你更希望获得哪种年投资汇报，9%还是10%？
不出意外的话，所有人都倾向于获得10%的回报。但是在计算年投资回报率时，计算方法不同也会产生惊人的差异。在本文中，我们将向您展示年投资回报率的计算和这些计算方式对投资者对他们投资回报认知的影响。
经济现实
仅仅通过指出年回报率方法的异同，我们发现一个问题：哪种方法能最佳反映出真实情况呢？这里的真实我们指的是经济现实。因此那个方法将预定投资者最后能获得多少收入。
在所有选项中，集合平均数（也被称为综合平局数）能最好的反映投资回报的真实情况。为了说明这一点，假设你三年的投资回报如下：
第一年：15%
第二年：-10%
第三年：5%
要计算综合平均回报，我们首先在每个年回报率上加1，得到1.15、0.9和1.05.我们将这三个数字相乘用得到的结果除以三，最后得到的是综合回报。计算如下：
1.15*0.9*1.05/3=1.0281
最后将这个数字转换成百分比，我们可以发现我们在这三年中获得年回报率为2.81%%。
这个回报能反映真实情况吗？我么用标价法来做个简单测试:
假设初始资金为100美元：
第一年回报率为15%，也就是15美元
第二年初始资金为115美元
第二年回报率-10%，也就是-11.5美元
第三年初始资金就是103.5美元
第三年回报率5%，5.17美元
最后资金总额为108.67美元
常用计算的缺点
更常见的计算方法被成为算数平均法，或者简单平均数。对于许多计算来说，简单平均数准确且易于使用。如果我们想要计算某个月的日平均降雨量、棒球运动员的平均击球数或者你支票账户的日均余额，简单平均数都是一个非常适合的方式。
用年回报率2.81计算如下：
第一年：（1+2.81%）*100=102.81
第二年：（1+2.81%）*102.81=105.70
第三年：（1+2.81%）*105.70=108.67
然而，如果我们想要得到年平均回报率，简单平均数并不准确。回到我们原来的例子，简单平均数得到的结果如下：
15% + -10% + 5% = 10%
10%/3 = 3.33%
正如我们之前看到的，投资者实际上没有获得与3.33相同的资金回报。这说明简单平均数不能准确的获得经济现实。
年回报3.33%和2.81看起来差异不是很大，但是在我们上面的例子中将会夸大收入1.66美元，或者说1.5%。如果是10年，差异将会变大为6.83美元，或者5.2%。
波动性因素
简单平均数和综合平均数之间的差异也受波动的影响。让我们假设三年周期投资组合回报如下：
第一年：25%
第二年：-25%
第三年：10%
在这个例子中，简单平均回报仍然是3.33%，而综合平均收益实际上是1.03%。两者之间差异增大可以用詹森不等式解释，即简单平均回报和综合平均回报间的差异增加，真实回报就会下降。思考这一问题的另一个方式是，如果我们损失50%的投资资本，那么我们需要一个100%的回报达到盈亏平衡。
反之亦然，如果波动性减小，简单平均回报和综合平均回报的差距也会减小。如果我们在三年中获得年回报率相同，那么简单平均收益和综合平均收益则相同。
复合和你的回报
像詹森不等式这样含糊不清的东西的实际应用是什么？你过去三年投资的平均回报是什么？你知道他们是如何计算出来的吗？
让我们举一个投资经理讲述的营销案例来阐述简单和综合平均数是如何被扭曲的。在一个换灯片中，该经理表示因为他的基金提供波动性比标准普尔500低，选择投资他的基金的投资者将会获得更多的回报，实际上他们获得相同的假设回报。该投资经理植入了一个令人印象深刻的图形帮助潜在投资者发现最终获得资金的差异。
现实检查：投资者可能获得相同的简单平均回报率，但是实际上呢？他们肯定没有收到一个相同的与资金直接相关的综合平均回报。
总结
综合平均回报反映透支的真实经济现实。了解你投资绩效评估的细节是个人财务管理的关键，使你能更好的评估你的经纪商、理财经理或者基金经理的能力。
你更倾向于获得哪个投资回报呢，9%还是10%？答案应该取决于哪个能将更多的钱放入你的腰包。
How To Calculate Your Investment Return
By Stephen Carr
Which annual investment return would you prefer to have: 9% or 10%?
All things being equal, of course, anyone would rather earn 10% than 9%. But when it comes to calculating annualized investment returns, all things are not equal and differences between calculation methods can produce striking dissimilarities over time. In this article, we'll show you annualized returns can be calculated and how these calculations can skew investors' perceptions of their investment returns.
A Look at Economic Reality
Just by noting that there are dissimilarities among methods of calculating annualized returns, we raise an important question: Which option best reflects reality? By reality, we mean economic reality. So which method will determine how much extra cash an investor will actually have in his or her pocket at the end of the period?
Among the alternatives, the geometric average (also known as the "compound average") does the best job of describing investment return reality. To illustrate, imagine that you have an investment that provides the following total returns over a three-year period:
Year 1: 15%
Year 2: -10%
Year 3: 5%
To calculate the compound average return, we first add 1 to each annual return, which gives us 1.15, 0.9, and 1.05, respectively. We then multiply those figures together and raise the product to the power of one-third to adjust for the fact that we have combined returns from three periods.
Numerically this gives us:
(1.15)*(0.9)*(1.05)^1/3 = 1.0281
Finally, to convert to a percentage, we subtract the 1, and multiply by 100. In doing so, we find that we earned 2.81% annually over the three-year period.
Does this return reflect reality? To check, we use a simple example in dollar terms:
Beginning of Period Value = $100
Year 1 Return (15%) = $15
Year 1 Ending Value = $115
Year 2 Beginning Value = $115
Year 2 Return (-10%) = -$11.50
Year 2 Ending Value = $103.50
Year 3 Beginning Value = $103.5
Year 3 Return (5%) = $5.18
End of Period Value = $108.67
If we simply earned 2.81% each year, we would likewise have:
Year 1: $100 + 2.81% = $102.81
Year 2: $102.81 + 2.81% = $105.70
Year 3: $105.7 + 2.81% = $108.67
Disadvantages of the Common Calculation
The more common method of calculating averages is known as the arithmetic mean, or simple average. For many measurements, the simple average is both accurate and easy to use. If we want to calculate the average daily rainfall for a particular month, a baseball player's batting average, or the average daily balance of your checking account, the simple average is a very appropriate tool.
However, when we want to know the average of annual returns that are compounded, the simple average is not accurate. Returning to our earlier example, let's now find the simple average return for our three-year period:
15% + -10% + 5% = 10%
10%/3 = 3.33%
As we saw above, the investor does not actually keep the dollar equivalent of 3.33% compounded annually. This shows that the simple average method does not capture economic reality.
Claiming that we earned 3.33% per year compare to 2.81% may not seem like a significant difference. In our three-year example, the difference would overstate our returns by $1.66, or 1.5%. Over 10 years, however, the difference becomes larger: $6.83, or a 5.2% overstatement.
The Volatility Factor
The difference between the simple and compound average returns is also impacted by volatility. Let's imagine that we instead have the following returns for our portfolio over three years:
Year 1: 25%
Year 2: -25%
Year 3: 10%
In this case, the simple average return will still be 3.33%. However, the compound average return actually decreases to 1.03%. The increase in the spread between the simple and compound averages is explained by the mathematical principle known as Jensen's inequality; for a given simple average return, the actual economic return - the compound average return - will decline as volatility increases. Another way of thinking about this is to say that if we lose 50% of our investment, we need a 100% return to get back to breakeven.
The opposite is also true; if volatility declines, the gap between the simple and compound averages will decrease. And if we earned the exact same return each year for three years - say with two different certificates of deposit - the compound and simple average returns would be identical.
Compounding and Your Returns
What is the practical application of something as nebulous as Jensen's inequality? Well, what have your investments' average returns been over the past three years? Do you know how they have been calculated?
Let's consider the example of a marketing piece from an investment manager that illustrates one way in which the differences between simple and compound averages get twisted. In one particular slide, the manager claimed that because his fund offered lower volatility than the S&P 500, investors who chose his fund would end the measurement period with more wealth than if they invested in the index, despite the fact that they would have received the same hypothetical return. The manager even included an impressive graph to help prospective investors visualize the difference in terminal wealth.
Reality check: the two sets of investors may have indeed received the same simple average returns, but so what? They most assuredly did not receive the same compound average return - the economically relevant average.
Conclusion
Compound average returns reflect the actual economic reality of an investment decision. Understanding the details of your investment performance measurement is a key piece of personal financial stewardship and will allow you to better assess the skill of your broker, money manager or mutual fund managers.
Which annual investment return would you prefer to have: 9% or 10%? The answer is: It depends on which return really puts more money in your pocket.
本文翻译由兄弟财经提供
文章来源:http://www.investopedia.com/articles/08/annualized-returns.asp?rp=i