[机器学习]Kaggle:Hull Tactical - Market Prediction-标普500

个人主页星柚程精选文章《MATLAB多目标优化》《Kaggle:CV、Public LB 》、《我的第一次 Kaggle》、《C构造传参》、《蛇形机械臂的模拟退火优化》️专栏建设|深度学习|、|Python量化|、|C学习|、|数据结构|流水不争先争得是涛涛不绝。Excess Returns: Meaning, Risk, and FormulasBy James ChenFull BioJames Chen, CMT is an expert trader, investment adviser, and global market strategist.Learn about our editorial policiesUpdated February 11, 2026Reviewed by Margaret JamesFact checked byYarilet PerezDefinitionExcess returns refer to the financial gains generated by an investment that surpass the expected rate of return established by standard financial models or market averages.Key TakeawaysExcess returns will depend on a designated investment return comparison for analysis.The riskless rate and benchmarks with similar levels of risk to the investment being analyzed are commonly used in calculating excess return.Alpha is a type of excess return metric that focuses on performance return in excess of a closely comparable benchmark.Excess return is an important consideration when using modern portfolio theory which seeks to invest with an optimized portfolio.What Are Excess Returns?Excess returns are returns achieved above and beyond the return of a representation of the stock market. They depend on a designated investment return comparison for analysis. Some of the most basic return comparisons include a riskless rate and benchmarks with levels of risk that mirror the investment being analyzed.Understanding Excess ReturnsExcess returns are an important metric that helps an investor gauge performance in comparison to other investment alternatives. In general, all investors hope for positive excess return because it provides an investor with more money than they could have achieved by investing elsewhere.Excess return is identified by subtracting the return of one investment from the total return percentage achieved in another investment. When calculating excess return, multiple return measures can be used. Some investors may wish to see excess return as the difference in their investment over a risk-free rate.Other times, an excess return may be calculated in comparison to a closely comparable benchmark with similar risk and return characteristics. Using closely comparable benchmarks is a return calculation that results in an excess return measure known as alpha.In general, return comparisons may be either positive or negative. A positive excess return shows that an investment outperformed its comparison, while a negative difference in returns occurs when an investment underperforms. Investors should keep in mind that purely comparing investment returns to a benchmark provides an excess return that does not necessarily take into consideration all of the potential trading costs of a comparable proxy.For example, using the SP 500 as a benchmark provides an excess return calculation that does not typically take into consideration the actual costs required to invest in all 500 stocks in the Index or management fees for investing in an SP 500 managed fund.Excess Return vs. Riskless RatesRiskless and low-risk investments are often used by investors seeking to preserve capital for various goals. U.S. Treasuries are typically considered the most basic form of riskless securities. Investors can buy U.S. Treasuries with maturities of one month, two months, three months, six months, one year, two years, three years, five years, seven years, 10 years, 20 years, and 30 years.12Each maturity will have a different expected return found along the U.S. Treasury yield curve.3 Other types of low-risk investments include certificates of deposits, money market accounts, and municipal bonds.Investors can determine excess return levels based on comparisons to risk-free securities. For example, if the one-year Treasury has returned 2.0% and the technology stock Meta (formerly Facebook) has returned 15%, then the excess return achieved for investing in Meta is 13%.AlphaOftentimes, an investor will want to look at a more closely comparable investment when determining excess return. That’s where alpha comes in. Alpha is the result of a more narrowly focused calculation that includes only a benchmark with comparable risk and return characteristics to an investment. Alpha is commonly calculated in investment fund management as the excess return a fund manager achieves over a fund’s stated benchmark.Broad stock return analysis may look at alpha calculations in comparison to the SP 500 or other broad market indexes like the Russell 3000. When analyzing specific sectors, investors will use benchmark indexes that include stocks in that sector. The Nasdaq 100 for example can be a good alpha comparison for large-cap technology.In general, active fund managers seek to generate some alpha for their clients over a fund’s stated benchmark. Passive fund managers will seek to match the holdings and return of an index.Consider a large-cap U.S. mutual fund that has the same level of risk as the SP 500 index. If the fund generates a return of 12% in a year when the SP 500 has only advanced 7%, the difference of 5% would be considered as the alpha generated by the fund manager.Excess Return vs. Risk ConceptsAs discussed, an investor has the opportunity to achieve excess returns beyond a comparable proxy. However, the amount of excess return is usually associated with risk. Investment theory has determined that the more risk an investor is willing to take the greater their opportunity for higher returns. As such, several market metrics can help an investor to understand if the returns and excess returns they achieve are worthwhile.BetaBeta is a risk metric quantified as a coefficient in regression analysis that provides the correlation of an individual investment to the market (usually the SP 500). A beta of one means that an investment will experience the same level of return volatility from systematic market moves as a market index.A beta above one indicates that an investment will have higher return volatility and therefore higher potential for gains or losses. A beta below one means an investment will have less return volatility and therefore less movement from systematic market effects with less potential for gain but also less potential for loss.Beta is an important metric used when generating an Efficient Frontier graph to develop a Capital Allocation Line which defines an optimal portfolio. Asset returns on an Efficient Frontier are calculated using the following Capital Asset Pricing Model:RaRrfβ×(Rm−Rrf)where:RaExpected return on a securityRrfRisk-free rateRmExpected return of the marketβBeta of the securityRm−RrfEquity market premium​Ra​Rrf​β×(Rm​−Rrf​)where:Ra​Expected return on a securityRrf​Risk-free rateRm​Expected return of the marketβBeta of the securityRm​−Rrf​Equity market premium​Beta can be a helpful indicator for investors when understanding their excess return levels. Treasury securities have a beta of approximately zero. This means that market changes wont effect on the return of a Treasury, and the 2.0% earned from the one-year Treasury in the example above is riskless.Meta on the other hand, has a beta of approximately 1.29 so systematic market moves that are positive will lead to a higher return for Meta than the SP 500 Index overall and vice versa.4Jensen’s AlphaIn active management, fund manager alpha can be used as a metric for evaluating the performance of a manager overall. Some funds provide their managers a performance fee which offers extra incentive for fund managers to exceed their benchmarks. In investments, there is also a metric known as Jensen’s Alpha. Jensen’s Alpha seeks to provide transparency around how much of a manager’s excess return was related to risks beyond a fund’s benchmark.Jensen’s Alpha is calculated by:Jensen’s AlphaRi−(Rfβ(Rm−Rf))where:RiRealized return of the portfolio or investmentRfRisk-free rate of return for the time periodβBeta of the portfolio of investmentwith respect to the chosen market indexRmRealized return of the appropriate market index​Jensen’s AlphaRi​−(Rf​β(Rm​−Rf​))where:Ri​Realized return of the portfolio or investmentRf​Risk-free rate of return for the time periodβBeta of the portfolio of investmentwith respect to the chosen market indexRm​Realized return of the appropriate market index​A Jensen’s Alpha of zero means that the alpha achieved exactly compensated the investor for the additional risk taken on in the portfolio. A positive Jensen’s Alpha means the fund manager overcompensated its investors for the risk and a negative Jensen’s Alpha would be the opposite.Sharpe RatioIn fund management, the Sharpe Ratio is another metric that helps an investor understand their excess return in terms of risk.The Sharpe Ratio is calculated by:Sharpe RatioRp−RfPortfolio Standard Deviationwhere:RpPortfolio returnRfRiskless rate​Sharpe RatioPortfolio Standard DeviationRp​−Rf​​where:Rp​Portfolio returnRf​Riskless rate​The higher the Sharpe Ratio of an investment the more an investor is being compensated per unit of risk. Investors can compare Sharpe Ratios of investments with equal returns to understand where excess return is more prudently being achieved. For example, two funds have a one year return of 15% with a Sharpe Ratio of 2 vs. 1. The fund with a Sharpe Ratio of 2 is producing more return per one unit of risk.Special ConsiderationsCritics of mutual funds and other actively managed portfolios contend that it is next to impossible to generate alpha consistently over the long term. Investors are then theoretically better off investing in stock indexes or optimized portfolios that provide them with a level of expected return and a level of excess return over the risk-free rate.This helps to make the case for investing in a diversified portfolio that is risk-optimized to achieve the most efficient level of excess return over the risk-free rate based on risk tolerance.This is where the Efficient Frontier and Capital Market Line can come in. The Efficient Frontier plots a frontier of returns and risk levels for a combination of asset points generated by the Capital Asset Pricing Model. An Efficient Frontier considers data points for every available investment an investor may wish to consider investing in. Once an efficient frontier is graphed, the capital market line is drawn to touch the efficient frontier at its most optimal point.With this portfolio optimization model developed by financial academics, an investor can choose a point along the capital allocation line for which to invest based on their risk preference. An investor with zero risk preference would invest 100% in risk-free securities.The highest level of risk would invest 100% in the combination of assets suggested at the intersect point. Investing 100% in the market portfolio would provide a designated level of expected return with excess return serving as the difference from the risk-free rate.As illustrated by the Capital Asset Pricing Model, Efficient Frontier, and Capital Allocation Line, an investor can choose the level of excess return they wish to achieve above the risk-free rate based on the amount of risk they wish to take on.在金融数学里阿尔法和贝塔来自一个简单的线性回归方程你的收益 α β × 市场收益 残差随机误差把它想象成画散点图拟合一条直线· 纵轴Y你的股票/基金收益率。· 横轴X大盘指数比如沪深300的收益率。· 把每天或每月的数据点画上去用最小二乘法画一条“最贴切”的直线。这条直线的数学特征就拆解出了三个核心信息1. 贝塔β 直线的“斜率”它衡量你跟着大盘波动的敏感度。· β 1.2大盘涨1%你的平均涨1.2%波动比市场大β 0.8大盘涨1%你只涨0.8%比市场稳。· 回归视角看β就是这条直线的陡峭程度代表系统性风险躲不掉。2. 阿尔法α 直线的“截距”这才是回归视角最精妙的地方α是当市场收益X为0时你的起点位置。· 如果α 0.02即2%意味着哪怕大盘一年不涨不跌0%你这条直线在Y轴上的截距也是正的说明你硬生生比大盘多赚了2%。· 关键认知回归的过程本质上就是把市场波动β×市场的影响“剥离”掉剩下的那个“纯纯的、靠本事硬扛出来的”平均超额收益就是α。3. 残差 点到直线的“上下波动”这是回归线没能解释的部分代表纯粹的运气或突发事件比如突发的黑天鹅。真正的α必须足够大且统计上显著P值小否则那些正收益很可能只是残差运气造成的假象。