## The joy of stock returns

August 5, 2018

A good reason for investing in stocks is to earn more money than the interest paid by a bank account or savings bonds.  Some investors ignore their stocks until it’s time to cash in.  Most prefer to watch the growth of returns, in which case they need to know the total return and holding period.

### Total Return

Stock profits depend on the capital gains and dividends.  A capital gain is the amount earned when the current stock price exceeds its purchase price.  A capital loss is the amount lost when a current price is below the purchase price.  The capital gain (or loss) is “unrealized” if the investor doesn’t sell the stock or “realized” if the investor sold the stock.  Some companies make occasional cash payments, called dividends, to their stock holders.

Total Return is the total profit from your stock investment.  It represents the stock’s change in market value combined with all dividends you received from the company.  In equation 1, the change in market value is equal to “market value – total cost”.

total return = market value – total cost + dividends

• Market value is the combined value of all shares owned at the current market price (market value = current price * volume; volume is the number of shares).
• Total cost is the value of all shares purchased (cost = purchase price * volume) during the holding period
• Holding period is the period of stock ownership.

Trading fees are ignored in order to simplify this discussion. In actual transactions, trading fees reduce the market value and increase the cost by small amounts so as to reduce the total return by a small amount.  The impact of trading fees on profits is lower in larger transactions.  For example, a \$5.00 trading fee is 5% of a \$100 purchase compared to 0.5% of a \$1,000 purchase.

### Return on Investment (ROI)

ROI is the basic measurement of profitability (ref 1).  It is the ratio of total return to total cost (equation 2).

ROI = total return/total cost

Significance: the ROI shows how much profit you earned from every invested dollar.  If the ROI were 0.2/1, which is 20%, then you earned 20 cents per invested dollar.

“Price performance” (equation 3) may not measure the ROI.  Price performance = (price2 – price1)/price1, where price1 is the earlier number and price2 is the later number.  Price 1 may neither be the purchase price nor the only purchase price.  ROI includes all purchase prices.

### Rate of Return

The rate of return measures profitability with respect to time.  Think of if it as the ROI for the holding period (equation 4).

Rate of Return = ROI/holding period

Don’t forget that the ROI compares profit to cost when the time period is anchored to the date of the initial purchase.

ISSUE: The rate of return is most precise when there is just one purchase.  Serial purchases require a more complicated calculation of the “annual return”.

### Annual return

The annual return is a number that represents the average rate of growth per year of the holding period.  The annual return has several important properties:

1. It doesn’t change during the holding period.
2. It’s a geometric average, not an arithmetic average.  The graph of geometric growth is a curved line (“exponential”) rather than a straight line (“linear”).
3. The geometric average represents the phenomenon of compounded growth known as “compounded interest”.

Stock investors are interested in 2 types of compounded growth:

1. The compound annual growth rate (“CAGR”) of a single purchase.
2. The internal rate of return (“IRR”) for a series of purchases.

There are free calculators which are posted online to determine the CAGR (ref 2) and the IRR (ref 3).

ISSUE: The annual return is usually inaccurate during the first year of compounded growth and becomes more accurate over longer time periods.

### Rule of 72

The payback period is something to celebrate!  It’s the point when the investment doubles your money.  Payback is measured by the ratio of total cost to the rate of return.  Or, it can be estimated by the Rule of 72 (equation 5).

Rule of 72 = 72/assumed annual return

Significance: the Rule of 72 is used to forecast the holding period needed to double your money (ref. 4).  For example, assume that your total return will grow at a constant rate of 10% per year [approximately the same rate as the growth of the U.S. Stock Market].  The expected payback period is 7.2 years (7.2 = 72/10).

### Summary

The total profit of your stock investment is called the total return.  The simplest way to measure profitability is to calculate the ROI with equation 2.  The ROI is insensitive to time until you calculate the rate of return with equation 4, which allows you to compare the profitability of several stocks in your portfolio.  The annual return of compounded growth is a refined measurement of your calculated rate of return.  After a holding period beyond one year [to avoid the chance of considerable inaccuracy], the annual return may be determined with an online calculator for a single investment (CAGR) or serial investments (IRR).

### References

1. Return on Investment (ROI). https://www.investopedia.com/terms/r/returnoninvestment.asp

2. Compound annual growth rate (CAGR) calculator.  http://www.moneychimp.com/calculator/discount_rate_calculator.htm  .

3. XIRR calculator to calculate IRR of non-periodic cash flows. https://www.free-online-calculator-use.com/xirr-calculator.html   .

4. Brian Beers. What is the Rule of 72? https://www.investopedia.com/ask/answers/what-is-the-rule-72/ , 1/2/2018.

November 6, 2011

## Summary

Financial success is typically discussed in terms of return, rate of return, and performance.  The compound annual growth rate (CAGR) is a good measure of an investment’s rate of return.  An investment “outperforms” or “underperforms” a market index according to the difference in CAGRs.

## Return

Investors hope to earn a profit called the return.  The two main types of return are cash distributions and capital gains.  Cash distributions include dividends and interest.  A realized capital gain(loss) is the actual return earned from an increase(decrease) in market value between times of purchase and sale.   An unrealized capital gain(loss) is an imaginary return calculated by the increase(decrease) in market value of an unsold investment.  The total return from an investment is the sum of its cash distributions, realized capital gains, and unrealized capital gains 1Significance: It’s important to know whether a market index measures the total return of the market (e.g., S&P 500 Total Return) or the price return of the market (e.g., S&P 500) 2.

## Rate of return

The rate of return is a change in value with respect to time.  Annualized return is a return, or rate of return, from any time period that’s converted to an annual value 3.  Annualization may or may not account for the effects of compounding.  Example #1:  To annualize any monthly or quarterly rate of return without concern for compounding the returns, multiply the unannualized rate of return by 12 months/year or 4 quarters/year as appropriate for the time period.  This method provides only an estimation of the annual rate of return.  Example #2:  The annual rate of return is simply the ratio of yearly return to initial investment expressed as a percentage.  Example#3:  To annualize a compounded return over several preceding years, compute the compound annual growth rate (CAGR4,5.  VIDEO: CAGR

Figure 4 illustrates the use of CAGR to describe the 5-year growth of a market Index called the S&P United States 500 Total Return 1988 (SPTR).

Fig. 4 Rate of return measured by CAGR

Each datum (blue dot) is a spot value of the Index at the end of the last trading day of the year.  Lines connecting the data form peaks and valleys on the graph to illustrate the dynamic nature of the stock market.  The dashed line represents a smoothed continuum of Index values as if the Index grew at the appropriate CAGR of 2.29%.  Significance: Net growth of value occurs when CAGR is a positive value and net loss occurs when CAGR is a negative value.

## Performance

Investment performance is best determined by comparing the investment return to an impartial standard value.  The standard value is either a numerical goal or the value of a market index.  The comparison is only meaningful when the investment return and standard return are based on the same,

• class of financial assets
• type of return (e.g., total return, price appreciation) 2
• units of return (e.g., percentage)
• time interval (e.g., annual)

Outperform” means that the investment return exceeds the standard return and “underperform” means that the investment return lags the standard return.  Investment performance is often measured by comparing the CAGR of an investment portfolio to the CAGR of an appropriate market index1.  For example, Fig. 5 shows that a Fund’s portfolio underperformed its market index.

Fig. 5 Performance measured by CAGR

Figure 5 compares the SPDR S&P 500 ETF Trust’s 5-year portfolio return (NAV) to the benchmark return of the S&P United States 500 Total Return 1988 (SPTR).  The 5-year returns were measured as CAGRs and performance was measured by the difference in CAGRs.

## Concluding trivia

• CAGR is a statistic that’s calculated as the geometric mean for a series of annual percentage returns.
• The graph of CAGR is an exponential curve defined by the formula Y = X(1+CAGR)N . Y is the final value, X is the initial value, CAGR is a decimal number, and N is the number of years.
• The generic rate of return (R ) applies to unannualized growth rates and financial applications such as the calculation of future values.