The compound interest formula tells the fate over time of a single deposited amount, but another common question that arises in finance is: What size deposit do you need to make regularly in an account with a fixed rate of interest, to have a specified amount at a particular time in the future?
This question is important in planning for a major purchase in the future or accumulating a retirement nest egg. In Chapter 22, we apply the same concepts and formula to paying off a mortgage and making installment payments on a car.
EXAMPLE 8 A Savings Plan
A graduate at her first job saves $100 per month, deposited directly into her credit union account on payday, the last day of the month. The account earns 1.8% per year, compounded monthly. How much will she have at the end of five years, assuming that the credit union continues to pay the same interest rate?
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She makes the first deposit at the end of the first month and the last deposit at the end of the 60th month. The monthly interest rate is .
It’s easier to look at the deposits in reverse time order. The last deposit is made on the last day of the five years, so it earns no interest and contributes just $100 to the total.
The second-last deposit earns interest for one month, contributing . Similarly, the third-last contribution is on deposit for two months, contributing .
Continuing in the same way, we find that the first deposit earns interest for 59 months and contributes . The total of all of the contributions is
We will return to this example after developing a formula needed for the solution.
The expression in Example 8 for the total of all contributions is known as a geometric series because the successive terms have geometric growth: Each succeeding term is a constant—in this case, ()—times the preceding term. For the sum of such a series with general ratio , we have the following rule.
Sum of a Geometric Series RULE
The sum of the geometric series with first term and common ratio
is
provided .
The formula is easy to derive. Multiply the sum by , getting
and then subtract that quantity from itself:
With most terms canceling, we are left with
with the last step of dividing by () allowed only if . (If , what is each of the terms of the series?)
That this formula works can be confirmed by multiplying both sides by () and watching terms on the left cancel. (You should do this confirmation for .) Why doesn’t this approach confirm that the formula should work for ?
In the case of periodic deposits at periodic interest rate , we have , and the formula becomes
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We return to the graduate at her first job. Saving $100 per month to an account earns 1.8% per year, compounded monthly, for five years. We have , months, and monthly interest rate , so . The total accumulation after five years is
Suppose that the annual interest rate is 1.2% per year, compounded monthly, and our graduate saves for 4 years. How much will she have?
We can generalize the result of the example to a periodic deposit of per compounding period (deposited at the end of the period) and an interest rate per period. The amount accumulated after compounding periods is given by the following savings formula.
Savings Formula RULE
where
Algebra Review Appendix
Arithmetic and Geometric Sequences and Series
The expression on the right gives the amount accumulated in terms of the nominal annual interest rate , the number of compounding periods per year, and the number of years, using the relations and .
The savings formula involves four quantities: , , , and . If any three are known, the fourth can be found. A common situation is for , , and to be known, with (the regular payment) to be found.
Since we often want to find , we solve the savings formula algebraically once and for all for to get the following payment formula.
Payment Formula RULE
Sometimes the purpose of saving is to accumulate a fixed sum by a particular date. Such a savings plan is called a sinking fund because you sink money into it. The opposite of a sinking fund is an annuity, where an amount is paid out regularly.
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Sinking Fund DEFINITION
A sinking fund is a savings plan to accumulate a fixed sum by a particular date, usually through equal periodic deposits.
Annuity DEFINITION
An annuity is a sequence of periodic payments.
EXAMPLE 9 Sinking Fund
Suppose that your parents had started saving for your college education when you were born. How much would they have had to save each month to accumulate $15,000 (to pay for tuition, room, and board for just your first year of college!) over 18 years, with an account earning a steady 5% per year, compounded monthly?
Applying the payment formula with , monthly rate , and , we get
Suppose that you start saving for your child’s college education when your child is born, but you can earn only a steady 3% interest per year (about what 30-year U.S. Treasury bonds are currently earning). How much would you have to save each month to accumulate $15,000 when the child turns 18?
Under $50 sounds like a manageable amount to contribute, but it doesn’t take into account inflation, costs beyond the first year, the higher cost of a private college, or putting a younger brother or sister through college, too. In the next section, we investigate how to take inflation into account.
Saving for Retirement (Why It’s Never Too Early to Start)
Financial advisers stress the importance of beginning early to save for retirement. Many firms offer a 401(k) plan (named after a section of law regulating pensions), which allows an employee to make monthly contributions to a retirement account. Often, the employer also contributes. The plan has the advantage that income tax on the contributions is deferred until the employee withdraws the money during retirement. Meanwhile, the account accumulates earnings on what would have been taxed.
That means, for example, that an employee making a $100 monthly contribution may see a reduction in take-home pay of only $75 or less, since taxes are not withheld on the contribution.
Sometimes a company’s pension plan consists of just contributing company stock to the employee’s individual 401(k) account. In 2002, the bankruptcy of Enron Corporation resulted in thousands of its employees losing almost their entire retirement savings. Those savings consisted largely of Enron stock contributed by Enron, which fell from $90 per share to under $1 per share in just a couple of months. The Enron bankruptcy illustrated how unwise it is for most of an employee’s retirement fund to consist of stock in just one company, particularly if-as was the case for Enron-the employee is not free to sell the stock. Even more people lost retirement savings and jobs when the stock of WorldCom declined more than 99% in 2002, after news of financial fraud by its management.
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EXAMPLE 10 Retirement Fund Annuity Savings
Suppose that you start a 401(k) plan when you turn 23 and contribute $50 at the end of each month until you turn 65 and retire. Suppose that you put your contributions into a very safe long-term investment that returns a steady 5% annual interest compounded monthly. How much will be in your fund at retirement?
Apply the savings formula with . We get
At first glance, that may seem like a lot of money, but it is not so much if that’s all you have to live off for the rest of your life. (of course, there may also be Social Security payments.) In the exercises later in this chapter, we explore the effects of saving more each month, getting a higher interest rate, saving on taxes, and—especially—having I inflation erode the value of your savings.
Suppose that you wait until you are 45 to start saving. How much would you have?
Annuities are a common way for retirees to receive funds saved up for retirement. We examine an example in Section 22.4 (page 929), where we turn the savings formula around to get a formula (the amortization formula) to relate the amount of savings to a regular payout. In that section, you can find out how much monthly income for a fixed period—or for life—$86,000 could buy.
The winner of a grand prize in a lottery often has a choice between an immediate cash payment and an annuity of a fixed number of annual payments. Lottery winners almost elect the immediate cash payment, but the annual payments are an attractive alternative for a few.
For most major U.S. lotteries today, the annuity’s annual payments increase by a certain percentage each year. The lottery uses the immediate cash amount to purchase a contract with an insurance company to make the payments. The starting payment amount is determined so that the present value of the entire stream of payments is the same as the immediate cash value. So we next look into the meaning of present value.
Present Value
Suppose that you want to make a one-time deposit of amount that will grow to a specific amount in compounding periods from now by earning interest at a rate per period. The quantities , , , and are related through the compound interest formula, . The quantity is called the present value of the amount A that is to be paid compounding periods in the future.
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Present Value DEFINITION
The present value of an amount to be paid at a specific time in the future is what that future payment would be worth today, as determined from a given interest rate and compounding period.
Present Value RULE
The present value of an amount to be paid years in the future, earning in the meantime a nominal rate of interest compounded times per year—that is, after compounding periods at a rate per compounding period—is
If the interest is instead compounded continuously, then the present value is
The present value takes into account that the amount would grow under compound interest, using the compound interest formula (page 875), to the amount A at the end of the years.
EXAMPLE 11 Certificates of Deposit
A certificate of deposit (CD) pays a fixed rate of interest for a term specified in advance, which may range from 1 month to 10 years. The best local rate in June 2015 was for a 60-month CD at 0.747%, compounded daily. How much would need to have been set aside in such a CD to have $12,000 in 60 months?
We find the present value of receiving $12,000 60 months from now, with , and , so that . (Fine point: We use despite the fact that one or two leap days will occur during the 5 years. For a leap year, the bank may apply a daily rate of either 1/366 or 1/365 of the interest rate for the year. We neglect those complications, which would make little difference.) The present value formula gives
An alternative in June 2015 was a 24-month CD at 0.300%, also compounded daily. How much would need to have been set aside in such a CD to have $12,000 at the end of the 24-month term? (Neglect leap days and use a 365-day year.) ?
Spotlight 21.3 shows how to use a spreadsheet to do various kinds of financial calculations.
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Using a Spreadsheet for Financial Calculations Spotlight 21.3
Both commercial (e.g., Microsoft Excel) and open- source (e.g., openoffice) spreadsheets have commands that use the formulas developed in this chapter in situations of compound interest:
where
Both and will be negative if they correspond to payment by you. The guess can be omitted. Also, for simplicity, all problems in this chapter have payments at the end of the period, so ; since that is the default for the spreadsheet, it too can be omitted. ?(Note: openoffice uses semicolons for separators instead of commas.)
We show how some of our previous examples can be solved by using a spreadsheet:
Example 3 (compound interest, page 873), but with monthly compounding: We want the amount accumulated in an account of $1000 after 10 years at 10% interest compounded monthly. We have , (since we make no payments after depositing the principal), and . Put into a cell in the spreadsheet ) and see $2707.04 emerge, as in Table 21.2, page 873.
Example 8 (savings plan, page 882): We want to determine how much a regular $100 per month deposit will amount to after 5 years at 1.8% per year, compounded monthly. We have (remember, payments are outlays), and . Put into a ) and see $6273.37 emerge.
Example 9 (sinking fund, page 885): Your parents want to save $15,000 over 18 years at 5% compounded monthly. Put into a cell and see ($42.96)—it’s in red because it is , meaning that this amount must be deposited monthly.
Example 11 (certificate of deposit, page 887): We want to deposit an amount that will accumulate $12,000 in at 0.747% interest compounded daily. Put in and see ($11,560.07), which is in red because it is a deposit.