When considering any investment opportunity, classical literature tells us to always compare it to the risk-free rate. While this is never a bad idea, this article will study the S&P 500 for use as a comparable investment. The everyday investor now has access to the S&P 500 as an investment vehicle through ETFs (Exchange Traded Funds) managed by companies such as SPDR (SPY), iShares (IVV), and Vanguard (VOO). For this analysis, we will focus on long-term investing, so we will assume returns through the ETFs mentioned will be essentially equal to the returns on the S&P 500 Index.

We often hear the term year-over-year (abbreviated **YoY**) when discussing return distributions on financial assets. The YoY return is the percentage change from the closing one year prior to the current closing price. For example, the YoY return on the S&P 500 on December 8th, 2016 is 8.84%. The price on 12/08/2016 was $2246.19, and the price on 12/08/2015 was $2063.59, reflecting an 8.84% increase. We will extrapolate this concept to reflect varying time periods.

We will detail our discussion on the return distribution of the S&P 500 with analyses of the 1-Year (YoY), 2-Year, and 3-Year (and so on) returns, measured daily. Using this information, we will get to one of the core questions underlying every individual’s investment decisions: *If I invest today, what are my chances of breaking even in n years?* This is an important question when weighing the value of highly qualitative investments (real estate, startups, and classic cars) against highly quantitative investments (financial trading, derivatives, corporate lending).

## Analyzing Return Series

I took daily closing prices of the S&P 500 from 1950 through 2016 and measured the **n**-over-**n** returns for many different values of **n**. The results inspire a lot of confidence about investing in the S&P 500 for the long haul. The table below gives various percentiles for **n**-over-**n** returns for **n** specified in the** YRS** column. The rightmost column **B.E. PROB** gives the break-even probability for the time period specified in **YRS**. The break-even probability is the probability that any given investment of time-length specified in **YRS** will make a greater-than-0% return.

### Return Distribution for S&P 500 Investments (Measured Daily)

yrs | 1% | 5% | 10% | 25% | 50% | mean | 75% | 90% | 95% | 99% | B.E. prob |
---|---|---|---|---|---|---|---|---|---|---|---|

0.083 | -11.5 | -6.3 | -4.4 | -1.7 | 1.0 | 0.7 | 3.3 | 5.4 | 6.9 | 10.7 | 60.80 |

0.25 | -19.9 | -10.2 | -6.7 | -1.9 | 2.4 | 2.1 | 6.7 | 10.4 | 13.0 | 18.7 | 65.39 |

0.50 | -26.5 | -13.7 | -9.3 | -2.1 | 4.7 | 4.3 | 10.9 | 17.7 | 21.2 | 29.1 | 69.46 |

1 | -35.6 | -18.5 | -12.7 | -1.3 | 9.7 | 8.7 | 19.5 | 28.6 | 33.6 | 42.5 | 72.97 |

2 | -40.5 | -23.3 | -12.0 | 4.4 | 16.8 | 17.9 | 32.2 | 48.9 | 60.2 | 76.1 | 81.05 |

3 | -35.8 | -23.8 | -12.2 | 8.5 | 26.2 | 27.5 | 44.3 | 65.8 | 89.7 | 108.6 | 83.47 |

5 | -23.9 | -15.4 | -9.1 | 9.6 | 46.6 | 49.3 | 76.0 | 113.5 | 141.8 | 193.1 | 81.67 |

8 | -27.7 | -3.8 | 2.7 | 26.8 | 70.8 | 82.9 | 137.8 | 175.2 | 210.3 | 257.6 | 92.06 |

10 | -28.6 | -7.4 | 2.9 | 46.1 | 102.9 | 113.2 | 181.5 | 218.8 | 285.7 | 335.9 | 91.98 |

12 | -8.2 | 3.7 | 13.7 | 43.2 | 133.2 | 148.5 | 232.9 | 304.5 | 352.3 | 450.2 | 96.35 |

15 | 20.0 | 32.5 | 43.7 | 64.6 | 163.0 | 212.4 | 299.3 | 408.1 | 600.0 | 736.9 | 100 |

20 | 71.0 | 89.9 | 106.9 | 154.2 | 277.5 | 358.6 | 466.6 | 805.8 | 1020.4 | 1203.4 | 100 |

## Observations and Stylized Facts

We can draw some very interesting analytic conclusions from the above table.

- A
**15-year**investment in the S&P 500 any day in the years 1950 through 2001 had a**100% chance of yielding a positive return**. The mean 15-year return over these years was 299%. - The same principle holds for a 20-year investment, with the mean return being 467%.
- A
**2-year**investment in the S&P 500 any day in the years 1950 through 2014 had an**81% chance of yielding a positive return**, with the mean return being 17.9%. - A
**1-month**investment in the S&P 500 any day in the years 1950 through 2016 had an**60.8% chance of yielding a positive return**, with the mean return being 0.7%.

## Another Point of Comparison

The most rock-solid and oft-referenced point of comparison for any investment is the rate on the 90-day treasury bill. The bill is easy to obtain, virtually risk free, and offers a small return. Just as every investor should have the annualized return of the 90-day t-bill in the back of their mind, every investor should be familiar with the return distribution of investment in S&P 500 ETFs.

I often discuss plans for retirement investing with wealthy individuals that have their saving invested in mutual funds. These individuals commonly fear that, when funds fail to beat the S&P 500, they have not done so through adequately low-risk investments. One way to gauge the performance of a fund is to compare its performance on your intended time horizon to the information in the above table. For example, a fund that under-performs against the S&P 500 over 10 years but has a break-even probability of 99% may be a great investment, because they have adequately reduced risk of loss in exchange for lower overall returns. These are very basic principles, but we often lack the data to make such comparisons.

## Scripts and Data

I have attached the R code and data set used in this analysis. The scripts offers opportunity to customize analysis and create some plots not shown below. Credits to Yahoo! Finance for supplying the data.

## Notable Plots

Some interesting plots generated in this analysis (click to enlarge):

**S&P 500 Return Series #1**

**S&P 500 Return Series #2**

**S&P 500 Return Series #3**

**S&P 500 CDFs #1**

**S&P 500 CDFs #2**

**S&P 500 CDFs #3**

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Frank says

Hi Chris! I found your article very interesting. I work in the health-care industry so economics is not my area of activity. I’d like to ask you if in your calculations you included inflation ‘costs’ and currency devaluation?

Chris Conlan says

Hi Frank,

The analysis did not include any of those costs. It is a fairly raw analysis on S&P 500 prices.

Kit Eizenga says

Hi Chris have you been able to put any of this data into action with trading at all?

Chris Conlan says

Absolutely. We’re doing a lot of financial data work at my business. See Conlan Scientific Website.

I recommend you check out the blog there if you found this interesting.