How much active return per unit of active risk?

Sharpe asks whether it was worth leaving cash. The information ratio asks a sharper question: was it worth leaving the index? For anyone judging active management, that is the question that matters.

The formula

Information ratio = (portfolio return − benchmark return) ÷ tracking error

Both pieces are already familiar from Unit 1. The numerator is the annualised active return. The denominator is the annualised standard deviation of that active return — the tracking error. The whole ratio reads: excess return earned per unit of deviation risk taken.

Two funds, one headline, two stories

Both funds beat their benchmark by +2.0 points last year.

• Fund A ran a tracking error of 4%. IR = 2 ÷ 4 = 0.50
• Fund B ran a tracking error of 8%. IR = 2 ÷ 8 = 0.25

Fund B took twice the deviation risk for the same reward. Same press release, half the efficiency. Nothing in the headline number shows this; only the ratio does.

Annualising it correctly

Same rule as Sharpe: the numerator scales with time, the denominator with its square root. From monthly active returns:

• Annualised active return = mean monthly active return × 12
• Annualised tracking error = standard deviation of monthly active returns × √12
• IR = monthly IR × √12

Worked example. Mean monthly active return 0.15%, monthly standard deviation of active returns 1.30%.

• Annualised active return: 0.15% × 12 = 1.80%
• Annualised tracking error: 1.30% × √12 = 4.50%
• IR = 1.80 ÷ 4.50 = 0.40

Breadth: where an information ratio comes from

There is a well-known piece of intuition in active management, usually stated as:

IR ≈ skill per decision × √(number of independent decisions)

Two routes to the same ratio, then. A manager can be highly accurate on a handful of calls, or mildly accurate across hundreds of genuinely independent ones. The word independent does the heavy lifting: two hundred positions that all rise and fall with the same sector bet are close to one decision wearing two hundred hats.

This is not a formula to apply numerically — the inputs are unobservable. It is a lens. When a fund reports a strong information ratio, the useful follow-up is how many separate bets produced it?, because that determines whether the record is a sample of a repeatable process or a single lucky call recorded two hundred times.

The bridge to Unit 4

The information ratio has one more use, and it is the least comfortable one. It sets how long a track record must run before it can be distinguished from chance — a relationship of roughly t ≈ IR × √years. An IR of 0.40 needs a great many years to clear the usual statistical bar. That arithmetic is the subject of the final unit.

Try it now

1. A fund and the index you would defend as its fair benchmark are below. Switch both to Monthly. Note before you compute anything that the pairing itself is the first decision this ratio depends on — a different benchmark gives a different answer with the same fund.

2. Build both monthly return series, subtract to get the monthly active return, then take its mean and standard deviation and annualise (× 12 and × √12 respectively). Divide to get the information ratio.
3. Now Measure the first half of the window and the second half separately on both charts, and recompute the ratio on each half. Note how different the two answers are. That instability is the honest headline of this lesson, and no amount of decimal places fixes it.