你更希望獲得哪種年投資匯報，9%還是10%？
不出意外的話，所有人都傾向於獲得10%的回報。但是在計算年投資回報率時，計算方法不同也會產生驚人的差異。在本文中，我們將向您展示年投資回報率的計算和這些計算方式對投資者對他們投資回報認知的影響。
經濟現實
僅僅通過指出年回報率方法的異同，我們發現一個問題：哪種方法能最佳反映出真實情況呢？這裡的真實我們指的是經濟現實。因此那個方法將預定投資者最後能獲得多少收入。
在所有選項中，集合平均數（也被稱為綜合平局數）能最好的反映投資回報的真實情況。為了說明這一點，假設你三年的投資回報如下：
第一年：15%
第二年：-10%
第三年：5%
要計算綜合平均回報，我們首先在每個年回報率上加1，得到1.15、0.9和1.05.我們將這三個數字相乘用得到的結果除以三，最後得到的是綜合回報。計算如下：
1.15*0.9*1.05/3=1.0281
最後將這個數字轉換成百分比，我們可以發現我們在這三年中獲得年回報率為2.81%%。
這個回報能反映真實情況嗎？我麼用標價法來做個簡單測試:
假設初始資金為100美元：
第一年回報率為15%，也就是15美元
第二年初始資金為115美元
第二年回報率-10%，也就是-11.5美元
第三年初始資金就是103.5美元
第三年回報率5%，5.17美元
最後資金總額為108.67美元
常用計算的缺點
更常見的計算方法被成為算數平均法，或者簡單平均數。對於許多計算來說，簡單平均數準確且易於使用。如果我們想要計算某個月的日平均降雨量、棒球運動員的平均擊球數或者你支票賬戶的日均餘額，簡單平均數都是一個非常適合的方式。
用年回報率2.81計算如下：
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，如果我們想要得到年平均回報率，簡單平均數並不準確。回到我們原來的例子，簡單平均數得到的結果如下：
15% + -10% + 5% = 10%
10%/3 = 3.33%
正如我們之前看到的，投資者實際上沒有獲得與3.33相同的資金回報。這說明簡單平均數不能準確的獲得經濟現實。
年回報3.33%和2.81看起來差異不是很大，但是在我們上面的例子中將會誇大收入1.66美元，或者說1.5%。如果是10年，差異將會變大為6.83美元，或者5.2%。
波動性因素
簡單平均數和綜合平均數之間的差異也受波動的影響。讓我們假設三年週期投資組合回報如下：
第一年：25%
第二年：-25%
第三年：10%
在這個例子中，簡單平均回報仍然是3.33%，而綜合平均收益實際上是1.03%。兩者之間差異增大可以用詹森不等式解釋，即簡單平均回報和綜合平均回報間的差異增加，真實回報就會下降。思考這一問題的另一個方式是，如果我們損失50%的投資資本，那麼我們需要一個100%的回報達到盈虧平衡。
反之亦然，如果波動性減小，簡單平均回報和綜合平均回報的差距也會減小。如果我們在三年中獲得年回報率相同，那麼簡單平均收益和綜合平均收益則相同。
複合和你的回報
像詹森不等式這樣含糊不清的東西的實際應用是什麼？你過去三年投資的平均回報是什麼？你知道他們是如何計算出來的嗎？
讓我們舉一個投資經理講述的營銷案例來闡述簡單和綜合平均數是如何被扭曲的。在一個換燈片中，該經理表示因為他的基金提供波動性比標準普爾500低，選擇投資他的基金的投資者將會獲得更多的回報，實際上他們獲得相同的假設回報。該投資經理植入了一個令人印象深刻的圖形幫助潛在投資者發現最終獲得資金的差異。
現實檢查：投資者可能獲得相同的簡單平均回報率，但是實際上呢？他們肯定沒有收到一個相同的與資金直接相關的綜合平均回報。
總結
綜合平均回報反映透支的真實經濟現實。了解你投資績效評估的細節是個人財務管理的關鍵，使你能更好的評估你的經紀商、理財經理或者基金經理的能力。
你更傾向於獲得哪個投資回報呢，9%還是10%？答案應該取決於哪個能將更多的錢放入你的腰包。
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.
本文翻譯由兄弟財經提供
文章來源:http://www.investopedia.com/articles/08/annualized-returns.asp?rp=i