CTM Measures of Return

Understanding these measures is important because each measure tells a different story about the performance of an investment. For example, given a $1,000 buy price, $2,000 sell price and a 10-year holding period you can say that you:

Each statement is correct but each requires a different interpretation. Wall Street often uses profits and losses, percentage returns and average annual returns because these measures give higher numeric values than annualized returns so they make investments sound more appealing.

Annualized return is the most "complete" measure of performance because it is the only measure that includes buy and sell price, time held and the effect of compound interest. Therefore, the annualized return standardizes investments of different sizes and holding periods.

Before you buy an investment, be sure you understand what performance measure is being use to promote the investment.

All examples use the CTM results for the S&P 500 from 1982, the start of the 1982 - 2000 bull market, to March 2003. For the period there were 30,826 trades based on monthly closes.

For each measure one or more charts that show the measure of return and time held (years) are displayed and discussed in detail. The primary message of each chart is the variability of the returns present in the chart.

The simplest measure of return is the dollar difference between the sale price and the buy price of a CTM trade. When the difference is positive, you have a profit from a winning trade. If the difference is zero or negative, you didn't make money on the trade.

The first chart shows the profits and losses for the 30,876 CTM trades. Each data point represents the dollar return for one CTM trade. So there are 30,876 data points on the chart.

Each data point represents the dollar return and corresponding holding period for one CTM trade. The vertical axis of the chart shows the dollar return. The horizontal axis shows the numbers of years the investment was held - the difference in years between the buy date and sell date.

Since the period starts in July 1982, a full one-year holding period represents a sale in July 1983, a full two-year holding period is for a July 1984 sale and so forth.

The red horizontal line separates the winning trades (above the line) from losing trades (below the line). Of the 30,876 CTM trades, 93% were winners and 7% were losers, with the losing trades occurring for a holding period of six or less years.

The darker areas on the chart denote a concentration of returns, meaning that many like returns occurred for a given holding period.

At first glance the most striking pattern in the dollar profit chart is the variability of the dollar returns. Since the chart shows the profit or loss for each of the 30,876 CTM trades, you would expect lots of variability in the outcomes. Some trades are held for years while others just one or two months.

Trades from purchases at low prices and sold later at very high prices produced large profits. Other trades bought at high prices and later sold at lower prices result in losses. For example, the maximum profit was $1,410.59 and occurred at 18.08 years held for the trade purchased at the July 1982 starting low price and sold at the peak in August 2000. The largest loss was $702.40 for the trade purchased at the peak and sold at the September 2002 low. Therefore, the collection of 30,876 results gives a scattered visual pattern. But the pattern is not random - there is order to it.

The chart shows a general upward trend of profits and years held. So in general, the longer you hold the investment the higher your profit. But this is just a generalization and it isn't true for all trades. For any given holding period there is a very large variation of dollar returns. For example, the profits range from $412.92 to $1,195.12 for the 10 to 11-year holding period. For the 5 to 6-year holding period the returns range from -$279.76 and $1,063.99.

If you fix the dollar return at an amount, for example $500 - $600, the chart shows that you can achieve these dollar returns with many holding period. Simply place one horizontal line at $500 and another at $600 and you'll see the holding periods begin at 1.3 years and end at 17 years. Therefore, there is a band of like returns over many holding periods.

The downward sloping pattern of return on the right side of chart indicates that returns are falling with the decline in prices after the price peak in August 2000. The data point furthest to right on the chart represents a $714.90 profit for the trade purchased July 1982 and sold March 2003, the longest holding period. Not surprisingly the chart and numbers confirm that holding an investment in a declining market means your profits will decrease.

What was your chance of making a very large profit, for example over $1,000? Look at the top of the chart and locate the data points greater than $1,000 profit. There are 2,978 of them, which is 9.6% of the total 30,876 buy/sell combinations. So you had about a 1 in 10 chance of making more than $1,000.

CTM computes the percent return for each trade for a price series and stores the returns in the CTM Trade Table. The percentage return is computed using the following equation:

The following chart, which is a visual representation of the CTM Trade Table, shows the percentage return for the 30,876 CTM trades.

The vertical axis shows the percentage returns. The maximum percentage return is 1,317 percent and results from the July 1882 purchase date and the August 2000 sell date. The largest loss is -46.28 percent for the peak price purchase in August 2000 and the sale in March 2003. Only 151 (0.49%) of the trades were huge winners with a percentage return greater than or equal to 900 percent, a 10-fold increase on your money. These large gains occurred between 15.7 and 19.8-year holding periods.

Notice the dark band of data points that slope upward from 10 to 16 years and then declines. The shape of the band follows the general shape of S&P prices from 1982 to 2003. If you look closely at the data point patterns, you can see several other bands of data points with a similar increasing and decreasing pattern.

Why does this up and down pattern occur? Remember that CTM selects a buy price and then computes the percentage return for all sell prices given that buy price. Since the sell prices move up over time and then decline after the price peak, the general pattern of percentage returns is to increase and then decrease as holding period increases (sell dates get farther from the buy date) for all buy prices before the peak price.

The pattern of percent returns for buy prices after the peak is decreasing because prices are declining. Therefore, the chart is a series of overlays of data points where each overlay includes the percentage returns for all sell prices for each buy price in the price series. The duration of a increasing/decreasing band depends on the date of the buy price. The further the buy date is from July 1982, the first buy date, the shorter the duration of the band.

The next chart displays four bands of percentage returns to illustrate the overlay concept. Each band results from a different buy price. Buy prices in 1982, 1987 and 1992 occur on the upside price pattern of the S&P 500 before the August 2000 peak. The fourth band is for downside buy prices in 2000 after the peak price.

The upper band of blue data points represents percentage returns for buy dates in 1982. These returns rose, peaked at the 18-year holding period, which corresponds to the S&P 500 August 2000 price peak and then declined. The band of brown data points is for 1987 purchase dates. This band peaks around the 12-year holding period, which again corresponds to the August 2000 price peak. The green band is for three 1992 buy dates and peaks around the eight-year holding period, which is the August 2000 peak price. The red band is for 2000 buy prices just after the August peak. Therefore, these buy/sell combinations are on the downside and the returns are negative.

The average annual return is a percentage return that includes a measure of time in its computation. The following equation computes the percentage return for a buy/sell combination and then divides the return by the number of years held to obtain the average annual return:

The next chart displays the average annual returns between -50% and 100% for the S&P 500 from July 1982 to March 2003. The entire range is -98% to 300%.

The red horizontal line separates the positive and negative average annual returns. There were 93% positive and 7% negative returns. The left side of the chart shows the average annual returns for brief holding periods. Notice the returns with less than one year holding period have a very wide range.

Returns for short duration trades like weeks or a few months are very large because the equation divides the percentage return by the time held. If the time held is a decimal value less than one, the average annual return is a large number. As the holding period approaches two to three years, the returns settle into a narrower range.

You can see the dark bands of returns the track the shape of the 1982 to 2003 S&P price curve. This chart, like the percentage return chart, is a series of overlays of returns for different buy prices so you will see the pattern of up and down bands.

This chart shows the variability of returns for a given holding period and the variability in holding periods for a given average annual return.

You compute the annualized return using the buy and sell prices and the total time you held the investment. Using the mathematical routine called a power function with a spreadsheet or computer program, you compute the interest rate that if compounded for the holding period gives the final investment value for the initial amount invested. That rate is called the annualized rate of return.

Suppose you started with $1,000, and ended with $2,000 after holding the investment for 10 years. For another investment you paid $500 and sold it for $750 in two years. Which investment performed the best? For the above two investments the annualized returns are 7.78 percent and 14.47 percent respectively. Therefore, the $500 investment yielded a larger annualized return than the $1,000 investment.

Because the annualized return standardizes the holding period to one year, you can compare returns for investment with different holding periods. Most investment professionals use annualized returns to evaluate the performance of their investments.

The following chart shows the annualized returns for the 30,876 CTM trades for the S&P 500 from July 1982 to March 2003.

The vertical axis of the chart shows the annualized return in percent. The red horizontal line separates the positive returns from negative returns. Of the 30,876 CTM trades, 93% were winners and 7% were losers, with the losing trades occurring for a holding period of six or less years. The returns for the short-term holding periods on the left side of the chart are extremely variable. The positive gains can be very large for short-term trades of a few weeks or months. When the value is annualized, it explodes to a large number. These large annualized returns are not too important because they do not occur frequently.

Also notice the negative annualized values (losing trades) are grouped for holding periods less than six years. After six years all annualized returns are positive.

The next macro pattern is that returns tend to converge toward a somewhat narrower band that fluctuates around the median annualized return as the holding period increases.

The close-up view chart includes only the data points for positive annualized returns from zero to 30 percent. This magnified view displays sharper detail of the data point patterns than the previous chart.

Notice the dense arc pattern of data points from eight to 19 years held. In this band the concentration of data points (dark area) shows an increase in annualized returns from holding periods 8 to fifteen years and then a decrease in returns. These patterns coincide with increasing and decreasing prices of the S&P 500.

Conclusions and Recommendations

Each measure of return gives a different performance result for an investment.

Profit and loss dollar amounts tell you how much money you made or lost in a holding period.

Simple percent returns tell you the percentage gain or loss for an investment for the entire holding period.

Average annual returns include time and let you standardize your return to a unit of time. Average annual return is usually higher than annualized return for the same holding period.

Annualized return includes time and the concept of compound interest. It is very useful in comparing the performance of different long-term investments. For instance, if you are deciding among different long-term investment options, select the option that you think will give the highest annualized return.