Practical Portfolio Construction

Prerequisites: Chapter 9 on convex quadratic programming and KKT conditions, Chapter 15 on multi-factor risk models, Chapter 18 on volatility estimation, and Chapter 20 on modern portfolio theory and the unconstrained analytical solution for the mean–variance tangency portfolio. After completing this chapter, you will be able to estimate covariance matrices from a training window using Ledoit–Wolf shrinkage and check conditional positive definiteness and out-of-sample risk error; transform long-only maximum-Sharpe optimization, whose objective is not concave, exactly into an equivalent convex quadratic program (convex QP); solve for an equal risk contribution (ERC) portfolio; combine market equilibrium and subjective views with the Black–Litterman model; derive the Kelly criterion's log-growth rate and explain fractional-Kelly constraints; calculate asset drift, turnover, and trading costs; and objectively evaluate portfolio out-of-sample performance after costs.

1. Chapter Objectives

After studying this chapter, you should be able to:

1. Calculate estimation error in sample means and covariance matrices, and quantify why uncertainty in expected returns can damage optimized weights more severely than the covariance matrix;
2. Derive the optimal shrinkage intensity toward a scaled identity matrix in Ledoit–Wolf (2004), then use training-window data to compute a positive-definite covariance matrix for out-of-sample use;
3. Calculate two-sided rebalancing turnover, drifted portfolio weights, and trading-friction drag, and maintain an accounting ledger for gross and post-cost net returns;
4. Distinguish dollar-neutral, beta-neutral, and sector-neutral portfolios; explain how absolute sector neutrality across mutually exclusive and exhaustive sectors implies dollar neutrality but generally does not imply beta neutrality; and explain that Euclidean projection minimizes only weight distance;
5. Derive the Euler decomposition of portfolio volatility, calculate marginal risk contributions and relative risk shares, and solve for ERC weights using damped Newton's method with a Newton-decrement stopping rule;
6. Restate the long-only maximum-Sharpe variable substitution proved in Chapter 9, which converts the problem into a convex quadratic program, and implement a solution workflow with a feasibility precheck for weight caps;
7. Derive the Black–Litterman reverse-optimization prior, the block-Gaussian posterior mean and posterior-mean covariance, and verify the orthogonality property of view tilts;
8. Derive the binary-bet log-growth optimum under the Kelly criterion, its discrete second-order expansion, and its diffusion-limit quadratic proxy; analyze the tail-ruin mechanism under full Kelly and the risk–return trade-off under half Kelly;
9. Use a multi-factor risk model to decompose portfolio variance and volatility into factor and specific-risk contributions;
10. Using the French 10 industry historical snapshot and Project 3's 440-month rolling backtest, objectively evaluate each strategy's post-cost rolling out-of-sample performance and statistical uncertainty on the currently revised historical snapshot, without treating it as point-in-time (PIT) tradable evidence.

You have completed the chapter when you can independently reproduce all numerical examples and pass the code assertions, understand the assumptions and failure boundaries of every model, complete all exercises in Sections 15–17, and check the full derivations and code in Section 18.

2. Why Quants Need Practical Portfolio Construction

In Chapter 20's mean–variance theory (Modern Portfolio Theory, MPT), tangency-portfolio weights are given by the analytical expression $\mathbf{w}^* \propto \Sigma^{-1}\boldsymbol{\mu}_{excess}$. In live quantitative investing, however, directly substituting the historical sample mean $\hat{\boldsymbol{\mu}}$ and sample covariance matrix $S$ into this expression (the plug-in estimator) usually produces extreme and unstable weights.

This failure arises from three unavoidable practical challenges.

First, the inversion step sharply amplifies parameter-estimation error. Financial time series are full of low signal-to-noise-ratio observations. The annualized standard error of an expected-return estimate is often on the same order as the mean itself, while the smallest eigenvalues of a high-dimensional covariance matrix are often close to zero. Inverting a noisy matrix greatly magnifies the directions associated with these small eigenvalues, leading the optimizer to take large long and short positions in directions with the most in-sample noise. In quantitative finance, this phenomenon is sometimes called an “error maximizer.”

Second, the theoretical tangency portfolio does not model real trading frictions or constraints. The analytical solution assumes unrestricted short selling, no borrowing-rate penalty, no market-impact or liquidity cost, and no trading commissions. In institutional investing, however, managers face strict long-only rules, position limits, sector and market-risk budgets, and slippage caused by turnover. If these real-world boundaries are ignored, apparently strong paper returns can be eroded by monthly rebalancing costs.

Finally, diversification is often misunderstood in practice. A nominally equal-weighted allocation is not necessarily diversified by risk. If a portfolio contains highly volatile assets, a small number of them may dominate total portfolio risk. Likewise, a long–short portfolio described as “market neutral” may still carry substantial systematic market beta or sector-tilt risk if it satisfies only dollar neutrality.

The goal of practical portfolio construction is to use mathematically rigorous shrinkage estimation, convex-optimization transformations, risk budgeting, and Bayesian prior constraints to turn noisy forecasts into real holdings that are implementable out of sample, have controlled risk, and can withstand trading frictions.

3. Intuition: From Sample Noise to Implementable Holdings

To see why unconstrained optimization is fragile, consider a deliberately simple thought experiment.

Suppose the market has two assets, each with a true monthly excess return of $1.0\%$, monthly volatility of $4.0\%$, and a correlation of $0.7$ (a constructed teaching example). The two assets are perfectly symmetric, so the true optimal tangency portfolio clearly holds $50\%$ of each.

Now suppose a researcher estimates the inputs from the past 60 months. Due to sampling variation, asset 1's sample mean happens to be high at $1.3\%$, while asset 2's happens to be low at $0.7\%$. This small sampling perturbation of $0.3\%$ is entirely plausible (the standard error is about $4.0\% / \sqrt{60} \approx 0.52\%$).

Substituting these estimates into $\Sigma^{-1}\boldsymbol{\mu}_{excess}$, the two assets' equal volatilities imply that $\Sigma^{-1}$ is proportional to $\begin{bmatrix}1&-0.7\\-0.7&1\end{bmatrix}$. The resulting direction is proportional to $(1.3-0.7\times0.7,\ 0.7-0.7\times1.3)=(0.81,-0.21)$, giving normalized weights of roughly $(135\%,-35\%)$, even though the true optimum is $(50\%,50\%)$. Evaluated using the true parameters, both portfolios have expected excess returns of 1.0% per month, but monthly volatility is $4\%\times\sqrt{1.35^2+0.35^2-2(0.7)(1.35)(0.35)}=4.53\%$ for $(135\%,-35\%)$, compared with $4\%\times\sqrt{0.85}=3.69\%$ for the equal-weight portfolio. The monthly Sharpe ratio falls from 0.271 to 0.221. The higher the correlation, the more the same mean-estimation error is amplified (at $\rho=0.5$, the weights are about $(95\%,5\%)$).

This simple example reveals the core logic of practical portfolio construction:

1. Shrinkage guards against covariance ill-conditioning: With a limited sample, the extreme eigenvalues of the sample covariance matrix can be badly distorted. Shrinking the matrix toward a structured target, such as a diagonal matrix or factor model, at a theoretically chosen rate can dampen very small eigenvalues and keep the inverse from turning local noise into extreme positions.
2. Constraints are the first line of defense against mean-estimation errors: Long-only constraints rule out short positions and mathematically cap one mechanism by which estimated means create extreme weights. Although long-only constraints invalidate the analytical solution, they restrict optimization to the simplex and can prevent extreme hedges.
3. Risk parity avoids relying on unreliable expected-return estimates: When expected returns cannot be estimated precisely, a portfolio that uses the covariance matrix to equalize assets' contributions to total risk avoids one source of estimation error. It still depends on a risk model, so any improvement in return or risk must be compared out of sample.
4. Trading frictions shape the turnover boundary: Price changes cause asset weights to drift away from their targets. If the portfolio is forced back to the exact target every month, commissions and market-impact costs from turnover accumulate quickly. A balance is needed between tracking error from drift and turnover costs.

Simulation of Estimation Error and Comparison of Strategy Distributions

• Horizontal axis: Annualized population Sharpe ratio realized when each strategy's estimated weights are applied to the true population (multiplied by $\sqrt{12}$);
• Vertical axis: The six strategies compared (plug-in tangency, GMV sample, GMV Ledoit–Wolf, long-only max Sharpe (LW), ERC (LW), and 1/N);
• Units: Dimensionless Sharpe ratio;
• Sample and setup: The full-sample monthly first and second moments of the French 10 industries are treated as a fixed “pseudo-population.” With random seed 20261021, 500 independent Gaussian Monte Carlo samples are drawn, each 60 months long (representing a five-year rolling window);
• How to read the figure and what to look for:
  1. The theoretical maximum true tangency Sharpe ratio is marked by the red vertical line (about $0.6262$ annualized);
  2. The unconstrained plug-in tangency portfolio performs worst, with a median annualized Sharpe ratio of just $0.2812$ and a 10th percentile of about $0.079$ (the whiskers show the 5th–95th percentiles). In addition, 57 samples failed outright because the denominator was non-positive; those failures were excluded from the plotted distribution, making the box plot substantially optimistic. The long-only max-Sharpe (LW) strategy had another 11 samples with no positive-excess-return portfolio in the training sample, triggering the input-feasibility failure described in Section 12.3; those were also excluded from the box plot;
  3. Adding a long-only constraint raises the median for long-only max Sharpe (LW) to $0.4859$, reducing dispersion caused by estimation error;
  4. Strategies that do not rely on expected-return estimates (GMV LW, ERC LW, and 1/N) are the most stable: the median annualized Sharpe is $0.5215$ for GMV (LW), $0.5057$ for ERC (LW), and a stable $0.4983$ for 1/N;
• Limitation: This figure shows a statistical simulation of sampling error only. All asset returns are strictly Gaussian; real-market jumps, fat tails, and structural breaks are not included.

4. Mathematical Foundations and Notation

Before turning to the derivations, we standardize notation, matrix dimensions, and units throughout the chapter:

• Number of assets and sample periods: The number of assets is denoted by $n$ (or by $p$ for the cross-sectional dimension), and the number of periods in the historical training window by $T$.

• Return vector: The vector of single-period simple returns at time $t$ is $\mathbf{r}_t = (r_{1,t}, \dots, r_{n,t})^\top \in \mathbb{R}^n$. This chapter uses monthly decimal returns by default. Explicitly labeled annualized teaching examples are exceptions and must not be mixed with monthly empirical inputs; for example, a monthly return of $1\%$ is written as $0.01$.

• Expected-excess-return vector: $\boldsymbol{\mu}_{excess} = \boldsymbol{\mu} - r_f \mathbf{1} \in \mathbb{R}^n$, where $\boldsymbol{\mu} = E[\mathbf{r}]$, $r_f$ is the risk-free rate, and $\mathbf{1} = (1, \dots, 1)^\top$ is the all-ones vector.

• Covariance matrix: $\Sigma \in \mathbb{R}^{n \times n}$ is the single-period return covariance matrix and is positive semidefinite (PSD). If invertibility is required, it must be positive definite (PD), denoted by $\Sigma \succ 0$.

• Portfolio-weight vector: $\mathbf{w} = (w_1, \dots, w_n)^\top \in \mathbb{R}^n$. $w_i$ is the fraction of portfolio net asset value (NAV) invested in asset $i$. Fully invested means $\mathbf{1}^\top \mathbf{w} = 1$, and long-only means $\mathbf{w} \ge \mathbf{0}$.

• Matrix Frobenius-norm convention: Following Ledoit & Wolf (2004), this chapter normalizes the Frobenius norm of a $p \times p$ matrix $A$ by its dimension $p$:

  $$\|A\|^2 = \frac{1}{p}\operatorname{tr}(AA^\top) = \frac{1}{p}\sum_{i=1}^p \sum_{j=1}^p a_{ij}^2,$$

  The matrix inner product is defined as $\langle A, B \rangle = \frac{1}{p}\operatorname{tr}(AB^\top)$. Unless stated otherwise, $\|\cdot\|$ denotes this normalized norm.

• Time and information set: Maintain strict causality. In the idealized month-boundary test, if a rebalance decision is made and executed at the end of month $t-1$, the historical training window available for that decision is restricted to $[t-W, t-1]$ (or to $[t-W-1, t-2]$ when a one-month data-publication lag is assumed). The current-period return $\mathbf{r}_t$ is never known before the end of month $t$. Observing returns, calculating weights, and executing at the same month-end does not guarantee real-world availability; a live system must record when data are received and when the next trade can occur.

5. Formal Definitions: Estimation Error, Neutrality Constraints, and Risk Budgets

5.1 Mean-Estimation Error and Covariance Ill-Conditioning

Suppose asset returns are independently and identically distributed, with true mean $\boldsymbol{\mu}$ and true covariance $\Sigma$. The sample-mean estimator is $\hat{\boldsymbol{\mu}} = \frac{1}{T}\sum_{t=1}^T \mathbf{r}_t$, and its estimation-error standard deviation is:

If an asset has monthly volatility $\sigma = 4\%$ (about $13.86\%$ annualized), then with $T = 120$ months (10 years of history), the standard error of its monthly mean is:

For a hypothetical stock, if a researcher assumes an annualized excess return of only $5\% \sim 8\%$ (a teaching assumption, not a market statistic verified in this chapter), even 10 years of data may not narrow the expected-return confidence interval enough to determine its sign reliably.

The sample covariance matrix, centered by the sample mean and divided by $T$, is defined as:

Equation (12) in Section 2.2 of Ledoit and Wolf (2004) rewrites Lemma 2.1 in terms of eigenvalues: in expectation, the dispersion of sample eigenvalues around their common mean equals the dispersion of the true eigenvalues plus $E\lVert S-\Sigma\rVert^2$. This shows that estimation noise inflates eigenvalue dispersion in the population sense; it does not guarantee that each sample maximum or minimum eigenvalue moves in a particular direction. The distortion becomes more pronounced as $p/T$ increases. When $p > T$, $S$ is rank-deficient: at least $p-T+1$ eigenvalues are exactly zero under exact arithmetic (after centering, the rank is at most $T-1$), so the sample inverse $S^{-1}$ does not exist. This does not imply that the population matrix $\Sigma$ is singular.

5.2 Three Definitions of Portfolio Neutrality

Quantitative investors often seek to eliminate exposure to specific macroeconomic or systematic risks. Common forms of “neutrality” include:

1. Dollar neutrality / capital neutrality:

   $$\mathbf{1}^\top \mathbf{w} = \sum_{i=1}^n w_i = 0.$$

   The total market value of long positions equals that of short positions, leaving net exposure equal to 0.

2. Beta neutrality:
   Let $\boldsymbol{\beta} = (\beta_1, \dots, \beta_n)^\top \in \mathbb{R}^n$ be the vector of regression slopes of the assets relative to a benchmark index, such as the S&P 500. A beta-neutral portfolio satisfies:

   $$\boldsymbol{\beta}^\top \mathbf{w} = \sum_{i=1}^n w_i \beta_i = 0.$$

   At the margin, the portfolio has no direct linear exposure to movements in the overall benchmark return.

3. Sector neutrality:
   Suppose the market is partitioned into $K$ mutually exclusive sectors, and let $I_k$ be the set of asset indices in sector $k$. Sector neutrality requires net holdings to be zero in every sector:

   $$\sum_{i \in I_k} w_i = 0, \quad \forall k \in \{1, \dots, K\}.$$

These conditions are distinct, but they are not independent. If the mutually exclusive sectors cover all assets, summing across sectors gives

Therefore, the absolute sector neutrality used in this chapter necessarily implies dollar neutrality, but generally does not imply beta neutrality. Dollar neutrality generally implies neither sector nor beta neutrality, and beta neutrality generally implies neither of the other two. For example, within one sector, weights $(0.5,-0.5)$ and betas $(1.5,0.5)$ produce zero net sector weight and zero net dollar exposure, but beta exposure is $0.5$. Conversely, weights $(0.5,-1.0)$ with betas $(1,0.5)$ have zero beta exposure but net weight of $-0.5$. If “sector neutrality” instead means an active-weight constraint relative to a fully invested benchmark $\mathbf w_b$, namely $Z(\mathbf w-\mathbf w_b)=0$, it implies $\sum_iw_i=\sum_iw_{b,i}$, not zero absolute net weight. Here $Z$ is a $K\times n$ matrix whose rows are sector indicator vectors; this benchmark-relative constraint should be distinguished from the absolute neutrality defined in this section.

5.3 Risk Contributions and the Euler Decomposition

Portfolio volatility is the function $\sigma(\mathbf{w}) = \sqrt{\mathbf{w}^\top \Sigma \mathbf{w}}$. Because $\sigma(a\mathbf{w}) = a\sigma(\mathbf{w})$ for any positive scalar $a > 0$, portfolio volatility is homogeneous of degree 1 in the weight vector $\mathbf{w}$.

The derivatives and risk shares below require $\sigma(\mathbf w)>0$ and a symmetric positive-semidefinite matrix $\Sigma$. By Euler's theorem for homogeneous functions:

This gives rise to three core definitions in the risk-contribution framework:

• Marginal risk contribution (MRC):

  $$\operatorname{MRC}_i = \frac{\partial \sigma(\mathbf{w})}{\partial w_i} = \frac{(\Sigma \mathbf{w})_i}{\sqrt{\mathbf{w}^\top \Sigma \mathbf{w}}} = \frac{(\Sigma \mathbf{w})_i}{\sigma(\mathbf{w})}.$$

  The economic meaning of $\operatorname{MRC}_i$ is the rate of change in total portfolio volatility when the weight of asset $i$ increases by a small amount. It has the same units as return volatility (monthly decimal units here). This is a partial derivative that holds all other coordinates fixed. If increasing asset i requires selling asset j to preserve the budget, the change along the self-financing direction is MRC_i−MRC_j; it should not be read as MRC_i alone.

• Absolute risk contribution (RC):

  $$\operatorname{RC}_i = w_i \operatorname{MRC}_i = \frac{w_i (\Sigma \mathbf{w})_i}{\sigma(\mathbf{w})}.$$

  By Euler's theorem, the absolute risk contributions of all assets sum exactly to total portfolio volatility: $\sum_{i=1}^n \operatorname{RC}_i = \sigma(\mathbf{w})$.

• Risk fraction / relative risk contribution:

  $$p_i = \frac{\operatorname{RC}_i}{\sigma(\mathbf{w})} = \frac{w_i (\Sigma \mathbf{w})_i}{\mathbf{w}^\top \Sigma \mathbf{w}}.$$

  Relative risk shares are dimensionless and sum to one: $\sum_{i=1}^n p_i = 1$. They are not probabilities. If a weight and its marginal risk contribution have opposite signs, $p_i$ can be negative, meaning that the position offsets some risk in the current portfolio. A long position can also have a negative risk contribution; a short position multiplied by a negative marginal contribution has a positive contribution.

5.4 Equal Risk Contribution (ERC) Portfolios

An equal risk contribution (ERC) portfolio requires every asset to contribute equally in absolute terms to total portfolio volatility, over the fully invested long-only simplex:

Thus, each asset carries exactly $1/n$ of the relative risk share: $p_i = 1/n$.

Check the two-asset case before memorizing an ERC formula. In this annualized-decimal teaching example, $\sigma_1=0.2,\sigma_2=0.3,\rho=0.2$, so

Set the two risk contributions equal, first canceling their common positive denominator $\sigma_p$:

Combining this with $w_1+w_2=1$ gives $(w_1,w_2)=(0.6,0.4)$. Substituting these weights back into the risk calculation:

Volatility is 18.59%, and the two risk shares are 50% each even though the capital weights are 60% and 40%. The two-asset cross terms cancel, so when volatility is positive and $\rho\in(-1,1)$, ERC weights do not depend on correlation; correlation still changes total volatility. With three or more assets, these pairwise cancellations do not occur, so inverse-volatility weights are not a general substitute for ERC.

The horizontal axis shows the two assets and the vertical axis shows shares (%). Capital weights are 60%/40%, while risk shares are 50%/50%. This figure checks only the annualized example above; it does not show that ERC has higher returns. Section 6.3 explains how to solve for a specified risk budget under a general covariance matrix.

6. Core Derivations: Shrinkage, Convex-QP Transformation, and Risk Decomposition

6.1 Deriving the Ledoit–Wolf Linear Shrinkage Estimator

Ledoit & Wolf (2004) seek a linear combination $\Sigma^* = \rho_1 I + \rho_2 S$. In the theoretical derivation, the shrinkage target is $\mu I$, where $\mu = \langle \Sigma, I \rangle = \frac{1}{p}\operatorname{tr}(\Sigma)$ is the mean of the true eigenvalues. The feasible estimator later replaces $\mu$ with its sample counterpart, $m = \langle S, I \rangle$. Let the true, unknown covariance matrix be $\Sigma$. The theoretical objective is to minimize expected squared loss under the normalized Frobenius norm:

Using an orthogonal decomposition and expanding the expectation, define three key geometric distance parameters. In this subsection, $\delta^2$ denotes only the total dispersion relevant to shrinkage; it is not the square of the risk-aversion coefficient $\delta$ in Section 6.4:

1. Expected squared distance from the sample covariance to the target: $\delta^2 = E\left[ \|S - \mu I\|^2 \right]$;
2. Squared distance from the true covariance to the target: $\alpha^2 = \|\Sigma - \mu I\|^2$;
3. Expected estimation error of the sample covariance relative to the true covariance: $\beta^2 = E\left[ \|S - \Sigma\|^2 \right]$.

First derive the result under the theoretical benchmark $E[S]=\Sigma$. Under this condition, the Pythagorean structure holds in expectation. Centered outer products satisfy the condition when the true mean is known. In a finite sample, a matrix centered using the sample mean and divided by $T$ instead has $E[S]=(T-1)\Sigma/T$, so the following relation is not an unconditional finite-sample identity. The practical shrinkage estimator uses an asymptotic construction:

The proof of Lemma 2.1 uses only $E[S]=\Sigma$: when expanding $E\lVert S-\Sigma+\Sigma-\mu I\rVert^2$, the cross term $2\langle E[S-\Sigma],\,\Sigma-\mu I\rangle=0$. Rewrite $\Sigma^*$ as the convex combination $\Sigma^* = \kappa (\mu I) + (1 - \kappa) S$ (the original paper first proves that the optimal target scale is exactly $\mu$), and substitute it into the squared-loss objective:

Expanding the expectation and using the first-moment unbiasedness that makes the cross term vanish gives a quadratic in the shrinkage intensity $\kappa$:

Differentiate with respect to $\kappa$ and set the derivative to zero:

Thus, the theoretically optimal shrinkage intensity is exactly the share of total dispersion attributable to estimation-error variance, $\beta^2/\delta^2$. Because the population matrix $\Sigma$ is unknown in practice, Ledoit & Wolf construct asymptotically unbiased feasible estimators for the corresponding quantities:

• Estimate of the target scale: $m = \frac{1}{p}\operatorname{tr}(S)$;

• Estimate of total sample dispersion: $d^2 = \|S - mI\|^2 = \frac{1}{p}\sum_{i=1}^p \sum_{j=1}^p (s_{ij} - m\delta_{ij})^2$;

• Asymptotically unbiased estimate of estimation error: Let the rows of the centered sample matrix be $\mathbf{x}_t \in \mathbb{R}^p$, with the outer product for each period $\mathbf{x}_t \mathbf{x}_t^\top$. Since $S$ is the average of these $T$ outer products, the error variance of the average is approximately the dispersion of a single outer product multiplied by $1/T$. Lemma 3.4 in the original paper therefore uses $\bar b^2=\frac{1}{T^2}\sum_t\lVert\mathbf{x}_t\mathbf{x}_t^\top-S\rVert^2$ to estimate $\beta^2$. Expanding with $\sum_t\mathbf{x}_t^\top S\mathbf{x}_t=T\lVert S\rVert_F^2$ gives:

  $$\bar{b}^2 = \frac{1}{T^2 p}\left( \sum_{t=1}^T \|\mathbf{x}_t\|_2^4 - T \sum_{i=1}^p \sum_{j=1}^p s_{ij}^2 \right);$$

• Truncated shrinkage intensity: $b^2 = \min(\max(\bar{b}^2, 0), d^2)$, giving the feasible shrinkage intensity:

  $$\hat{\kappa} = \frac{b^2}{d^2} \in [0, 1].$$

The resulting feasible shrinkage estimator is positive semidefinite; it is positive definite under the following condition:

Because the target matrix $mI$ is positive definite as long as at least one asset has nonzero variance, when $\hat{\kappa} > 0$, the smallest eigenvalue of $S^*$ is strictly positive even if $S$ is rank-deficient because $T < p$. The shrunk matrix is therefore numerically invertible.

6.2 Convex-QP Reformulation of Long-Only Maximum Sharpe Ratio

The original long-only optimization problem for maximizing a portfolio's Sharpe ratio is:

The objective is a ratio, not a concave function, so it cannot be passed directly to CVXPY under Disciplined Convex Programming (DCP) rules. Section 7 of Chapter 9 and exercise 9-C01 already prove the variable substitution below and its conditions. We restate it here as implemented in this chapter; Lesniewski's Baruch MFE Lecture 5 gives the same transformation.

Proof: The discussion below concerns a finite Sharpe ratio and requires every feasible portfolio with positive excess return to have positive variance; $\Sigma\succ0$ is a simple sufficient condition. If a zero-risk direction has positive excess return, the Sharpe denominator is zero and the QP objective may be zero. This is a degenerate or infinite-Sharpe case, not an ordinary finite optimum.

Keep three separate gates in view. Budget feasibility requires $nw_{\max}\ge1$. Feasibility of positive excess return requires $\max_w\mu_{excess}^\top w>0$ under the weight cap. A finite Sharpe ratio also requires the risk denominator of every compared portfolio to be nonzero. $\Sigma\succ0$ guarantees this for nonzero weights. If the matrix is only positive semidefinite and a feasible $w$ satisfies both $w^\top\Sigma w=0$ and $\mu_{excess}^\top w>0$, it lies on a zero-risk, positive-excess-return boundary where ordinary finite Sharpe is undefined; a QP optimum of 0 cannot be interpreted as a finite maximum Sharpe ratio. A singular matrix does not necessarily create this boundary, but the risk domain must be checked explicitly. The Project 3 implementation requires numerically positive-definite input and does not introduce a separate zero-risk strategy.

Assume that at least one portfolio in the feasible set has strictly positive expected excess return, that is, $\max_{\mathbf{w}} \boldsymbol{\mu}_{excess}^\top \mathbf{w} > 0$. Define the scalar transformation factor:

and define the scaled weight vector:

Since $\mathbf{w} \ge \mathbf{0}$ and $\kappa > 0$, it follows that $\mathbf{y} \ge \mathbf{0}$. The constraints map algebraically as follows:

1. Expected excess return is normalized:

   $$\boldsymbol{\mu}_{excess}^\top \mathbf{y} = \kappa (\boldsymbol{\mu}_{excess}^\top \mathbf{w}) = 1;$$

2. The investment-budget constraint becomes:

   $$\mathbf{1}^\top \mathbf{y} = \kappa (\mathbf{1}^\top \mathbf{w}) = \kappa;$$

3. The weight-cap constraint becomes:

   $$w_i \le w_{\max} \iff y_i \le w_{\max} \kappa.$$

Consider the reciprocal of the squared objective:

Thus, maximizing the Sharpe ratio is equivalent to minimizing its squared reciprocal:

subject to the linear constraints:

When $\Sigma \succeq 0$, $\mathbf{y}^\top \Sigma \mathbf{y}$ is convex; when $\Sigma\succ0$, it is strictly convex and the solution is unique. All constraints are affine equalities or inequalities, so this is a convex quadratic program. The strict inequality $\kappa>0$ need not be passed to the solver: $\boldsymbol{\mu}_{excess}^\top\mathbf{y}=1$ forces $\mathbf{y}\ne\mathbf{0}$, and because $\mathbf{y}\ge\mathbf{0}$, we have $\kappa=\mathbf{1}^\top\mathbf{y}>0$. The reverse mapping $\mathbf{w}=\mathbf{y}/\kappa$ sends any feasible $(\mathbf{y},\kappa)$ back to a feasible portfolio with positive excess return. The feasible solutions of the two problems are in one-to-one correspondence, and their objective values equal $1/\operatorname{SR}^2$. After solving for the optimum $(\mathbf{y}^*, \kappa^*)$, recover the optimal portfolio weights for the original problem directly using the inverse transformation:

This completes the proof.

6.3 Convex Log-Barrier Formulation and Damped Newton Algorithm for Risk Budgeting (ERC)

Section 3.4 of Maillard, Roncalli, and Teiletche (2009) formulate ERC as a minimum-volatility problem subject to $\sum_i\ln x_i\ge c$. The unconstrained log-barrier form below corresponds to that formulation, with the constraint multiplier absorbed into the objective; the derivation and implementation here are developed in this chapter. Consider the log-barrier function:

where $b_i > 0$ are the specified risk-budget weights, satisfying $\sum b_i = 1$ (for ERC, $b_i = 1/n$). The gradient of this objective is:

The Hessian matrix is:

As long as $\Sigma \succeq 0$ and $b_i > 0$, the diagonal terms are strictly positive and $\nabla^2 f(\mathbf{y}) \succ 0$. The objective is therefore strictly convex on $\mathbf{y}>\mathbf{0}$, and any minimizer, if it exists, is unique. The condition $\Sigma\succ0$ guarantees existence, but is not necessary. For example, although $\Sigma=\begin{bmatrix}1&1\\1&1\end{bmatrix}$ is singular, with $b_1=b_2=1/2$ it still has a unique minimizer $\mathbf y=(1/2,1/2)^\top$, which normalizes to equal weights. The key danger is a nonnegative zero-risk direction: if some nonzero $\mathbf{v}\ge\mathbf{0}$ satisfies $\mathbf{v}^\top\Sigma\mathbf{v}=0$, then along $\mathbf y=\mathbf y_0+t\mathbf v$, with $\mathbf y_0>0$ and $t\to\infty$, positive semidefiniteness implies $\Sigma\mathbf v=0$. The quadratic term stays constant while at least one logarithmic term tends to negative infinity, so no minimizer exists. The core function therefore requires a numerically positive-definite covariance matrix. At a first-order optimum, the gradient is zero:

Because the variance contribution $y_i(\Sigma\mathbf y)_i$ is homogeneous of degree 2 in $\mathbf{y}$ (while the volatility contribution $y_i(\Sigma\mathbf y)_i/\sigma(\mathbf y)$ is homogeneous of degree 1), normalize $\mathbf{y}^*$ onto the simplex:

Normalization does not change the relative proportions of the assets' risk contributions:

The denominator also needs to be checked. Let $s=\sum_j y_j^*$. Summing the first-order conditions gives $\mathbf y^{*\top}\Sigma\mathbf y^*=\sum_i b_i=1$, so

This proves the risk-budget proportions, not the budget amounts. In the two-asset example above, set $\mathbf y=(0.6,0.4)^\top/\sqrt{0.03456}$; the quadratic form is 1 and each contribution is 0.5. Normalization then recovers the original weights. Finding a minimizer, reaching the algorithm's stopping threshold, obtaining a sufficiently small budget residual, and deciding that a strategy is worth trading are four separate judgments.

Damped Newton Algorithm and Stopping Rule:

1. Initial point: start at the inverse-diagonal-standard-deviation point $\mathbf{y}^{(0)} = (1/\sqrt{\Sigma_{11}}, \dots, 1/\sqrt{\Sigma_{nn}})^\top$, then normalize so that $\mathbf{y}^\top \Sigma \mathbf{y} = 1$.

2. Compute the Newton direction: $\Delta \mathbf{y} = -[\nabla^2 f(\mathbf{y})]^{-1}\nabla f(\mathbf{y})$.

3. Compute the Newton decrement (Boyd & Vandenberghe, Section 9.5):

   $$\lambda^2 = -\nabla f(\mathbf{y})^\top \Delta \mathbf{y}.$$

   $\lambda^2/2$ is a quadratic-model estimate of $f(\mathbf{y}) - f^*$ (Boyd–Vandenberghe, Section 9.5.1), not a general strict upper bound. Stop the iteration when $\lambda^2 \le 10^{-14}$, then check the risk-share deviations.

4. Backtracking line search: use the Armijo condition to choose a step size $\alpha \in (0, 1]$, ensuring that the new point remains in the feasible region $\mathbf{y} + \alpha \Delta \mathbf{y} > \mathbf{0}$.

6.4 Bayesian Derivation of the Black–Litterman Model and Its Gain Form

The Black–Litterman model combines market-equilibrium implied returns and investor views into an expected-return vector. The original Black and Litterman (1992) paper was not available for this chapter; the setup and formulas below are based on Appendix B of He and Litterman (1999).

1. Reverse optimization determines the prior: Assume that investors as a whole hold the market portfolio $\mathbf{w}_{mkt}$. By the first-order condition for mean–variance optimization, the market-equilibrium expected-return vector $\boldsymbol{\Pi}$ must satisfy:

   $$\boldsymbol{\Pi} = \delta \Sigma \mathbf{w}_{mkt},$$

   Here $\delta$ is the aggregate risk-aversion coefficient for the market as a whole (He & Litterman 1999 commonly use $\delta = 2.5$).

2. Prior distribution: Treat the expected-return vector itself as a random variable with a prior normal distribution centered on equilibrium returns:

   $$\boldsymbol{\mu} \sim N(\boldsymbol{\Pi}, \tau \Sigma),$$

   The scalar $\tau > 0$ scales confidence by the prior variance relative to the variance of realized returns. All returns here are expected excess returns at the same frequency; total returns would also include the cash return over the same period.

3. View system: Suppose the investor holds $k$ linear views, expressed in matrix form as:

   $$P \boldsymbol{\mu} = \mathbf{Q} + \boldsymbol{\varepsilon}, \quad \boldsymbol{\varepsilon} \sim N(\mathbf{0}, \Omega),$$

   Here $P \in \mathbb{R}^{k \times n}$ is the view-selection matrix, $\mathbf{Q} \in \mathbb{R}^k$ is the vector of view returns, and $\Omega \in \mathbb{R}^{k \times k}$ is the covariance matrix of view errors. He and Litterman assume view errors are independent of prior expected-return errors and that errors across views are independent, which is why the view-error covariance is usually taken to be diagonal. The first independence assumption is also needed for this chapter's block-Gaussian covariance derivation.

4. Deriving the posterior distribution (block-normal conditional update): The joint distribution of $\boldsymbol{\mu}$ and the observed views $\mathbf{v} = P\boldsymbol{\mu} + \boldsymbol{\varepsilon}$ is:

   $$\begin{bmatrix} \boldsymbol{\mu} \\ \mathbf{v} \end{bmatrix} \sim N\left( \begin{bmatrix} \boldsymbol{\Pi} \\ P\boldsymbol{\Pi} \end{bmatrix}, \begin{bmatrix} \tau \Sigma & \tau \Sigma P^\top \\ P \tau \Sigma & P \tau \Sigma P^\top + \Omega \end{bmatrix} \right).$$

   Applying the conditional-normal formula gives the posterior expected-return mean $\boldsymbol{\mu}_{BL}$ and posterior-mean covariance $V_{BL}$:

   $$\boldsymbol{\mu}_{BL} = \boldsymbol{\Pi} + \tau \Sigma P^\top (P \tau \Sigma P^\top + \Omega)^{-1}(\mathbf{Q} - P\boldsymbol{\Pi}),$$
   $$V_{BL} = \tau \Sigma - \tau \Sigma P^\top (P \tau \Sigma P^\top + \Omega)^{-1} P \tau \Sigma.$$

   Define the Kalman-form gain matrix $M = \tau \Sigma P^\top (P \tau \Sigma P^\top + \Omega)^{-1}$. The update can then be written compactly as:

   $$\boldsymbol{\mu}_{BL} = \boldsymbol{\Pi} + M(\mathbf{Q} - P\boldsymbol{\Pi}).$$

   This gain form does not require $\Omega$ itself to be invertible, but it does require $S_v=P\tau\Sigma P^\top+\Omega$ to be invertible. When $\Omega=0$, $\Sigma\succ0$, and $\tau>0$, the rows of $P$ must be linearly independent. Repeating the same certain view makes $S_v$ singular; assigning two different values to that duplicated view is even contradictory. The core implementation raises an explicit error rather than hiding the issue with an inverse or pseudoinverse. $V_{BL}$ measures posterior uncertainty about the unknown mean; it is not the covariance of next-period asset returns. If we separately model $R=\mu+\varepsilon$, with $\operatorname{Cov}(\varepsilon)=\Sigma$, and assume the new return noise is independent of the posterior mean, the law of total variance gives the posterior predictive covariance $\Sigma+V_{BL}$. The portfolio tilt below uses $\Sigma$ as the return-risk penalty, preserving the original model interface. If predictive risk is used instead, the objective and weights must both be recomputed; the two covariance matrices cannot be treated as interchangeable.

5. Orthogonality of the optimal portfolio tilt: Substitute the posterior mean into the mean–variance optimal allocation:

   $$\mathbf{w}^* = (\delta \Sigma)^{-1}\boldsymbol{\mu}_{BL} = (\delta \Sigma)^{-1}\left[\delta \Sigma \mathbf{w}_{mkt} + \tau \Sigma P^\top (P \tau \Sigma P^\top + \Omega)^{-1}(\mathbf{Q} - P\boldsymbol{\Pi})\right].$$

   Expanding and rearranging gives:

   $$\mathbf{w}^* = \mathbf{w}_{mkt} + P^\top \left[ \frac{\tau}{\delta}(P \tau \Sigma P^\top + \Omega)^{-1}(\mathbf{Q} - P\boldsymbol{\Pi}) \right] = \mathbf{w}_{mkt} + P^\top \boldsymbol{\lambda}.$$

   This expression shows that, without constraints, the deviation of portfolio weights from market weights lies in the column space of $P^\top$, meaning it is a linear combination of the view portfolios. If a column of $P$ is all zeros (no view concerns that asset), its portfolio weight remains at the market weight, although its posterior mean can still change through correlations (asset 2 in Section 8.5). This property generally no longer holds after adding constraints such as long-only or weight caps.

6.5 The Kelly Criterion and Deriving Log Growth

Kelly (1956) studied capital-allocation rules that maximize the geometric growth rate of wealth over the long run. Discrete binary bet: Suppose an investor bets a fraction $f \in [0, 1)$ of capital. The probability of winning is $p$, and the net odds are $b$: a one-unit stake earns $b$ units net if it wins and loses the one-unit stake if it loses. Expected log growth in one period is:

Differentiate with respect to $f$:

Set $g'(f) = 0$:

Solving gives the well-known Kelly-optimal bet fraction:

With symmetric odds ($b = 1$), the optimal fraction simplifies to $f^* = 2p - 1$. When $bp\le 1-p$ (expected net return is non-positive), $g'(0)=pb-(1-p)\le0$ and $g$ is concave, so every $f>0$ is no better than not betting. Therefore, $f^*=\max\{(bp-(1-p))/b,\,0\}$; the core function kelly_binary_fraction applies this truncation. Notation note: On printed page 920 of Kelly's (1956) paper in the Bell System Technical Journal, the letter $q$ denotes the probability that a signal is transmitted correctly (a win), $p$ denotes the probability of a transmission error (a loss), and the bet fraction is denoted by $\ell$. This chapter follows modern financial convention, using $p$ for the probability of winning, $1-p$ for the probability of losing, and $f$ for the bet fraction.

Second-order expansion and quadratic proxy for discrete asset allocation: For assets with continuous returns, let single-period excess return be $\mathbf{r} - r_f \mathbf{1}$, with mean $\mathbf{m}$ and covariance $\Sigma$. A second-order Taylor expansion of single-period portfolio log growth is:

Let $a=E[r_p]=r_f+\mathbf w^\top\mathbf m$. The cash return $r_f$ is deterministic over this holding period, so $E[r_p^2]=\operatorname{Var}(r_p)+a^2=\mathbf w^\top\Sigma\mathbf w+a^2$. Taking the expectation of the Taylor polynomial gives the full discrete second-order expression:

The term $a^2$ cannot be removed from the second-order term without explanation. For example, if $r_p$ is equally likely to be 0.1 or 0.2, then $a=0.15$, the variance is 0.0025, and the full second-order mean is 0.1375; omitting the squared mean would give 0.14875. This polynomial approximation also assumes returns stay within a region where the log expansion is valid, and cannot replace a check of the ruin boundary.

The code below uses a local model that also ignores the squared drift term:

This is an additional approximation. A continuous-time diffusion limit is justified only when, over a time step $\Delta t$, the mean is $O(\Delta t)$ and the variance is $O(\Delta t)$, making $a^2=O(\Delta t^2)$. Fixed monthly data do not automatically satisfy this limit. quadratic_log_growth calculates only $g_q$, and gaussian_kelly_weights optimizes only this explicit proxy objective. For the full discrete second-order expression, the gradient is $(1-a)\mathbf m-\Sigma\mathbf w$, and the unconstrained solution should be $(1-r_f)(\Sigma+\mathbf m\mathbf m^\top)^{-1}\mathbf m$, which generally differs from $\Sigma^{-1}\mathbf m$. Taking the first-order condition with respect to the weight vector $\mathbf{w}$:

Here $\mathbf{w}^*$ gives the nominal weights of the risky assets as fractions of net asset value; there is no fully invested constraint. The remainder, $1-\mathbf{1}^\top\mathbf{w}^*$, is held in the risk-free asset; a negative remainder represents borrowing at the risk-free rate (in 21-E08, $w^*=3.2$ means borrowing 2.2 times net asset value). If we introduce a fractional-Kelly scaling parameter $c \in (0, 1]$ and set the portfolio weights to $\mathbf{w}_c = c \mathbf{w}^*$, then:

• Growth-rate ratio: $g(c\mathbf{w}^*) - r_f = c \mathbf{w}^{*\top}\mathbf{m} - \frac{1}{2}c^2 \mathbf{w}^{*\top}\Sigma \mathbf{w}^* = (2c - c^2)\left[ \frac{1}{2}\mathbf{m}^\top \Sigma^{-1}\mathbf{m} \right]$;

• Return-variance ratio: $\operatorname{Var}(c\mathbf{w}^*) = c^2 \operatorname{Var}(\mathbf{w}^*)$. Under half Kelly ($c = 0.5$):

• Under the quadratic proxy objective above, growth above the cash baseline is $2(0.5) - 0.5^2 = 75\%$ of full Kelly;

• With the same input covariance, variance is $0.5^2 = 25\%$ of full Kelly. Under the known-moment quadratic approximation, half Kelly gives up one quarter of the growth rate in exchange for a three-quarter reduction in variance. In an ideal wealth stochastic differential equation with bounded coefficients, no jumps, continuous rebalancing without limit, and no margin liquidation, the stochastic exponential can keep wealth positive; the current discrete monthly model offers no such guarantee. Strictly Gaussian simple returns have an unbounded left tail, so any position with nonzero variance has positive probability of $r_p\le-1$, making exact $E\ln(1+r_p)$ ill-defined. The quadratic proxy can still return a number, but that does not make it a valid exact Kelly solution. Even if the asset price cannot fall below zero, a discrete 3.2-times position (ignoring the cash return) would exhaust net asset value after a single-period asset loss greater than 31.25%. Jumps, margin, and financing restrictions widen this model error. If estimated $\mathbf{m}$ is too high, “full Kelly” already exceeds true Kelly; fractional Kelly provides protection against that error.

Kelly Growth-Rate and Variance Trade-Off Curves

• Horizontal axes: Bet fraction $f$ ($\%$) in the left panel and Kelly multiplier $c$ in the right panel;
• Vertical axes: Expected log growth per period in the left panel; growth-rate and variance multiples relative to full Kelly in the right panel;
• How to read the figure and what to look for:
  1. In the left panel, for $p = 0.55, b = 1$, log growth peaks at $f^* = 10\%$, with a maximum of $0.005008$. At twice the Kelly position ($f = 20\%$), expected log growth turns negative. Under this example's assumptions of an IID bet, continued proportional betting, and infinitely many repetitions, wealth almost surely tends to zero;
  2. In the Gaussian approximation shown in the right panel, half Kelly (the gray dashed line at $c = 0.5$) achieves $75\%$ of the theoretical peak growth rate while its variance is only $25\%$ of full Kelly;
• Limitation: The right panel shows only a quadratic growth proxy with known moments and the squared-mean term omitted; the 75% refers to proxy growth above the cash baseline. Gaussian simple returns also have an unbounded left tail and cannot guarantee that exact log utility exists. Fat tails, jumps, financing, and margin constraints further change the position size that can be sustained.

6.6 Factor Risk Decomposition

The factor model from Chapter 15, $\mathbf{r}=B\mathbf{f}+\boldsymbol{\varepsilon}$, gives $\Sigma=B\Sigma_fB^\top+\Sigma_\varepsilon$. Here $B$ is an $n\times K$ loading matrix, $\operatorname{Cov}(\mathbf{f})=\Sigma_f$, the specific covariance $\Sigma_\varepsilon$ is diagonal (consistent with Chapter 15 notation; in this chapter $\Omega$ is reserved for the Black–Litterman view covariance), and factor and specific returns are uncorrelated. Portfolio exposure is $\boldsymbol{\beta}_p=B^\top\mathbf{w}$ (a $K\times1$ vector), so

Asset $i$'s variance contribution, $w_i(\Sigma\mathbf{w})_i$, can also be split into a factor component, $w_i(B\Sigma_f\boldsymbol{\beta}_p)_i$, and a specific component, $w_i^2(\Sigma_\varepsilon)_{ii}$. Consider a constructed single-factor example with $\boldsymbol{\beta}=(1.1,0.9)$, $\Sigma_f=0.03$, $\Sigma_\varepsilon=\operatorname{diag}(0.02,0.01)$, and $\mathbf{w}=(0.5,0.5)$ (annualized decimal units). Then $\beta_p=1.0$; factor variance is $0.03$, specific variance is $0.25(0.02)+0.25(0.01)=0.0075$, total variance is $0.0375$, and volatility is 19.36%. The factor accounts for 80% of variance. By asset, asset 1 contributes $0.5(1.1)(0.03)(1.0)=0.0165$ from the factor and $0.005$ from specific risk; asset 2 contributes $0.0135$ and $0.0025$, respectively. The four components sum to $0.0375$, and asset 1's risk share is 57.3%. Equal capital weights do not imply equal risk; factor exposures determine most of the risk here. This decomposition depends on the factor-model specification, and parameter-estimation error in loadings and factor covariance also flows into the contribution estimates.

7. Financial Interpretation: Leverage, Costs, and the Diversification Illusion

7.1 Leverage and Position-Exposure Measures

Long–short quantitative strategies must distinguish two measures of portfolio exposure:

• Net exposure: $\sum_{i=1}^n w_i$. This measures net capital or net market-value exposure; market beta exposure must be calculated separately as the sum of each weight times its asset beta. For a market-neutral strategy, net exposure is usually anchored near 0.
• Gross exposure: $\sum_{i=1}^n |w_i|$. This measures the total capital invested as a leverage multiple. For example, with net exposure equal to 0, gross exposure of 2.0 (or $200\%$) means that, on a net asset value of 100 million yuan, the portfolio holds 100 million yuan long and 100 million yuan short.
• Leverage: When weights are already normalized by net asset value, leverage is gross exposure, $\sum_i|w_i|$; do not divide by net asset value again. If the inputs are amounts, first divide the sum of the absolute long and short market values by net asset value. With asset returns fixed and risky positions scaled by a common factor, their gross-return contribution and fees under a linear proportional-cost model scale by that factor. Financing costs, market impact, and tail liquidation risk can change nonlinearly; proportional scaling is not guaranteed.

7.2 Trading Frictions, Portfolio Drift, and Turnover Drag

Over a full investment period, asset weights change for two reasons: passive price changes create “drift,” while active portfolio rebalancing creates “turnover.” Suppose beginning-of-period weights are $\mathbf{w}$ (with $\sum w_i = 1$) and realized returns during the period are $\mathbf{r} = (r_1, \dots, r_n)^\top$. At the end of the period, the naturally evolved weight of asset $i$ as a share of total portfolio value (the pre-trade drifted weight) is:

If next period's target weights are $\mathbf{w}^{target}$, the normalized weight-turnover proxy for buys and sells combined (not divided by 2) is:

If the one-way trading-cost rate is $c$ (for example, 10 basis points, or $c = 0.0010$), the trading-cost drag for this period is $c \times \operatorname{Turnover}$. Under the model, this proxy cost is deducted at the start of the holding period, and the target weights are applied to the remaining capital. The resulting modeled net return is:

This is not an exact self-financing execution ledger: actual trade amounts also depend on post-cost net asset value, so the fee fraction $q$ should satisfy $q=c\sum_i|(1-q)w_i^{target}-w_i^+|$. This chapter and Project 3 use $q\approx c\operatorname{Turnover}$ as a low-cost approximation and do not separately charge the additional adjustment needed to fund fees; see Section 3 of the Project 3 report. Annualized average fees, $12E[q]$, differ from the annualized average gross-to-net return difference, $12E[q(1+r_{gross})]$. Higher turnover increases costs, but whether those costs erase an advantage must be calculated; high turnover alone does not prove that a strategy loses money.

7.3 The Diversification Illusion and the Failure of Geometric Projection

Intuitively, holding more assets may appear to spread risk. But when risk contributions are extremely uneven, investors fall into the diversification illusion. Constructed example: stocks have 16% annualized volatility, bonds 5%, and their correlation is 0.2; capital weights are 60% stocks and 40% bonds. Let subscripts s and b denote stocks and bonds: $(\Sigma\mathbf{w})_{s}=0.6(0.0256)+0.4(0.2)(0.16)(0.05)=0.016$, $(\Sigma\mathbf{w})_{b}=0.4(0.0025)+0.6(0.0016)=0.00196$, and portfolio variance is $0.6(0.016)+0.4(0.00196)=0.010384$. Stocks account for $0.0096/0.010384=92.4\%$ of risk: the 40% bond allocation carries only about 7.6% of risk.

Another common conceptual error is to try to satisfy neutrality constraints using a “Euclidean projection.” Suppose an alpha model produces weights $\mathbf{w}$ that do not satisfy a neutrality constraint $A\mathbf{w} = \mathbf{0}$. The unweighted Euclidean projection onto the nearest feasible point solves:

Its closed-form solution is:

The serious financial flaw is that Euclidean-distance minimization takes place in an abstract mathematical space with no financial dimensions. It is completely unaware of differences in asset volatility, correlations, or liquidity frictions. When constraint rows are correlated, the projection may greatly shrink the original long and short weights (in Section 8.3, gross exposure falls from 1.6 to 0.2). Whether the projection preserves the source of expected returns must be reevaluated using a risk model and return forecasts; a more common approach is to include neutrality conditions as constraints in an optimization with a risk term.

8. Numerical Examples

This section gives five benchmark numerical examples with explicit inputs, step-by-step hand calculations, and clearly specified units.

8.1 Ledoit–Wolf Covariance-Shrinkage Example by Hand (4×2 Matrix, Percentage Units)

8.2 Hand Calculation of Drift, Turnover, and the Cost Ledger

8.2.1 Checking Exact Self-Financing in Currency Amounts

8.3 Implications Among the Three Neutralities and a Geometric-Projection Failure Example

8.4 Four-Asset Risk-Budget Example from Maillard, Roncalli, and Teiletche (2009)

8.5 Black–Litterman Two-Asset Example by Hand

9. Python Implementation and Assertion Checks

Default Standalone Run Entry Point

From the root of the companion reader package, create an isolated environment following the package instructions and run:

.venv/bin/python run_notebook.py notebooks/ch21_portfolio_construction.ipynb

By default, RUN_FRENCH=False. The Notebook checks LW shrinkage, MRT four-asset risk, BL, Kelly, neutrality constraints, and drift in sequence, and checks boundary cases such as finite Sharpe ratios, ERC existence, posterior-mean covariance, and the quadratic growth proxy. It creates only the chapter's RUN_DIR, writes no CSV or PNG files, and skips the historical figures. Figures in Sections 3, 5.4, 6.5, 8.5, and 11 are existing reader materials, not outputs of this run. The chapter run directory differs from the separate runs/execution_ch21_portfolio_construction_<uuid>/ directory created by the runner, which stores an execution copy and status.

This chapter's 21-E08 uses a monthly excess mean of 0.008 and variance of 0.0025, giving a quadratic-proxy weight of 3.2. The Notebook instead uses 0.006 and 0.0016, giving a weight of 3.75; these are different constructed inputs. The exact binary half-Kelly growth ratio is about 74.9268%; the quadratic growth proxy gives 75% at half of its own optimal risky-asset weight, and this ratio should not be treated as a general result for exact binary growth. The neutrality constraint should be assessed as described in Section 5.2: absolute sector neutrality across mutually exclusive and exhaustive sectors implies dollar neutrality and generally does not imply beta neutrality.

For Project 3's standalone entry point, constructed outputs, and historical report, see the project guide. The call and assertions for the shared implementation are shown below. The course workspace root must have the course package and its dependencies installed; a bare Python interpreter in an unconfigured environment is not a complete reader entry point.

Calling the Shared Implementation

The code below shows how to call the core module src/quant_course/portfolio_construction.py and uses assertions to check Sections 8.1–8.3 and 8.5. Full code for the four-asset ERC example in Section 8.4 is in the answer to 21-P01. The code requires the course package and its dependencies.

import numpy as np
from quant_course.portfolio_construction import (
    ledoit_wolf_identity, risk_contributions, risk_budget_weights,
    max_sharpe_long_only, black_litterman, he_litterman_view_covariance,
    kelly_binary_fraction, binary_log_growth, gaussian_kelly_weights,
    quadratic_log_growth, exposure_report, project_to_constraints,
    drift_weights, turnover,
)

# 1. Ledoit-Wolf 4x2 percentage-matrix shrinkage hand-check
x_hand = np.array([[3.0, 2.0], [-1.0, -2.0], [2.0, 3.0], [-4.0, -3.0]])
lw_res = ledoit_wolf_identity(x_hand)
assert np.isclose(lw_res.target_scale, 7.0)
assert np.isclose(lw_res.intensity, 8.0 / 42.5)
np.testing.assert_allclose(lw_res.sample_covariance, [[7.5, 6.5], [6.5, 6.5]], atol=1e-12)
np.testing.assert_allclose(
    lw_res.covariance,
    [[7.40588235, 5.27647059], [5.27647059, 6.59411765]],
    atol=1e-6,
)

# 2. Black-Litterman two-asset hand-check
cov_bl = np.array([[0.04, 0.02], [0.02, 0.09]])
w_mkt = np.array([0.6, 0.4])
p_view = np.array([[1.0, 0.0]])
q_view = np.array([0.10])
tau_param = 0.05
omega_view = he_litterman_view_covariance(p_view, cov_bl, tau=tau_param)
bl_res = black_litterman(
    cov_bl, w_mkt, risk_aversion=2.5, tau=tau_param,
    view_matrix=p_view, view_returns=q_view, view_covariance=omega_view,
)
np.testing.assert_allclose(bl_res.equilibrium_returns, [0.08, 0.12], atol=1e-12)
np.testing.assert_allclose(bl_res.posterior_mean, [0.09, 0.125], atol=1e-12)
np.testing.assert_allclose(bl_res.weights, [0.7, 0.4], atol=1e-12)

# 3. Kelly log-growth hand-check (p=0.55, b=1)
f_star = kelly_binary_fraction(0.55, 1.0)
assert np.isclose(f_star, 0.10)
g_full = binary_log_growth(0.10, 0.55, 1.0)
g_half = binary_log_growth(0.05, 0.55, 1.0)
g_double = binary_log_growth(0.20, 0.55, 1.0)
assert np.isclose(g_full, 0.0050083668, atol=1e-6)
assert np.isclose(g_half, 0.0037526078, atol=1e-6)
assert g_double < 0  # Double Kelly: expected log growth is negative; long-run wealth almost surely tends to 0

# 4. Neutrality and projection checks
w_demo = np.array([0.5, 0.3, -0.4, -0.4])
beta_demo = np.array([1.2, 0.8, 1.0, 0.6])
sectors_demo = ["A", "B", "A", "B"]
rep = exposure_report(w_demo, beta_demo, sectors_demo)
assert np.isclose(rep.net, 0.0)
assert np.isclose(rep.beta, 0.2)
a_mat = np.vstack([np.ones(4), beta_demo, [1.0, 0.0, 1.0, 0.0]])
proj_w = project_to_constraints(w_demo, a_mat)
np.testing.assert_allclose(a_mat @ proj_w, 0.0, atol=1e-12)

# 5. Drift and turnover ledger checks
w_init = np.array([0.5, 0.5])
r_month = np.array([0.10, -0.10])
w_drift = drift_weights(w_init, r_month)
np.testing.assert_allclose(w_drift, [0.55, 0.45], atol=1e-12)
turnover_val = turnover(w_drift, w_init)
assert np.isclose(turnover_val, 0.10, atol=1e-12)

10. Visualization

This section examines four core charts generated in advance. All figures were produced from fixed seeds and frozen data.

The figures appear near the relevant discussion: the estimation-error simulation in Section 3, Kelly growth in Section 6.5, BL view confidence in Section 8.5, and French industry risk shares in Section 11. Their original inputs, axes, units, and limitations are retained. The new two-asset figure in Section 5.4 distinguishes capital shares from risk shares.

11. Empirical Example Using Real-Market Data

The empirical example uses a frozen snapshot from the Kenneth R. French Data Library:

• Asset universe: Value-weighted monthly returns for 10 U.S. equity industries (NoDur, Durbl, Manuf, Enrgy, HiTec, Telcm, Shops, Hlth, Utils, Other), plus the Fama–French three-factor risk-free rate (RF);
• Sample period: 1926-07 to 2026-08 (1,202 calendar months);
• Raw-file hash: SHA-256 for industry_raw_zip: ebd937a4b995d821867aeb956a96f09a666386f38fa82516b7d1767ee4a49ae5;
• Data status: The French data are a currently revised academic version, not a point-in-time (PIT) historical snapshot. Monthly data are published with a lag, historical values may be revised, and the local manifest does not preserve historical versions. A zero-month lag is an idealized month-boundary decision test.

For the 120-month period from 2010-01 to 2019-12, the observed performance of four representative strategies using the Ledoit–Wolf covariance estimator is recorded in outputs/ch21/ch21_results.json:

The data show that the long-only minimum-variance portfolio has the lowest volatility but pays for it with highly concentrated holdings (the two largest industries account for more than $70\%$). Under the fully invested constraint, the ERC portfolio balances volatility reduction with broader market coverage.

Risk Decomposition Across the French 10 Industries in the Historical Window

• Horizontal axis: French 10 industry codes (NoDur, Durbl, Manuf, Enrgy, HiTec, Telcm, Shops, Hlth, Utils, Other);
• Vertical axis: Nominal portfolio weights ($\%$) in the upper panel and relative risk shares ($\%$) in the lower panel;
• Units: Percent ($\%$);
• Sample: 2010-01 to 2019-12 (120 months), using a Ledoit–Wolf shrunk covariance matrix;
• How to read the figure and what to look for:
  1. In the equal-weighted portfolio (1/N), every industry has a weight of exactly $10.0\%$, but utilities (Utils) account for only $4.52\%$ of risk in the lower panel, while durable goods (Durbl) and energy (Enrgy) account for $15.09\%$ and $12.97\%$, respectively;
  2. The long-only minimum-variance portfolio concentrates its weights in defensive industries (Utils $47.45\%$, NoDur $23.31\%$, and Hlth $15.27\%$); its risk shares match its weights exactly;
  3. In the ERC portfolio, each industry's risk share in the lower panel is $10.0\%$. To compensate for the low volatility of Utils, ERC raises its portfolio weight to $17.62\%$ and reduces Durbl to $6.41\%$;
  4. Monthly volatility comparison: minimum variance ($2.73\%$) < maximum Sharpe ($2.88\%$) < ERC ($3.24\%$) < 1/N ($3.51\%$);
• Limitation: This figure is a static, ex-post description of one historical window, 2010–2019. It is not a rolling out-of-sample result or a live-strategy conclusion.