## On Variation and Investing for the Short Term

Investment return data are usually easy for investors to understand but measures of volatility are harder to understand and interpret. Fund performance summaries often provide year to date returns and annualized returns for some fixed periods such as those over a year, 3 years, 5 years, 10 years or since inception. Measures of variation are typically not provided. Understanding variation is essential to helping the investor make better choices about short, medium- and long-term investments. We will, further down in this page, provide assessments of very low to very high annualized returns in short to long holding periods, using prediction intervals. Financial analysts may not discuss measures of variation such as the standard deviation of returns possibly due to difficulties with interpretation and the likelihood of overly risk-averse reactions from investors. Let’s briefly look at these measures and some notes on baskets (usually called a mutual fund or ETF) of securities constructed from individual securities (stocks and/or bonds), before we return to our calculator and a discussion to aid individual investors understand variation.

Variation as well as correlations can be used to help strike a good balance between the return and the risk of a collection of securities. Often these assessments of balance are limited to portfolios constituting a collection of mutual funds. These mutual funds (the ones trading when financial markets are open are called ETFs) in investor portfolios are increasingly funds following market indices, thus having low expense ratios, instead of managed funds which require additional research on constituent securities. Further, managed funds have not been able to provide better returns than index funds catering to similar classes of securities. Efficient markets, it is argued, adjust very quickly to information, allowing few opportunities for incremental gain by managed fund personnel. While this argument continues to find support, there have been some recent concerns that inefficiencies are likely being generated by increased index tracking transactions without the corrections that transactions based on valuations by fund managers and other investors could produce. Even the late Mr. Bogle, a strong early proponent of Index funds, was advocating managed funds as necessary to help make markets efficient, given the strong market penetration of index funds. There are similar mechanical index tracking investments for volatility. Perhaps we will see drops in the share of managed mutual funds levelling off as fund managers exploit these inefficiencies arising from programmed trading in securities. We will now look at a calculator and use that to better understand volatility.

## Calculator Assessing Variation in Annualized Returns in Short to Long Term Holding Periods

The default entries in the calculator are entries for annualized returns for a stock mutual fund. Such annualized returns for mutual funds are reported for periods ending on the date the investor checks on his fund performance or for periods ending on some landmark date such as the end of the year or quarter. The annualized returns can be entered in the second blue row of the calculator. The first row has the periods, lower to higher from left to right, over which the annualized return occurred. In the default example the return over the past year since about early December 2018 was 6.5%, the returns over the past 3 years were 12.4%, over the past 5 years were 11.3% and that over 10 years was 14.10% (numbers altered slightly to de-identify). Some funds may provide annualized returns over other periods. For instance, a relatively new mutual fund may report annualized data for 1, 3 and 5 years and since inception (6 years, 6 months ago). In this case you would enter data corresponding to 1, 3, 5 and 6.5. Year to date returns are usually not annualized and should not be used in the calculator. Where only 3 periods are reported, leave the two blue cells in the last column blank. Generally, the assessment of variation will be poorer with less data and the calculator will assess a higher likelihood of extreme low and high returns for that holding period.

Calculations in Table 1 assume independence across disjoint yearly intervals and a lognormal distribution (see DeFusco et al, 2001) for the price ratios of the prices at the end of yearly intervals to those at the start. Details on the calculations are in this hyperlinked document. The price ratio is a 100% plus the annual return. For the default data in this table the estimated expected annualized return for the fund is 14.1%. Not surprisingly, it computes as the annualized return based on the entire 10-year period. For one-year periods, we can see in the first column, that we can expect a low annualized return of -1.91% or lower, or a high return of 32.73% or higher, once in about 20 years. This comes out of a 90% prediction interval (1 in 20 or 5% probability at each end). The 2.7% or lower, or the 26.76% or higher, come from an 80% prediction interval (1 in 10 or 10% probability at each end). The typical low and the typical high of 7.5% and 21.5% respectively, come from a 60% prediction interval.

Note that the anticipated highs and lows are in much smaller ranges as we look at longer time frames. If one is looking at a 1-year time frame for cashing an investment, there is some chance of losing 1.91% or more on this investment. If you have a 3-year time frame, there is some probability that you will not make what you expected but may still come up ahead with yearly gains averaging about 3.77% over the three years. For a 10-year time-frame, the pessimistic to optimistic range shrinks to 6.98% to 21.7%, compared to the -1.91% to 32.73% for the 1-year withdrawal time-frame. Note that we made an assumption of an independence of returns about some common mean return and this table can help assess if there is some downward or upward trend. The actual return for the current 1-year period of 6.5%, falling below and close to what the calculator deems a ‘typical’ low return. This should trigger some increased scrutiny to see if the fund performance is on a decreasing trend. Note as well that the calculator is trained to data over the holding period in the right-most column and may be somewhat off if the holding period is somewhat unusual, like an entirely recessive phase for the fund class. The data in the table corresponds to returns for a US growth mutual fund from about December 2008 to December 2018, a period of relatively large growth following the recession that ended a little after Obama was elected for his first term. The expected annualized returns and the range of variation in other periods could be to 5 to 10% lower.

## Assessing Variation in Monthly Returns in Sub-Annual Holding Periods

The bottom table provides assessments of the variability of returns in short holding periods. We consider a month, a quarter, 2 quarters and 3 quarters. Here we make the more stringent assumption of independence across disjoint monthly intervals and a lognormal distribution (see DeFusco et al, 2001) for the price ratios of the prices at the end of monthly intervals to those at the start. Further the calculations derive from the same annualized returns in the top of table 1. Hence the estimates in this table are somewhat cruder. For one-month periods, we can see in the first column, that we can expect a low monthly return of -1.96% or lower, or a high return of 4.26% or higher, once in about 12 months. This comes out of a prediction interval with 8.33% (1 in 12) probability at each end. The typical low of – 0.84% or a typical high of 3.09%, come from intervals with 16.7% (1 in 6) probability in the tails. For half year periods in the third column, variation diminishes to effective monthly returns in the period of between about an unusual low of -0.18% to an unusual high of 2.41%.

## Investing for the Short Term

Given vagaries of the job market and unanticipated large medical and other expenses, investors are often advised to keep 4 to 8 months of income in liquid assets. Banks offer very little interest and one good investment vehicle for part of this emergency cash (or much more if you are saving for an impending mortgage down payment) is in a tax-free intermediate municipal bond fund. These have less downside risk as a large component of the income is dividend income compared to the share prices growth component and an emergency withdrawal usually would not have too much impact even when taxed as ordinary income. Additional benefit arises from dividends which are free of federal taxes, and often state taxes as well, unlike bank income. For a discussion and evaluation of tax free investments and other taxable and tax saving investments please check this link.

As a default, I have entered returns of 1.3%, 2.36%, 4.05% and 5.05% for a tax-free intermediate term bond fund in the second tab of the calculator (numbers altered slightly to de-identify). These returns, as for the stock fund in the first tab of the calculator, are for a 10-year period ending December 31 2018. This period had a series of interest rate increases by the Federal Reserve resulting in a noticeable downward trend and increased variability. The expected annualized returns and the range of variation in other periods could be to a few percentage points higher.

**Edit the blue cells in the spreadsheet and enter your data and the calculations in the bottom boxes of the spreadsheet will refresh. **