複合年增長率 (CAGR) 是多少？

複合年增長率 (CAGR) 複合年增長率是一個指標，表示投資在特定時期內從初始值增長到最終值所需的恆定回報率，考慮到複合年回報率的影響。它提供了投資隨時間的成長率的簡化衡量標準，便於不同投資之間的比較，無論年回報率如何波動。

What is Compound Annual Growth Rate (CAGR)?

The Compound Annual Growth Rate represents o income necessary for an investment to evolve from its initial value to its final value, considering that the gains were reapplied at the end of each year of the investment period.

«返回字典索引什麼是複合年增長率 (CAGR)？複合年增長率代表一項投資從其初始值發展到其最終值所需的收入，考慮到收益在每年年底重新應用投資期限。

How to Calculate Compound Annual Growth Rate (CAGR)

To determine the CAGR of an investment, follow the steps below:

如何計算複合年增長率 (CAGR) 要確定投資的複合年增長率，請按照以下步驟操作：

• Calculate the ratio between the final value of the investment and its initial value.
• Raise this ratio to the power of one over the number of years in the period.
• Deduce one from the result found.
• Transform the final result into a percentage by multiplying it by 100.

Meaning of CAGR

Although CAGR does not reflect the true rate of return, it does offer a representative measure. It works as an index that indicates the speed at which an investment would have expanded if it had maintained a constant rate of annual growth, with income reinvested at the end of each year.

計算投資的最終價值與其初始價值之間的比率。將該比率乘以該期間年數的 1 次方。從找到的結果中減去 1。將最終結果乘以百分比，將其轉換為百分比100.複合年增長率的含義雖然複合年增長率並不反映真實的回報率，但它確實提供了一個具有代表性的衡量標準。它作為一個指數，表明如果投資保持恆定的年增長率，並且收入在每年年底進行再投資，那麼投資將擴大的速度。

Due to actual annual performance variations, achieving this uniform growth is atypical. However, CAGR serves to level returns, making it easier to understand in contrast to other methods.

由於實際的年度業績差異，實現這種均勻成長是不典型的。然而，複合年增長率有助於平衡回報，與其他方法相比更容易理解。

Practical example of using CAGR

Consider an initial investment of $10.000 in a portfolio that yielded the following returns:

使用 CAGR 的實際範例考慮投資組合的初始投資為 10.000 美元，該投資組合產生以下回報：

• From January 1, 2018 to January 1, 2019, the portfolio value increased to $13.000, representing 30% growth in the first year.
• As of January 1, 2020, the portfolio was worth $14.000, which is a 7,69% increase from January 2019 to January 2020.
• As of January 1, 2021, the final value was $19.000, marking 35,71% growth from January 2020 to January 2021.

The portfolio's annual returns varied significantly, depending on the percentages indicated. However, by applying CAGR, a unified view of investment performance is obtained, disregarding annual fluctuations. In this case, the CAGR for the three-year period was 23,86%, providing a simplified comparison basis for the investor to evaluate different resource allocation options or project future values.

2018年1月1日至2019年1月1日，投資組合價值增加至13.000美元，第一年成長30%。截至2020年1月1日，投資組合價值14.000美元，較1月份成長7.69% 2019 年至2020 年1 月。截至2021 年1 月1 日，最終價值為19.000 美元，從2020 年1 月到2021 年1 月增長了35.71%。投資組合的年回報率差異很大，取決於所示的百分比。然而，透過應用複合年增長率，可以獲得投資績效的統一視圖，忽略年度波動。在這種情況下，三年期間的複合年增長率為23.86%，為投資者評估不同的資源配置方案或預測未來價值提供了簡化的比較基礎。

For example, when comparing two distinct investments, where one may appreciate while the other depreciates in a given year, as would be the case when comparing high-yield bonds with stocks or real estate investments with emerging markets, the CAGR helps to level out annual returns, simplifying the comparison between alternatives.

例如，在比較兩種不同的投資時，一種投資可能在某一年升值，另一種則貶值，就像將高收益債券與股票或房地產投資與新興市場進行比較時的情況一樣，複合年成長率有助於平衡年增長率回報，簡化了替代方案之間的比較。

Another example would be an investor who purchased 55 shares of Amazon.com in December 2017 for $1.180 each, for a total investment of $64.900. Three years later, in December 2020, the shares appreciated to $3.200 each, bringing the value of the investment to $176.000.

另一個例子是，一位投資者在 2017 年 12 月以每股 1.180 美元的價格購買了 55 股 Amazon.com 股票，總投資為 64.900 美元。三年後，即 2020 年 12 月，每股股票升值至 3,200 美元，投資價值達到 176,000 美元。

What is CAGR?

To understand CAGR, let’s consider a practical example with the following data:

什麼是複合年增長率？為了理解複合年增長率，讓我們考慮一個包含以下數據的實際範例：

• Final investment value: $176.000
• Initial investment value: $64.900
• Duration of investment: 3 years

Applying this information to the CAGR formula, we have:

最終投資價值：176.000 美元初始投資價值：64.900 美元投資期限：3 年將此資訊應用於 CAGR 公式，我們有：

Additional CAGR Applications

CAGR is useful in determining the average annual growth of an investment. This proves especially valuable in volatile market scenarios, where the return on an investment can vary significantly from year to year.

[($176.000 / $64.900) ^ (1/3)] – 1 = 39,5%。其他複合年增長率應用複合年增長率可用於確定投資的平均年增長率。事實證明，這在波動的市場情況下尤其有價值，因為投資回報率每年都可能存在很大差異。

For example, it may happen that an investment appreciates by 8% in one year, depreciates by 2% in the following year, and appreciates again by 5% in the following year. CAGR helps normalize returns, providing a more stable and understandable view of the growth trajectory, even when these rates are volatile and inconsistent.

例如，一項投資可能在一年內升值8%，第二年貶值2%，第二年又升值5%。複合年增長率有助於回報正常化，即使這些比率波動且不一致，也能提供更穩定、更易於理解的成長軌跡。

Investment Comparison

CAGR provides a means of comparing the performance of different types of investments. Imagine an investor who, in 2015, invested $10.000 in a savings account with fixed annual interest of 1%, and an equal amount in a stock mutual fund, whose return tends to be more irregular over time.

投資比較複合年增長率提供了一種比較不同類型投資績效的方法。想像一下，一位投資者在 2015 年將 10,000 美元投資於儲蓄帳戶，固定年利率為 1%，並將同等金額投資於股票共同基金，隨著時間的推移，其回報往往更加不規則。

At the end of five years, suppose the savings account has reached a balance of $10.510,10 and the stock fund, despite uneven growth, has reached $15.348,52. At first glance, the equity fund appears to be the most profitable investment, with a significantly higher return compared to the savings account.

五年結束時，假設儲蓄帳戶餘額達到 10.510,10 美元，股票基金儘管成長不平衡，但已達到 15.348,52 美元。乍一看，股票基金似乎是最賺錢的投資，與儲蓄帳戶相比，回報率明顯更高。

However, CAGR, by offering an average of returns, does not reveal the volatility or risk associated with the equity fund. Despite this limitation, CAGR can be incorporated into the MAR index, which adjusts results to risk, thus providing a more balanced analysis.

然而，複合年增長率透過提供平均回報率，並沒有揭示與股票基金相關的波動性或風險。儘管有此限制，CAGR 仍可以納入 MAR 指數，根據風險調整結果，從而提供更平衡的分析。

Performance Monitoring

CAGR is also applicable in monitoring the performance of different business aspects, whether of a single company or comparing several. For example, if the CAGR for big-box market share is 1,82% over five years, while the CAGR for customer satisfaction is -0,58% over the same period, this comparison reveals valuable insights into the strengths and weaknesses of the company.

績效監控CAGR也適用於監控不同業務方面的績效，無論是單一公司或比較多個公司。例如，如果五年內大型主機市場份額的複合年增長率為 1.82%，而同期客戶滿意度的複合年增長率為 -0.58%，則這種比較揭示了有關大型機的優勢和劣勢的寶貴見解。公司。

Assessment of Strengths and Weaknesses

Analyzing the CAGRs of competing companies' business operations offers an effective means of identifying relative advantages and disadvantages. A clear example of this can be seen when comparing Big-Sale's customer satisfaction CAGR with that of SuperFast Cable, which recorded a customer satisfaction CAGR of -6,31% in the same time frame. Such a comparison allows a detailed assessment of the strengths and weaknesses between competitors.

優勢和劣勢評估分析競爭公司業務營運的複合年增長率提供了識別相對優勢和劣勢的有效方法。將Big-Sale 的客戶滿意度複合年增長率與SuperFast Cable 的客戶滿意度複合年增長率進行比較即可看到一個明顯的例子，後者在同一時間範圍內的客戶滿意度複合年增長率為-6.31%。這種比較可以詳細評估競爭對手之間的優勢和劣勢。

Use of CAGR by Investors

Mastering the CAGR formula opens the door to several other applications used by investors to analyze past returns and project future profits. Algebraic manipulation of the CAGR formula can make it easier to calculate the present or future value of money, or determine a minimum required rate of return.

投資者使用複合年增長率掌握複合年增長率公式為投資者打開了通往其他幾種應用程式的大門，投資者可以使用這些應用程式來分析過去的回報並預測未來的利潤。 CAGR 公式的代數運算可以更輕鬆地計算貨幣的當前或未來價值，或確定最低所需回報率。

Consider, for example, an investor who needs to accumulate US$50.000 for a child's college education in 18 years, currently having US$15.000 to invest. To calculate the average rate of return necessary to achieve this objective, one can resort to calculating the CAGR, adapting the formula to find the required rate.

例如，一位投資者需要在 18 年內為孩子的大學教育累積 50,000 美元，目前有 15,000 美元可供投資。為了計算實現這一目標所需的平均回報率，可以透過計算複合年增長率，調整公式來找到所需的回報率。

This adaptation is nothing more than a rearranged version of the present value and future value equations. If an investor aims to achieve an amount of $50.000, assuming an expected annual return of 8% on the investment, he can use this version of the CAGR formula to calculate the initial amount needed to reach his goal.

這種調整隻不過是現值和未來值方程式的重新排列版本。如果投資者的目標是達到 50,000 美元，假設投資的預期年回報率為 8%，他可以使用此版本的 CAGR 公式來計算達到目標所需的初始金額。

Adjustments to the CAGR Formula

Investments are rarely made precisely at the beginning and sold precisely at the end of a calendar year. Suppose an investor wants to calculate the CAGR of an investment of US$10.000 made on June 1, 2013 and settled for US$16.897,14 on September 9, 2018.

CAGR FormulaInvestments 的調整很少在年初進行精確調整，並在日曆年年底精確出售。假設投資者想要計算 2013 年 6 月 1 日投資 10,000 美元並於 2018 年 9 月 9 日結算為 16,897,14 美元的複合年增長率。

To calculate the CAGR, it is initially necessary to determine the exact fraction of the investment period. The investment lasted 213 days in 2013, followed by full years until 2017, and a further 251 days in 2018, totaling an investment period of approximately 5,271 years, calculated as follows:

要計算複合年增長率，首先需要確定投資期間的準確比例。 2013年投資持續213天，隨後持續至2017年全年，2018年又持續251天，總計投資期間約5,271年，計算如下：

• 2013: 213 days
• 2014 to 2017: 365 days each year
• 2018: 251 days

The total number of days on which the investment was maintained totals 1.924. Dividing this total by 365, we obtain 5,271 years, the total duration of the investment.

2013年：213天2014年至2017年：每年365天2018年：251天投資維持天數總計1.924天。將這個總數除以 365，我們得到 5,271 年，即投資的總期限。

This exact period, expressed in years, is then applied to the denominator of the exponent in the CAGR formula to obtain the adjusted calculation.

然後，將此以年為單位的精確週期應用於 CAGR 公式中指數的分母，以獲得調整後的計算結果。

Moderation in Growth Rate with CAGR

The main restriction of CAGR lies in the fact that it calculates a moderate rate of growth over a specific period, disregarding volatility and assuming constant growth during that time. This contrasts with the reality of investments, whose returns tend to vary significantly, except for fixed income investments such as held-to-maturity bonds or bank deposits.

複合年增長率的適度增長率複合年增長率的主要限制在於，它計算特定時期內的適度增長率，忽略波動性並假設該時期內持續增長。這與投資的實際情況形成鮮明對比，投資的回報往往差異很大，但持有至到期債券或銀行存款等固定收益投資除外。

Additionally, the CAGR does not take into account any contributions or withdrawals of resources by the investor in their portfolio during the period analyzed. For example, if an investor were to make additional contributions to the portfolio over five years, the resulting CAGR could be overestimated. The formula calculates the rate of return based on the initial and final values ​​during the five-year period, unduly considering contributions as part of annual growth, which distorts reality.

此外，複合年增長率並未考慮投資者在分析期間對其投資組合的任何資源貢獻或提取。例如，如果投資者在五年內向投資組合做出額外貢獻，則由此產生的複合年增長率可能會被高估。該公式根據五年期間的初值和終值計算回報率，不當地將貢獻視為年度增長的一部分，扭曲了現實。

Other Limitations of CAGR

In addition to its growth smoothing characteristic, CAGR has other limitations. One of them is the impossibility of guaranteeing the continuity of past performance in the future. The reliability of the CAGR decreases the shorter the period considered for the analysis, reducing the chances that the projected CAGR corresponds to that actually realized, based on historical data.

CAGR 的其他限制 除了其成長平滑特性之外，CAGR 還有其他限制。其中之一是無法保證過去業績在未來的連續性。分析所考慮的時間越短，複合年增長率的可靠性就越低，從而減少了根據歷史數據預測的複合年增長率與實際實現的複合年增長率相對應的機會。

Another limitation concerns its representativeness. Considering an investment fund that was worth $100.000 in 2016, fell to $71.000 in 2017, declined further to $44.000 in 2018, rose to $81.000 in 2019, and reached $126.000 in 2020. If, in 2021, it were disclosed that the fund's CAGR over the past three years was a significant 42,01%, the information would be technically correct. However, it would be omitting crucial aspects of the fund's trajectory, such as the modest 4,73% rate over five years.

另一個限制在於其代表性。考慮到一隻投資基金在2016 年價值100,000 美元，2017 年跌至71,000 美元，2018 年進一步跌至44,000 美元，2019 年升至81,000 美元，2020 年達到126,000 美元。如果在2021 年披露該基金的複合年增長率超過過去三年顯著為 42.01%，資訊​​在技術上是正確的。然而，這將忽略該基金發展軌蹟的關鍵方面，例如五年內 4.73% 的溫和利率。

CAGR vs. IRR

While CAGR offers a measure of an investment's return over a given period, Internal Rate of Return (IRR) provides a more comprehensive view of investment performance and is more versatile than CAGR.

CAGR 與 IRR 雖然 CAGR 提供了在給定時期內投資回報的衡量標準，但內部報酬率 (IRR) 提供了更全面的投資績效視圖，並且比 CAGR 更通用。

The main distinction between the two lies in the simplicity of CAGR, which allows for manual calculation. On the other hand, for more complex investments or projects, with multiple inflow and outflow transactions, the IRR is more appropriate. Determining the IRR generally requires the use of a financial calculator, spreadsheets such as Excel, or specialized portfolio management systems.

兩者之間的主要區別在於 CAGR 的簡單性，允許手動計算。另一方面，對於更複雜的投資或項目，有多個流入和流出交易，IRR 更合適。確定 IRR 通常需要使用金融計算器、Excel 等電子表格或專門的投資組合管理系統。

Individuals interested in deepening their knowledge of CAGR and other financial topics can benefit from exploring some of the best investment courses available on the market today.

有興趣加深複合年增長率和其他金融主題知識的個人可以從探索當今市場上的一些最佳投資課程中受益。

Conclusion

The Compound Annual Growth Rate (CAGR) emerges as a fundamental analytical tool for investors and financial analysts, providing a clear, composite view of an investment's growth over time. When considering the effect of compounding, CAGR provides a smoothed and consistent measure that can be compared across different investment options, regardless of annual fluctuations.

結論複合年增長率 (CAGR) 已成為投資者和金融分析師的基本分析工具，為投資隨時間的成長提供清晰、全面的觀點。在考慮複利的影響時，複合年增長率提供了一種平滑且一致的衡量標準，可以在不同的投資選擇之間進行比較，無論年度波動如何。

While CAGR offers numerous advantages, such as the ability to level out the volatility of returns and provide a long-term perspective on growth, it is not without limitations. Its smoothing approach can hide periods of underperformance and does not account for capital movements during the analysis period, such as contributions or withdrawals. Furthermore, CAGR's effectiveness as an indicator of future financial performance is limited, as it cannot predict changes in market conditions or a company's management.

雖然複合年增長率具有許多優勢，例如能夠消除回報波動並提供長期成長視角，但它並非沒有限制。其平滑方法可以隱藏表現不佳的時期，並且不會考慮分析期間的資本流動，例如捐款或提款。此外，複合年增長率作為未來財務績效指標的有效性是有限的，因為它無法預測市場狀況或公司管理層的變化。

The distinction between CAGR and other metrics, such as Internal Rate of Return (IRR) and simple growth rates, highlights the importance of selecting the most appropriate analysis tool for each specific situation. By adjusting CAGR for risk factors, investors can obtain a more accurate assessment of the risk-adjusted performance of their investments.

CAGR 與內部報酬率 (IRR) 和簡單成長率等其他指標之間的差異凸顯了針對每種特定情況選擇最合適的分析工具的重要性。透過調整風險因素的複合年增長率，投資者可以更準確地評估其投資的風險調整後績效。

FAQ

What is an example of a Compound Annual Growth Rate (CAGR)?

CAGR is a key indicator used by investors to analyze the evolution of the value of an investment over time. The inclusion of the word “compound” reflects CAGR's ability to incorporate the effect of capitalization, that is, the reinvestment of earnings over a period. Imagine, for example, that a company saw an increase in revenue from $3 million to $30 million in ten years. In this case, the CAGR would be around 25,89%.

常見問題複合年增長率 (CAGR) 的範例是什麼？複合年增長率是投資者用來分析投資價值隨時間變化的關鍵指標。 「複合」一詞的加入反映了複合年增長率納入資本化效應的能力，即一段時間內收益的再投資。例如，假設一家公司的收入在十年內從 300 萬美元增加到 3,000 萬美元。在這種情況下，複合年增長率約為 25.89%。

What constitutes a good CAGR?

A “good” CAGR varies depending on the specific context, including factors such as opportunity cost and risk associated with the investment. For example, an annual growth rate of 25% could be considered modest in an industry where the average is 30%, but impressive in a market where the average is 10% to 15%. Generally speaking, a higher CAGR is preferable.

什麼是良好的複合年增長率？「良好」的複合年增長率取決於具體情況，包括機會成本和與投資相關的風險等因素。例如，在平均成長率為 30% 的行業中，25% 的年增長率可能被認為是適度的，但在平均成長率為 10% 至 15% 的市場中卻令人印象深刻。一般來說，複合年增長率越高越好。

Difference between CAGR and Growth Rate

The main distinction between CAGR and a simple growth rate lies in the fact that CAGR assumes a constant, annually compounded growth rate, whereas a standard growth rate assumes no such constancy. CAGR is often preferred by investors for its ability to even out fluctuations in growth rates from year to year, providing a more stable view of a company's growth over time, even if there are years of lower performance.

CAGR 和成長率之間的差異 CAGR 和簡單成長率之間的主要差異在於，CAGR 假定恆定的年複合成長率，而標準成長率則不假定這種恆定性。複合年增長率通常受到投資者的青睞，因為它能夠平衡每年增長率的波動，從而為公司隨時間的增長提供更穩定的視角，即使多年來業績較低。

Can CAGR be negative?

Yes, it is possible for the CAGR to be negative, which indicates a reduction in the value of the investment over the analyzed period, rather than an increase.

複合年增長率可以為負嗎？是的，複合年增長率有可能為負，這表明在分析期間投資價值減少而不是增加。

What does risk adjusted CAGR mean?

Investors looking to evaluate the performance of different investment options considering both returns and associated risks can turn to risk-adjusted CAGR. This calculation is performed by multiplying the CAGR by (1 – the standard deviation of the return on investment). If the standard deviation is zero, the risk-adjusted CAGR will be the same as the unadjusted CAGR. However, a higher standard deviation results in a lower risk-adjusted CAGR, reflecting the increased risk.

風險調整後複合年增長率是什麼意思？希望在考慮回報和相關風險的情況下評估不同投資選擇的表現的投資者可以轉向風險調整後複合年增長率。此計算方法是將複合年增長率乘以（1 – 投資報酬率的標準差）。如果標準差為零，則風險調整後的複合年增長率將與未調整的複合年增長率相同。然而，較高的標準差會導致較低的風險調整複合年增長率，反映了風險的增加。

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