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--- invest:basic:cagr [2020/04/21 01:31]
zhwiki created
+++ invest:basic:cagr [2021/02/27 04:07] (current)
zhwiki [CAGR计算公式]
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 ====== CAGR ======
+复合增长率（Compound annual growth rate, CAGR）是当投资收益在投资期每年末全部再投入时，从期初到期末总的投资回报率。
-    Compound Annual Growth Rate
+===== CAGR计算公式 =====
-    Formula and Calculation of CAGR
-    What CAGR Can Tell You
-    Example of How to Use CAGR
-    Additional CGAR Uses
-    Investor Use of CAGR
-    Modifying the CAGR Formula
-    Smooth Rate of Growth Limitation
-    Other CAGR Limitations
-    CAGR vs. IRR
-    Example of How to Use CAGR
-What Is Compound Annual Growth Rate – CAGR?
+设终值为EB(Ending balance)，初值为BB(Beginning balance)，年数为n(Number of years)，则根据
-Compound annual growth rate (CAGR) is the rate of return that would be required for an investment to grow from its beginning balance to its ending balance, assuming the profits were reinvested at the end of each year of the investment’s lifespan.
+$$
-Formula and Calculation of CAGR
+EB=BB(1+CAGR)^{n}
+$$
-CAGR=(EBBB)1n−1where:EB=Ending balanceBB=Beginning balancen=Number of years\begin{aligned} &CAGR=\left(\frac{EB}{BB}\right)^{\frac{1}{n}}-1\\ &\textbf{where:}\\ &EB = \text{Ending balance}\\ &BB = \text{Beginning balance}\\ &n = \text{Number of years} \end{aligned}CAGR=(BB
+到达CAGR的计算公式：
-EB)n
-−1where:EB=Ending balanceBB=Beginning balancen=Number of years
+$$
+\begin{aligned}
+CAGR &= \left(\frac{EB}{BB}\right)^{\frac{1}{n}}-1\\
+&=(TG+1)^{\frac{1}{n}}-1 \\
+\textbf{其中:}\\
+TG &= \text{总增长率(Total Growth)}\\
+\end{aligned}
+$$
+===== CAGR应用 =====
-To calculate the CAGR of an investment:
+计算投资复合增长率：用期末资本除以期初的资本，然后开1/n次方，结果减1。
-    Divide the value of an investment at the end of the period by its value at the beginning of that period.
-    Raise the result to an exponent of one divided by the number of years.
-    Subtract one from the subsequent result.
-Key Takeaways
-    CAGR is one of the most accurate ways to calculate and determine returns for anything that can rise or fall in value over time.
-    Investors can compare the CAGR of two alternatives in order to evaluate how well one stock performed against other stocks in a peer group or against a market index.
-    CAGR does not reflect investment risk.
-What CAGR Can Tell You
-The compound annual growth rate isn't a true return rate, but rather a representational figure. It is essentially a number that describes the rate at which an investment would have grown if it had grown the same rate every year and the profits were reinvested at the end of each year. In reality, this sort of performance is unlikely. However, CAGR can be used to smooth returns so that they may be more easily understood when compared to alternative investments.
-Example of How to Use CAGR
-Imagine you invested $10,000 in a portfolio with the returns outlined below:
-    From Jan 1, 2014, to Jan 1, 2015, your portfolio grew to $13,000 (or 30% in year one).
-    On Jan 1, 2016, the portfolio was $14,000 (or 7.69% from Jan 2015 to Jan 2016).
-    On Jan 1, 2017, the portfolio ended with $19,000 (or 35.71% from Jan 2016 to Jan 2017).
-We can see that on an annual basis, the year-to-year growth rates of the investment portfolio were quite different as shown in the parenthesis.
-On the other hand, the compound annual growth rate smooths the investment’s performance and ignores the fact that 2014 and 2016 were so different from 2015. The CAGR over that period was 23.86% and can be calculated as follows:
-CAGR=($19,000$10,000)13−1=23.86%CAGR=\left(\frac{\$19,000}{\$10,000}\right )^{\frac{1}{3}}-1=23.86\%CAGR=($10,000
-$19,000)3
-−1=23.86%
-The compound annual growth rate of 23.86% over the three-year investment period can help an investor compare alternatives for their capital or make forecasts of future values. For example, imagine an investor is comparing the performance of two investments that are uncorrelated. In any given year during the period, one investment may be rising while the other falls. This could be the case when comparing high-yield bonds to stocks, or a real estate investment to emerging markets. Using CAGR would smooth the annual return over the period so the two alternatives would be easier to compare.
-Additional CGAR Uses
-The compound annual growth rate can be used to calculate the average growth of a single investment. As we saw in our example above, due to market volatility, the year-to-year growth of an investment will likely appear erratic and uneven. For example, an investment may increase in value by 8% in one year, decrease in value by -2% the following year and increase in value by 5% in the next. CAGR helps smooth returns when growth rates are expected to be volatile and inconsistent.
-Compare Investments
-CAGR can be used to compare investments of different types with one another. For example, suppose in 2013 an investor placed $10,000 into an account for 5 years with a fixed annual interest rate of 1% and another $10,000 into a stock mutual fund. The rate of return in the stock fund will be uneven over the next few years so a comparison between the two investments would be difficult.
-Assume that at the end of the five-year period, the savings account’s balance is $10,510.10 and, although the other investment has grown unevenly, the ending balance in the stock fund was $15,348.52. Using CAGR to compare the two investments can help an investor understand the difference in returns:
-Savings Account CAGR=($10,510.10$10,000)15−1=1.00%\text{Savings Account CAGR} =\, \left ( \frac{\$ 10,510.10}{\$ 10,000} \right )^{\frac{1}{5}}-1 = 1.00\%Savings Account CAGR=($10,000
-$10,510.10)5
-−1=1.00%
-And:
-Stock fund CAGR=($15,348.52$10,000)15−1=8.95%\text{Stock fund CAGR} =\, \left ( \frac{\$ 15,348.52}{\$ 10,000} \right )^{\frac{1}{5}}-1 = 8.95\%Stock fund CAGR=($10,000
-$15,348.52)5
-−1=8.95%
-On the surface, the stock fund may look like a better investment with nearly nine times the return of the savings account. On the other hand, one of the drawbacks to CAGR is that by smoothing the returns, CAGR cannot tell an investor how volatile or risky the stock fund was.
-Track Performance
-CAGR can also be used to track the performance of various business measures of one or multiple companies alongside one another. For example, over a five-year period, Big-Sale Stores’ market share CAGR was 1.82%, but its customer satisfaction CAGR over the same period was -0.58%. In this way, comparing the CAGRs of measures within a company reveals strengths and weaknesses.
-Detect Weaknesses and Strengths
-Comparing CAGRs of business activities across similar companies will help evaluate competitive weaknesses and strengths. For example, Big-Sale’s customer satisfaction CAGR might not seem so low when compared with SuperFast Cable’s customer satisfaction CAGR of -6.31% during the same period.
-Investor Use of CAGR
-Understanding the formula used to calculate CAGR is an introduction to many other ways investors evaluate past returns or estimate future profits. The formula can be manipulated algebraically into a formula to find the present value or future value of money, or to calculate a hurdle rate of return.
-For example, imagine that an investor knows that they need $50,000 for a child’s college education in 18 years and they have $15,000 to invest today. How much does the average rate of return need to be in order to reach that objective? The CAGR calculation can be used to find the answer to this question as follows:
-Required Return=($50,000$15,000)118−1=6.90%\text{Required Return} =\, \left ( \frac{\$ 50,000}{\$ 15,000} \right )^{\frac{1}{18}}-1 = 6.90\%Required Return=($15,000
-$50,000)18
-−1=6.90%
-This version of the CAGR formula is just a rearranged present value and future value equation. For example, if an investor knew that they needed $50,000 and they felt it was reasonable to expect an 8% annual return on their investment, they could use this formula to find out how much they needed to invest to meet their goal.
-Modifying the CAGR Formula
-An investment is rarely made on the first day of the year and then sold on the last day of the year. Imagine an investor who wants to evaluate the CAGR of a $10,000 investment that was entered on June 1st, 2013 and sold for $16,897.14 on September 9th, 2018.
-Before the CAGR calculation can be performed, the investor will need to know the fractional remainder of the holding period. They held the position for 213 days in 2013, a full year in 2014, 2015, 2016, and 2017, and 251 days in 2018. This investment was held for 5.271 years, which calculated by the following:
-= 213 days
-= 365
-= 365
-= 365
-= 365
-= 251
-The total number of days the investment was held was 1,924 days. To calculate the number of years, divide the total number of days by 365 (1,924/365), which equals 5.271 years.
-The total number of years the investment was held can be placed in the denominator of the exponent inside CAGR’s formula as follows:
-Investment CAGR=($16,897.14$10,000)15.271−1=10.46%\text{Investment CAGR} =\, \left ( \frac{\$ 16,897.14}{\$ 10,000} \right )^{\frac{1}{5.271}}-1 = 10.46\%Investment CAGR=($10,000
-$16,897.14)5.271
-−1=10.46%
-Smooth Rate of Growth Limitation
-The most important limitation of CAGR is that because it calculates a smoothed rate of growth over a period, it ignores volatility and implies that the growth during that time was steady. Returns on investments are uneven over time, except bonds that are held to maturity, deposits, and similar investments.
-Also, CAGR does not account for when an investor adds funds to a portfolio or withdraws funds from the portfolio over the period being measured.
-For example, if an investor had a portfolio for five years and injected funds into the portfolio during the five year period, the CAGR would be inflated. The CAGR would calculate the rate of return based on the beginning and ending balances over the five years, and essentially count the deposited funds as part of the annual growth rate, which would be inaccurate.
-Other CAGR Limitations
-Beside the smoothed rate of growth, CAGR has other limitations. A second limitation when assessing investments is that, no matter how steady the growth of a company or investment has been in the past, investors cannot assume the rate will remain the same in the future. The shorter the time frame used in the analysis, the less likely it will be for realized CAGR to meet expected CAGR when relying on historical results.
-A third limitation of CAGR is a limitation of representation. Say that an investment fund was worth $100,000 in 2012, $71,000 in 2013, $44,000 in 2014, $81,000 in 2015 and $126,000 in 2016. If the fund managers represented in 2017 that their CAGR was a whopping 42.01% over the past three years, they would be technically correct. They would, however, be omitting some very important information about the fund’s history, including the fact that the fund’s CAGR over the past five years was a modest 4.73%.
-CAGR vs. IRR
-The CAGR measures the return on an investment over a certain period of time. The internal rate of return (IRR) also measures investment performance but is more flexible than CAGR.
-The most important distinction is that CAGR is straightforward enough that it can be calculated by hand. In contrast, more complicated investments and projects, or those that have many different cash inflows and outflows, are best evaluated using IRR. To back into the IRR rate, a financial calculator, Excel, or portfolio accounting system is ideal.
-Example of How to Use CAGR
-Let’s say an investor bought 100 shares of Amazon.com (AMZN) stock in December 2015 at $650 per share, for a total investment of $65,000. After 3 years, in December 2018, the stock has risen to $1,750 per share, and the investor’s investment is now worth $175,000. What is the compound annual growth rate?
-Using the CAGR formula, we know that we need the:
-    Ending Balance: $175,000
-    Beginning Balance: $65,000
-    Number of Years: 3
-So to calculate the CAGR for this simple example we'd enter that data into the formula as follows:
-CAGR for Amazon=($175,000$65,000)13−1=39.12%\text{CAGR for Amazon} =\, \left ( \frac{\$ 175,000}{\$ 65,000} \right )^{\frac{1}{3}}-1 = 39.12\%CAGR for Amazon=($65,000
-$175,000)3
-−1=39.12%
-This tells us that the compound annual growth rate for the investment in Amazon is 39.12%.
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