Calculating Arithmetic Mean Returns
You are considering investing in one of two mutual funds. The table below summarizes the annual holding period returns for the two funds (Fund B started operating at the end of 2011, so return data is not available for that fund in 2010 or 2011):
| Year | Fund A | Fund B |
|------|--------|--------|
| 2014 | 5% | 3% |
| 2013 | 4% | 10% |
| 2012 | 10% | 3% |
| 2011 | 8% | |
| 2010 | 2% | |
Given that the performance periods for the two funds are not the same, it is difficult to make direct comparisons. How can the performance of the two funds be made comparable?
Correct!
Incorrect, since average performance of each fund is not common to both funds.
Consider the asset return in the table below:
| Year | Asset Return (%) |
|------ |------------------ |
| 1 | -20 |
| 2 | 5 |
| 3 | 10 |
| 4 | 15 |
| 5 | 4 |
Try applying the formula above to calculate the mean return for the asset from this table. What is the mean return?
Incorrect.
The solution is:
$$\displaystyle \bar{R}_i=\frac{R_1+R_2+R_3+R_4+R_5}{T}=\frac{-20+5+10+15+4}{5}=2.8\%$$
which equates to 2.8% in this example.
In words, the mean return is calculated by taking the sum of the return observations and dividing the sum by the number of returns in the sample.
Correct.
The solution is:
$$\displaystyle \bar{R}_i=\frac{R_1+R_2+R_3+R_4+R_5}{T}=\frac{-20+5+10+15+4}{5}=2.8\%$$
In words, the mean return is calculated by taking the sum of the return observations and dividing the sum by the number of returns in the sample.
Suppose both funds returned 5.5% in 2015. Without performing any additional calculations, what would you expect to happen to the arithmetic mean returns for each fund over the new period of 2012 - 2015?
Yes!
In fact, fund A's average will decrease from 6.3% to:
$$\displaystyle \frac{10 \% + 4 \% + 5 \% + 5.5 \%}{4} = 6.125 \% $$
fund B's average will increase from 5.3% to:
$$ \frac{3 \% + 10 \% + 3 \% + 5.5 \%}{4} = 5.375 \% $$.
Incorrect.
Consider that 5.5% is above average for one fund, and below average for another.
To summarize:
[[summary]]
It is best to compare the common periods, as this way systematic factors are common for both funds.
Potentially, the market had a very strong year in 2011, and the return of 8% reflects market trends and not fund-level performance.
The __arithmetic or mean return__ is calculated as follows:
$$\displaystyle \bar{R}_i=\frac{R_{i1}+R_{i2}+...+R_{iT-1}+R_{iT}}{T}=\frac{1}{T}\displaystyle\sum_{t=1}^TR_{it}$$
where _R_ signifies return and _T_ signifies the number of observations in the sample.
This calculation is likely familiar, since it is the same as calculating an average which is done commonly in everyday life or in a basic statistics class. Going back to funds A and B, the mean return for Fund A is $$ \frac{10 \% + 4 \% + 5 \% }{3} = 6.3 \% $$ and for Fund B is $$ \frac{3 \% + 10 \% + 3 \%}{3} = 5.3 \% $$.
Incorrect.
Notice that fund A's mean is 6.3% prior to this below average year.
Comparison of the two funds is still difficult as there are multiple years of data. A useful method to aggregate multiple data points into one comparable metric is to calculate the __arithmetic mean return__.
Look only at the common period for both funds (2012-2014)
Utilize average performance of each fund given the available data
3.5%
2.8%
Mean return for fund A will fall, and mean return for fund B will increase.
Mean return for fund A will increase, and mean return for fund B will fall.
Mean returns for both funds will decrease.
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