# Discussion of investment returns in Moore & McCabe

Regarding Example 1.19 in Introduction to the Practice of Statistics, 3rd edition, by D. S. Moore and G. P. McCabe:

This examples compares yearly returns on US stocks and treasury bills, and concludes that the five-number summary is a better description of the distributions of returns than means and standard deviations.

The first problem with this example is that it looks at the wrong data. The returns need to be adjusted for inflation for their distributions to be of any use as a guide to investment decisions. Fortunately, the web site referenced has the inflation-adjusted data, from which the following stem plots can be obtained:

```Percent change, US stocks (S&P 500),    Percent change, US government bills,

N = 47   Median = 10.668  Mean = 9.444  N = 47   Median = 1.392  Mean = 1.159
Quartiles = -1.347646, 20.26834         Quartiles = 0.18678, 2.074247

Decimal point is 1 place to the         Decimal point is at the colon
right of the colon
-4 : 42
-3 : 4                                  -3 : 9
-2 : 2                                  -2 : 4
-1 : 433330                             -1 : 4420
-0 : 921110                             -0 : 330
0 : 1245567                             0 : 23467778
1 : 00123555778889                      1 : 0113444555577778
2 : 01266678                            2 : 1346
3 : 14                                  3 : 067
4 : 1                                   4 :
5 : 1                                   5 : 0037
6 : 7
```

One can see that the description given in the book of returns on treasury bills as being always positive and skewed to the right is not correct, when you look at the data that is actually relevant.

The second problem is that the returns are expressed in a misleading fashion, as the percent change in the value of the investment. The problem here is that if you lose 20% one year and gain 20% the next year, you don't come out even. Instead, you end up losing 4% of your investment. To avoid this, one can look at the log of the ratio of the value of the investment at the end of the year to its value at the end of the previous year. This gives the following stem plots:

```Log ratio, US stocks (S&P 500),         Log ratio, US government bills,

N = 47   Median = 0.101  Mean = 0.078   N = 47   Median = 0.014  Mean = 0.011
Quartiles = -0.01356809, 0.1845552      Quartiles = 0.00186606, 0.02053028

Decimal point is 1 place to the         Decimal point is 2 places to the
left of the colon                       left of the colon

-4 : 2                                  -4 : 53
-3 :                                    -3 : 9
-2 : 4                                  -2 : 4
-1 : 544440                             -1 : 4420
-0 : 921110                             -0 : 330
0 : 1244567                             0 : 23467778
1 : 0001244466667789                    1 : 0113444455567777
2 : 03334579                            2 : 13469
3 : 4                                   3 : 66
4 : 1                                   4 : 89
5 : 26
6 : 5
```

The third problem with this example is that it talks about "describing" the distribution without discussing what the purpose of the whole exercise is. For a long-term investor, the mean of the log of the ratio shown above is what is most relevant, since this indicates what return can be expected in the long run, if the future follows the pattern of the past. The means of 0.078 and 0.011 correspond to percentage returns after inflation of 8.1% and 1.1%. Looking at the median seriously overstates the return that can be expected from stocks, whose returns are revealed here (but not in book's plot) to be skewed to the left. Common sense should also indicate that for this purpose one would not want a "resistant" measure of the centre of the distribution of returns, such as the median, unless you're not concerned about having your investment wiped out now and then.

Of course, it is quite common to see figures on returns which have not been adjusted for inflation in advertisments for mutual funds, and in the financial press. One also sees returns of +50% and -50% compared on a basis that makes them appear to be of similar magnitude. Impressive looking plots of exponential growth on a linear scale that make it look like there has never been a better time to invest than now are also quite common. It is probably not a coincidence that all these errors make investing look more attractive than it really is, since these people have an interest in convincing you to invest your money.

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