COMPOUNDING

Let’s say we invested $100 today and ended up with $200 ten years from now. Yet, what if our colleague put the same $100 to work and

Single Period Return 27

ended up with $200 at the end of the first year? We both doubled our money, but clearly it would not be correct to credit both situations with the same performance. When we calculate returns, we take into account the time value of money.

Returns can be equated to the interest rates used in the calculation of the future value of a fixed income investment. Unlike returns, how- ever, interest rates may be known ahead of time, so we can project the future value at the beginning of the period. The future value of an investment equals the present value plus the interest and other gains earned over the period:

(2.5) where FV is the value at end of period, PV is the current value of the investment, Ris the rate of per period interest, and N is the number of valuation periods.

In return calculations, we calculate this rate R using observations of the beginning and ending market values. To calculate the MVE of an investment during a single period, we multiply the MVB by 1 plus the interest rate:

The difference between the start and end value is the income earned. In a Simple Interest scenario, the income earned is not reinvested to earn additional interest in the following periods. For example, if an MVB =

$1000 is put to work for four months at an interest rate = 5% per month, we calculate an ending value of $1200:

We use simple interest calculations if the investor withdraws the income earned at the end of each period. In this example, the total gain over the four months = $200. Divided by the $1000 invested gives a 20% return for the four-month period. This equals the monthly periodic dollar return multiplied by four.

Compoundingis the reinvestment of income to earn more income in subsequent periods. If the income and gains are retained within the investment vehicle or reinvested, they will accumulate and contribute to

FV = PV×(1+R)^N

Ending market value = Beginning market value×(1+Interest rate)

End value = Beginning value

1+(Rate in percent 100⁄ )×# of time periods invested

[ ]

×

1000×[1+(0.05×4)] = 1200

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the starting balance for each subsequent period’s income calculation.

Exhibit 2.6 shows that $100 invested at 7% for ten years, assuming yearly compounding, will result in an ending value of $196.72.

To illustrate the compounding process, we can step through the cal- culations for the first four years:

1. The original investment is $100.

2. $100 × (1 + 0.07) = $107 to invest at the start of the second year.

3. $107 × (1 + 0.07) = $114.49 to invest at the start of the third year.

4. $114.49 × (1 + 0.07) = $122.50 to invest at the start of the fourth year.

5. $122.50 × (1 + 0.07) = $131.08 to invest at the start of the fifth year.

Or $100 ×(1 + 0.07)⁴= $131.08. A rule of thumb to use when project- ing values when compounding income is that investments at 7% income earned per year double in ten years, before the addition of any more principal.

Let’s return to the question of evaluating the return earned on two investments with the same dollar gain over different time periods. If we had two investments both earning the same dollar gain of 100 on an investment of 100, but the first fund took ten years to accomplish what the second fund did in one, and we assume that the investment income and gains compounded yearly, we would ascribe an annual return rate of 7.18% to the first and 100% to the second fund as shown:

EXHIBIT 2.6 Compound Interest

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EXHIBIT 2.7 Interest on Interest

Standard investments industry performance calculations and presen- tations assume both reinvestment and compounding. With compound interest we assume the accumulation of gains earned in each period is reinvested in the successive period. Because of the reinvestment assump- tion, cash withdrawals, investment expenses, taxes, and other factors that impede this compounding process may result in lower realized returns than the return that is actually presented to investors. The rein- vestment assumption is not realistic for all investors. For example, any taxable investor investing outside a vehicle that is shielded from taxes, such as a 401(k)-plan account, will have to pay taxes on income distri- butions from the fund. The taxes reduce the income available for rein- vestment in the next period. Given this fact, one of the trends in performance measurement is to incorporate these factors that lower the reinvestment amount into the return calculation.

Factoring in taxes and expenses is important because the power of investing lies in the compound interest, the interest on the interest earned in prior periods. Given a 10-year investment earning a 7%

return, the interest on interest component comprises 14% of the ending value. Exhibit 2.7 shows that if we invest for 30 years at 7%, the inter- est on interest will approach 60% of ending value.

When interest earnings are withdrawn after each period, the simple interest calculation is a better measure of the situation. If income is left to earn more income, then the compound interest calculation is the bet- ter measure. Compound interest is assumed in almost all investment applications. With interest rates we usually assume that interest is rein- vested at the same interest rate for subsequent periods. The difference between working with returns instead of interest rates is that in return calculations, while we also assume that the income is reinvested, we rec- ognize that the periodic returns will fluctuate over time.

While we understand that earning a higher return over the holding period will increase the ending investment value, the frequency of com- pounding also impacts the ending value. Exhibit 2.8 shows that holding the rate the same; the more frequent the compounding within the period the higher the ending value.

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EXHIBIT 2.8 Future Value and Compounding Frequency

The inverse is also true; the more frequent the compounding assump- tion, the lower the return credited for the same pattern of market values.

Interest rates are usually quoted on a yearly or annualbasis. We can adjust the quoted annual interest rate to account for more frequent compounding:

(2.6)

where r is the periodic interest rate and m is the times per period that interest is paid or compounds.

For example, if a $100 investment yielded 3% for 6 months (i.e., MVB = 100 and MVE = 103), the value at the end of a year, assuming semiannual compounding and reinvestment of the interest, is $106.09:

As we continue to increase the compounding frequency m, the com- pounding formula converges on a limit where the returns are continuously compounded. Calculation of future value with continuous compounding simplifies the compounding formula to Equation (2.7).

(2.7) Wheree is the exponential constant 2.71828…