Category: Statistics

Fig 5. The impact of number of samples on the variance of the confidence distribution.

Understanding the Uncertainty of Correlation Estimates

March 9, 2025 / Thijs van den Berg

Correlation is everywhere in finance. It’s the backbone of portfolio optimization, risk management, and models like the CAPM. The idea is simple: mix assets that don’t move in sync, and you can reduce risk without sacrificing too much return. But there’s a problem—correlation is usually taken at face value, even though it’s often some form of an estimate based on historical data. …and that estimate comes with uncertainty!

This matters because small errors in correlation can throw off portfolio models. If you overestimate diversification, your portfolio might be riskier than expected. If you underestimate it, you could miss out on returns. In models like the CAPM, where correlation helps determine expected returns, bad estimates can lead to bad decisions.

Despite this, some asset managers don’t give much thought to how unstable correlation estimates can be. In this post, we’ll dig into the uncertainty behind empirical correlation, and how to quantify it.

Validating Trading Backtests with Surrogate Time-Series

December 29, 2020 / Thijs van den Berg

Back-testing trading strategies is a dangerous business because there is a high risk you will keep tweaking your trading strategy model to make the back-test results better. When you do so, you’ll find out that after tweaking you have actually worsened the ‘live’ performance later on. The reason is that you’ve been overfitting your trading model to your back-test data through selection bias.

In this post we will use two techniques that help quantify and monitor the statistical significance of backtesting and tweaking:

First, we analyze the performance of backtest results by comparing them against random trading strategies that similar trading characteristics (time period, number of trades, long/short ratio). This quantifies specifically how “special” the timing of the trading strategy is while keeping all other things equal (like the trends, volatility, return distribution, and patterns in the traded asset).
Second, we analyse the impact and cost of tweaking strategies by comparing it against doing the same thing with random strategies. This allows us to see if improvements are significant, or simply what one would expect when picking the best strategy from a set of multiple variants.

Gaussian Mixture Approximation for the Laplace Distribution

January 30, 2020 / Thijs van den Berg

The Laplacian distribution is an interesting alternative building-block compared to the Gaussian distribution because it has much fatter tails. A drawback might be that some nice analytical properties that Gaussian distribution gives you don’t easily translate to Laplacian distributions. In those cases, it can be handy to approximate the Laplacian distribution with a mixture of Gaussians. The following approximation can then be uses

$L(x) = \frac{1}{2}e^{-|x|} \approx \frac{1}{n} \sum_{i=1}^n N\left(x | \mu=0, \sigma^2=-2\ln \frac{1+2i}{2n}\right)$

def laplacian_gmm(n=4):
    # all components have the same weight
    weights = np.repeat(1.0/n, n)
    
    # centers of the n bins in the interval [0,1]
    uniform = np.arange(0.5/n, 1.0, 1.0/n)
    
    # Uniform- to Exponential-distribution transform
    sigmas = np.array(-2*np.log(uniform))**.5
    return weights, sigmas

def laplacian_gmm_pdf(x, n=4):
    weights, sigmas = laplacian_gmm(n)
    p = np.zeros_like(x)
    for i in range(n):
        p += weights[i] * norm(loc=0, scale=sigmas[i]).pdf(x)
    return p