Portfolio Management
coreTreynor Ratio
Builds onSharpe Ratio · Beta — if this page feels steep, start there.
- the Treynor ratio — excess return per unit of market (beta) risk
- the portfolio's return
- the risk-free rate
- the excess return: reward earned beyond doing nothing risky
- the portfolio's beta — its dose of undiversifiable market risk
Reading the notation
Why it must be true
The Treynor ratio is the Sharpe ratio's sibling with a different opinion about risk. Sharpe divides excess return by total volatility; Treynor divides by beta — only the market risk that diversification cannot remove.
Which sibling to trust depends on context. If the portfolio being judged is someone's entire wealth, total volatility hurts them and Sharpe is right. But if it is one sleeve inside a well-diversified whole, its private wiggles wash out against everything else — only its beta adds risk to the total, so beta is the fair denominator. Treynor asks: *how much excess return per unit of systematic risk contributed?*
The derivation
Start from the same reward as Sharpe — the excess over cash:
Then choose the risk that actually matters to a diversified holder. Idiosyncratic volatility cancels across the wider portfolio; the sleeve's lasting contribution to risk is its beta. Standardize by it:
Note the units differ from Sharpe: Treynor is "excess return per unit of beta" — comparable across funds, but not on the same scale as a Sharpe ratio.
When to reach for it
Ranking funds that are components of a diversified portfolio, where systematic (beta) risk is the relevant denominator rather than total volatility.
Listen for
Back-of-the-envelope
Estimate it in your head first — then the calculator only confirms.
- ≈
β = 1 anchor: the ratio is just the excess return itself. Beta above 1 shrinks it, below 1 inflates it — check your answer moved the right way.
- ≈
Same numerator as Sharpe: compute Rp − Rf once and reuse it if a question asks for both ratios (item sets love this).
- ≈
Cross-check with the market's own Treynor: it equals the market premium (Rm − Rf). Funds beating that on Treynor beat the market risk-adjusted.
Traps in applying it
- ✗Dividing by standard deviation — that's the Sharpe ratio; Treynor divides by BETA.
- ✗Forgetting to subtract the risk-free rate from the return first.
- ✗Comparing a Treynor number against a Sharpe number — the denominators have different units, so the magnitudes aren't comparable.
Limits & criticisms
Treynor inherits every weakness of beta: it is meaningless for undiversified holdings (their idiosyncratic risk is real but invisible to it), unstable when the beta estimate wobbles, and perverse for negative-beta portfolios where the sign flips. Like Sharpe it is backward-looking and silent about tail risk — and ranking on it assumes the CAPM's one-factor world.
Where it came from
Jack Treynor proposed the measure in 1965, a year before Sharpe's ratio, in one of the papers that quietly co-invented the CAPM (Treynor's unpublished 1962 manuscript anticipated much of it). His question was practical: mutual funds sit inside diversified household portfolios, so rank them on the risk they add, not the risk they contain. The ratio remains standard in performance attribution alongside Sharpe and Jensen's alpha, and the trio still frames every "did the manager earn their fee?" debate.
One identity, 1 questions
The exam can hide any variable. Each face below is the same equation solved for a different unknown — drill them separately.
Reward per unit of beta
The sleeve-ranking face: for funds inside a diversified whole, only their beta adds risk — so pay for excess return per unit of it.
On the BA II Plus
Worked example: A fund posts a return of 14.5% against T-bills at 1.5%, running a beta of 0.9. What Treynor measure should its factsheet report?
- 1.0.145 [−] 0.015 [=]excess return
- 2.[÷] 0.9 [=]per unit of beta
→ 0.1444