91Ó°ÊÓ

The future value \(V\) of a savings plan, where regular payments \(P\) are made \(n\) times to an account in which the interest rate, \(i,\) is compounded each payment period, can be calculated using the formula $$ V=P \cdot \frac{(1+i)^{n}-1}{i} $$ The total number of payments, \(n,\) equals the number of payments per year, \(m,\) times the number of years, \(t,\) so $$ n=m \cdot t $$ The interest rate per compounding period, \(i\), equals the annual interest rate, \(r,\) divided by the number of compounding periods a year, \(m,\) so $$ i=r / m $$ a. Substitute \(n=m \cdot t\) and \(i=r / m\) in the formula for \(V\), getting an expression for \(V\) in terms of \(m, t,\) and \(r\). b. If a parent plans to build a college fund by putting \(\$ 50\) a month into an account compounded monthly with a \(4 \%\) annual interest rate, what will be the value of the account in 17 years? c. Solve the original formula for \(P\) as a function of \(V, i,\) and \(n\). d. Now you are able to find how much must be paid in every month to meet a particular final goal. If you estimate the child will need \(\$ 100,000\) for college, what monthly payment must the parent make if the interest rate is the same as in part (b)?

Step by step solution

01

Substitute for n and i

Given the formulas for the total number of payments () and the interest rate per compounding period (), substitute these into the future value () formula:$$V=P \frac{(1+i)^{n}-1}{i}$$ becomes$$V=P \frac{(1+\frac{r}{m})^{m \t}-1}{r / m}$$.

02

Simplify the expression

Simplify the expression by multiplying the numerator and the denominator by m. This will give us:$$V=P \frac{m[(1+\frac{r}{m})^{m \t}-1]}{r}$$ Hence, the expression for in terms of m, , and \r is:$$V = P \frac{m[(1+\frac{r}{m})^{m \t}-1]}{r}$$.

03

Calculate future value (b)

For P = 50, r = 0.04, m = 12, and =17: First, find :$$n = m \t = 12 \times 17 = 204$$Then, find i:$$ i = \frac{r}{m} = \frac{0.04}{12} = \frac{0.003333}.$$ Now use the given formula:$$v= 50 \frac{(1+0.003333)^{204}-1}{0.003333}= 50\frac{2.00905-1}{0.003333}$$$$ v \tilde \frac{50*1.00905}{0.003333}= 15118.46\text{->\( } $$ Hence, the future value of the account will be approximately \)\text {18.46}

04

Solve for P (c)

Rearrange the original formula for \r:$$ P= \frac{V*i}{(1+i)^{n}-1}$$ Thus, the expression of as a function of i, and is obtained.

05

Find monthly payment (d)

To find the monthly payment necessary to meet the goal of \(100,000, use :$$ V=100000, r=0.04, m=12, and t=17:: n=12 *17= 204 classes$$$$=\frac{1+r}{m}=\frac{0.04}{12}=0.003333$$So$$=\frac{100000*0.003333}{(1+0.003333)^{204}-1}$$$$ =\frac {333.333}{60905}\times 9,i.e.,$$Thus P\tilde= 150.92\text \).

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Key Concepts

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

Compounded Interest

Compounded interest is the interest on a loan or deposit that is calculated based on both the initial principal and the accumulated interest from previous periods. This means you earn interest on the interest, which can lead to exponential growth of your savings over time.
When interest is compounded, it is added to the principal at regular intervals called compounding periods. The more frequently interest is compounded, the higher the amount of interest accumulated.
For instance, if interest is compounded monthly, it is calculated and added to the principal 12 times a year.

Future Value Formula

The future value formula helps to determine how much a series of regular payments will accumulate to over a period of time, given a specific interest rate.
The formula is:
\[ V = P \frac{(1+i)^n - 1}{i} \ \]where:

• \(V\) = Future value of the savings plan
• \(P\) = Regular payment amount
• \(i\) = Interest rate per compounding period
• \(n\) = Total number of payment periods

This formula takes into account the effect of compounded interest, making it a powerful tool for understanding how savings grow over time.

Payment Periods

Payment periods are intervals at which regular payments are made into a savings account. They can be monthly, quarterly, yearly, etc. The total number of payment periods is crucial in determining the future value of the savings.
This is calculated using the formula:
\[ n = m \times t \]where:

• \(n\) = Total number of payment periods
• \(m\) = Number of payments per year
• \(t\) = Number of years

So, if you are making monthly payments for 17 years, with 12 payments per year, the total number of payment periods \(n\) would be \(12 \times 17\) = 204.

College Fund

A college fund is a specific savings plan that parents set up to ensure they can meet the financial requirements for their child's college education. Using the future value formula, you can easily figure out how much you need to save and the impact of interest on your savings.
For example, if you start saving \(50 each month in an account compounded monthly at a 4% annual interest rate, you can calculate the future value of the account over 17 years using the given formula. This will help you plan effectively to reach your goal.
Additionally, by solving the formula for \(P\), you can determine the required monthly payment to meet a particular financial goal, like \)100,000 for college expenses.