Risk-adjusted return

Risk-adjusted return is a measure of investment performance that evaluates the return generated by an investment relative to the amount or type of risk undertaken to achieve that return. It allows investors to compare investments with different levels of risk rather than considering returns alone.

A simplified representation is:

$Risk-Adjusted Return = \frac{Investment Return}{Measure of Risk}$

The appropriate measure of risk depends on the specific risk-adjusted performance metric being used.

Concept

Two investments can generate the same return while exposing investors to different levels of risk. Risk-adjusted return measures seek to account for this difference.

For example, if two portfolios each generate a 10% return, but one experiences substantially greater volatility, the portfolio with lower volatility may have a higher risk-adjusted return.

Excess return

Many risk-adjusted return measures use return above a reference rate rather than the total investment return.

A simplified excess-return calculation is:

$Excess Return = R_{p} - R_{f}$

where:

$R_{p}$ = portfolio return;
$R_{f}$ = risk-free rate.

The risk-free rate provides a reference point for evaluating the additional return generated by taking investment risk.

Sharpe ratio

The Sharpe ratio is one of the most widely used measures of risk-adjusted return.

It is calculated as:

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

where:

$R_{p}$ = portfolio return;
$R_{f}$ = risk-free rate;
$σ (R_{p})$ = standard deviation of portfolio returns.

The Sharpe ratio therefore measures excess return relative to total return volatility.

Sortino ratio

The Sortino ratio is similar to the Sharpe ratio but focuses on downside risk rather than total volatility.

A simplified expression is:

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

where:

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

The Sortino ratio does not treat positive volatility in the same manner as the Sharpe ratio.

Information ratio

The Information ratio evaluates active investment performance relative to a benchmark.

It is commonly calculated as:

$Information Ratio = \frac{R_{p} - R_{b}}{σ (R_{p} - R_{b})}$

where:

$R_{p}$ = portfolio return;
$R_{b}$ = benchmark return;
$σ (R_{p} - R_{b})$ = standard deviation of active returns, also known as tracking error.

The information ratio therefore measures active return relative to the variability of active return.

Treynor ratio

The Treynor ratio measures excess return relative to beta, a measure of systematic risk.

It can be expressed as:

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

where:

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

Unlike the Sharpe ratio, the Treynor ratio does not use total volatility as its measure of risk.

Jensen's alpha

Jensen's alpha measures the difference between an investment's actual return and the return predicted by the Capital Asset Pricing Model (CAPM).

A simplified expression is:

$α_{p} = R_{p} - [R_{f} + β_{p} (R_{m} - R_{f})]$

where:

$α_{p}$ = Jensen's alpha;
$R_{p}$ = portfolio return;
$R_{f}$ = risk-free rate;
$β_{p}$ = portfolio beta;
$R_{m}$ = market return.

A positive alpha indicates that the portfolio generated a return above that predicted by the CAPM, while a negative alpha indicates a return below the model's prediction.

Risk measures

Different risk-adjusted return measures use different definitions of risk.

Common measures include:

Standard deviation — total volatility;
beta — systematic market risk;
Tracking error — variability of active returns;
downside deviation — downside volatility;
Value at risk — potential loss under specified assumptions;
Maximum drawdown — largest peak-to-trough decline.

The choice of risk measure can materially affect the resulting assessment of investment performance.

Risk-adjusted return and absolute return

Absolute return measures the investment's return without explicitly adjusting it for risk.

A simplified calculation is:

$Absolute Return = \frac{V_{1} - V_{0}}{V_{0}} \times 100 %$

where:

$V_{0}$ = initial investment value;
$V_{1}$ = final investment value.

Risk-adjusted return goes further by incorporating a measure of risk or a reference return.

Risk-adjusted return and benchmark performance

Risk-adjusted return measures can be used to compare portfolios with different risk characteristics.

For example, two portfolios may have:

similar absolute returns;
different levels of volatility;
different benchmark-relative performance.

A risk-adjusted measure can provide additional information about the efficiency with which each portfolio generated its return.

Importance

Risk-adjusted return is an important concept in portfolio management and investment analysis because it allows investors to evaluate returns in relation to the risks undertaken. Measures such as the Sharpe ratio, Sortino ratio, Treynor ratio, information ratio, and Jensen's alpha use different definitions of risk and therefore provide different perspectives on investment performance.

Limitations

Risk-adjusted return measures have several limitations.

These include:

reliance on historical data;
sensitivity to the measurement period;
dependence on the chosen risk measure;
sensitivity to the selected benchmark or reference rate;
assumptions concerning the statistical distribution of returns;
inability to capture every form of investment risk.

Different measures can therefore produce different conclusions about the same investment.

Concept

Excess return

Sharpe ratio

Sortino ratio

Information ratio

Treynor ratio

Jensen's alpha

Risk measures

Risk-adjusted return and absolute return

Risk-adjusted return and benchmark performance

Importance

Limitations

See also