Reward to variability

What Is Reward to Variability?

Reward to variability, commonly known as the Sharpe Ratio, is a key metric in portfolio theory that measures the risk-adjusted return of an investment or portfolio. It quantifies the amount of return an investor receives for each unit of risk taken, specifically using standard deviation as the measure of risk or volatility. A higher reward to variability ratio indicates a better risk-adjusted performance, meaning the investment is generating more excess return for the level of volatility endured.

This ratio helps investors understand whether the excess returns generated by a portfolio are due to shrewd investment decisions or simply the result of taking on excessive risk. It is widely used in investment performance evaluation, allowing for a standardized comparison of different investment opportunities.

History and Origin

The concept of reward to variability was developed by Nobel laureate William F. Sharpe in 1966. He initially referred to it as the "reward-to-variability ratio" in his seminal work. Sharpe's work built upon earlier foundational ideas in Modern Portfolio Theory and contributed significantly to the development of the Capital Asset Pricing Model (CAPM).³⁰, ³¹

Sharpe introduced this ratio as a means to measure the performance of an investment by adjusting for its risk, recognizing that higher returns alone do not necessarily signify a superior investment if they come with disproportionately higher risk. He was awarded the Nobel Memorial Prize in Economic Sciences in 1990 for his pioneering contributions to the theory of financial economics, including the CAPM, from which the reward-to-variability ratio emerged.²⁹

Key Takeaways

• Reward to variability, also known as the Sharpe Ratio, assesses an investment's risk-adjusted return.²⁸
• It measures the excess return earned per unit of total risk (volatility) assumed.²⁷
• A higher ratio generally indicates better risk-adjusted performance, implying more efficient returns for the level of risk.²⁵, ²⁶
• The ratio helps investors compare different investments by leveling the playing field regarding risk.²⁴
• It is a foundational tool in portfolio management and performance evaluation.²³

Formula and Calculation

The formula for the reward to variability ratio is as follows:

Where:

• ( R_p ) = Expected return of the portfolio or investment.
• ( R_f ) = Risk-free rate of return. This is typically the return on a very low-risk investment, such as a U.S. Treasury bond, that serves as a benchmark for returns without significant risk.²²
• ( \sigma_p ) = Standard deviation of the portfolio's or investment's excess return. This measures the total volatility or dispersion of returns around the average.²¹

The numerator ( (R_p - R_f) ) represents the "excess return" or "risk premium," which is the additional return generated above what could have been earned from a risk-free asset. The denominator, standard deviation, quantifies the total risk taken to achieve that excess return.

Interpreting the Reward to Variability

Interpreting the reward to variability ratio involves understanding that a higher value is generally preferred. This signifies that the investment is providing more compensation for the level of risk undertaken.²⁰

• Positive Ratio: A positive ratio means the portfolio has generated returns in excess of the risk-free rate, per unit of volatility.
• Negative Ratio: A negative ratio suggests that the risk-free rate outperformed the portfolio, or the portfolio's return was negative. In such cases, the investment might be considered less attractive than simply holding a risk-free asset.

While there aren't universally fixed "good" or "bad" thresholds, general guidelines for the reward to variability ratio are often cited:

• Less than 1.0: Sub-optimal, indicating insufficient reward for the risk taken.¹⁹
• 1.0 to 1.99: Acceptable to good risk-adjusted performance.¹⁸
• 2.0 to 2.99: Very good performance.¹⁷
• 3.0 or greater: Excellent and often rare in public markets, suggesting exceptional risk-adjusted return.¹⁶

It is crucial to compare the reward to variability of an investment against its peers or relevant benchmarks within the same asset class or investment strategy, rather than in isolation. A ratio of 1.0 might be considered good in one market environment or asset class but mediocre in another.

Hypothetical Example

Consider two hypothetical investment portfolios, Portfolio A and Portfolio B, over a year. The prevailing risk-free rate is 3% (e.g., based on the 3-month Treasury bill yield as found on the Federal Reserve Economic Data (FRED) website). [EXTERNAL_2]

• Portfolio A:

  • Annual return = 12%
  • Annualized standard deviation = 8%

  Calculation for Portfolio A:

  $$\text{Sharpe Ratio}_A = \frac{0.12 - 0.03}{0.08} = \frac{0.09}{0.08} = 1.125$$

• Portfolio B:

  • Annual return = 15%
  • Annualized standard deviation = 14%

  Calculation for Portfolio B:

  $$\text{Sharpe Ratio}_B = \frac{0.15 - 0.03}{0.14} = \frac{0.12}{0.14} \approx 0.857$$

In this example, Portfolio A has a higher reward to variability ratio (1.125) compared to Portfolio B (0.857). This suggests that while Portfolio B generated a higher absolute return (15% vs. 12%), Portfolio A provided a more efficient return for the amount of risk taken. An investor focused on risk-adjusted return would likely prefer Portfolio A.

Practical Applications

The reward to variability ratio serves as a fundamental metric with several practical applications in finance and investing:

• Portfolio Comparison: It is widely used to compare the investment performance of different portfolios, funds, or individual securities. By normalizing returns for risk, it allows investors to identify which options offer a better return for a given level of risk.¹⁴, ¹⁵
• Investment Selection: Investors can use the ratio to evaluate potential investments, such as mutual funds, exchange-traded funds (ETFs), or even individual stocks, and integrate it into their investment decision-making process.¹², ¹³ Funds with higher reward to variability ratios are often preferred as they demonstrate a more favorable risk-adjusted outcome.
• Portfolio Management and Optimization: Portfolio managers utilize this ratio to optimize asset allocation and adjust the mix of assets within a portfolio. The goal is often to maximize the reward to variability for a given level of acceptable risk.
• Performance Evaluation of Managers: The reward to variability ratio can be applied to assess the effectiveness of portfolio managers in generating returns relative to the risk they undertake. It helps differentiate between managers who achieve high returns through skill versus those who do so simply by taking on excessive risk.¹¹
• Risk Assessment: It helps investors understand the trade-off between return and risk, aiding in the assessment of a portfolio's risk exposure and how well it aligns with an investor's risk tolerance.⁹, ¹⁰

Limitations and Criticisms

Despite its widespread use, the reward to variability ratio has several limitations and criticisms:

• Assumption of Normal Distribution: The ratio assumes that investment returns are normally distributed, meaning that positive and negative deviations from the mean are equally likely. However, financial markets often exhibit "fat tails" (more frequent extreme positive or negative events) and skewness (asymmetric distribution of returns), which can lead to the standard deviation underestimating actual downside risk.⁷, ⁸
• Treats All Volatility Equally: The standard deviation, as a measure of volatility, does not distinguish between upside volatility (desirable) and downside volatility (undesirable). A portfolio with significant positive price swings could have a lower reward to variability ratio simply because its overall volatility is high, even if those swings are beneficial.⁶ This has led to the development of alternative metrics like the Sortino Ratio, which focuses only on downside deviation.⁵
• Susceptibility to Manipulation: Portfolio managers can potentially manipulate the reward to variability ratio to appear more favorable. For instance, lengthening the return measurement interval can artificially lower the annualized standard deviation, boosting the ratio.
• Reliance on Historical Data: The ratio is calculated using historical data, and past performance is not indicative of future results. Market conditions, economic cycles, and other factors can change, affecting the future risk-return profile of an investment.⁴
• Choice of Risk-Free Rate: The selection of the risk-free rate can influence the calculated ratio. While short-term government bonds are commonly used, different proxies can lead to different results.
• Comparability Issues: The ratio is most useful for comparing similar investments. Comparing the reward to variability of a highly diversified equity portfolio to a single commodity, for example, may not provide meaningful insights due to fundamental differences in their risk profiles.

These limitations suggest that while the reward to variability ratio is a powerful tool, it should be used in conjunction with other metrics and qualitative analysis for a comprehensive assessment of investment performance. A detailed article from Morningstar also elaborates on these points in the context of the Sharpe Ratio. [EXTERNAL_3]

Reward to Variability vs. Sharpe Ratio

The terms "Reward to variability" and "Sharpe ratio" are synonymous and refer to the exact same financial metric. William F. Sharpe, the Nobel laureate who developed the ratio, initially coined it the "reward-to-variability ratio." Over time, the more concise "Sharpe Ratio" became the universally accepted and widely used name for this measure of risk-adjusted return. Therefore, when discussing "reward to variability," one is inherently referring to the Sharpe Ratio. Both terms describe the measurement of excess return per unit of standard deviation of an investment.

FAQs

What does a high reward to variability ratio mean?

A high reward to variability ratio indicates that an investment or portfolio has generated a greater amount of return for each unit of risk taken, relative to a risk-free asset. This generally signifies more efficient risk-adjusted performance.², ³

Can the reward to variability ratio be negative?

Yes, the reward to variability ratio can be negative. This occurs if the portfolio's return is less than the risk-free rate, or if the portfolio experiences a negative absolute return. A negative ratio implies that investing in the risk-free asset would have yielded a better outcome.¹

Is the reward to variability ratio suitable for all types of investments?

The reward to variability ratio is most appropriate for comparing diversified portfolios or investments where standard deviation is an adequate measure of total risk. While it can be applied to individual securities, its effectiveness might be limited for investments with highly skewed returns or non-normal distributions, or for those with specific types of non-diversifiable risk that beta would better capture. For instance, the Bogleheads forum provides useful discussions on applying risk-adjusted return metrics. [EXTERNAL_4]

How does diversification affect the reward to variability ratio?

Diversification can potentially increase the reward to variability ratio of a portfolio. By combining assets with low correlations, investors can reduce the overall volatility of the portfolio without necessarily sacrificing return. This reduction in risk (denominator) for the same or similar excess return (numerator) would lead to a higher ratio.