Treynor Ratio Equation, Derivation & Rearrangements

Core idea

Overview

The Treynor Ratio measures the excess return earned per unit of systematic risk, as represented by beta. It is a performance metric that evaluates how well an investor is compensated for taking on risk that cannot be diversified away.

When to use: This ratio is best applied when evaluating well-diversified portfolios where unsystematic risk has been eliminated. It is specifically used to compare different portfolios or fund managers against a market benchmark to see who provided the best return relative to market volatility.

Why it matters: It allows investors to distinguish between returns generated through high-risk market exposure and returns generated through skilled management. By focusing only on beta, it provides a clearer picture of a portfolio's performance within the context of the broader market movement.

Symbols

Variables

T = Treynor Ratio, R_p = Portfolio Return, R_f = Risk-free Rate, \beta_p = Portfolio Beta

T

Treynor Ratio

V a r iab l e

R_{p}

Portfolio Return

0.08

R_{f}

Risk-free Rate

0.01

β_{p}

Portfolio Beta

0.8

Walkthrough

Derivation

Definition: Treynor Ratio

The Treynor Ratio measures risk-adjusted return per unit of systematic (market) risk, making it suitable for comparing well-diversified portfolios.

Portfolio is well-diversified, so only systematic risk (beta) is relevant.
Higher Treynor Ratio indicates better risk-adjusted performance.

1

Compute excess return over the risk-free rate:

The excess return is what the portfolio earns above a riskless investment.

R_{p} - R_{f}

2

Divide by portfolio beta:

Unlike the Sharpe Ratio (which uses total volatility σ), Treynor uses only beta, ignoring diversifiable risk. It is most meaningful when comparing fully diversified funds.

T = \frac{R _{p} - R _{f}}{β _{p}}

Result

T = \frac{R _{p} - R _{f}}{β _{p}}

Source: University Finance — Portfolio Performance Measurement

Free formulas

Rearrangements

Solve for $T$

Make T the subject

T = \frac{R _{p} - R _{f}}{β _{p}}

Exact symbolic rearrangement generated deterministically for T.

Difficulty: 3/5

Solve for $R_{p}$

Make Rp the subject

R_{p} = T β_{p} + R_{f}

Exact symbolic rearrangement generated deterministically for Rp.

Difficulty: 2/5

Solve for $R_{f}$

Make Rf the subject

R_{f} = - T β_{p} + R_{p}

Exact symbolic rearrangement generated deterministically for Rf.

Difficulty: 2/5

Solve for $β_{p}$

Make beta the subject

β_{p} = \frac{R _{p} - R _{f}}{T}

Exact symbolic rearrangement generated deterministically for beta.

Difficulty: 3/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph is a hyperbolic curve when plotting the Treynor Ratio (T) against systematic risk (Beta), assuming a constant risk premium. As Beta approaches zero, the ratio increases toward a vertical asymptote, while it flattens toward a horizontal asymptote as Beta grows larger.

Graph type: hyperbolic

Why it behaves this way

Intuition

A financial picture where the Treynor Ratio represents the slope of a portfolio's characteristic line, illustrating the reward-to-risk trade-off by showing how much additional return an investor gets for each unit of

T

The Treynor Ratio, a measure of excess return per unit of systematic risk.

A higher Treynor Ratio indicates better risk-adjusted performance, meaning more return for each unit of market risk taken.

R_{p}

The total return generated by the investment portfolio over a specific period.

This is the overall gain or loss from the portfolio, including both income and capital appreciation.

R_{f}

The risk-free rate of return, typically approximated by the return on short-term government securities.

This represents the minimum return an investor could expect without taking any market risk.

R_{p} - R_{f}

The excess return of the portfolio, which is the return earned above the risk-free rate.

This is the compensation received for taking on any risk, both systematic and unsystematic.

β_{p}

The portfolio's beta, a measure of its systematic risk or sensitivity to overall market movements.

A beta of 1 means the portfolio moves in line with the market; a beta greater than 1 indicates higher volatility, and less than 1 indicates lower volatility relative to the market. It quantifies non-diversifiable risk.

Signs and relationships

R_p - R_f: This term calculates the 'excess return' of the portfolio beyond what could be earned from a risk-free asset. A positive value indicates the portfolio generated more return than the risk-free rate, justifying the risk
/\beta_p: Dividing the excess return by beta normalizes it by the portfolio's systematic risk. A higher beta (more systematic risk) in the denominator means a lower Treynor Ratio for the same excess return, implying less efficient

Free study cues

Insight

Canonical usage

The Treynor Ratio is a dimensionless performance metric. Returns are typically expressed as decimals for calculation, though often reported as percentages.

Common confusion

A common mistake is using percentage values directly in the formula (e.g., '10' for 10%) instead of converting them to their decimal equivalents (e.g., '0.10'), leading to incorrect results.

Dimension note

The Treynor Ratio is a dimensionless performance metric because it is a ratio of excess return (a difference of dimensionless rates) to systematic risk (beta, which is also dimensionless).

Unit systems

$R_{p}$ dimensionless (decimal or percentage) · Represents the portfolio's total return. Must be expressed consistently with R_f (e.g., both as decimals or both as percentages) for calculation. Decimals are preferred for calculation.

$R_{f}$ dimensionless (decimal or percentage) · Represents the risk-free rate of return. Must be expressed consistently with R_p (e.g., both as decimals or both as percentages) for calculation. Decimals are preferred for calculation.

$β_{p}$ dimensionless · Represents the portfolio's systematic risk, measuring its sensitivity to market movements.

$T$ dimensionless · The Treynor Ratio itself, representing excess return per unit of systematic risk.

Ballpark figures

Quantity:

One free problem

Practice Problem

An investment fund reports an annual return of 12% during a period where the risk-free rate is 3%. If the portfolio's beta is measured at 1.2, calculate the Treynor Ratio.

Portfolio Return0.12 0.08

Risk-free Rate0.03 0.01

Portfolio Beta1.2 0.8

Solve for: T

Hint: Subtract the risk-free rate from the portfolio return to find the excess return before dividing by beta.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

An investment fund produces a return of 12% in a year where the risk-free rate is 2%. If the fund has a beta of 1.25, its Treynor Ratio is 0.08 (or 8%), representing the excess return earned per unit of market risk.

Study smarter

Tips

Use only for diversified portfolios; for non-diversified ones, the Sharpe Ratio is preferred.
A higher Treynor Ratio indicates a more favorable risk-adjusted return.
Ensure the timeframe for returns and the risk-free rate are consistent.
Negative beta values can make the ratio misleading; exercise caution with such inverse funds.

Avoid these traps

Common Mistakes

Confusing Beta with Standard Deviation (which is used in the Sharpe Ratio).
Using inconsistent time periods for the portfolio return and the risk-free rate.
Applying the ratio to undiversified portfolios where unsystematic risk is still significant.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

The Treynor Ratio measures risk-adjusted return per unit of systematic (market) risk, making it suitable for comparing well-diversified portfolios.

This ratio is best applied when evaluating well-diversified portfolios where unsystematic risk has been eliminated. It is specifically used to compare different portfolios or fund managers against a market benchmark to see who provided the best return relative to market volatility.

It allows investors to distinguish between returns generated through high-risk market exposure and returns generated through skilled management. By focusing only on beta, it provides a clearer picture of a portfolio's performance within the context of the broader market movement.

Confusing Beta with Standard Deviation (which is used in the Sharpe Ratio). Using inconsistent time periods for the portfolio return and the risk-free rate. Applying the ratio to undiversified portfolios where unsystematic risk is still significant.

An investment fund produces a return of 12% in a year where the risk-free rate is 2%. If the fund has a beta of 1.25, its Treynor Ratio is 0.08 (or 8%), representing the excess return earned per unit of market risk.

Use only for diversified portfolios; for non-diversified ones, the Sharpe Ratio is preferred. A higher Treynor Ratio indicates a more favorable risk-adjusted return. Ensure the timeframe for returns and the risk-free rate are consistent. Negative beta values can make the ratio misleading; exercise caution with such inverse funds.

References

Sources

Zvi Bodie, Alex Kane, Alan J. Marcus, Investments, McGraw-Hill Education
Jack L. Treynor, How to Rate Management of Investment Funds, Harvard Business Review, 1965
Wikipedia: Treynor ratio
Bodie, Zvi; Kane, Alex; Marcus, Alan J. (2021). Investments (12th ed.). McGraw-Hill Education.
Zvi Bodie, Alex Kane, Alan J. Marcus, Investments, 12th ed., McGraw-Hill Education, 2021
University Finance — Portfolio Performance Measurement

Treynor Ratio

Overview

Variables

Derivation

Compute excess return over the risk-free rate:

Divide by portfolio beta:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Beta Coefficient

CAPM (Expected Return)

Frequently Asked Questions

Sources