Large Cap portfolios was 6.73%. The 5-year average return for the S&P 500 was 10.70%. If we were an investor in the Magellan fund, we could use these statistics to evaluate whether or not we are happy with its per- formance. But what do these numbers mean? How can we adjust the numbers to account for the timing of the investor’s investments into the fund? How can we calculate the returns for our own portfolios? These are some of the topics covered by the chapters in Part I of this book.
RETURN
We can best explain the concept of return with a simple example. Sup- pose we invest $100 in a fund. Our investment goes up in value and we get $130 back. What was the return on this investment? We’ve gained
$30. Taking this dollar returnand dividing it by the $100 invested, and multiplying the decimal result 0.3 by 100 gives us the return expressed as a percentage:
1Source: Fidelity Investments, www.fidelity.com.
A
30 100---
×100 = 30%
16 RETURN MEASUREMENT
Arate of return is the benefit we have received from an investment over a period of time expressed as a percentage. Returns are a ratio relating:
How much was gained or lost given
How much was risked
We interpret a 30% return as a gain over the period equal to almost 1/3 of the original $100 invested. Although it would seem that no spe- cial knowledge of investments is required to calculate and interpret rates of return, several complications make the subject worthy of further attention:
■ Selection of the proper inputs to the return calculation.
■ Treatment of additional client contributions and withdrawals to and from the investment account.
■ Adjusting the return to reflect the timing of these contributions and withdrawals.
■ Differentiating between the return produced by the investment man- ager and the return experienced by the investor.
■ Computing returns spanning multiple valuation periods.
■ Averaging periodic rates of return.
Why do we use rates of return rather than absolute dollar gains to describe the performance of an investment? There are several reasons that returns are the preferred statistic for representing investment per- formance:
■ A return summarizes a lot of information into a single statistic. This includes data on the market value, income earned, and transactions made on all of the investments in the fund.
■ Returns are ratios, and it is usually faster and easier for us to inter- pret a proportion between two things than to use the underlying data.
■ Returns are unaffected by the relative size of portfolios. For example, if we put $100 at work and gain $10 we have earned the same return as the investor that put $1 million at work and ended up with $1.1 million. Returns are much more useful for comparing the perfor- mance of different funds, funds to indices, and managers to other managers.
■ Returns calculated for different periods are comparable; we can com- pare the returns earned in one year to those earned in prior years.
Single Period Return 17
■ The return earned on two investments can be compared to show the relative gains earned over the period. For example, a fund that earned 10% in a year produced twice as much gain as a fund that earned 5%
during the year.
■ The interpretation of the rate of return is intuitive. Return is the value reconciling the beginning investment value to the ending value over the time period we are measuring. We can take a reported return and use it to determine the amount of money we would have at the end of the period given the amount invested:
For example, if we were to invest $100 at a return of 10% we would have $110 at the end of the period:
Adding one to the decimal return before multiplying gives a result equal to the beginning value plus the amount earned over the period.
Multiplying the investment made by the return of 0.1 will give the amount earned over the period ($10).
Let’s look closer at the calculation of return. In our introductory example we earned a $30 gain on an investment of $100. By dividing the gain by the amount invested we derive the 30% return using
(2.1)
Suppose that instead of investing and then getting our money back within a single period, we held an investment worth $100 at the begin- ning of the period and we still held on to it at the end of the period when it was valued at $130. We can calculate the return by:
1. Taking the ratio of the ending value to the beginning (130/100 = 1.3) and
2. Subtracting one from the ratio to take away the portion representing the original investment. This leaves the relative growth over the period (1.3 – 1 = 0.3).
3. Multiplying this result by 100 transforms the decimal fraction into a percentage gain (0.3 × 100 = 30%).
Investment made×(1+Decimal return) = Accumulated value
$100×(1.10) = $110
Return in percent Gain or loss Investment made ---
×100
=
18 RETURN MEASUREMENT
We calculate the same return whether we buy and then liquidate an investment within a period or we carry an investment over from a prior period and hold it. The smallest unit of time we use to measure return is called a single measurement period, or simply period. When we measure the return on an investment we buy and hold across periods, we treat the beginning market value as if it were a new investment made at the beginning of the period and the ending market value like it were the proceeds from the sale of the investment at the end of the period.
It does not matter which of the two forms of return calculation pre- sented so far we use because the two methods are equivalent:
We can prove they are the same by deriving the second form from the first:
where MVE = market value at the end of the measurement period and MVB = market value at the beginning of the measurement period.
Using the first form, the numerator of the rate of return calculation is the unrealized gain or loss: the difference between the starting and ending market value. In either form the denominator is the investment madeorinvestment base. The amount in the denominator represents the money at risk, or principal, invested during the period. In the first period, the investment made is equal to the amount originally invested in the fund. In subsequent periods, it is equal to the ending market value of the previous period. The market value at the end of the investment period plus the income earned over the period equals the accumulated value for the period. Exhibit 2.1 shows the calculation of monthly return where we invest $100 on December 31 and it grows to $110 at the end of January and then $120 at the end of February.
Gain or Loss Investment made ---
×100 Current value
Investment made ---
–1 ×100
=
30% 130 100– ---100
×100 130
100---
–1 ×100
= =
MVE MVB– ---MVB
×100 MVE
MVB--- MVB MVB---
–
×100 MVE
MVB---–1
×100
→ →
Single Period Return 19
EXHIBIT 2.1 Percentage Return versus Dollar Return
Notice that even though the dollar return is the same $10 in each monthly period, the percent return is lower in the second month (10/
110 = 9.09%) than it was in the first month (10/100 = 10.00%). The reason for the lower return in the second month is that the amount at risk in the fund for the second month equals not only the original investment of $100 but also the additional $10 gained in the first month. Given the same dollar gain, but more money put at risk, the lower the return that will be credited to the investment.
Now that we have looked at the basic calculation of returns, in the next three sections we step back to look deeper into the component inputs to the return calculation, the market value and cash flows into or out of the portfolio.