Treynor ratio

Treynor ratio is a risk-adjusted return measure that evaluates the excess return generated by an investment relative to its systematic risk. Unlike the Sharpe ratio, which uses total volatility as its measure of risk, the Treynor ratio uses beta to measure sensitivity to systematic market risk.

Calculation

The Treynor ratio is commonly calculated as:

$Treynor Ratio = \frac{R_{p} - R_{f}}{β_{p}}$

where:

$R_{p}$ = portfolio or investment return;
$R_{f}$ = risk-free rate;
$β_{p}$ = portfolio beta.

The numerator represents the portfolio's excess return over the risk-free rate, while beta represents its systematic risk.

Example

Suppose a portfolio has an annual return of 12%, the risk-free rate is 4%, and the portfolio beta is 1.2.

$Treynor Ratio = \frac{12 % - 4 %}{1.2} = 6.67 %$

The portfolio therefore generated 6.67 percentage points of excess return for each unit of systematic risk as measured by beta.

Interpretation

A higher Treynor ratio generally indicates that an investment generated greater excess return relative to its systematic risk.

A lower Treynor ratio indicates less excess return relative to systematic risk.

A negative Treynor ratio indicates that the investment's return was below the risk-free rate over the measurement period, assuming a positive beta.

The ratio is most useful when comparing diversified portfolios whose systematic risk is appropriately represented by beta.

Beta

The Treynor ratio uses beta rather than total return volatility.

A simplified beta calculation is:

$β_{p} = \frac{Cov (R_{p}, R_{m})}{Var (R_{m})}$

where:

$R_{p}$ = portfolio return;
$R_{m}$ = market return;
$Cov (R_{p}, R_{m})$ = covariance between portfolio and market returns;
$Var (R_{m})$ = variance of market returns.

Beta therefore measures the portfolio's sensitivity to movements in the market.

Treynor ratio and Sharpe ratio

The Sharpe ratio measures excess return relative to total volatility:

$Sharpe Ratio = \frac{R_{p} - R_{f}}{σ (R_{p})}$

The Treynor ratio measures excess return relative to systematic risk:

$Treynor Ratio = \frac{R_{p} - R_{f}}{β_{p}}$

where:

$σ (R_{p})$ = standard deviation of portfolio returns;
$β_{p}$ = portfolio beta.

The key distinction is that the Sharpe ratio considers total volatility, whereas the Treynor ratio considers only systematic risk.

Treynor ratio and Sortino ratio

The Sortino ratio focuses on downside risk rather than systematic market risk.

A simplified Sortino ratio is:

$Sortino Ratio = \frac{R_{p} - R_{t}}{σ_{d}}$

where:

$R_{t}$ = target or minimum acceptable return;
$σ_{d}$ = downside deviation.

The three measures therefore use different risk concepts:

Measure	Excess return	Risk measure
Sharpe ratio	$R_{p} - R_{f}$	Standard deviation
Treynor ratio	$R_{p} - R_{f}$	Beta
Sortino ratio	$R_{p} - R_{t}$	Downside deviation

Portfolio management

The Treynor ratio can be used in portfolio management to evaluate the performance of diversified portfolios relative to their systematic market risk.

It can be considered alongside:

Limitations

The Treynor ratio has several limitations.

These include:

dependence on the accuracy of beta estimates;
sensitivity to the selected market benchmark;
limited usefulness for poorly diversified portfolios;
reliance on historical returns and risk estimates;
inability to account directly for unsystematic risk;
sensitivity to the measurement period.

Because the Treynor ratio uses beta as its risk measure, it may provide an incomplete assessment of risk for portfolios that are not well diversified.

Importance

The Treynor ratio is an important risk-adjusted performance measure for evaluating diversified portfolios. By relating excess return to systematic market risk, it helps investors assess how effectively a portfolio has historically compensated investors for the market risk undertaken.

Treynor ratio

Contents

Calculation

Example

Interpretation

Beta

Treynor ratio and Sharpe ratio

Treynor ratio and Sortino ratio

Portfolio management

Limitations

Importance

See also