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Forecasting Tangency Portfolios and Investing in the Minimum Euclidean Distance Portfolio to Maximize Out-of-Sample Sharpe Ratios

This paper proposes a novel asset allocation model that forecasts the future tangency portfolio by decomposing the efficient frontier's functional form into interpretable coefficients and then invests in the minimum Euclidean distance portfolio to achieve superior out-of-sample Sharpe ratios, thereby addressing the limitations of traditional methods that rely on stationary return and covariance estimates.

Original authors: Nolan Alexander, William Scherer

Published 2026-04-07

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Nolan Alexander, William Scherer

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to navigate a ship across a vast, unpredictable ocean. Your goal is to reach a destination (maximum profit) while avoiding storms (risk).

In the world of investing, there's a famous map called the Efficient Frontier. Think of this map as a curved line showing all the possible "best" routes you could take. Some routes are fast but dangerous; others are slow but safe. The "perfect" spot on this map is called the Tangency Portfolio. It's the sweet spot where you get the most speed for the least amount of danger.

The Problem: The Map is a Lie

The old way of finding this perfect spot is like looking at yesterday's weather report to predict tomorrow's storm. Investors usually assume that what happened in the past (historical data) will happen exactly the same way in the future.

But the market is like the ocean: it changes. A calm sea today can become a hurricane tomorrow. When investors rely on old data, they often get the coordinates wrong. They think they are steering toward the perfect spot, but they end up crashing into a reef because the map was outdated.

The New Solution: Forecasting the Destination

The authors of this paper, Nolan Alexander and William Scherer, propose a clever new way to navigate. Instead of trying to guess the exact wind speed and wave height (returns and volatility) for tomorrow, they decided to forecast the shape of the map itself.

Here is how they did it, broken down into simple steps:

1. Simplifying the Map (The Three Magic Numbers)

Drawing a complex curve for every single day is hard. So, the authors realized that this "Efficient Frontier" curve can be described by just three simple numbers (coefficients):

The Lowest Point: The safest, calmest spot on the map.
The Height: How high the curve goes (how much return you can get).
The Curvature: How "bendy" the map is. A very bendy map means you have to take big risks for small rewards; a flatter map means you get great rewards for small risks.

They call these numbers $r_{min}$ , $\sigma_{min}$ , and $u$ . Think of them as the GPS coordinates, the altitude, and the terrain type of your investment map.

2. Predicting the Future Shape

Instead of guessing tomorrow's stock prices, they used a computer model (called VARX) to predict how these three numbers will change next month.

Analogy: Imagine you don't know exactly where the storm will hit, but you can predict that the ocean will become "rougher" (curvature changes) and the "calmest spot" will move slightly to the left. You are predicting the shape of the ocean, not every single wave.

3. The "Minimum Distance" Strategy

Here is the tricky part. Once they predict where the "perfect spot" (Tangency Portfolio) should be next month, they realize that spot might not actually exist on today's map. The market is too chaotic.

So, instead of trying to force the ship to a spot that doesn't exist, they ask: "What is the closest safe harbor we can reach right now that looks like our predicted destination?"

They calculate the Minimum Euclidean Distance.

Analogy: Imagine you are aiming for a target on a moving wall. You can't hit the exact center because the wall is shaking. So, you aim for the spot on the wall that is physically closest to your target. You don't hit the bullseye, but you hit the closest possible thing to it, minimizing your error.

Why This Works Better

The paper tested this method against four other popular strategies (like buying the whole market, or a standard 60/40 mix) over 20+ years.

The Result: Their "Minimum Distance" ship sailed smoother and faster. It made more money per unit of risk (a higher Sharpe Ratio) and suffered less damage during big market crashes (like 2008 and 2020).
The Secret Sauce: By focusing on the shape of the investment possibilities rather than guessing specific stock prices, they avoided the trap of over-optimizing based on old data. They acknowledged that the future is uncertain, so they chose the "closest possible match" to their prediction rather than a perfect but impossible one.

In a Nutshell

Most investors try to predict the exact future (which is impossible).
These authors predicted the general shape of the future and then steered their ship to the closest safe spot they could find today. It's a smarter, more humble way to invest that admits we can't see the future perfectly, but we can still navigate it better than the competition.

1. Problem Statement

Traditional Modern Portfolio Theory (MPT) and Mean-Variance Optimization rely heavily on estimating future expected returns and covariance matrices based on historical data. This approach assumes that future market parameters are stationary (i.e., they will resemble historical data). However, in practice, markets are non-stationary, leading to significant estimation errors.

The Limitation: As noted by Dickinson [4] and Kan and Zhou [5], misestimating these parameters results in unstable portfolio weights and poor out-of-sample performance.
Existing Solutions: Models like Black–Litterman, Covariance Matrix Shrinkage, and DCC MV-GARCH attempt to improve parameter estimation. However, they often require complex inputs (e.g., investor views), fail to estimate return vectors effectively, or are difficult for investors to interpret and trust.
The Gap: There is a need for a method that bypasses direct estimation of returns and covariances to forecast the optimal portfolio directly, thereby improving out-of-sample Sharpe ratios.

2. Methodology

The authors propose a novel two-step framework: forecasting the tangency portfolio via efficient frontier decomposition, and investing in the minimum Euclidean distance portfolio relative to that forecast.

A. Decomposition of the Efficient Frontier

Instead of optimizing weights directly, the model decomposes the efficient frontier's functional form.

Merton's Derivation: The efficient frontier is traditionally defined as a square root second-order polynomial with coefficients $A, B, C$ derived from the covariance matrix ( $V$ ) and return vector ( $r$ ).
Novel Coefficients: The authors derive three interpretable coefficients to replace $A, B, C$ $A, B, C$ :
1. $r_{min}$ : The return of the Minimum Variance Portfolio (MVP).
2. $\sigma_{min}$ : The standard deviation of the MVP.
3. $u$ : The rate of curvature of the efficient frontier (representing the utility of the asset set for mean-variance optimization).
- Interpretation: $r_{min}$ reflects market return demands; $\sigma_{min}$ reflects baseline risk; $u$ indicates the quality of risk-return tradeoffs (higher $u$ implies better tradeoffs).
- Mathematical Insight: The coefficient $u$ is shown to be a function of the Mahalanobis distance of the return vector to the zero vector and the cosine similarity between the return vector and the vector of ones.

B. Forecasting the Tangency Portfolio

Model: The authors use an online Vector Autoregression with Exogenous variables (VARX) with a lag order of 1.
Inputs: The model forecasts the future values of the three coefficients ( $r_{min}, \sigma_{min}, u$ ) using their own lags and the historical equal-weighted return moving average.
Tangency Calculation: Using the forecasted coefficients and the risk-free rate ( $r_f$ ), the model analytically solves for the return ( $r_{tan}$ ) and volatility ( $\sigma_{tan}$ ) of the forecasted tangency portfolio without needing the future return vector or covariance matrix.

C. The Minimum Euclidean Distance Portfolio

Since the forecasted tangency portfolio likely does not exist on the current efficient frontier (due to market non-stationarity), the model cannot simply invest in it.

Strategy: The model identifies the portfolio on the current efficient frontier that has the minimum Euclidean distance to the forecasted tangency portfolio.
Optimization: This is formulated as minimizing the distance function:
$\min \sqrt{(\hat{r}_{tan} - r)^2 + (\hat{\sigma}_{tan} - \sigma(r))^2}$
This is solved using Newton's Method to find the target return, after which standard Mean-Variance Optimization is used to derive the specific asset weights.

3. Key Contributions

Novel Coefficients: Introduction of three interpretable coefficients ( $r_{min}, \sigma_{min}, u$ ) that decompose the efficient frontier, allowing for direct forecasting of the frontier's shape rather than raw parameters.
Forecasting Mechanism: A parsimonious online VARX approach that forecasts these coefficients using limited features (lags and moving averages) to prevent overfitting.
Investment Strategy: A robust selection method (Minimum Euclidean Distance) that bridges the gap between a forecasted optimal state and the current feasible market state, avoiding the instability of direct parameter estimation.
Empirical Validation: Demonstration that this approach outperforms standard benchmarks across diverse asset universes.

4. Empirical Results

The model was tested on two distinct asset universes using data from Yahoo Finance and Fama-French factors:

Universe 1 (GVMC): 6 mutual funds representing Growth/Value and Large/Small caps + 1 bond fund. Tested out-of-sample (2000–2022).
Universe 2 (Sectors): 9 S&P Sector ETFs + 1 bond fund. Tested out-of-sample (2008–2022).

Benchmarks Compared:

Standard Tangency Portfolio (1-month rolling window).
Equal-Weighted Portfolio.
S&P 500 Total Return.
60/40 Stock/Bond Portfolio.

Performance Metrics:

Sharpe Ratio: The proposed "Minimum Distance Portfolio" achieved the highest Sharpe ratios in both universes (e.g., 1.00 vs. 0.67 for the standard tangency in GVMC; 0.76 vs. 0.01 for Sectors).
Drawdowns: The model exhibited significantly lower maximum drawdowns during market crashes (e.g., -18.7% vs. -25.0% for GVMC; -29.3% vs. -51.8% for S&P 500 in the sector test).
Statistical Significance: Alpha regressions confirmed that the proposed model statistically significantly outperformed all benchmarks (p-values < 0.05, often < $10^{-4}$ ).
Visuals: Cumulative return charts showed the proposed model providing consistent returns with lower volatility compared to benchmarks.

5. Significance and Conclusion

This paper offers a paradigm shift in asset allocation by moving away from the difficult task of forecasting raw returns and covariances. Instead, it forecasts the geometry of the efficient frontier itself.

Robustness: By investing in the portfolio closest to the forecasted optimum, the model adapts to non-stationary markets without requiring complex Bayesian priors (Black-Litterman) or heavy regularization (Shrinkage).
Interpretability: The use of $r_{min}, \sigma_{min},$ and $u$ provides clear economic intuition regarding market conditions (risk levels, return demands, and curvature).
Practicality: The model is computationally efficient, requires fewer inputs than Black-Litterman, and demonstrated superior risk-adjusted returns over a 20+ year period across different market regimes.

Future Work: The authors plan to test the model on broader market universes, analyze the impact of leverage constraints more deeply, and explore incorporating technical indicators and non-linear models (e.g., LASSO, decision trees) to enhance the forecasting accuracy of the coefficients.