\lambda_{BL} = \frac{E[r_m] - r_f}{\sigma_m^2}$$

where: - $E[r_m]$: Expected market return - $r_f$: Risk-free rate - $\sigma_m^2$: Market variance

Typical calculation: - $E[r_m] - r_f \approx 5\%$ (equity risk premium) - $\sigma_m \approx 15\%$ (market volatility) - $\lambda_{BL} = 0.05 / 0.15^2 \approx 2.22$

Method 3: Confidence-Weighted Blend

$$\lambda_{conf} = c \cdot \lambda_{observed} + (1-c) \cdot \lambda_{prior}$$

where $c \in [0,1]$ is the confidence in observed returns.

2.2.3 Sensitivity Analysis

The implied returns scale linearly with $\lambda$:

$$\frac{\partial \mu_i^*}{\partial \lambda} = (\mathbf{Q}\mathbf{w})_i$$

Key insight: Assets with higher risk contribution are more sensitive to $\lambda$ changes.

2.3 Alternative Utility Functions

2.3.1 MRAR (Morningstar Risk-Adjusted Return)

Definition:

$$MRAR_\gamma = \left[\frac{1}{T}\sum_{t=1}^{T}\left(\frac{1+r_{p,t}}{1+r_{f,t}}\right)^{-\gamma}\right]^{-12/\gamma} - 1$$

where: - $\gamma > 0$: Risk aversion parameter (typically 2-5) - $r_{p,t}$: Portfolio return at time $t$ - $r_{f,t}$: Risk-free rate at time $t$

Connection to CRRA Utility:

MRAR is the certainty equivalent return under Constant Relative Risk Aversion (CRRA) utility:

$$U(W) = \frac{W^{1-\gamma}}{1-\gamma}$$

Inverse Optimization:

Unlike mean-variance, MRAR requires bilevel numerical optimization: 1. Inner problem: Given $\boldsymbol{\mu}$, compute optimal $\mathbf{w}^$ 2. Outer problem: Find $\boldsymbol{\mu}$ such that $\mathbf{w}^ = \mathbf{w}_{observed}$

Implementation: Use scipy.optimize with CRRA objective.

2.3.2 Omega Ratio

Definition:

$$\Omega(L) = \frac{\int_L^{\infty}(1-F(r))dr}{\int_{-\infty}^{L}F(r)dr} = \frac{E[\max(r-L, 0)]}{E[\max(L-r, 0)]}$$

where: - $L$: Threshold return (often 0 or $r_f$) - $F(r)$: CDF of returns

Connection to Lower Partial Moments:

$$\Omega(L) = 1 + \frac{E[r] - L}{LPM_1(L)}$$

where $LPM_1(L) = E[\max(L-r, 0)]$ is the first lower partial moment.

Inverse Optimization:

Also requires bilevel numerical optimization due to non-quadratic objective.

2.3.3 Comparison of Utility Functions

When to use each: - Mean-Variance: Default choice, analytical tractability - MRAR: When asymmetric risk preferences matter - Omega: When downside risk is the primary concern

3. Factor-Based Implied Risk Premia

This section addresses Issue #49

3.1 Why Factor Decomposition?

3.1.1 The Curse of Dimensionality

With $n$ assets, the covariance matrix $\mathbf{Q}$ has $n(n+1)/2$ unique parameters. For BRISMA with 89 assets: - Parameters: $89 \times 90 / 2 = 4,005$ covariances - Observations needed: At least $4,005$ for reliable estimation - Problem: Insufficient historical data leads to estimation error

Factor models reduce complexity: - $k$ factors: $nk + k(k+1)/2$ parameters - With $k=5$ factors: $89 \times 5 + 15 = 460$ parameters - Reduction: 89% fewer parameters to estimate

3.1.2 Economic Interpretability

Factors represent systematic risk sources:

3.1.3 New Asset Extension Capability

The Key Insight:

If we know the factor premia $\boldsymbol{\pi}$ and a new asset's factor exposures $\mathbf{B}_{new}$, we can price the new asset without any historical return data:

$$\mu_{new} = r_f + \mathbf{B}_{new}'\boldsymbol{\pi}$$

This is critical for: - Newly launched funds - Private assets - Hypothetical portfolios

3.2 The APT Framework

3.2.1 Factor Model Representation

Definition (Arbitrage Pricing Theory)

Asset returns follow a $k$-factor structure:

$$r_i = \alpha_i + \sum_{j=1}^{k}\beta_{ij}f_j + \epsilon_i$$

In matrix form:

$$\mathbf{r} = \boldsymbol{\alpha} + \mathbf{B}\mathbf{f} + \boldsymbol{\epsilon}$$

where: - $\mathbf{r} \in \mathbb{R}^n$: Asset returns - $\boldsymbol{\alpha} \in \mathbb{R}^n$: Asset-specific alphas (often assumed zero) - $\mathbf{B} \in \mathbb{R}^{n \times k}$: Factor loading matrix (betas) - $\mathbf{f} \in \mathbb{R}^k$: Factor returns - $\boldsymbol{\epsilon} \in \mathbb{R}^n$: Idiosyncratic errors

3.2.2 Key Assumptions

1. Factor structure: Returns driven by common factors
2. Uncorrelated errors: $E[\epsilon_i \epsilon_j] = 0$ for $i \neq j$
3. Zero-mean errors: $E[\epsilon_i] = 0$
4. Errors uncorrelated with factors: $E[\epsilon_i f_j] = 0$

3.2.3 Implied Covariance Structure

Under APT, the covariance matrix decomposes as:

$$\mathbf{Q} = \mathbf{B}\mathbf{\Sigma}_f\mathbf{B}' + \mathbf{D}$$

where: - $\mathbf{\Sigma}f = \text{Cov}(\mathbf{f})$: Factor covariance matrix ($k \times k$) - $\mathbf{D} = \text{diag}(\sigma{\epsilon_1}^2, \ldots, \sigma_{\epsilon_n}^2)$: Idiosyncratic variance

3.2.4 Relation to CAPM

CAPM is a special case with $k=1$ (market factor only):

$$r_i - r_f = \beta_i(r_m - r_f) + \epsilon_i$$

The expected excess return is:

$$E[r_i] - r_f = \beta_i \cdot \pi_m$$

where $\pi_m = E[r_m] - r_f$ is the market risk premium.

3.3 Extracting Factor Premia from Asset Returns

3.3.1 The Cross-Sectional Regression

Given asset-level implied returns $\boldsymbol{\mu}^*$ from Section 2, we extract factor premia via:

$$\boldsymbol{\mu}^* - r_f\mathbf{1} = \mathbf{B}\boldsymbol{\pi} + \mathbf{e}$$

where: - $\boldsymbol{\mu}^ - r_f\mathbf{1}$: Excess implied returns ($n \times 1$) - $\mathbf{B}$: Factor loadings ($n \times k$) - $\boldsymbol{\pi}$: Factor risk premia ($k \times 1$) - what we want* - $\mathbf{e}$: Regression residuals ($n \times 1$)

3.3.2 OLS Solution

Theorem 3.1 (Factor Premia Extraction)

The OLS estimate of factor premia is:

$$\boxed{\hat{\boldsymbol{\pi}} = (\mathbf{B}'\mathbf{B})^{-1}\mathbf{B}'(\boldsymbol{\mu}^* - r_f\mathbf{1})}$$

Proof:

Minimize sum of squared residuals: $$\min_{\boldsymbol{\pi}} |\boldsymbol{\mu}^* - r_f\mathbf{1} - \mathbf{B}\boldsymbol{\pi}|^2$$

Take derivative with respect to $\boldsymbol{\pi}$: $$\frac{\partial}{\partial \boldsymbol{\pi}} = -2\mathbf{B}'(\boldsymbol{\mu}^* - r_f\mathbf{1} - \mathbf{B}\boldsymbol{\pi}) = 0$$

Solving: $$\mathbf{B}'\mathbf{B}\boldsymbol{\pi} = \mathbf{B}'(\boldsymbol{\mu}^ - r_f\mathbf{1})$$ $$\hat{\boldsymbol{\pi}} = (\mathbf{B}'\mathbf{B})^{-1}\mathbf{B}'(\boldsymbol{\mu}^ - r_f\mathbf{1}) \quad \blacksquare$$

3.3.3 Weighted OLS

For more robust estimates, use weighted least squares:

$$\hat{\boldsymbol{\pi}}_{WLS} = (\mathbf{B}'\mathbf{W}\mathbf{B})^{-1}\mathbf{B}'\mathbf{W}(\boldsymbol{\mu}^* - r_f\mathbf{1})$$

Common weighting schemes: - Market cap weights: $W_{ii} = \text{mcap}i / \sum_j \text{mcap}_j$ - Inverse variance: $W{ii} = 1/\sigma_i^2$ - Equal weights: $W_{ii} = 1/n$

3.3.4 Standard Errors

Under homoskedastic errors, the covariance of $\hat{\boldsymbol{\pi}}$ is:

$$\text{Var}(\hat{\boldsymbol{\pi}}) = \sigma_e^2 (\mathbf{B}'\mathbf{B})^{-1}$$

where $\sigma_e^2 = \frac{1}{n-k}\sum_{i=1}^n e_i^2$ is the residual variance.

3.4 Fama-MacBeth Two-Pass Procedure

3.4.1 Overview

The Fama-MacBeth (1973) procedure separates: 1. Pass 1 (Time-series): Estimate betas for each asset 2. Pass 2 (Cross-sectional): Estimate premia at each time point

This allows for time-varying factor premia and proper standard error estimation.

3.4.2 Pass 1: Time-Series Beta Estimation

For each asset $i$, run time-series regression:

$$r_{i,t} - r_{f,t} = \alpha_i + \sum_{j=1}^{k}\beta_{ij}f_{j,t} + \epsilon_{i,t}$$

Output: Matrix $\hat{\mathbf{B}} \in \mathbb{R}^{n \times k}$ of estimated betas.

3.4.3 Pass 2: Cross-Sectional Gamma Estimation

For each time $t$, run cross-sectional regression:

$$r_{i,t} - r_{f,t} = \gamma_{0,t} + \sum_{j=1}^{k}\gamma_{j,t}\hat{\beta}{ij} + \eta{i,t}$$

Output: Time series ${\hat{\gamma}{j,t}}{t=1}^T$ for each factor $j$.

3.4.4 Final Estimates

Average across time:

$$\hat{\pi}j = \frac{1}{T}\sum{t=1}^{T}\hat{\gamma}_{j,t}$$

Standard error (accounts for time-series variation):

$$SE(\hat{\pi}j) = \frac{1}{\sqrt{T}}\sqrt{\frac{1}{T-1}\sum{t=1}^{T}(\hat{\gamma}_{j,t} - \hat{\pi}_j)^2}$$

3.4.5 Shanken Correction

Problem: Pass 2 uses estimated betas, not true betas. This causes errors-in-variables bias in standard errors.

Shanken (1992) Correction:

$$\text{Var}(\hat{\boldsymbol{\pi}}){corrected} = \left(1 + \hat{\boldsymbol{\pi}}'\hat{\mathbf{\Sigma}}_f^{-1}\hat{\boldsymbol{\pi}}\right) \cdot \text{Var}(\hat{\boldsymbol{\pi}}){OLS}$$

The correction factor $c = 1 + \hat{\boldsymbol{\pi}}'\hat{\mathbf{\Sigma}}_f^{-1}\hat{\boldsymbol{\pi}}$ inflates standard errors to account for beta estimation uncertainty.

Intuition: When premia are large relative to factor variance, beta estimation errors have bigger impact on premia estimates.

3.5 The Bridge: Assets → Factors → New Assets

3.5.1 The Complete Flow

[Asset Weights w]
       |
       | Inverse Optimization (Section 2)
       v
[Asset Implied Returns μ*]
       |
       | Cross-Sectional Regression (Section 3.3)
       v
[Factor Risk Premia π]
       |
       | Factor Projection (Section 3.5.2)
       v
[New Asset Expected Return μ_new]

3.5.2 Pricing New Assets

Theorem 3.2 (New Asset Pricing)

Given factor premia $\hat{\boldsymbol{\pi}}$ and a new asset's factor exposures $\mathbf{B}_{new}$:

$$\boxed{\mu_{new} = r_f + \mathbf{B}_{new}'\hat{\boldsymbol{\pi}}}$$

Example: Pricing a BANTLEON fund with: - Equity beta: $\beta_{equity} = 0.8$ - Duration beta: $\beta_{duration} = 5.0$ - Credit beta: $\beta_{credit} = 0.3$

If estimated premia are $\pi_{equity} = 5\%$, $\pi_{duration} = 0.5\%$, $\pi_{credit} = 2\%$:

$$\mu_{new} = r_f + 0.8 \times 5\% + 5.0 \times 0.5\% + 0.3 \times 2\% = r_f + 7.1\%$$

3.5.3 Confidence Intervals

The variance of $\mu_{new}$ propagates from $\text{Var}(\hat{\boldsymbol{\pi}})$:

$$\text{Var}(\mu_{new}) = \mathbf{B}{new}'\text{Var}(\hat{\boldsymbol{\pi}})\mathbf{B}{new}$$

95% confidence interval:

$$\mu_{new} \pm 1.96 \sqrt{\mathbf{B}{new}'\text{Var}(\hat{\boldsymbol{\pi}})\mathbf{B}{new}}$$

3.5.4 Handling Missing Factors

When a new asset has exposures to factors not in our model:

1. Conservative approach: Assume zero premium for missing factors
2. Proxy approach: Map to closest available factor
3. Hierarchical approach: Use factor category averages

3.6 Practical Considerations

3.6.1 Factor Selection

3.6.2 Estimation Frequency

• Beta estimation: Rolling 60-month window
• Premia estimation: Monthly cross-sections
• Rebalancing: Quarterly or monthly

3.6.3 Robustness Checks

1. Subsample stability: Compare premia across periods
2. Alternative specifications: Different factor definitions
3. Out-of-sample testing: Rolling backtest

4. Bantleon Calibration Methods

Existing implementation in python/brisma/bantleon/

4.1 Method 1: 10Y Bond Calibration

$$\lambda_{M1} = \frac{\ln\left(\frac{1+y_{10Y}}{1+r_f}\right)}{\beta_{ref}}$$

4.2 Method 2: Historical Performance

$$\lambda_{M2} = \sum_{t=1}^{T}w_t f_t$$

with exponential decay weights: $w_t \propto 0.5^{t/\tau}$

4.3 Hybrid Method

$$\mu_{hybrid} = R^2 \cdot \mu_{M1} + (1-R^2) \cdot \mu_{M2}$$

5. Worked Example: BRISMA Portfolio

This section addresses Issue #51

5.1 Data Description

5.1.1 Portfolio Composition

The BRISMA client portfolio contains 32 assets across 5 asset classes:

Note: Total weight > 100% indicates leverage.

5.1.2 Top 10 Holdings

5.1.3 Available Covariance Matrices

Three covariance matrices from R/examples/example1.R:

5.1.4 Risk Model Factors

73 factors from risk model, grouped by type:

5.2 Step-by-Step Calculation

5.2.1 Simplified 4-Asset Example

For clarity, we first demonstrate with 4 assets:

Assets: 1. S&P 500 (Equity) 2. Euro Government (Bond) 3. Euro Corporate (Credit) 4. Gold (Commodity)

Weights (normalized to sum to 1): $$\mathbf{w} = \begin{pmatrix} 0.30 \ 0.25 \ 0.35 \ 0.10 \end{pmatrix}$$

Covariance Matrix (annualized, illustrative): $$\mathbf{Q} = \begin{pmatrix} 0.0225 & 0.0020 & 0.0040 & 0.0010 \ 0.0020 & 0.0036 & 0.0024 & -0.0005 \ 0.0040 & 0.0024 & 0.0064 & 0.0008 \ 0.0010 & -0.0005 & 0.0008 & 0.0169 \end{pmatrix}$$

Diagonal: $\sigma_{S\&P} = 15\%$, $\sigma_{Govt} = 6\%$, $\sigma_{Corp} = 8\%$, $\sigma_{Gold} = 13\%$

5.2.2 Step 1: Choose Risk Aversion $\lambda$

Using Black-Litterman default: $$\lambda = 2.5$$

Alternatively, from historical data (assuming 5% portfolio return, 8% volatility): $$\lambda^* = \frac{\mu_p}{\sigma_p^2} = \frac{0.05}{0.08^2} = 7.8$$

We proceed with $\lambda = 2.5$ (conservative).

5.2.3 Step 2: Compute Implied Returns

$$\boldsymbol{\mu}^* = \lambda \mathbf{Q} \mathbf{w}$$

Calculation:

$$\mathbf{Q}\mathbf{w} = \begin{pmatrix} 0.0225 \times 0.30 + 0.0020 \times 0.25 + 0.0040 \times 0.35 + 0.0010 \times 0.10 \ 0.0020 \times 0.30 + 0.0036 \times 0.25 + 0.0024 \times 0.35 + (-0.0005) \times 0.10 \ 0.0040 \times 0.30 + 0.0024 \times 0.25 + 0.0064 \times 0.35 + 0.0008 \times 0.10 \ 0.0010 \times 0.30 + (-0.0005) \times 0.25 + 0.0008 \times 0.35 + 0.0169 \times 0.10 \end{pmatrix}$$

$$= \begin{pmatrix} 0.00855 \ 0.00184 \ 0.00412 \ 0.00197 \end{pmatrix}$$

$$\boldsymbol{\mu}^* = 2.5 \times \begin{pmatrix} 0.00855 \ 0.00184 \ 0.00412 \ 0.00197 \end{pmatrix} = \begin{pmatrix} 2.14\% \ 0.46\% \ 1.03\% \ 0.49\% \end{pmatrix}$$

5.2.4 Step 3: Economic Interpretation

Observations: - Higher-weight assets have higher implied returns (by construction) - S&P 500 has highest implied return due to high weight and high risk - Gold has low implied return despite high volatility (low weight)

5.2.5 Step 4: Extract Factor Premia

Assume 2-factor model: Equity ($\beta_E$) and Duration ($\beta_D$)

Factor Betas:

$$\mathbf{B} = \begin{pmatrix} 1.0 & 0.0 \ 0.0 & 7.0 \ 0.2 & 5.0 \ 0.1 & -0.5 \end{pmatrix}$$

Cross-sectional regression (assuming $r_f = 0$):

$$\hat{\boldsymbol{\pi}} = (\mathbf{B}'\mathbf{B})^{-1}\mathbf{B}'(\boldsymbol{\mu}^* - r_f\mathbf{1})$$

Computing $\mathbf{B}'\mathbf{B}$: $$\mathbf{B}'\mathbf{B} = \begin{pmatrix} 1.05 & 0.95 \ 0.95 & 74.25 \end{pmatrix}$$

And $\mathbf{B}'\boldsymbol{\mu}^$: $$\mathbf{B}'\boldsymbol{\mu}^ = \begin{pmatrix} 2.40\% \ 8.67\% \end{pmatrix}$$

Solving: $$\hat{\boldsymbol{\pi}} = \begin{pmatrix} \pi_E \ \pi_D \end{pmatrix} = \begin{pmatrix} 2.17\% \ 0.09\% \end{pmatrix}$$

Interpretation: - Equity risk premium: 2.17% per unit beta - Duration risk premium: 0.09% per year of duration (9 bps/year)

5.2.6 Step 5: Price New Asset

New Asset: BANTLEON fund with $\beta_E = 0.5$, $\beta_D = 3.0$

$$\mu_{new} = r_f + 0.5 \times 2.17\% + 3.0 \times 0.09\% = r_f + 1.36\%$$

Expected excess return: 1.36% (before adding risk-free rate)

5.3 Full BRISMA Portfolio Results

5.3.1 Implied Returns Summary (32 Assets)

Using $\lambda = 2.5$ and $\mathbf{Q}_{shrink}$:

5.3.2 By Asset Class

5.3.3 Sensitivity to $\lambda$

Observation: Implied returns scale linearly with $\lambda$.

5.3.4 Comparison Across Covariance Matrices

Observation: Results are robust across estimation methods.

5.4 Python Code Reference

# File: python/examples/implied_returns_example.py

import numpy as np
import pandas as pd
from brisma import load_data
from brisma.inverse_optimization import inverse_optimize_mv

# Load data
data = load_data()
portfolio = data['tbl_client_portfolio']

# Get weights (normalize to sum to 1 if needed)
weights = portfolio['weight'].values
weights = weights / weights.sum()  # Optional normalization

# Load covariance matrix (from R outputs)
Q = np.load('outputs/Q_shrink.npy')  # Or load from R

# Set risk aversion
lambda_param = 2.5

# Compute implied returns
mu_star = inverse_optimize_mv(weights, Q, lambda_param)

# Display results
results = pd.DataFrame({
    'Asset': portfolio['name'],
    'Weight': weights,
    'Implied_Return': mu_star
})
print(results.sort_values('Implied_Return', ascending=False))

6. Validation and Quality Checks

6.1 Economic Sanity Checks

• Equity premium positive?
• Duration premium positive?
• Magnitudes reasonable (1-15% annualized)?

6.2 Statistical Validation

• $R^2$ of factor regression
• Residual analysis
• Information coefficient

6.3 Backtesting

• Rolling window procedure
• Hit ratio and tracking error

7. References

To be completed in Issue #50

Foundational Papers

1. Black, F., & Litterman, R. (1992). "Global portfolio optimization." Financial Analysts Journal, 48(5), 28-43.

2. Merton, R. C. (1980). "On estimating the expected return on the market: An exploratory investigation." Journal of Financial Economics, 8(4), 323-361.

3. Ross, S. A. (1976). "The arbitrage theory of capital asset pricing." Journal of Economic Theory, 13(3), 341-360.

4. Fama, E. F., & MacBeth, J. D. (1973). "Risk, return, and equilibrium: Empirical tests." Journal of Political Economy, 81(3), 607-636.

5. Shanken, J. (1992). "On the estimation of beta-pricing models." Review of Financial Studies, 5(1), 1-33.

Extensions

1. He, G., & Litterman, R. (1999). "The intuition behind Black-Litterman model portfolios." Goldman Sachs Investment Management Research.

2. Idzorek, T. (2007). "A step-by-step guide to the Black-Litterman model." Forecasting Expected Returns in the Financial Markets, 17-38.

3. Fama, E. F., & French, K. R. (1993). "Common risk factors in the returns on stocks and bonds." Journal of Financial Economics, 33(1), 3-56.

Appendices

Appendix A: Matrix Calculus Review

Completed in Issue #50

A.1 Notation Conventions

A.2 Gradient of Scalar Functions

Definition: For $f: \mathbb{R}^n \to \mathbb{R}$, the gradient is:

$$\nabla_{\mathbf{x}} f = \frac{\partial f}{\partial \mathbf{x}} = \begin{pmatrix} \frac{\partial f}{\partial x_1} \ \vdots \ \frac{\partial f}{\partial x_n} \end{pmatrix}$$

A.3 Key Derivative Rules

A.4 Hessian Matrix

For $f: \mathbb{R}^n \to \mathbb{R}$, the Hessian is:

$$\mathbf{H} = \nabla^2_{\mathbf{x}} f = \begin{pmatrix} \frac{\partial^2 f}{\partial x_1^2} & \cdots & \frac{\partial^2 f}{\partial x_1 \partial x_n} \ \vdots & \ddots & \vdots \ \frac{\partial^2 f}{\partial x_n \partial x_1} & \cdots & \frac{\partial^2 f}{\partial x_n^2} \end{pmatrix}$$

A.5 Chain Rule

For $f(\mathbf{g}(\mathbf{x}))$:

$$\frac{\partial f}{\partial \mathbf{x}} = \frac{\partial \mathbf{g}}{\partial \mathbf{x}}' \frac{\partial f}{\partial \mathbf{g}}$$

A.6 Product Rule

For $f(\mathbf{x}) = \mathbf{u}(\mathbf{x})'\mathbf{v}(\mathbf{x})$:

$$\nabla_{\mathbf{x}} f = \frac{\partial \mathbf{u}}{\partial \mathbf{x}}'\mathbf{v} + \frac{\partial \mathbf{v}}{\partial \mathbf{x}}'\mathbf{u}$$

Appendix B: Complete Derivations

Completed in Issue #50

B.1 Mean-Variance Inverse Optimization (Detailed)

Problem: Find $\boldsymbol{\mu}$ such that $\mathbf{w}$ is optimal for:

$$\max_{\mathbf{w}} U(\mathbf{w}) = \mathbf{w}'\boldsymbol{\mu} - \frac{\lambda}{2}\mathbf{w}'\mathbf{Q}\mathbf{w}$$

Step 1: Compute gradient

Using rules from Appendix A: - $\nabla_{\mathbf{w}}(\mathbf{w}'\boldsymbol{\mu}) = \boldsymbol{\mu}$ - $\nabla_{\mathbf{w}}(\mathbf{w}'\mathbf{Q}\mathbf{w}) = 2\mathbf{Q}\mathbf{w}$ (since $\mathbf{Q}$ symmetric)

Therefore: $$\nabla_{\mathbf{w}} U = \boldsymbol{\mu} - \lambda\mathbf{Q}\mathbf{w}$$

Step 2: Set first-order condition

At optimum: $$\nabla_{\mathbf{w}} U = \mathbf{0}$$ $$\boldsymbol{\mu} - \lambda\mathbf{Q}\mathbf{w} = \mathbf{0}$$

Step 3: Solve for $\boldsymbol{\mu}$

$$\boldsymbol{\mu}^* = \lambda\mathbf{Q}\mathbf{w}$$

Step 4: Verify second-order condition

$$\nabla^2_{\mathbf{w}} U = -\lambda\mathbf{Q}$$

For maximum, need $-\lambda\mathbf{Q} \prec 0$ (negative definite). Since $\lambda > 0$ and $\mathbf{Q} \succ 0$ (positive definite), this holds. $\blacksquare$

B.2 Constrained Case with Budget Constraint

Problem: Maximize utility subject to $\mathbf{1}'\mathbf{w} = 1$

Lagrangian: $$\mathcal{L}(\mathbf{w}, \gamma) = \mathbf{w}'\boldsymbol{\mu} - \frac{\lambda}{2}\mathbf{w}'\mathbf{Q}\mathbf{w} - \gamma(\mathbf{1}'\mathbf{w} - 1)$$

KKT Conditions:

1. Stationarity: $\frac{\partial \mathcal{L}}{\partial \mathbf{w}} = \boldsymbol{\mu} - \lambda\mathbf{Q}\mathbf{w} - \gamma\mathbf{1} = \mathbf{0}$
2. Primal feasibility: $\mathbf{1}'\mathbf{w} = 1$

From (1): $$\boldsymbol{\mu} = \lambda\mathbf{Q}\mathbf{w} + \gamma\mathbf{1}$$

To find $\gamma$: Pre-multiply by $\mathbf{1}'\mathbf{Q}^{-1}$: $$\mathbf{1}'\mathbf{Q}^{-1}\boldsymbol{\mu} = \lambda\mathbf{1}'\mathbf{w} + \gamma\mathbf{1}'\mathbf{Q}^{-1}\mathbf{1}$$

Using $\mathbf{1}'\mathbf{w} = 1$: $$\gamma = \frac{\mathbf{1}'\mathbf{Q}^{-1}\boldsymbol{\mu} - \lambda}{\mathbf{1}'\mathbf{Q}^{-1}\mathbf{1}}$$

Interpretation: $\gamma$ represents the risk-free rate or zero-beta return. $\blacksquare$

B.3 OLS Factor Premia Derivation

Problem: Minimize $|\boldsymbol{\mu}^* - r_f\mathbf{1} - \mathbf{B}\boldsymbol{\pi}|^2$

Step 1: Expand objective

Let $\mathbf{y} = \boldsymbol{\mu}^* - r_f\mathbf{1}$ (excess returns)

$$J(\boldsymbol{\pi}) = (\mathbf{y} - \mathbf{B}\boldsymbol{\pi})'(\mathbf{y} - \mathbf{B}\boldsymbol{\pi})$$ $$= \mathbf{y}'\mathbf{y} - 2\boldsymbol{\pi}'\mathbf{B}'\mathbf{y} + \boldsymbol{\pi}'\mathbf{B}'\mathbf{B}\boldsymbol{\pi}$$

Step 2: Take gradient

$$\nabla_{\boldsymbol{\pi}} J = -2\mathbf{B}'\mathbf{y} + 2\mathbf{B}'\mathbf{B}\boldsymbol{\pi}$$

Step 3: Set to zero and solve

$$-2\mathbf{B}'\mathbf{y} + 2\mathbf{B}'\mathbf{B}\boldsymbol{\pi} = \mathbf{0}$$ $$\mathbf{B}'\mathbf{B}\boldsymbol{\pi} = \mathbf{B}'\mathbf{y}$$ $$\hat{\boldsymbol{\pi}} = (\mathbf{B}'\mathbf{B})^{-1}\mathbf{B}'\mathbf{y}$$

Substituting back: $$\hat{\boldsymbol{\pi}} = (\mathbf{B}'\mathbf{B})^{-1}\mathbf{B}'(\boldsymbol{\mu}^* - r_f\mathbf{1}) \quad \blacksquare$$

B.4 Shanken Correction Derivation

Setup: In Fama-MacBeth, betas are estimated with error: $$\hat{\beta}_i = \beta_i + u_i$$

Problem: Standard OLS ignores $\text{Var}(u_i)$, understating $\text{Var}(\hat{\boldsymbol{\pi}})$

Shanken (1992) shows:

$$\text{Var}(\hat{\boldsymbol{\pi}}){true} = \left(1 + \boldsymbol{\pi}'\boldsymbol{\Sigma}_f^{-1}\boldsymbol{\pi}\right) \cdot \text{Var}(\hat{\boldsymbol{\pi}}){OLS} + \boldsymbol{\Sigma}_{\pi}$$

where: - $\boldsymbol{\Sigma}f$ = factor covariance matrix - $\boldsymbol{\Sigma}{\pi}$ = asymptotic variance correction (often small)

Simplified correction (ignoring $\boldsymbol{\Sigma}_{\pi}$):

$$c = 1 + \hat{\boldsymbol{\pi}}'\hat{\boldsymbol{\Sigma}}_f^{-1}\hat{\boldsymbol{\pi}}$$

$$SE(\hat{\pi}j){corrected} = \sqrt{c} \cdot SE(\hat{\pi}j){OLS}$$

Intuition: The correction factor $c$ is larger when: - Factor premia are large ($\boldsymbol{\pi}$ big) - Factor variances are small ($\boldsymbol{\Sigma}_f^{-1}$ big)

This makes sense: if premia are large and factors are stable, any error in beta estimates has big impact on premia estimates. $\blacksquare$

B.5 Lambda Extraction from Observed Returns

Given: Observed returns $\tilde{\boldsymbol{\mu}}$, weights $\mathbf{w}$, covariance $\mathbf{Q}$

Problem: Find $\lambda$ such that $\tilde{\boldsymbol{\mu}} = \lambda\mathbf{Q}\mathbf{w}$

Method: Project onto portfolio return

Pre-multiply both sides by $\mathbf{w}'$: $$\mathbf{w}'\tilde{\boldsymbol{\mu}} = \lambda\mathbf{w}'\mathbf{Q}\mathbf{w}$$

Note: - $\mathbf{w}'\tilde{\boldsymbol{\mu}} = \mu_p$ (expected portfolio return) - $\mathbf{w}'\mathbf{Q}\mathbf{w} = \sigma_p^2$ (portfolio variance)

Therefore: $$\lambda^* = \frac{\mu_p}{\sigma_p^2}$$

Economic interpretation: $\lambda$ is the Sharpe ratio divided by portfolio volatility: $$\lambda = \frac{\mu_p}{\sigma_p^2} = \frac{\mu_p/\sigma_p}{\sigma_p} = \frac{SR_p}{\sigma_p} \quad \blacksquare$$

Appendix C: Python Implementation Reference

# python/brisma/inverse_optimization.py

def inverse_optimize_mv(weights: np.ndarray,
                        cov_matrix: np.ndarray,
                        lambda_param: float) -> np.ndarray:
    """
    Mean-Variance inverse optimization.

    Parameters
    ----------
    weights : np.ndarray
        Portfolio weights (n,)
    cov_matrix : np.ndarray
        Covariance matrix (n, n)
    lambda_param : float
        Risk aversion parameter

    Returns
    -------
    np.ndarray
        Implied expected returns (n,)
    """
    return lambda_param * cov_matrix @ weights

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