Part III

Relative performance and risk

A return only means something next to a benchmark and next to the risk taken to earn it.

7

Against the benchmark, and at what risk

A 9% return is great in a year the market made 3%, and poor in a year it made 20%. So every portfolio gets a benchmark: an index representing what the client could have had passively. The difference is the active return, also called excess return.

Arithmetic active = R − B
Geometric active = 1 + R1 + B − 1

Return alone isn't the whole story. Tracking error measures how far the portfolio strays from the benchmark: the volatility of the active return. The information ratio divides active return by tracking error. It answers "how much extra return did we get per unit of extra risk?"

Other risk measures you'll meet

Volatility
The standard deviation of returns, annualized. How bumpy the ride is, in either direction.
Maximum drawdown
The biggest fall from a peak to a later low. It's the number investors actually feel: "at worst, how much was I down?"
Beta
How much the portfolio tends to move when the benchmark moves 1%. A beta of 1.2 means it typically moves 1.2%.
Jensen's alpha
The return beyond what the portfolio's beta alone would have earned. It separates skill from simply taking more market risk.
Sortino ratio
Like Sharpe, but only counts downside volatility, since few investors complain about upside surprises.
Up and down capture
What share of the benchmark's rises and falls the portfolio captured. 90% up capture with 70% down capture is an attractive, defensive profile.
In plain termsPicture the benchmark as the centre of a lane. Tracking error is how much the car wobbles. A small wobble that still gets you ahead beats wild swerving that ends up in the same place.

Skill or luck?

Set the manager's true skill and how much risk they take. Then draw new 3-year samples and see how often luck hides, or fakes, that skill.

PortfolioBenchmarkGap
Portfolio, per year
Benchmark, per year
Active return (arithmetic)
Active return (geometric)
Tracking error
Wobble around the benchmark
Information ratio
Active return ÷ tracking error
Volatility
Sharpe ratio (rf 3%)
Maximum drawdown
Worst peak-to-trough fall
Beta
Jensen's alpha
Sortino ratio
Up / down capture

For practitionersAn information ratio around 0.5 sustained over years is considered good; 1.0 is exceptional. With 36 months of data the standard error of an annualized IR is roughly 1/√3 ≈ 0.58, so one 3-year sample says little on its own. Sharpe uses total volatility and the risk-free rate; IR uses tracking error and the benchmark. Arithmetic excess is common in US reporting; geometric excess is common in Europe and compounds consistently across periods and currencies.

All numbers are illustrative. The models are simplified for teaching: annual coupons, Black-Scholes options, simplified fee mechanics and no intra-period trading. Real systems add transaction-based returns, daily valuation, tax and corporate action processing, and reconciliation to the official TWR.