What returns have you earned on investments in securities? To answer this question, you should consider the purchase price of the security, the sale price, the flow of income (such as dividends or interest), and how long you owned the asset. The easiest (and perhaps the most misleading) return is the holding period return (HPR). It is derived by dividing the gain (or loss) plus any income by the price paid for the asset. That is,
HPR5P11D2P0
P0 , 10.1
holding period return (HPR)
Income plus price appreciation during a specified time period divided by the cost of the investment.
in which P1 is the sale price, D is the income, and P0 is the purchase price. If an investor buys a stock for $40, collects dividends of $2, and sells the stock for $50, the holding period return is
HPR5 $501$22$40
$40 530%.
The holding period return has a major weakness because it fails to consider how long it took to earn the return. This problem is immediately apparent if the information in the previous example had been a stock that cost $40, paid annual dividends of $1, and was sold at the end of the second year for $50. Given this information, what is the return? Although the holding period return remains the same, 30 percent is obviously higher than the true annual return. If the time period is greater than a year, the holding period return overstates the true annual return. (Conversely, for a period that is less than a year, the holding period return understates the true annual return.)
Because the holding period return is easy to compute, it is frequently used, produc- ing misleading results. Consider the following example. You buy a stock for $10 per share and sell it after ten years for $20. What is the holding period return on the invest- ment? This simple question can produce several misleading answers. You may respond by answering, “I doubled my money!” or “I made 100 percent!” That certainly sounds impressive, but it completely disregards the length of time needed to double your money.
You may compute the arithmetic average and assert that you made 10 percent annually (100% 4 10 years). This figure is less impressive than the claim that the return is 100 percent, but it is also misleading because it fails to consider compounding. Some of the return earned during the first year in turn earned a return in subsequent years, which was not taken into consideration when you averaged the return over the ten years.
The correct way to determine the rate of return or internal rate of return (IRR) that was earned is to phrase the question as follows: “At what rate does $10 grow to $20 after ten years?” You should recognize this as another example of the time value of money. The equation used to answer this question is
P0111r2n5Pn,
in which P0 is the cost of the security, r is the rate of return per period, n is the number of periods (e.g., years), and Pn is the price at which the security is sold. When the values are substituted in the equation, the answer is
$10111r2105$20, 111r21052,
r 5 10!22151.07182157.18%,
so the annual rate of return is 7.18 percent. The correct rate of return on the investment (excluding any dividend income) is considerably less impressive than “I doubled my money!” or “I averaged 10 percent each year.”
The inclusion of income makes the calculation of a rate of return more difficult.
Consider the example in which you bought a stock for $40, collected $2 in dividends,
rate of return (internal rate of return or IRR) The discount rate that equates the cost of an investment (cash out- flows) with the cash inflows generated by the investment.
Calculator Solution Function key Data
Input
PV = 210
FV = 20
PMT = 0
N = 10
I = ?
Function key Answer I = 7.18
and then sold the stock for $50 after two years. What is the rate of return? The holding period return is overstated because it fails to consider the time value of money. If you compute the rate of growth and consider only the original cost and the terminal value, the rate of return is understated because the dividend payments are excluded.
These problems are avoided by computing an investment’s internal rate of return that equates the present value of all an investment’s future cash inflows with the present cost of the investment. The general equation for the internal rate of return (r) for a stock is
P05 D1
111r2 1c1 Dn
111r2n 1 Pn
111r2n, 10.2
in which D is the annual dividend received in n years, and Pn is the price received for the stock in the nth year. The same equation is used to determine the yield to maturity in Chapter 14. The yield to maturity is the internal rate of return on an investment in a bond that is purchased today and redeemed at maturity.
If the internal rate of return were computed for the previous illustration of a stock that cost $40, paid an annual dividend of $1, and was sold at the end of the second year for $50, the equation to be solved is
$405 $1
111r2 1 $1
111r221 $50
111r22.
Notice that there are three cash inflows: the dividend received each year and the sale price. The internal rate of return equates all cash inflows with the cost of the invest- ment. These cash inflows include periodic payments as well as the sale price. (The calculation for the holding period return combined the dividend plus the capital gain on the investment and treated them as occurring at the end as a single cash inflow.)
Solving this equation is tedious, especially if there is a large number of years. Select a rate (e.g., 12 percent) and substitute it into the equation. If the results equate both sides of the equation, the internal rate of return has been determined. If the sides are not equal, select another rate and repeat the process. For example, if 12 percent is selected, then
$40 5 $1 3 (interest factor for the present value of an annuity at 12 percent for two years) 1 $50 3 (interest factor for the present value of $1 at 12 percent for two years)
5 $1(1.690) 1 $50(0.797) 5 $41.54.
Since the two sides are not equal, 12 percent is not the internal rate of return. Since
$41.54 exceeds $40, the rate is too small, so a greater rate would be selected and the process repeated.
This tedious process is made considerably easier with the use of a financial calcula- tor or software. When the data are entered into the calculator, the internal rate of return on the investment, 14.17 percent, is readily determined. This 14.17 percent is the true, annualized rate of return on the investment. (The use of a financial calculator facilitates the computation of the internal rate of return, but calculators do have weaknesses. In this illustration, the yearly payments are equal and are entered into the calculator as an an- nuity. If the yearly payments were unequal, each payment would have to be individually Calculator Solution
Function key Data Input
PV = 240
FV = 50
PMT = 1
N = 2
I = ?
Function key Answer I = 14.17
entered. Because calculators limit the number of individual entries, they may not be used to determine the internal rate of return for problems with large numbers of cash inflows.)
The internal rate of return has two potential problems. The first concerns the reinvest- ment of cash inflows received by the investor. The internal rate of return assumes that cash inflows are reinvested at the investment’s internal rate. In the preceding illustration that means the $1 received in the first year is reinvested at 14.17 percent. If the dividend pay- ment is reinvested at a lower rate or not reinvested (e.g., it is spent), the true annualized return on the investment will be less than the rate determined by the equation. Conversely, if the investor earns more than 14.17 percent when the $1 is reinvested, the true return on the investment will exceed the internal rate of return determined by the equation.
The second problem occurs when you make more than one purchase of the secu- rity. Although the problem is not insurmountable, it makes the calculation more diffi- cult. Suppose you buy one share for $40 at the beginning of the first year, buy a second share for $42 at the end of the first year, and sell both shares at the end of the second year for $50 each. The firm pays an annual dividend of $1, so $1 is collected at the end of year 1 and $2 at the end of year 2. What is the return on the investment?
To answer this question using the internal rate of return, you must equate the pres- ent value of the cash inflows and the cash outflows. The cash flows are as follows:
Time Year 0 End of Year 1 End of Year 2
Cash outflow $40 $42 —
Cash inflow — $ 1 $2 1 $100
There are two cash outflows (the purchases of $40 and $42) that occur in the present (year 0) and at the end of year 1. There are two cash inflows, the $1 dividend received at the end of year 1 and the $2 dividend at the end of year 2 plus cash from the sale of the shares ($100) at the end of year 2. The equation for the internal rate of return is
$401 42
111r2 5 $1
111r2 121100
111r22 and the internal rate of return is 16.46 percent.
In this example, you own one share during the first year and two shares during the second year. The return in the second year has more impact on the overall return than the rate earned during the first year when you owned only one share. Since the number of shares and hence the amount invested differ each year, this approach to determining rates of return is sometimes referred to as a dollar-weighted rate of return.
An alternative to the dollar-weighted or internal rate of return is the time-weighted return, which ignores the amount of funds invested during each time period. This tech- nique computes the return for each period and averages the results. In effect, it com- putes the holding period return for each period and averages them. In the illustration, the initial price was $40; the investor collected $1 in dividends and had stock worth
$42 at the end of the year. The return for the first year was ($42 1 1 2 40) 4 40 5 7.5.
During the second year, a share rose from $42 to $50 and paid a $1 dividend. The return was
($50 1 1 2 42) 4 42 5 21.43%.
dollar-weighted rate of return
The rate that equates the present value of cash inflows and cash outflows; the internal rate of return.
time-weighted rate of return
Geometric average of individual holding period returns.
The simple average return is
(7.5% 1 21.43) 4 2 5 14.47%, and the geometric average return is
"11.0752 11.21432 215"1.30542151.142521514.25%.
As discussed earlier, the geometric average is the true compound rate, while the simple average tends to overstate the true annual rate of return.
In this illustration, the dollar-weighted return (i.e., the internal rate of return) is greater than the time-weighted return. This is the result of the stock performing better in the second year when the investor owned more shares. The results would have been reversed if the stock had performed better the first year than during the second year (i.e., 21.4 percent in year 1 and 7.5 percent in year 2). In that case, the larger amount