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Chapter 6: Problem 40

You're prepared to make monthly payments of \(\$ 95,\) beginning at the end of this month, into an account that pays 10 percent interest compounded monthly. How many payments will you have made when your account balance reaches \(\$ 18,000 ?\)

Short Answer

Expert verified
To reach an account balance of $18,000$ with monthly payments of $95$ and an interest rate of $10\%$ compounded monthly, it will take approximately $108$ payments.

Step by step solution

01

1. Understand the given information

The given information is that monthly payments of \(95 are made into an account that pays 10% interest compounded monthly. We want to know the number of monthly payments required for the account balance to reach \)18,000.
02

2. Identify the annuity formula

We will use the future value of an annuity formula to solve this problem. The future value (FV) of an annuity can be calculated using the formula: \(FV = P\frac{(1 + r)^n - 1}{r}\), where P is the monthly payment, r is the interest rate per period, and n is the number of periods.
03

3. Convert the annual interest rate to a monthly rate

The annual interest rate is 10%, so the monthly rate (r) is \(\frac{10\%}{12} = \frac{1}{12} = 0.1 / 12 = 0.0083333\).
04

4. Substitute the given values and solve for n

Now we will substitute the given values into the annuity formula and solve for n. We have \(FV = 18,000\), P = 95, and r = 0.0083333. Plug these values into the formula: \(18,000 = 95\frac{(1 + 0.0083333)^n - 1}{0.0083333}\)
05

5. Rearrange the equation to solve for n

To solve for n, rearrange the equation and isolate n on one side. Divide both sides by 95: \(\frac{18,000}{95} = \frac{(1 + 0.0083333)^n - 1}{0.0083333}\) Next, multiply both sides by 0.0083333: \(\frac{18,000}{95} * 0.0083333 = (1 + 0.0083333)^n - 1\) Now, add 1 to both sides: \(\frac{18,000}{95} * 0.0083333 + 1 = (1 + 0.0083333)^n\) Finally, take the natural logarithm of both sides to get n: \(n = \frac{\ln(\frac{18,000}{95} * 0.0083333 + 1)}{\ln(1 + 0.0083333)}\)
06

6. Calculate the value of n

Using a calculator, compute the value of n: \(n = \frac{\ln(\frac{18,000}{95} * 0.0083333 + 1)}{\ln(1 + 0.0083333)} \approx 107.3\)
07

7. Interpret the result

Since we cannot have a fraction of a payment, we must round the result up to the nearest whole number. Therefore, the number of payments required is 108.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Interest Rate Conversion
When dealing with financial calculations, particularly those involving annuities, understanding how to convert interest rates is crucial. Often, interest rates are given annually, but investments or loans might require calculations on a monthly basis.
To convert an annual interest rate to a monthly rate, divide the annual rate by the number of months in a year. For example, if the annual interest rate is 10%, as in the exercise, you would calculate the monthly interest rate as:
  • First, convert the percentage to a decimal by dividing by 100, giving us 0.1.
  • Then, divide this by 12, the number of months in a year, resulting in a monthly rate of approximately 0.008333.
This conversion is necessary because annuity payments in the problem are monthly. Adopting this approach ensures that the interest applied to each payment period aligns with the payment schedule.
Compounded Interest
Compounded interest plays a significant role in how investments grow over time. It's the interest on a loan or deposit calculated based on both the initial principal and the accumulated interest from previous periods.
In the exercise, the interest is compounded monthly, meaning that interest is added to the principal balance at the end of each month. This compounding effect allows the investment to grow not just from the initial deposits but also from interest that accumulates on top of interest.
To see this compounding effect, use the formula for the future value of an annuity:
  1. Identify your periodic payment amount, which, in this case, is $95.
  2. Use the monthly interest rate you calculated earlier as 0.008333.
  3. Plug these into the future value of annuity formula to consider how each month's contributions and interest accumulate over n periods.
As each payment is made, the compound interest grows exponentially, showing why compounding can significantly impact the growth of savings over time.
Payment Calculation
Determining how many payments are necessary to reach a financial goal involves using the future value of an annuity formula. This formula helps calculate the total value of a series of regular payments over time considering a fixed interest rate.
The key steps involve:
  • Substitute the desired future balance (e.g. $18,000) for the future value (FV) in the formula.
  • Include the monthly payment amount ( P = 95).
  • Utilize the monthly interest rate, previously calculated as 0.008333.
After setting up the equation: \[18,000 = 95 \cdot \frac{(1 + 0.008333)^n - 1}{0.008333}\]simplify and solve for the number of periods ( n ) using algebraic techniques such as taking logarithms.
This calculation involves determining exactly how many total payments you'll need to make to hit your financial target, rounding up to ensure partial payments are not considered. This understanding allows for precise financial planning, ensuring you meet your financial goals efficiently.

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Most popular questions from this chapter

A 10-year annuity pays \$1,500 per month, and payments are made at the end of each month. If the interest rate is 15 percent compounded monthly for the first four years, and 12 percent compounded monthly thereafter, what is the present value of the annuity?

A financial planning service offers a college savings program. The plan calls for you to make six annual payments of \(\$ 5,000\) each, with the first payment occurring today, your child's 12 th birthday. Beginning on your child's 18 th birthday, the plan will provide \(\$ 15,000\) per year for four years. What return is this investment offering?

The Perpetual Life Insurance Co. is trying to sell you an investment policy that will pay you and your heirs \(\$ 5,000\) per year forever. If the required return on this investment is 9 percent, how much will you pay for the policy?

Friendly's Quick Loans, Inc., offers you "three for four or I knock on your door." This means you get \(\$ 3\) today and repay \(\$ 4\) when you get your paycheck in one week (or else). What's the effective annual return Friendly's earns on this lending business? If you were brave enough to ask, what APR would Friendly's say you were paying?

This problem illustrates a deceptive way of quoting interest rates called add- on interest. Imagine that you see an advertisement for Crazy Judy's Stereo City that reads something like this: "\$1,000 Instant Credit! \(14 \%\) Simple Interest! Three Years to Pay! Low, Low Monthly Payments!" You're not exactly sure what all this means and somebody has spilled ink over the APR on the loan contract, so you ask the manager for clarification. Judy explains that if you borrow \(\$ 1,000\) for three years at 14 percent interest, in three years you will owe: $$\$ 1,000 \times 1.14^{3}=\$ 1,000 \times 1.48154=\$ 1,481.54$$ Now, Judy recognizes that coming up with \(\$ 1,481.54\) all at once might be a strain, so she lets you make "low, low monthly payments" of \(\$ 1,481.54 / 36=\) \(\$ 41.15\) per month, even though this is extra bookkeeping work for her. Is this a 14 percent loan? Why or why not? What is the APR on this loan? What is the EAR? Why do you think this is called add-on interest?
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