Alpha

Excess return above what the benchmark explains -- Jensen's alpha formula, interpretation, and intermittent-exposure caveats.

What is Alpha?

Imagine you buy a stock and it goes up 10% over the year. You made money, but did you do anything clever? Not really -- the stock went up on its own. Anyone who held it would have made the same 10%.

Now imagine you used a trading strategy on that same stock and made 14%. That extra 4% didn't come from the stock going up -- it came from your strategy's timing decisions. That extra return is roughly what we call alpha.

In short, alpha answers one question: did my strategy add value beyond what the market gave for free?

The strategy made less than the benchmark (24% vs 32%) but more than expected given its exposure (24% vs 18.6%). The gap is alpha.

The Key Thing to Understand

Most backtested strategies are not in the market 100% of the time. Your strategy waits for the right conditions, enters a trade, exits, and waits again. It might only be in a position 60% of the time.

Alpha accounts for this. It does not compare your strategy's return to the full buy-and-hold return of the asset. Instead, it asks: given how much time your strategy actually spent in the market, did it do better or worse than expected? The "expected" part comes from beta, which measures how much your strategy participates in the asset's moves overall.

This leads to a result that surprises many people:

Why is my alpha positive when I made less than the benchmark?

This is one of the most common questions. If the asset returned 32% but your strategy only returned 24%, it looks like you underperformed. But if your strategy was only in the market about 60% of the time, alpha asks a different question: given that level of exposure, did your strategy do better or worse than expected? If beta says you'd expect around 19% given your exposure, then your 24% return represents roughly 5% of genuine added value -- positive alpha.

How It's Calculated

Simula calculates alpha by comparing your strategy's returns to what you'd expect based on its beta. Think of beta as a number that captures how much your strategy participates in the asset's moves -- factoring in both how it trades when positioned and how much time it spends out of the market.

Alpha

alpha = strategy return - (beta * asset return)

In plain English: take your strategy's return, subtract what you'd expect given your beta, and whatever is left over is your alpha.

A simplified view

The full formula includes something called the "risk-free rate" (what you'd earn with zero risk, like a savings account). Simula sets this to zero, which keeps things simple and focuses the comparison purely on your strategy vs the asset.

Alpha is Annualized

Alpha is always expressed as a yearly figure, regardless of how long your backtest ran or what candle interval you used. This is important because it means alpha won't directly match the raw gap between your strategy's return and the benchmark's return.

For example, if your backtest covers 2 years, the annualized alpha converts the total excess return into a per-year rate. If it covers only 3 months, the figure is scaled up to a yearly equivalent. Either way, the number you see represents a yearly rate, not the raw total.

Worked Examples

Simple example

A 1-year stock backtest where the strategy is always in the market:

Your strategy made 18%
The stock went up 12%
Beta is 1.2 (the strategy moved about 20% more than the stock on average)

Step by step:

Expected return = 1.2 * 12% = 14.4%
Alpha = 18% - 14.4% = 3.6%

The strategy earned 3.6% more than you'd expect from its market exposure alone.

Intermittent exposure example

A 2-year forex backtest where the strategy trades in and out:

Your strategy made 24% total
The asset went up 32% total
Exposure: 62% (strategy was in a trade 62% of the time)
Beta: 0.58

Step by step:

Expected return = 0.58 * 32% = 18.6%
Raw excess = 24% - 18.6% = 5.4%
Annualized over 2 years: ((1 + 0.054) ^ (1/2) - 1) * 100 = 2.66% per year

At first glance the strategy "lost" to the benchmark (24% vs 32%). But given that it was only in the market 62% of the time, it actually outperformed what that level of exposure would predict. The 2.66% annualized alpha reflects that added value on a per-year basis.

Why your number may differ from a hand calculation

These worked examples annualize a single total-return figure to keep the arithmetic followable. Simula computes alpha the rigorous way -- a candle-by-candle regression of your strategy's returns against the asset's -- so the exact figure in your results comes from that regression, not from plugging totals into the formula above. Treat these examples as intuition for what alpha captures, not a recipe that reproduces the number.

Reading Your Results

Positive alpha

Your strategy added value beyond what its market exposure would predict. The higher, the better.

Zero alpha

Your strategy performed about as well as you'd expect given its exposure. The trading rules didn't add or subtract value.

Negative alpha

Your strategy underperformed relative to its market exposure.

Always check beta alongside alpha

Alpha and beta are two halves of the same picture. A high alpha paired with a very high beta means your strategy is taking on more market risk to achieve that result. Look at both numbers together to understand whether the added return comes with added risk.

Things to Watch Out For

Alpha is not total return

Because alpha is annualized and adjusts for market exposure, it can look much smaller than the raw return gap between your strategy and the benchmark. A strategy that beat the benchmark by 15% over 3 years might show an annualized alpha of just a few percent. This doesn't mean the strategy performed poorly -- it means alpha is expressing that outperformance on a standardized yearly scale.

Short strategies

If your strategy profits by betting against the asset (going short), its relationship to the benchmark will be different from a long strategy. The beta reflects this relationship, and alpha still measures whether the strategy added value beyond what that relationship would predict.

To check whether your alpha persists beyond the specific period you tested on, walk-forward analysis tests your strategy across multiple time windows rather than a single backtest.

Beta -- measures how much your strategy participates in the asset's moves. Alpha and beta are always calculated together.

Exposure -- the percentage of time your strategy is in a trade. Helps explain why your beta (and therefore your expected return) might be lower than the benchmark.

Sharpe Ratio -- another way to measure performance, but instead of comparing to a benchmark it looks at return relative to volatility.

Total & Annualized Return -- your raw return numbers before any adjustment for market exposure.