Calculating the Risk Adjusted Rate of Return involves using specific metrics to evaluate an investment’s performance relative to the level of risk taken.
Unlike simple return, which only looks at the gain or loss, risk-adjusted return helps investors compare different assets by considering the volatility or systematic risk associated with them.
The three most common and key metrics for calculating risk-adjusted return are the Sharpe Ratio, the Treynor Ratio, and Jensen’s Alpha.
1. Sharpe Ratio
The Sharpe Ratio measures the excess return per unit of total risk (volatility) of an investment. It is the most widely used risk-adjusted performance measure.
![\[\text{Sharpe Ratio} = \frac{R_p - R_f}{\sigma_p}\]](https://www.SuperBusinessManager.com/wp-content/ql-cache/quicklatex.com-d066d15f9af2b7e62f11d0b6d2e34514_l3.png "Rendered by QuickLaTeX.com")
Formula Components:
: The Return of the portfolio or investment.
: The Risk-free rate of return (e.g., the return on a short-term U.S. Treasury bill).
: The Standard Deviation of the portfolio’s excess return, which represents its total volatility or total risk.
Interpretation:
- A higher Sharpe Ratio indicates better risk-adjusted performance.
- A ratio above 1.0 is generally considered good, suggesting returns that outweigh the risks taken.
- A negative ratio means the risk-free asset outperformed the investment.
Example (Real Business Context)
Suppose Fund A has an annual return () of 15% and a standard deviation () of 10%. The risk-free rate () is 3%.
![\[\text{Sharpe Ratio (Fund A)} = \frac{0.15 - 0.03}{0.10} = \frac{0.12}{0.10} = 1.2\]](https://www.SuperBusinessManager.com/wp-content/ql-cache/quicklatex.com-12462600a5165da9f4fbd129b80e6b88_l3.png "Rendered by QuickLaTeX.com")
If Fund B has a higher return of 20% but a higher standard deviation of 18%, its Sharpe Ratio would be:
![\[\text{Sharpe Ratio (Fund B)} = \frac{0.20 - 0.03}{0.18} = \frac{0.17}{0.18} \approx 0.94\]](https://www.SuperBusinessManager.com/wp-content/ql-cache/quicklatex.com-96f22dcd95b875a1aebd9372d4581e19_l3.png "Rendered by QuickLaTeX.com")
In this example, Fund A is the better investment on a risk-adjusted basis, as it provided a higher excess return for each unit of risk taken (1.2 vs. 0.94).
2. Treynor Ratio
The Treynor Ratio is similar to the Sharpe Ratio, but it measures the excess return per unit of systematic risk (market risk) instead of total risk. It is most suitable for evaluating well-diversified portfolios where unsystematic (specific) risk is largely eliminated.
![\[\text{Treynor Ratio} = \frac{R_p - R_f}{\beta_p}\]](https://www.SuperBusinessManager.com/wp-content/ql-cache/quicklatex.com-178d8310272f857b93db1c84bc0e23f9_l3.png "Rendered by QuickLaTeX.com")
Formula Components:
and : Same as in the Sharpe Ratio.
: The Beta of the portfolio, which measures its sensitivity to overall market movements (systematic risk).
Interpretation:
- A higher Treynor Ratio signifies better risk-adjusted performance, meaning a greater return was earned for each unit of systematic risk.
Example (Real Business Context)
Using the same risk-free rate of 3%, assume Portfolio C has a return of 14% and a Beta () of 0.8.
![\[\text{Treynor Ratio (Portfolio C)} = \frac{0.14 - 0.03}{0.8} = \frac{0.11}{0.8} = 0.1375\]](https://www.SuperBusinessManager.com/wp-content/ql-cache/quicklatex.com-a1529e74ae384326ecf240c4fbd1f39d_l3.png "Rendered by QuickLaTeX.com")
If Portfolio D has a return of 18% but a higher Beta of 1.4:
![\[\text{Treynor Ratio (Portfolio D)} = \frac{0.18 - 0.03}{1.4} = \frac{0.15}{1.4} \approx 0.1071\]](https://www.SuperBusinessManager.com/wp-content/ql-cache/quicklatex.com-e76706f51209da5b76b4cde5fa659827_l3.png "Rendered by QuickLaTeX.com")
Portfolio C has the higher Treynor Ratio, suggesting it provided a better return for the systematic risk it carried compared to Portfolio D.
3. Jensen’s Alpha (
)
Jensen’s Alpha calculates the abnormal return of an investment—the return it generated above or below the return predicted by the Capital Asset Pricing Model (CAPM), given its systematic risk (Beta).
![\[\alpha = R_p - [R_f + \beta_p(R_m - R_f)]\]](https://www.SuperBusinessManager.com/wp-content/ql-cache/quicklatex.com-4d8784dde4655ffd22332f6a0bb7cbaf_l3.png "Rendered by QuickLaTeX.com")
Formula Components:
, , : Same as in the Treynor Ratio.
: The Market Return (e.g., the return on a broad market index like the S&P 500).
The term ![[R_f + \beta_p(R_m - R_f)]](https://www.SuperBusinessManager.com/wp-content/ql-cache/quicklatex.com-da53fa10bcab5da16357931596463eb2_l3.png "Rendered by QuickLaTeX.com")
is the Expected Return predicted by CAPM.
Interpretation:
(Positive Alpha): The portfolio outperformed the market, earning a return greater than expected for the level of risk taken. This is often attributed to superior fund management or stock selection skill.
(Negative Alpha): The portfolio underperformed the market relative to its risk.
(Zero Alpha): The portfolio earned the return exactly expected for its level of risk.
Example (Real Business Context)
Assume the Risk-free rate () is 4% and the Market Return () is 10%.
Hedge Fund Z realized an actual return () of 16% with a Beta () of 1.2.
First, calculate the Expected Return:
![\[\text{Expected Return} = 0.04 + 1.2 \times (0.10 - 0.04) = 0.04 + 1.2 \times 0.06 = 0.04 + 0.072 = 0.112 \text{ or } 11.2\%\]](https://www.SuperBusinessManager.com/wp-content/ql-cache/quicklatex.com-f5a7baf6ae4899730f1810beb08563f3_l3.png "Rendered by QuickLaTeX.com")
Now, calculate Jensen’s Alpha:
![\[\alpha = 0.16 - 0.112 = 0.048 \text{ or } 4.8\%\]](https://www.SuperBusinessManager.com/wp-content/ql-cache/quicklatex.com-1f0fc4d6ec0fc2ac1817ba22a1f68a10_l3.png "Rendered by QuickLaTeX.com")
Hedge Fund Z generated an alpha of 4.8%, meaning it delivered 4.8 percentage points more return than what would have been expected based on its systematic risk and the market’s performance.