Sharpe Ratio | Comprehensive Guide with Excel Examples

ByDheeraj Vaidya

source: Zacks.com

Sharpe ratiois a critical component for marking the overall returns on a portfolio. It is the average return earned in excess of the risk-free return compared to the total amount of risk borne. It is a way to examine the performance of an investment by adjusting for its risk component. The Sharpe ratio characterizes how well the return of an asset compensates the investor for the risk taken. When comparing two assets versus a common benchmark, the one with a higher Sharpe ratio provides is indicated as a favorable investment opportunity at the same level of risk.

夏普比率在衡量投資組合的整體回報方面是個很重要的指標，是投資者在承擔了風險的情況下，所賺取的超過無風險收益率的平均收益率。通過對所承擔的風險要素進行調整，該比率可用來衡量投資的回報表現。夏普比率描繪的是一項資產的回報補償投資者所承擔風險的情況。如果把所投資的資產與公認的業績標準進行對比，在風險水平一致的情況下，夏普比率更高的投資被認為是更好的投資選項。

If you look at the table above, you will see that PRWCX has the higher Sharpe Ratio of 1.48 and is the best fund in its group.

看看上面的圖表就知道，PRWCX的夏普比率更高，為1.48，是該組中回報最好的基金。

Sharpe Ratio, like any other mathematical model relies on the accuracy of the data which needs to be correct. While examining the investment performance of assets with smoothening of returns, the Sharpe ratio would be derived from the performance of the underlying assets rather than the fund returns. This ratio along with Treynor Ratios and Jeson』s Alphas are often used to rank the performance of various portfolios or Fund managers.

夏普比率像許多其他的數學模型一樣高度依賴所提供的數據的準確性。如果把資產的回報進行平滑處理，那麼夏普比率應來自於標的資產的回報表現而不是基金的回報。該比率與特雷諾比率以及詹森阿爾法一樣經常被用來給基金經理所管理的各類投資組合的回報進行排名。（譯者註：Treynor 比率是由美國經濟學家「傑克特雷諾」（Jack Treynor) 發明的測算投資回報的指標。用於在系統風險基礎之上對投資的收益風險進行調整。該指標反映基金承擔單位系統風險所獲得的超額收益。指數值越大，承擔單位系統風險所獲得的超額收益越高。

詹森阿爾法（又稱詹森指數）被用來確定來自某一證券或投資組合超過理論預期收益的超額收益。詹森阿爾法最初於1968年被邁克爾·詹森用於評估共同基金經理表現。資產的理論預期收益率與衡量該資產風險的β值之間存在正相關關係，而風險較高的資產相較風險較低的資產有更高的理論預期收益率。如果一個資產的實際收益率高於風險調整後的理論預期收益率，這樣的資產就被稱為有「正的α」或者「超額收益」。投資者時刻在尋找著有較高阿爾法值的投資品種。）

In this article, we discuss Sharpe Ratio in detail –

Sharpe Ratio Formula

Sharpe Ratio Example

Calculating Sharpe Ratio in Excel

Advantages of Using Sharpe Ratio

Criticisms of Sharpe Ratio

Ex-Ante and Ex-Post Sharpe Ratio

Conclusion – Sharpe Ratio

在本文中，我們將從以下幾個方面探討夏普比率：

夏普比率計算公式

夏普比率的案例

用Excel表計算夏普比率

夏普比率的好處

反對夏普比率的聲音

事前夏普比率和事後夏普比率

對夏普比率的總結

Sharpe Ratio Formula

夏普比率計算公式

In 1966, William Sharpe developed this ratio which was originally called it the 「reward-to-variability」 ratio before it began being called the Sharpe ratio by subsequent academics and financial operators. It was defined in multiple ways till ultimately it was charted as below:

1966年，威廉·夏普首先提出來夏普比率，剛開始的時候被稱為「回報一波動性比率」，後來被學者和金融界人士稱為夏普比率，該比率曾經有多個不同的表述方式，最終的版本如下：

Sharpe Ratio Formula = (Expected Return – Risk-Free rate of return) / Standard Deviation (Volatility)

夏普比率=（投資組合的預期回報率-無風險回報率）/投資組合的標準差（即波動率）

Some of the concepts which we require to understand are:

我們需要先了解幾個概念：

Returns– The returns could be of various frequencies such as daily, weekly, monthly or annually as long as the distribution is spread normally since these returns can be annualized to arrive at precise results. Abnormal situations like higher peaks, skewness on the distribution can be a problem area for the ratio as standard deviation does not possess the same effectiveness when these issues exist.

回報：只要回報呈現正態分布，回報產生的頻率可以是多樣的，比如每天，每周，每月或每年，因為這些回報率可以被年化統計以得出精確的結果。非正常的情況，比如回報突然大漲及回報分布上的偏離，對於夏普比率來講很麻煩，因為只要這些不正常現象存在，標準差的效力就會大打折扣。

Risk-Free Rate of Return –This is used to assess if one is being correctly compensated for the additional risk borne because of the risky asset. Traditionally, the rate of return with no financial loss is the Government securities with the shortest duration (e.g. US Treasury Bill). While such a variant of security has the least amount of volatility, it can be argued that such securities should match with other securities of equivalent duration.

無風險回報率：無風險回報率被用來衡量投資了高風險資產的投資者是否因所承擔的額外風險而獲得了恰當的補償。傳統上，零信用風險的回報率來自於短久期的政府債券，如短期美國國債。由於此類證券的波動率最低，可以說此類證券可以用來與久期相等的其他證券進行比較。

Standard Deviation –It is a quantity which expresses how many units from a given set of variables differ from the Mean average of the group. Once this excess return over the risk-free return is computed it has to be divided by the Standard deviation of the risky asset being measured. Greater the number, attractive will the investment appear from a risk/return perspective. However, unless the standard deviation is substantially large, the leverage component may not impact the ratio. Both the numerator (return) and denominator (standard deviation) could be doubled with no problems.

標準差：標準差是一種量化的指標，用來表示一系列的變數偏離一組數據的均值的程度。在這種高於無風險利率的超額回報被統計之時，需要用超額回報除以高風險資產的標準差。結果值越大，從風險收益的角度來看這筆投資的吸引力似乎就越高。但是，除非標準差大得很明顯，投資組合中的槓桿因素對夏普比率不會有什麼影響。不管是作為分子的回報率還是作為分母的標準差都可以毫不費力地在數值上翻一番。

Sharpe Ratio Example

夏普比率的案例

Client 『A』 currently is holding a $450,000 invested in a portfolio with an expected return of 12% and a volatility of 10%. The efficient portfolio has an expected return of 17% and a volatility of 12%. The risk free rate of interest is 5%. What is the Sharpe Ratio?

客戶A投資了450,000美元，預期回報率為12%，波動率為10%。而實際回報率為17%，波動率為12%，無風險利率為5%，求夏普比率？

Sharpe Ratio Formula = (Expected Return – Risk-Free rate of return) / Standard Deviation (Volatility)

夏普比率=（投資組合的預期回報率-無風險回報率）/投資組合的標準差（即波動率）

Sharpe Ratio = (0.12-0.05)/0.10 = 70% or 0.7x

夏普比率=(0.12-0.05)/0.10 = 70% or 0.7

Calculating Sharpe Ratio in Excel

用Excel表計算夏普比率

Now that we know how the formula works, let us calculate Sharpe Ratio in excel.

前面已經知道公式了，下面開始用Excel表計算夏普比率。

Step 1 – Get the returns in the tabular format

步驟1——用列表列出回報率

The first step involves arranging for the returns of the portfolio or the mutual fund that you want to analyze. The time period can be monthly, quarterly or annual. Below table provides annual returns of a mutual fund.

第一步是列出所研究的投資組合或共同基金的投資回報率，統計頻率可以是月度，季度或年度，下表顯示了某共同基金的年回報率。

Step 2 – Get Risk Free Return Details in the table

步驟2——在表中列出無風險回報率

In this table below, I have made an assumption that the risk free return is 3.0% throughout the span of 15 years. However, risk free rate may change each year and you need to put that number here.

在下表中，我們假設無風險利率為3%，總期限為15年。但是實際上，每年的無風險利率都應不同才對，並需將不同的無風險回報率填在各自對應的年份欄。

Step 3 – Find Excess Return

步驟3——算出超額回報率

The third step in calculating Sharpe ratio in excel is to find the excess returns of the portfolio. In our case, excess return is the Yearly Returns – Risk-Free Return.

第三步是用 excel 計算各年度的超額回報率，超額回報率等於年化回報率減去無風險利率。

Step 4 – Find the average of Annual Returns.

步驟4——算出年化回報率的均值

The fourth step in calculating Sharpe ratio in excel is to find the average of the yearly returns. You can use excel formula AVERAGE to find the average of the portfolio. In our example, we get the average return of 12.09%.

第四步為是用excel 計算各年度年化回報率的均值，用excel表的求均值功能計算出整個投資組合的年化回報率，在本例中，年回報率的均值為12.09%。

Step 5 – Find Standard Deviation of the Excess Returns

步驟5——算出額外回報率的標準差

In order to find the standard deviation of excess returns, you can use the excel formula STDEV as given below.

用excel表的STDEV功能計算出整個投資組合超額回報率的標準差。

Step 6 – Calculate Sharpe Ratio

步驟6——算出夏普比率

The final step to calculating sharpe ratio in excel is to divide the Average Returns by Standard deviation. We get the ratio = 12.09% / 8.8% = 1.37x

最後一步是用年回報率的均值除以整個投資組合超額回報率的標準差，結果為12.09% / 8.8% = 1.37

Advantages of Using Sharpe Ratio

夏普比率的優勢

#1 – Sharpe Ratio helps in comparing and contrasting new asset addition

一、可以幫助投資者對新的投資選項進行比較鑒別

It is used to compare the variance of a portfolio』s overall risk-return features whenever a new asset or a class of asset is added to it.

當新的投資或某一類新的資產類型被加入到投資組合中時，夏普比率被用來對投資組合的整體風險回報特徵的變動狀況進行比較。

For instance, a portfolio manager is considering the addition of a commodities fund allocation to his existing 80/20 investment portfolio of stocks having a Sharpe ratio of 0.81.

舉個例子，某投資組合管理人考慮要在其現有的股票投資組合中加入大宗商品投資，組合現有的夏普比率為0.81

If the new portfolio』s allocation is 40/40/20 stocks, bonds and a debt fund allocation, the Sharpe ratio increases to 0.92.

如果新的投資組合的資產配置比率分別為股票四成，債券四成，大宗商品基金為兩成，新組合的夏普比率為0.92

This is an indication that although the commodities fund investment is volatile as a stand alone exposure, in this case, it actually leads to an improvement of the risk-return characteristic of the combined portfolio, and thus adds a benefit of diversification into another asset class to the existing portfolio. There has to be an involvement of careful analysis that the fund allocation may have to be altered at a later stage if it is having a negative effect on the health of the portfolio. If the addition of the new investment is leading to a reduction in the ratio, it should not be included in the portfolio.

結果顯示，儘管大宗商品基金本身的波動率很高，但在本例中，投資組合中加入了大宗商品基金卻使整個投資組合的風險回報情況得到改善，因此新的資產類別給現有的投資組合帶來了好處。但這需要在事前認真地分析研究，因為如果新加入的資產類型給投資組合帶來負面後果，只能在事後對其進行調整。如果新投資導致夏普比率降低，那麼該項投資就不應進入投資組合。

#2 – Sharpe Ratio helps in Risk Return Comparision

二、夏普比率可以幫助投資者進行風險收益比衡量

This ratio can also provide guidance whether the excessive returns of a portfolio are due to careful investment decision making or a result of undue risks taken. Although an individual fund or portfolio can enjoy greater returns than its peers, it is only a reasonable investment if those higher returns do not come with undue risks. The greater the Sharpe ratio of a portfolio, the better its performance has been factoring the risk component. A negative Sharpe ratio indicates that the lesser riskier asset would perform better than the security being analyzed.

夏普比率也有助於投資者了解一個投資組合的額外收益是來自於認真思考後做出的投資決策還是來自於承擔了過高的風險。儘管某個基金或投資組合有可能會比同類型的基金或組合收益更高，但只有當這些高收益並不是來自於過高的風險的時候這個投資才是恰如其分的。一個投資組合的夏普比率越高，其計入了風險因素的回報表現越好。夏普比率如果為負說明風險較低的資產會產生比所分析的資產更好的回報。

Let us take an example for the Risk Return Comparision.

關於風險收益比衡量方面的一個案例

Assume portfolio A had or is expected to have a 12% rate of return with a standard deviation of 0.15. Assuming a benchmark return of about 1.5%, the rate of return (R) would be 0.12, Rf will be 0.015 and 『s』 will be 0.15. The ratio will be read as (0.12 – 0.015)/0.15 which computes to 0.70. However, this number will make sense when it is compared to another portfolio say Portfolio 『B』

假設預期投資組合A的收益率為12%，標準差0.15，再假設投資參照基準的回報率約為1.5%，那麼投資回報率R為0.12， Rf 為0.015，s等於0.15，則夏普比率=(0.12 – 0.015)/0.15=0.70。但是，這個數據只有和另一個投資組合B進行比較才有意義。

If portfolio 『B』 shows more variability than Portfolio 『A』, but has the same return, it will have a greater standard deviation with the same rate of return from the portfolio. Assuming the standard deviation for Portfolio B is 0.20, the equation would be read as (0.12 – 0.015) / 0.15. The Sharpe ratio for this portfolio will be 0.53 which is lower in comparison to Portfolio 『A』. This may not be an astonishing result, taking into consideration the fact that both the investments were offering the same return, but 『B』 had a greater quantum of risk. Obviously, the one which has less risk offering the same return will be a preferred option.

投資組合B的波動率A高很多，但回報率相同，這樣的話就造成同樣的收益率，但標準差更高一些。假設投資組合B的標準差等於0.20，則夏普比率=(0.12 – 0.015)/0.2=0.53，低於投資組合A。考慮到這兩個投資組合的回報率相同，而組合B的風險值更高，因此這個結果並不令人感到吃驚。很顯然，收益率相同但風險更低的投資是個更好的選擇。

Criticisms of Sharpe Ratio

反對夏普比率的聲音

The Sharpe ratio utilises the Standard deviation of returns in the denominator as an alternative to the overall portfolio risks, with an assumption that returns are evenly distributed. Past testing has shown that returns from certain financial assets may deviate from a normal distribution, resulting in relevant interpretations of the Sharpe ratio to be misguiding.

夏普比率的定價公式中用收益率的標準差而不是整個投資組合的風險值作為分母，是在假設組合的回報會呈現均勻分布。而回測結果顯示某類金融資產的回報情況並非正態分布，這就會造成夏普比率對投資組合的績效所做的解釋容易出現誤導。

This ratio can be improved by various fund managers attempting to boost their apparent risk-adjusted return which can be executed as below:

試圖大幅提升風險調整後收益的基金經理可以從以下幾個方面對夏普比率進行改進：

Increasing the Time Duration to be measured: This will result in a lesser probability of volatility. For instance, the annualized standard deviation of daily returns is generally higher than of weekly returns, which in turn is higher than that of the monthly returns. Greater the time duration, clearer picture one has to exclude any one-off factors which can impact the overall performance.延長衡量投資績效的時間跨度。這樣做的結果是減少投資組合回報率發生大幅波動的概率。舉個例子，每日回報率的年化標準差通常會高於每周回報率的年化標準差，而每周回報率的年化標準差又高於每月回報率的年化標準差。時間跨度越長，對整體回報表現會產生影響的偶發性因素就會更清晰地暴露出來。

Compounding of the monthly returnsbut computing the standard deviation excluding this recently calculated compounded monthly return.

對月度投資回報率進行複利計算，但是計算標準差時需要先剔除掉最近統計的月度回報率。

Writing out-of-the money sell and buy decision of a portfolio:Such strategy can potentially increase the returns by collecting the options premium without paying off for a number of years. Strategies which involve challenging the default risk, liquidity risk or other forms of wide-spreading risks possess the same ability to report an upwardly biased Sharpe ratio.

賣出以投資組合中的某些資產為標的的價外看漲或看跌期權：通過收取期權費的方式，該策略有可能不需等上很多年就會提升投資組合的回報率。對違約風險，流動性風險或以其他形式廣泛存在的風險進行挑戰的投資策略同樣會產生較高的夏普比率。

Smoothening of returns:Using certain derivative structures, irregular marking to market of less liquid assets or utilising certain pricing models which underestimate monthly profits or losses, can reduce the expected volatility.

將回報率進行平滑處理。使用某些衍生品交易結構，對投資組合中流動性缺失的資產並定期的做盯市估值，或使用某些可以對月度損益狀況進行一定程度低估的定價模型均可以降低預期波動率的均值。

Eliminating Extreme returns:Too high or too low returns can increase the reported standard deviation of any portfolio since it is distance from the average. In such a case, fund manager may choose to eliminate the extreme ends (best and the worst) monthly returns each year to reduce the standard deviation and affect the results since such a one-off situation can impact the overall average.

剔除在極端情況下的回報記錄。過高或過低的回報率會造成任何一個投資組合的標準差出現升高，因為其會大幅偏離均值水平。在此情況下，基金經理會選擇剔除某年度中最好或最差的月度極端收回報情況，以降低整個投資組合的標準差並對投資績效進行調整，因為偶發性的極端收益會影響整個投資組合的平均回報率。

Ex-Ante and Ex-Post Sharpe Ratio

事前夏普比率和事後夏普比率

The Sharpe ratio has been revised multiple times but two general forms which have been used are ex-ante (prediction of future return and variance) and ex-post (analysis of past return variance).

夏普比率有過多次修正，但有兩個常見的形式分別為：事前夏普比率，以預測未來的回報率和回報的波動率，事後夏普比率，用於分析過往投資回報率的波動率。

Ex-ante sharpe ratiopredictions are simpleestimate patternsafter observations of past performance of similar investment activities.

事前夏普比率：如果觀察類似的投資活動的過往回報表現，那麼做出預測就會變得簡單

The Ex-post Sharpe Ratiomeasures how high the returns were, versus how varied those returns were over a given time period. More specifically, it is the ratio of the differential returns ( the difference between the returns on investment and a benchmark investment) versus the historical variability (standard deviation) of those returns.

事後夏普比率：衡量的是在一段時間內回報率有多高以及回報的波動幅度有多大。具體的說就是，衡量的是在這些回報的歷史波動率，即標準差不同的情況下，不同投資的回報之間的比率，具體的投資回報與所參考的投資基準回報率之間的差異。

Conclusion

結論

The Sharpe ratio is a standard measure of the performance of the portfolio. Due to its simplicity and ease of interpretation, it is one of the most popular indexes. Unfortunately, most of the users forget the assumptions what results in an inappropriate outcome. You should consider checking the distribution of the returns or validation of the results with equivalent performance measures before arriving at a decision on the market.

夏普比率是用來衡量投資組合回報表現的一個標準量度，因其簡單明了而成為最常用的業績衡量指標。不幸的是，大多數使用者忽視的是如果計算夏普比率的前提假設出了問題會導致不理想的投資結果。投資者在做出投資決策之前應好好審視投資回報率的分布情況或類似的業績衡量指標得出來的結果的有效性。

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