Spectral Risk Measure

A spectral risk measure (SRM) refers to a mathematical tool devised to quantify the risk concerning the financial portfolio where the weighted average of outcomes is taken, and the weights depend upon risk aversion. It serves the purpose of providing a more clear understanding of risk considering all factors like skewness and risk.

It also enables capturing the comprehensive risk assessment through consideration of full distribution potential outcomes, which lies beyond conventional risk measures such as value-at-risk. Investors and securities managers use it to balance returns and risks as per their risk preferences. Investors assess and handle the risk of their portfolios effectively by applying to different asset classes in the securities market. 

• A spectral risk measure is a mathematical instrument designed to measure the level of risk associated with a financial portfolio. It calculates the weighted average of outcomes, and the weights depend upon one's risk aversion.
• The primary objective of spectral risk measure is to enhance risk comprehension by considering factors such as skewness and risk.
• Its characteristics include coherence, tail sensitivity, distributional awareness, customization, portfolio diversification, and risk decomposition.
• Its drawbacks include parameter selectivity, computational complexity, reliance on outdated information, and insufficient universality.

Spectral Risk Measure (SRM) is defined as an approach of risk quantification to integrate any risk aversion parameter with the joint distribution of portfolio returns. It tries to present a comprehensive perspective of risk by considering the full spectrum of potential investors' risk preferences and its outcome.

Spectral risk measure involves multiple steps.

• First, historical data and stimulation technique gets deployed to estimate the return of a portfolio having joint distribution.
• Second, a chosen risk aversion parameter reflects the investor's preference for risk appetite.
• Third, the integrated joint distribution of return and chosen risk aversion parameter computes the SRM.
• Finally, the result of the computation gives the measure representing the portfolio's risk exposure.

It offers a detailed and in-depth picture of risks by using skewness and tail risk, unlike conventional Value at Risk (VaR). Investors can customize their risk appetite preferences to optimize risk-return trade-offs. Risk evaluation of a diversified portfolio becomes possible along with its benefits. 

But there are some cons too. It remains sensitive to the selection of risk aversion parameters varying within investors leading to variable risk assessment. Moreover, estimating the joint distribution and integrating it with the risk aversion parameter becomes highly complex.

Strategies involving SRM encompass optimizing asset allocation based on risk, mitigating extreme market events through tail risk hedging, and aiding diversification and risk management decisions through risk decomposition. These strategies enable investors to optimize asset allocation, protect against extreme market events, and make informed diversification and risk management decisions.

Hence, SRM assists professionals and investors in understanding portfolio risk, making wise judgments, and maximizing strategies.

Due to certain characteristics, spectral risk measures are useful for evaluating and managing portfolio risk. It happens because they provide a more thorough and sophisticated approach than standard risk measurements. Those characteristics have been listed here:

• Coherence: They satisfy specific mathematical properties that align with rational risk preferences termed coherent. Coherence ensures consistent risk measures and obeying various principles like positive homogeneity, sub-additivity, and translation invariance.
• Tail Sensitivity: It shows sensitivity to a portfolio's tail risk. They provide deep insights into the possible losses during rare events or market downturns through the capture of the severity of extreme events and their probability. 
• Distributional Awareness: These measures cover the distribution of potential outcomes instead of a particular quantile, such as value at risk (VaR). Hence, it results in a more in-depth comprehension of the portfolio risk characteristics, kurtosis, and skewness. 
• Customization: Investors' risk choosing allows for customizing spectral risk measures. Investors tailor risk measures according to their risk appetite, balancing the return and risk of their portfolio as per their preferences by incorporating risk aversion parameters.
• Portfolio Diversification: Investors diversify their risk portfolio using spectral measures. They do it by first considering the joint distribution returns across various assets. They capture the benefits of the diversification and finally gauge it concerning the risk of the entire portfolio.
• Risk Decomposition: Spectral risk measurements can break down a portfolio's overall risk into the contributions of various assets or risk factors. Consequently, investors get information on the relative significance of various assets or risk variables in determining the portfolio's overall risk.

Furthermore, spectral risk measures provide comprehensive, nuanced portfolio risk assessment.

Let us use a few examples to understand the topic in detail.

Example # 1

Suppose Michael, a seasoned investor from Prosperity Heights, trusts Prosperity Capital Management with his investments in speculative equities, growth-oriented funds, and cryptocurrencies. He uses spectral risk metrics to assess danger. Michael determines the spectral risk measure by examining prior returns and estimating the joint distribution. He picks a reasonable risk aversion parameter with a strong risk appetite. 

This thorough evaluation helps him manage potential tail risk and bias intrinsic in his bold investments inside Prosperity Heights' unpredictable market, optimize asset allocation and make strategic decisions.

Example # 2

Suppose Emily, a cautious investor from Harmonyville, entrusted Harmonious Investments with managing her collection of safe dividend-paying equities and government bonds. She assesses the risk attached to her assets using spectral risk measurements. Emily determines the spectral risk measure by calculating the joint distribution of returns and choosing a risk aversion parameter that aligns with her low-risk tolerance. 

With the help of this thorough assessment, she is better equipped to manage possible tail risk and skewness, align her portfolio to her risk preferences, and reach her long-term financial goals.

It finds usage in diverse portfolio optimization, provides risk management procedures and risk mitigation strategies facilitation, proves beneficial in creating financial models and stimulation generating risk-aware scenarios, and allows evaluation of different risk elements' effects on portfolio performance.

Spectral risk measurements provide benefits but also drawbacks, such as:

• Selectivity of the parameter: These remain dependent on the parameter chosen for risk aversion. It becomes subjective to select an appropriate parameter when different investors have dynamic risk appetites. Consequently, the measures may give outcomes sensitive to such parameters leading to various optimum portfolio allocations and risk assessments. 
• The complexity of computation: Spectral risk measure becomes complex, particularly for large portfolios with complex dependencies amongst assets. Moreover, it may require significantly large computational resources to integrate it into risk aversion parameters and to forecast the joint distribution or returns. As a result, it takes more time taking and resource hogging compared to simple risk measurements. 
• Reliance on old information or presumptions: These risk measures rely heavily on historical data and different assumptions regarding underlying asset returns distribution. Therefore, such reliance induces potential limitations and bias, especially during changing market conditions and extreme events which deviate from historical trends. Consequently, any deviation, inadequate, or inaccurate data leads to ineffectiveness and unreliable spectral risk measures.
• Insufficient universality: it may not be able to resolve particular risk factors or idiosyncrasies related to special asset classes and investment strategies. Hence, extra risk measures may be needed to offer a better-detailed risk assessment per the nature of the investment. 

Nevertheless, spectral risk measurements and other risk management strategies must be combined for a comprehensive picture of portfolio risk.

This has been a guide to what is Spectral Risk Measure. Here, we explain its properties, limitations, examples, strategies, and applications. You can learn more about it from the following articles –

• Distortion Risk Measure
• Deviation Risk Measure
• Risk Premium Formula