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3

SO that a(0) = i- Thus, substituting back we have the familiar result

a{t) = \ + it forr > 0. (1.5)

The above derivation does not depend on t being a positive integer, and is valid for all t > 0.

Example 1.1 Find the accumulated value of $2000 invested for four years, if the rate of simple interest is 8% annum.

The answer is

2000[1 + (.08)(4)] = $2640.

Note that the amount of interest earned is 2640 - 2000 = $640. This could also have been obtained as 2000(.08)(4), or, in general, as /4(0) it. In different notation, this becomes the familiar result from elementary and secondary school

I = Prt

which states that the amount of interest is equal to the product of the amount of principal, the rate of interest, and the period of time.

1.5 COMPOUND INTEREST

Simple interest has the property that the interest is not reinvested to earn additional interest. For example, consider an investment of $100 for two years at 10% simple interest. Under simple interest the investor will receive $10 at the end of each of the two years. However, in reality, for the second year the investor has $110 which could have been invested. Clearly, it would be advantageous to invest the $110 at 10%, since the investor would then receive $11 in interest for the second year instead of $10.

The theory of compound interest handles this problem by assuming that the interest earned is automatically reinvested. The word "compound" refers to the process of interest being reinvested to earn additional interest. With compound interest the total investment of principal and interest earned to date is kept invested at all times.

It is now desired to find the accumulation function for compound interest. Consider the investment of 1 which accumulates to 1 + i at the end of the first period. This balance of 1 + can be considered as principal at the beginning of the second period and will earn interest of (1 + i) during the second period. The balance at the end of the second period is(l -1- i) + i(l + 0 = (1 + 0. Similarly, the balance of (1 -t- i) can be considered as principal at the beginning of the third period and will earn interest of /(1 + i) during the third period. The balance at the end of the third period is (1 + 0 + 1(1 + 0 = (1 + 0- Continuing this process indefinitely, we obtain

ait) = (1+0 for integral > 0. (1-8)

for integral > 1, which is a decreasing function of n. Thus, a constant rate of simple interest implies a decreasing effective rate of interest.

The accumulation function for simple interest has been defined only for integral values of > 0. However, it is natural to extend the definition to nonintegral values of r > 0 as well. This is equivalent to the crediting of interest proportionally over any fraction of a period. If this is the case, then the amount function can be represented by Figure 1.1 (a). If interest is accrued only for completed periods with no credit for fractional periods, then the amount function becomes a step function with discontinuities as illustrated by Figure 1.1 {d). Unless stated otherwise, it will be assumed that interest is accrued proportionally over fractional periods under simple interest.

A more rigorous mathematical approach to the definition of a{f) for nonintegral values of t can be developed by starting with the following property we would want simple interest to possess:

ait + 5) = a{t) + a{s) - 1 for r > 0 and 5 > 0. (1.7)

In essence, formula (1.7) says that under simple interest the amount of interest earned by an initial investment of one unit over t + s periods is equal to the amount of interest earned over t periods plus the amount of interest earned over s periods. The -1 is necessary in formula (1.7), since otherwise there would be an investment of two units on the right-hand side of the equation.

Assuming a(t) is differentiable, from the definition of the derivative we have

a\t) = lim + -

= lim [g(0 + - 1] - Q(0

i-0 s

= iim£(£L:

= a(0), a constant.

Replacing t by r and integrating both sides between the limits 0 and t, we have

a\r)dr= { 0 Jo

a(0 -a(0) = / a(0)

a(r) = 1 + r-a(0).

If we let f = 1 and remember that a(l) = 1 -I- i, we have

a{\) = 1 + j = 1 +a(0)



. a{n) - a{n - 1) (1 + 0" - (1 + 0

= (1 + 0 - 1. =

( -1) (1+0"- 1

which is independent of n. Thus, although defined differently, a rate of compound interest and an effective rate of interest are identical.

The result just derived can be compared with the result obtained in Section 1.4; namely, that a constant rate of simple interest implies a decreasing effective rate of interest. This result should be intuitively clear, since simple interest becomes progressively less favorable to the investor as the period of investment increases.

The accumulation function for compound interest has been defined only for integral values of > 0. It is necessary to develop the accumulation function for nonintegral values of > 0. It is possible to address this question on a basis analogous to that used in Section 1.4 for simple interest.

We start with the following property we would want compound interest to possess:

ait + s) = a{t) a{s) for r > 0 and s > Q.

(1.9)

In essence, formula (1.9) says that under compound interest the amount of interest earned by an initial investment of one unit over t + s periods is equal to the amount of interest earned if the investment is terminated at the end of t periods and the accumulated value at that point is immediately reinvested for an additional s periods.

Assuming a(/) is differentiable, from the definition of the derivative we have

ait) = lim SLtlLL

ajt) ajs) - a(t)

= a(01im = ait) aiQ).

Thus

= ,o,,»(=.(0). Replacing t by r and integrating both sides between the limits 0 and t, we have

. loggair)dr = {aiO)dr 0 dr Jo

loga(r) - loga(O) = t-aiO) log,a(/) = t-aiO)

since loga(O) = 0. If we let = 1 and remember that ail) = 1 + /, we have log,a(l) = log,(l +0 = aiO).

Thus, substituting back we have the result

log,ait) = t log,(l + 0 = logl + 0

ait) = (1 + O for t > 0. (1-8)

The above derivation does not depend on t being a positive integer, and is valid for all t > 0.

Unless stated otherwise, it will be assumed that interest is accrued over fractional periods according to formula (1.8) under compound interest. Thus the amount function is exponential and can be represented by Figure l.l(fc). The exponential form of the amount function should not be unexpected, since many growth curves encountered in the natural sciences are exponential.

Some readers may be troubled, on the one hand, by the statement that interest is paid at the end of the period and, on the other hand, by the statement that interest is accruing continuously. At first glance, the two statements appear contradictory. However, there is no inconsistency as long as interest is accrued over fractional periods as well as completed periods. When this is the case, the accumulated values at any point in time are equal from either perspective.

It is clear that simple and compound interest produce the same result over one measurement period. Over a longer period, compound interest produces a larger accumulated value than simple interest while the opposite is true over a shorter period. The proofs of these results are left as exercises.

Another insight into the difference between simple interest and compound interest can be seen from the two growth patterns involved. Under simple interest, it is the absolute amount of growth that is constant over equal periods of time; while under compound interest, it is the relative rate of growth that is constant. In terms of symbols, under simple interest

ait + s)- ait) is independent of t, while under compound interest

ait + s) - ait)

is independent of t.

ait)

It can be shown that a constant rate of compound interest implies a constant effective rate of interest and, moreover, that the two are equal. Let / be the rate of compound interest and let be the effective rate of interest for the nth period, as defined in Section 1.3. Then, we have



1 + J

(1.10)

The term v is often called a discount factor, since it "discounts" the value of an investment at the end of a period to its value at the beginning of the period.

We can generalize the above result to periods of time other than one period, i.e. to find the amount which a person must invest in order to accumulate an amount of 1 at the end of t periods. The answer is the reciprocal of the accumulation function a"(0, since the accumulated value of this amount at the end of t periods is aii) a(t) = 1. We will call a"(0 the discount function.

Thus, we obtain the following results for / > 0:

Simple Interest: a~{t) =

1 + If

Compound Interest: a \t) = -- = v

(1 + 0

(1.11) (1.12)

As specified before, we will use compound interest unless stated otherwise.

In a sense, accumulating and discounting are opposite processes. The term (1 + i) is said to be the accumulated value of 1 at the end of t periods. The term v is said to be the present value (or discounted value) of 1 to be paid at the end of t periods.

The term "accumulated value" as defined above seems to refer strictly to payments made in the past, while the term "present value" seems to refer strictly to payments to be made in the future. This is the sense in which we will use the terms. Some writers have used "present value" to refer to either past or future payments. We will use the term current value for this .

It is interesting to relate v to the accumulation function for compound interest from an alternate viewpoint. It is immediately clear that the values of v extend the definition of the accumulation function to negative values of Thus, the accumulation function for compound interest has meaning for all values of t. The graph of this function is shown in Figure 1.2.

a(/) = (I + /)

Figure 1.2 Accumulation function for compound interest

Example 1.3 Find the amount which must be invested at a rate of simple interest of 9% per annum in order to accumulate $1000 at the end of three years.

The answer is

1000

00

H-(.09)(3) 1.27

= $787.40.

Example 1.4 Rework Example 1.3 using compound interest instead of simple interest.

Compound interest is used almost exclusively for financial transactions covering a period of one year or more and is often used for shorter term transactions as well. Simple interest is occasionally used for short-term transactions and as an approximation for compound interest over fractional periods. This latter use of simple interest will be examined in more detail in Section 2.2. Hereafter, unless stated otherwise, we will use compound interest instead of simple interest.

There is an implicit assumption in this section that the interest earned under compound interest is reinvested at the same rate as the original investment. Although this is often the case, many situations do arise in practice in which the interest earned is reinvested at a different rate. Section 5.4 contains an analysis of the results when the reinvestment rate for the interest earned differs from the rate on the original principal.

Example 1.2 Rework Example 1.1 using compound interest instead of simple interest. The answer is

2000(1.08) = $2720.98.

This answer is in contrast with the answer of $2640 using simple interest. The extra $80.98 is the result of compounding of interest.

1.6 PRESENT VALUE

We have seen that an investment of 1 will accumulate to 1 + / at the end of one period. The term 1 + i is often called an accumulation factor, since it accumulates the value of an investment at the beginning of a period to its value at the end of the period.

It is often necessary to determine how much a person must invest initially so that the balance will be 1 at the end of one period. The answer is (1 + 0" since this amount will accumulate to 1 at the end of one period. We now define a new symbol v, such that



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