Problem 57
Question
Karl has two years to save \(\$ 10,000\) to buy a used car when he graduates. To the nearest dollar, what would his monthly deposits need to be if he invests in an account offering a \(4.2 \%\) annual interest rate that compounds monthly?
Step-by-Step Solution
Verified Answer
Karl needs to deposit approximately $407 per month.
1Step 1: Understanding the Problem
We need to find out how much Karl should deposit every month to reach his goal of $10,000 in two years, considering the account offers a compounded monthly interest rate.
2Step 2: Converting Annual Interest Rate to Monthly Rate
The annual interest rate is 4.2%. To find the monthly interest rate, we divide this by 12, giving us \[ \frac{4.2}{12} \% = 0.35\% \]. In decimal, this is \(0.0035\).
3Step 3: Calculating Total Number of Periods
Since Karl aims to reach his goal in two years with monthly deposits, the total number of periods will be \(2 \times 12 = 24\) months.
4Step 4: Using the Future Value of Annuity Formula
The future value of an annuity formula is \[ FV = P \left(\frac{(1+r)^n - 1}{r}\right) \] where \(FV\) is the future value, \(P\) is the monthly deposit, \(r\) is the monthly interest rate, and \(n\) is the total number of deposits. We set \(FV = 10,000\), \(r = 0.0035\), and \(n = 24\).
5Step 5: Solving for Monthly Deposit
Rearrange the formula to solve for \(P\): \[ P = \frac{FV \times r}{(1+r)^n - 1} \] Substituting known values: \[ P = \frac{10000 \times 0.0035}{(1+0.0035)^{24} - 1} \].
6Step 6: Calculating the Result
Calculate \((1+0.0035)^{24}\) first, which is approximately 1.08609. Then, \((1.08609 - 1)\) gives approximately 0.08609. Substitute these into the formula: \[ P = \frac{10000 \times 0.0035}{0.08609} \approx 406.84 \].
7Step 7: Rounding to the Nearest Dollar
Round \(406.84\) to the nearest dollar to find that Karl needs to deposit approximately $407 each month.
Key Concepts
Understanding AnnuityConcept of Monthly DepositsCalculating Future ValueInterest Rate Conversion
Understanding Annuity
An annuity is a series of equal payments made at regular intervals. When we talk about an annuity in the context of saving or investing, it refers to consistent deposits or payments made into an account over a period of time. For Karl's case, making monthly deposits into an interest-bearing account fits the model of an annuity.
By planning monthly deposits, Karl is essentially building up a future sum, which is often termed as the "future value" of an annuity. Each deposit Karl makes earns interest, which contributes to the growth of the total sum over time.
The process of accumulating interest on these regular payments makes annuities a very effective way to save towards a specific financial goal. Whether it's saving for a car, a house, or retirement, understanding how annuities work can help optimize savings.
By planning monthly deposits, Karl is essentially building up a future sum, which is often termed as the "future value" of an annuity. Each deposit Karl makes earns interest, which contributes to the growth of the total sum over time.
The process of accumulating interest on these regular payments makes annuities a very effective way to save towards a specific financial goal. Whether it's saving for a car, a house, or retirement, understanding how annuities work can help optimize savings.
Concept of Monthly Deposits
Monthly deposits refer to the amount of money put into a savings or investment account on a monthly basis.
For individuals like Karl who aim to save a specific sum within a certain timeframe, having a structured deposit plan is critical. Regular monthly deposits ensure consistency and take advantage of compounding interest over time.
This consistency aids not only in reaching financial goals but also in forming disciplined saving habits. In Karl’s scenario, the use of monthly deposits combined with a compounding interest account helps maximize his savings efficiently.
For individuals like Karl who aim to save a specific sum within a certain timeframe, having a structured deposit plan is critical. Regular monthly deposits ensure consistency and take advantage of compounding interest over time.
This consistency aids not only in reaching financial goals but also in forming disciplined saving habits. In Karl’s scenario, the use of monthly deposits combined with a compounding interest account helps maximize his savings efficiently.
Calculating Future Value
Future value refers to the value of a current asset at a specified date in the future based on an assumed rate of growth. In the context of Karl's savings goals, future value is the amount he aims to accumulate, which is $10,000.
To calculate future value in an annuity context, the future value formula is employed. This takes into account the regular deposits, interest rate, and the total number of deposit periods. By inputting these variables into the formula, one can determine how much the account will grow over time, assuming the interest rate remains constant.
Future value helps individuals like Karl plan and understand how today's savings can grow into tomorrow’s financial goals.
To calculate future value in an annuity context, the future value formula is employed. This takes into account the regular deposits, interest rate, and the total number of deposit periods. By inputting these variables into the formula, one can determine how much the account will grow over time, assuming the interest rate remains constant.
Future value helps individuals like Karl plan and understand how today's savings can grow into tomorrow’s financial goals.
Interest Rate Conversion
Interest rate conversion involves converting an annual interest rate into a rate that can be applied to shorter periods, such as monthly.
This conversion is important because interest rates are often given on an annual basis, but calculations for monthly savings or payments require a monthly rate. For Karl, the annual interest rate of 4.2% must be divided by 12 to obtain the monthly rate of 0.35%.
Understanding how to convert interest rates is essential for accurate financial planning, ensuring that calculations reflect the true cost or growth of money over shorter periods. This plays a pivotal role in making informed decisions about investments, savings, and loans.
This conversion is important because interest rates are often given on an annual basis, but calculations for monthly savings or payments require a monthly rate. For Karl, the annual interest rate of 4.2% must be divided by 12 to obtain the monthly rate of 0.35%.
Understanding how to convert interest rates is essential for accurate financial planning, ensuring that calculations reflect the true cost or growth of money over shorter periods. This plays a pivotal role in making informed decisions about investments, savings, and loans.
Other exercises in this chapter
Problem 56
At which term does the sequence \(\left\\{\frac{1}{2187}, \frac{1}{729}, \frac{1}{243}, \frac{1}{81} \quad \ldots\right\\}\) begin to have integer values?
View solution Problem 57
Use this data for the exercises that follow: In \(2013,\) there were roughly 317 million citizens in the United States, and about 40 million were elderly (aged
View solution Problem 57
For which term does the geometric sequence \(a_{\mathrm{n}}=-36\left(\frac{2}{3}\right)^{n-1}\) first have a non-integer value?
View solution Problem 57
Find the first five terms of the sequence \(a_{1}=\frac{87}{111}\), \(a_{n}=\frac{4}{3} a_{n-1}+\frac{12}{37} .\) Use the \(>\) Frac feature to give fractional
View solution