5.4: Performance Evaluation
5.4：绩效评估

Evaluating the performance of portfolios based on average returns is not very useful because returns must be adjusted for risk before they can be compared meaningfully.
根据平均回报评估投资组合的表现并不是很有用，因为回报必须根据风险进行调整，然后才能进行有意义的比较。

Portfolio Evaluation 投资组合评估

A simple manner to evaluate the performance of an investment is to consider its arithmetic and geometric average rate of return. We have already learnt how to calculate it here already on Canvas page C.3: Expected Return
评估投资绩效的一个简单方法是考虑其算术平均回报率和几何平均回报率。我们已经在Canvas 页面 C.3 上学习了如何计算它：预期回报.

The average return can be compared to a benchmark, such as an appropriate market index or the median return of funds in a comparison group.
平均回报可以与基准进行比较，例如适当的市场指数或比较组中基金的中值回报。

Alpha, in the previous page, is not the only method for evaluating how well an investment is doing. This falls into a more general category called risk-adjusted performance evaluation.
上一页中的 Alpha 并不是评估投资表现的唯一方法。这属于一个更一般的类别，称为风险调整绩效评估。

Risk-adjusted performance evaluation methods were proposed soon after the Capital Asset Pricing Model (CAPM) theory was developed. We will go over three measures here: Sharpe Ratio, Treynor Ratio, Jensen's Alpha. Like the arithmetic versus geometric return, one measure is not better than the other, rather each tells you something different about an investment.
资本资产定价模型（CAPM）理论发展后不久，就提出了风险调整绩效评估方法。我们将在这里讨论三个衡量标准：夏普比率、特雷诺比率、詹森阿尔法。就像算术回报与几何回报一样，一种衡量标准并不比另一种更好，而是每种衡量标准都告诉您有关投资的不同信息。

If you are interested, you can read more about these academic papers here (note that these papers are very dense and are not required reading but useful historical references for these topics):
如果您有兴趣，可以在这里阅读有关这些学术论文的更多信息（请注意，这些论文非常密集，不是必读内容，而是这些主题的有用历史参考资料） ：

Sharpe Ratio 夏普比率

$S h a r p e r a t i o = \frac{E R_{p} - r_{f}}{σ_{p}}$

$E R_{p} = Average return on the portfolio r_{f} = Risk-free rate σ_{p} = Standard deviation of portfolio returns$

We have seen this metric before on Canvas page 4.6: Optimal Portfolios II as this helped us select the tangency portfolio with the efficient frontier. The market portfolio (M) on the efficient frontier, has the highest possible Sharpe ratio of any portfolio on the efficient frontier. One benefit of the Sharpe ratio is that it does not rely on an asset pricing model like the CAPM.
我们之前在Canvas 第 4.6 页：最佳投资组合 II上看到过这个指标，因为这帮助我们选择了具有有效边界的切线投资组合。有效前沿上的市场投资组合 (M) 在有效前沿上的任何投资组合中具有尽可能高的夏普比率。夏普比率的好处之一是它不依赖于 CAPM 等资产定价模型。

Again, because the Sharpe Ratio is a ratio of excess returns over the standard deviation of returns, it tells the amount of returns an investment is providing given a unit of standard deviation, which is a very natural way of measuring the amount of returns given risk. The main benchmark value that investment professionals compare to is the Sharpe ratio for the market.
同样，由于夏普比率是超额收益与收益标准差的比率，因此它表明了给定标准差单位的投资所提供的收益额，这是衡量给定风险的收益额的一种非常自然的方法。投资专业人士比较的主要基准值是市场的夏普比率。

All else equal, a higher Sharpe ratio is better because it states that the investment is providing a higher return for a given risk level. When used among investments or funds with similar characteristics, it can tell you which one did better.
在其他条件相同的情况下，夏普比率越高越好，因为它表明投资在给定的风险水平下提供了更高的回报。当在具有相似特征的投资或基金中使用时，它可以告诉您哪一个表现更好。

Treynor Ratio 特雷诺比率

$T r e y n o r r a t i o = \frac{E R_{p} - r_{f}}{β_{p}}$

$E R_{p} = Average return on the portfolio r_{f} = Risk-free rate β_{p} = Beta of portfolio$

The Treynor ratio is similar to the Sharpe ratio but uses a different benchmark, the stock market as a whole. It does this by using the portfolio beta instead of the standard deviation in the Sharpe ratio.
特雷诺比率与夏普比率相似，但使用不同的基准，即整个股票市场。它通过使用投资组合贝塔而不是夏普比率中的标准差来实现这一点。

That is, the numerator is the same as the Sharpe ratio but here we are considering only the $β$ (systematic risk) of the investment instead.
也就是说，分子与夏普比率相同，但这里我们只考虑 $β$ （系统性风险）的投资。

Since the market has a beta of 1, superior performance is indicated where the Treynor ratio exceeds the market risk premium (MRP).
由于市场的贝塔值为 1，当特雷诺比率超过市场风险溢价 (MRP) 时，表明表现优异。

Jensen's Alpha 詹森的阿尔法

$J e n s e n^{'} s a l p h a = E R_{p} - [r_{f} + β (E R_{m} - r_{f})]$

$E R_{p} = Average return of portfolio r_{f} = Risk-free rate β = Beta of the portfolio E R_{m} = Expected return of the market$

Jensen's alpha is the average return on the portfolio over and above that predicted by an asset pricing model. In the formula presented, this would be the CAPM.
Jensen 的阿尔法是投资组合的平均回报超过资产定价模型预测的回报。在给出的公式中，这就是 CAPM。

In practice, this could be other asset pricing models, such as the Arbitrage-Pricing Theory (APT) model, which will be discussed at the end of this module. The basic idea is that a model (like CAPM) is used as a benchmark and everything over or under this prediction is alpha. The alpha of an investment was already discussed in detail on Canvas page 5.3: Alpha.
实际上，这可以是其他资产定价模型，例如套利定价理论（APT）模型，该模型将在本模块末尾讨论。基本思想是使用模型（如 CAPM）作为基准，高于或低于该预测的所有内容都是 alpha。投资的 alpha 已在Canvas 第 5.3 页：Alpha中详细讨论。

Superior performance is indicated with positive alpha whilst under-performance negative alpha. Jensen’s alpha relies upon the security market line and the CAPM being the correct model.
优秀的表现用正阿尔法表示，而表现不佳则用负阿尔法表示。 Jensen 的阿尔法依赖于证券市场线和 CAPM 是正确的模型。

The performance measures are summarised below:
绩效衡量指标总结如下：

	Definition	Application
Sharpe ratio	$\frac{E x c e s s r e t u r n}{S t a n d a r d d e v i a t i o n}$	Choosing among portfolios competing for the overall risky portfolio
Treynor ratio	$\frac{E x c e s s r e t u r n}{B e t a}$	Ranking many portfolios that will be combined together to form the overall risky portfolio
Jensen's alpha	$A c t u a l a v e r a g e r e t u r n - r e t u r n p r e d i c t e d b y C A P M$	Measuring performance compared to the market portfolio

Which measure is appropriate? Depends on your investment assumptions!

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5.4: Performance Evaluation5.4：绩效评估

Portfolio Evaluation 投资组合评估

Sharpe Ratio 夏普比率

Treynor Ratio 特雷诺比率

Jensen's Alpha 詹森的阿尔法

5.4: Performance Evaluation
5.4：绩效评估