**Introduction Compound Interest**

Yet a third illustration of exponential growth is in the compounding of interest. If principal *P* is put in the bank at *p* percent simple interest per year then after one year the account has

dollars. [Here we assume, of course, that all interest is reinvested in the account.] But if the interest is *compounded n* times during the year then the year is divided into *n* equal pieces and at each time interval of length 1/ *n* an interest payment of percent *p/n* is added to the account. Each time this fraction of the interest is added to the account, the money in the account is multiplied by

Since this is done *n* times during the year, the result at the end of the year is that the account holds

dollars at the end of the year. Similarly, at the end of *t* years, the money accumulated will be

Let us set

and rewrite (*) as

It is useful to know the behavior of the account if the number of times the interest is compounded per year becomes arbitrarily large (this is called *continuous compounding of interest* ). Continuous compounding corresponds to calculating the limit of the last formula as *k* (or, equivalently, *n* ), tends to infinity.

We know from the discussion in Subsection 6.2.3 that the expression (1 + 1/ *k* ) ^{k} tends to *e* . Therefore the size of the account after one year of continuous compounding of interest is

*P · e* ^{p /100} .

After *t* years of continuous compounding of interest the total money is

**Examples**

**Example 1**

If $6000 is placed in a savings account with 5% annual interest compounded continuously, then how large is the account after four and one half years?

**Solution 1**

If *M* ( *t* ) is the amount of money in the account at time *t* , then the preceding discussion guarantees that

*M* ( *t* ) = 6000 · *e* ^{5 t /100} .

After four and one half years the size of the account is therefore

*M* (9/2) = 6000 · *e* ^{5·(9/2)/100} ≈ $7513.94.

**Example 2**

A wealthy woman wishes to set up an endowment for her nephew. She wants the endowment to pay the young man $100,000 in cash on the day of his twenty-first birthday. The endowment is set up on the day of the nephew’s birth and is locked in at 11% interest compounded continuously. How much principal should be put into the account to yield the desired payoff?

**Solution 2**

Let *P* be the initial principal deposited in the account on the day of the nephew’s birth. Using our compound interest equation (**), we have

100000 = *P · e* ^{11·21/100} ,

expressing the fact that after 21 years at 11% interest compounded continuously we want the value of the account to be $100,000.

Solving for *P* gives

*P* = 100000 · *e* ^{−0.11·21} = 100000 · *e* ^{−2.31} = 9926.13.

The aunt needs to endow the fund with an initial $9926.13.

**You Try It** : Suppose that we want a certain endowment to pay $50,000 in cash ten years from now. The endowment will be set up today with $5,000 principal and locked in at a fixed interest rate. What interest rate (compounded continuously) is needed to guarantee the desired payoff?

Find practice problems and solutions for these concepts at: Transcendental Functions Practice Test.

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