The Marsaglia Polar Method is an efficient algorithm for generating normally distributed random numbers from uniform random inputs. It's the mathematical foundation that converts raw random numbers from the PRNG into the bell-curve returns used in each Monte Carlo iteration.
The Marsaglia Polar Method is an algorithm for generating pairs of independent standard normal random variables from uniform random numbers. It is the foundational building block of Retirement Lab's return generation pipeline — every simulated asset return starts as a Marsaglia Polar output before being transformed into Student-t or skewed distributions.
How It Works
- Sample two uniform random numbers u₁ and u₂ from [-1, 1] using the seeded PRNG
- Compute s = u₁² + u₂²
- Reject if s ≥ 1 or s = 0 (discard and resample — rejection sampling within a unit circle)
- Transform: multiply each u by
√(-2 ln(s) / s)to produce two independent standard normal values
The method is efficient because it avoids the trigonometric functions required by the Box-Muller transform, using only basic arithmetic and a square root. Each accepted sample produces two normal values, further improving throughput.
Why It Matters for Retirement Planning
While the algorithm itself is invisible to users, it underpins the entire simulation pipeline. In Retirement Lab:
- The PRNG produces uniform random numbers from a seed
- Marsaglia Polar transforms them into normal random variables
- Marsaglia-Tsang gamma sampling converts normals into Student-t variables
- The Fernandez-Steel transformation adds skewness
- Cholesky decomposition introduces correlation between asset classes
This pipeline runs millions of times per simulation. The deterministic seed ensures that identical inputs always produce identical results — making simulations reproducible and verifiable.
Frequently Asked Questions
- Why does Retirement Lab use the Marsaglia Polar Method instead of the Box-Muller transform?
- Both methods produce standard normal random variables from uniform inputs, but the Marsaglia Polar Method avoids expensive trigonometric function calls (sine and cosine), making it faster in practice. It uses only basic arithmetic and square root operations, which is important when generating millions of random numbers per simulation run.
- Does the random number generation method affect simulation results?
- No — all correct methods produce statistically identical normal distributions. The choice of algorithm affects computational speed, not the statistical quality of the results. What matters for reproducibility is the seeded PRNG that feeds uniform random numbers into the algorithm.