Mean-Variance Optimization for Indian Stocks

Introduction

Mean-Variance Optimization (MVO), developed by Nobel laureate Harry Markowitz, forms the cornerstone of Modern Portfolio Theory (MPT). Introduced in his groundbreaking paper, "Portfolio Selection," published in the Journal of Finance in 1952, this method revolutionized how investors approach portfolio construction.

Markowitz demonstrated that investors could optimize their portfolios by considering not just the expected returns of individual assets, but also how these assets move in relation to each other. This insight led to the formalization of diversification benefits through mathematical modeling, allowing investors to construct portfolios with superior risk-return characteristics.

Intuitive Explanation

Imagine going grocery shopping with a fixed budget. You want nutritious food (high returns) but also want to avoid overspending or wasting money on overpriced items (minimizing risk). MVO similarly helps investors select the best possible combination of assets (stocks, bonds, ETFs) that give maximum possible returns while controlling risk exposure efficiently.

The key insight of MVO is that combining assets that don't move in perfect sync (correlation less than 1) can actually reduce the overall risk of your portfolio while maintaining returns. This is the mathematical formalization of the old adage: "Don't put all your eggs in one basket."

Example: Consider two stocks: a solar energy company and an umbrella manufacturer. When it's sunny, the solar company performs well but umbrella sales drop. When it's rainy, umbrella sales surge while solar energy production declines. By investing in both companies, your portfolio becomes more stable across weather conditions, even though each individual company experiences significant fluctuations.

Detailed Mathematical Explanation

Mean-Variance Optimization mathematically balances expected returns against the volatility (risk) of a portfolio. The optimization problem can be precisely formulated using the following notation:

Notation:

\mu \in \mathbb{R}^n

: Expected returns vector for

n

assets

\Sigma \in \mathbb{R}^{n \times n}

: Covariance matrix of asset returns

w \in \mathbb{R}^n

: Portfolio weights vector

\mu^* \in \mathbb{R}

: Target portfolio return

\mathbf{1} \in \mathbb{R}^n

: Vector of ones

Portfolio Return

The expected return ( $\mu_p$ ) is calculated as a weighted sum of each asset's expected return:

\mu_p = w^T\mu = \sum_{i=1}^{n} w_i \mu_i

Portfolio Variance

Portfolio variance ( $\sigma_p^2$ ), a measure of risk, is computed by considering not only individual asset volatility but also how assets move together (covariance):

\sigma_p^2 = w^T\Sigma w = \sum_{i=1}^{n}\sum_{j=1}^{n} w_i w_j \sigma_{ij}

Where:

$w$ : Vector of portfolio weights ( $\sum w_i = 1$ )
$\mu$ : Vector of expected returns of assets
$\Sigma$ : Covariance matrix representing asset interrelationships
$\sigma_{ij}$ : Covariance between assets $i$ and $j$

Optimization Problem

The MVO problem mathematically stated is:

\begin{aligned} \min_{w} \quad & w^T \Sigma w \\ \text{subject to} \quad & w^T \mu \geq \mu^* \\ & w^T \mathbf{1} = 1 \\ & w_i \geq 0 \quad \forall i = 1,\ldots,n \end{aligned}

This optimization seeks the portfolio weights $w$ that minimize the portfolio variance (risk) while ensuring the expected return meets or exceeds the target $\mu^*$ , the weights sum to 1 (full investment), and no short-selling is allowed (weights are non-negative).

Efficient Frontier

The Efficient Frontier graphically represents the optimal portfolios achieving the highest possible return for any given level of risk or the lowest risk for a given return. By varying the target return $\mu^*$ , we can trace out this curve representing the set of optimal portfolios.

[Efficient Frontier Graph Placeholder]

The Efficient Frontier curve showing optimal portfolios

Interpretation

Points on the curve: Optimal portfolios that offer the best possible return for a given level of risk.
Points below: Suboptimal portfolios (too much risk for too little return).
Points above: Unachievable with the available assets (not feasible).

Tangency Portfolio and Capital Market Line (CML)

When a risk-free asset (e.g., Government Treasury) is available, the Efficient Frontier expands into a straight line called the Capital Market Line (CML).

Tangency Portfolio

The point where the CML touches (is tangent to) the Efficient Frontier.
Represents the portfolio with the highest Sharpe ratio (best return per unit of risk).

\text{Sharpe Ratio} = \frac{R_p - R_f}{\sigma_p}

Capital Market Line (CML)

Defines optimal portfolios combining the risk-free asset and the tangency portfolio.
All investors should hold combinations of the risk-free asset and the tangency portfolio according to their risk preferences.

R_p = R_f + \frac{R_m - R_f}{\sigma_m} \sigma_p

Where:

$R_p$ : Expected portfolio return
$R_f$ : Risk-free rate
$R_m$ : Expected return of the tangency portfolio
$\sigma_m$ : Standard deviation of the tangency portfolio
$\sigma_p$ : Standard deviation of the portfolio

[CML & Tangency Portfolio Graph Placeholder]

The Capital Market Line touching the Efficient Frontier at the Tangency Portfolio

Example Calculation

Consider a simple portfolio with two assets with the following characteristics:

Asset	Expected Return	Standard Deviation	Correlation
A	10%	8%	0.3
B	15%	15%	0.3

The covariance between A and B is calculated as:

\text{Cov}(A,B) = 0.3 \times 0.08 \times 0.15 = 0.0036

The covariance matrix $\Sigma$ is:

\Sigma = \begin{bmatrix} 0.0064 & 0.0036 \\ 0.0036 & 0.0225 \end{bmatrix}

Using the expected returns vector $\mu = [0.10, 0.15]^T$ , we can solve the MVO problem for various target returns.

For example, to find the minimum variance portfolio:

Step 1: Set up the portfolio variance formula:

\sigma_p^2 = w_A^2 \times 0.0064 + w_B^2 \times 0.0225 + 2 \times w_A \times w_B \times 0.0036

Step 2: Since $w_A + w_B = 1$ , we can substitute $w_B = 1 - w_A$ and minimize:

\sigma_p^2 = w_A^2 \times 0.0064 + (1-w_A)^2 \times 0.0225 + 2 \times w_A \times (1-w_A) \times 0.0036

Step 3: Take the derivative with respect to $w_A$ and set equal to zero to find the value of $w_A$ that minimizes portfolio variance.

The minimum variance portfolio in this example would have approximately 70% in Asset A and 30% in Asset B, resulting in a portfolio with expected return of:

\mu_p = 0.7 \times 0.10 + 0.3 \times 0.15 = 0.115 \text{ or } 11.5\%

And portfolio variance of:

\sigma_p^2 = 0.7^2 \times 0.0064 + 0.3^2 \times 0.0225 + 2 \times 0.7 \times 0.3 \times 0.0036 \approx 0.0054

Which gives a portfolio standard deviation of:

\sigma_p = \sqrt{0.0054} \approx 0.0735 \text{ or } 7.35\%

This is lower than the standard deviation of either asset individually (8% and 15%), demonstrating the power of diversification.

Advantages and Limitations

Advantages

Pioneering Framework: Provides a mathematical foundation for portfolio construction that has stood the test of time, earning Markowitz a Nobel Prize.
Diversification Benefits: Quantifies the risk-reduction benefits of combining assets with imperfect correlations.
Risk-Return Trade-off: Explicitly models the relationship between risk and return, allowing investors to select portfolios that match their risk tolerance.
Intuitive Visualization: The Efficient Frontier provides a clear graphical representation of portfolio optimization choices.

Limitations

Input Sensitivity: Highly sensitive to estimation errors in expected returns, variances, and covariances, which can lead to unreliable results.
Concentrated Portfolios: Often produces extremely concentrated portfolios, sometimes placing large weights on assets with estimation errors.
Normal Distribution Assumption: Assumes asset returns follow a normal distribution, which fails to capture fat tails and skewness in actual market returns.
Single-Period Model: Does not account for time-varying risk and return parameters or multi-period investment horizons.
Transaction Costs: Basic implementation ignores transaction costs, taxes, and liquidity constraints that exist in real markets.

Practical Improvements to Basic MVO

Challenge	Solution Approach
Input sensitivity	Robust optimization; shrinkage estimators (James-Stein, Ledoit-Wolf); Bayesian approaches (Black-Litterman model)
Extreme weights	Weight constraints; regularization penalties; portfolio resampling techniques
Non-normal returns	Higher-moment optimization; semi-variance optimization; historical simulation approaches
Single-period limitation	Multi-period optimization; dynamic programming approaches; rolling-window optimization
Market frictions	Transaction cost constraints; tax-aware optimization; turnover restrictions

References

Markowitz, H. (1952). "Portfolio Selection." The Journal of Finance, 7(1), 77-91.Access the paper
Markowitz, H. (1959). Portfolio Selection: Efficient Diversification of Investments. John Wiley & Sons.
Michaud, R. (1989). "The Markowitz Optimization Enigma: Is Optimized Optimal?" Financial Analysts Journal, 45(1), 31-42.
Black, F. & Litterman, R. (1992). "Global Portfolio Optimization." Financial Analysts Journal, 48(5), 28-43.
DeMiguel, V., Garlappi, L., & Uppal, R. (2009). "Optimal Versus Naive Diversification: How Inefficient is the 1/N Portfolio Strategy?" The Review of Financial Studies, 22(5), 1915-1953.

Mean-Variance Optimization (MVO)

The cornerstone of Modern Portfolio Theory