A traditional individual retirement account (IRA) is a special type of retirement account in which the money you invest is exempt from income taxes until you withdraw it. If you deposit $100 each month into an IRA earning 6% interest, how much will you have in the account after 20 years?

[latex]N = 20 \times 12 = 240[/latex] because you are making monthly deposits for 20 years.

[latex]I\% = 6[/latex] because the interest rate is 6%.

[latex]PV = 0[/latex] because you are saving money.

[latex]PMT = -100[/latex] because you are depositing $100 into the account.

[latex]FV[/latex] is what you want to know.

[latex]P/Y=12[/latex] because you are making 12 payments per year.

[latex]C/Y=12[/latex] because [latex]P/Y[/latex] and [latex]C/Y[/latex] are always the same number.

Plug all values into the TVM Solver, and click the [latex]\begin{array}{|c|}\hline \text{Solve for FV}\\ \hline \end{array}\;[/latex] button.

The account will grow to $46,204.09 in 20 years.

To find the interest, we can plug values for [latex]FV[/latex], [latex]N[/latex], and [latex]PMT[/latex] into the interest formula:

[latex]\text{Interest}=46204.09 - 240 \times 100=22,204.09[/latex]

So you made $22,204.09 in interest.

Return to Section 4.2

You want to have $200,000 in your account when you retire in 30 years. Your retirement account earns 8% interest. How much do you need to deposit each month to meet your retirement goal?

[latex]N = 30 \times 12 = 360[/latex] because you are making monthly deposits for 30 years.

[latex]I\% = 8[/latex] because the interest rate is 8%.

[latex]PV = 0[/latex] because you are saving money.

[latex]PMT[/latex] is what you want to know.

[latex]FV = 200,000[/latex] because you want to have $200,000 in the account in the future.

[latex]P/Y=12[/latex] because you are making 12 payments per year.

[latex]C/Y=12[/latex] because [latex]P/Y[/latex] and [latex]C/Y[/latex] are always the same number.

Plug all values into the TVM Solver, and click the [latex]\begin{array}{|c|}\hline \text{Solve for PMT}\\ \hline \end{array}\;[/latex]button.

So you would need to deposit $134.09 each month to have $200,000 in 30 years if your account earns 8% interest.

Notice that even though the TVM Solver gave us the answer as a negative number, when we talk about payments, we talk about them as positive numbers. So [latex]PMT[/latex] will always be negative when we plug [latex]PMT[/latex] into the TVM Solver, but it will be positive in the interest formula and when it is the solution to our problem.

Return to Section 4.2

If you invest $100 each month into an account earning 3% compounded monthly, how long will it take the account to grow to $10,000?

This is a savings annuity problem since we are making regular deposits into the account.

[latex]N[/latex] is what you want to know.

[latex]I\% = 3[/latex] because the interest rate is 3%.

[latex]PV = 0[/latex] because you are saving money.

[latex]PMT =-100[/latex] because you make payments of $100.

[latex]FV = 10,000[/latex] because you want to have $10,000 in the account in the future.

[latex]P/Y=12[/latex] because you are making 12 payments per year.

[latex]C/Y=12[/latex] because [latex]P/Y[/latex] and [latex]C/Y[/latex] are always the same number.

Plug all values into the TVM Solver, and click the [latex]\begin{array}{|c|}\hline \text{Solve for N}\\ \hline \end{array}\;[/latex] button.

We get that [latex]N = 89.37[/latex] months. Our solution is in months because [latex]N[/latex] tells us how many payments are made in total, and you are making monthly payments. If we wanted our solution in years, we could figure it out with a simple conversion.

[latex]89.37 \text{ months} \times \frac{1 \text{ year}}{12 \text{ months}}=7.45[/latex] years

Return to Section 4.2

After retiring, you want to be able to take $1,000 every month for a total of 20 years from your retirement account. The account earns 6% interest. How much will you need in your account when you retire?

[latex]N = 20 \times 12 = 240[/latex] because you are making monthly withdrawals for 20 years.

[latex]I\% = 6[/latex] because the interest rate is 6%.

[latex]PV[/latex] is what you want to know.

[latex]PMT =-1000[/latex] because you will make withdrawals of $1,000.

[latex]FV = 0[/latex] because after you’ve made all withdrawals, the account has no money in it.

[latex]P/Y=12[/latex] because you are making 12 withdrawals per year.

[latex]C/Y=12[/latex] because [latex]P/Y[/latex] and [latex]C/Y[/latex] are always the same number.

Plug all values into the TVM Solver, and click the [latex]\begin{array}{|c|}\hline \text{Solve for PV}\\ \hline \end{array}\;[/latex] button.

[latex]PV=139580.77[/latex], so you will need to have $139,580.77 in the account when you retire.

Using the interest formula, we find that [latex]\text{Interest}=240 \times 1000-139580.77 =100419.23[/latex]. This means that the interest earned in this account is $100,419.23.

Return to Section 4.3

You know you will have $500,000 in your account when you retire. You want to be able to take monthly withdrawals from the account for a total of 30 years. Your retirement account earns 8% interest. How much will you be able to withdraw each month?

[latex]N = 30 \times 12 = 360[/latex] because you are making monthly withdrawals for 30 years.

[latex]I\% = 8[/latex] because the interest rate is 8%.

[latex]PV=500000[/latex] because you start with $500,000 in the account when you retire.

[latex]PMT[/latex] is what you want to know.

[latex]FV = 0[/latex] because after you’ve made all withdrawals, the account has no money in it.

[latex]P/Y=12[/latex] because you are making 12 withdrawals per year.

[latex]C/Y=12[/latex] because [latex]P/Y[/latex] and [latex]C/Y[/latex] are always the same number.

Plug all values into the TVM Solver, and click the [latex]\begin{array}{|c|}\hline \text{Solve for PMT}\\ \hline \end{array}\;[/latex] button.

[latex]PMT=3668.82[/latex], so you would be able to withdraw $3,668.82 each month for 30 years.

Return to Section 4.3

You can afford $200 per month as a car payment. If you can get an auto loan at 3% interest for 60 months (5 years), how expensive of a car can you afford? In other words, what amount loan can you pay off with $200 per month?

[latex]N =60[/latex] because the auto loan is a 60-month loan.

[latex]I\% = 3[/latex] because the interest rate is 3%.

[latex]PV[/latex] is what you want to know.

[latex]PMT=-200[/latex] because you can afford car payments of $200.

[latex]FV = 0[/latex] because after you’ve made all payments, you owe no money on the car.

[latex]P/Y=12[/latex] because you are making 12 payments per year.

[latex]C/Y=12[/latex] because [latex]P/Y[/latex] and [latex]C/Y[/latex] are always the same number.

Plug all values into the TVM Solver, and click the [latex]\begin{array}{|c|}\hline \text{Solve for PV}\\ \hline \end{array}\;[/latex] button.

You can afford a $11,130.47 auto loan.

Using our interest formula, we can figure out how much interest you will have to pay.

[latex]\text{Interest}=60 \times 200-11130.47 =869.53[/latex], so you will pay $869.53 in interest.

Return to Section 4.4

You want to take out a $140,000 mortgage (home loan). The interest rate on the loan is 6%, and the loan is for 30 years. How much will your monthly payments be?

[latex]N =30 \times 12 = 360[/latex] because you will make monthly payments for 30 years.

[latex]I\% = 6[/latex] because the interest rate is 6%.

[latex]PV=140000[/latex] because you will owe $140,000 when you take out the loan.

[latex]PMT[/latex] is what you want to know.

[latex]FV = 0[/latex] because after you’ve made all payments, you owe no money on the home.

[latex]P/Y=12[/latex] because you are making 12 payments per year.

[latex]C/Y=12[/latex] because [latex]P/Y[/latex] and [latex]C/Y[/latex] are always the same number.

Plug all values into the TVM Solver, and click the [latex]\begin{array}{|c|}\hline \text{Solve for PMT}\\ \hline \end{array}\;[/latex] button.

You can afford payments of $839.37.

Using our interest formula, we can figure out how much interest you will have to pay.

[latex]\text{Interest}=360 \times 839.37-140000=162173.2[/latex], so you will pay $162,173.20 in interest.

Return to Section 4.4

Consider the $140,000 mortgage at 6% from the previous example. If the homeowner increased their payments to $1,000 per month, how long will it take them to pay off the loan?

[latex]N[/latex] is what you want to know.

[latex]I\% = 6[/latex] because the interest rate is 6%.

[latex]PV=140000[/latex] because the homeowner will owe $140,000 when they take out the loan.

[latex]PMT=-1000[/latex] because the homeowner will make payments of $1,000.

[latex]FV = 0[/latex] because after the homeowner has made all payments, no money is owed on the mortgage.

[latex]P/Y=12[/latex] because you are making 12 payments per year.

[latex]C/Y=12[/latex] because [latex]P/Y[/latex] and [latex]C/Y[/latex] are always the same number.

Plug all values into the TVM Solver, and click the [latex]\begin{array}{|c|}\hline \text{Solve for N}\\ \hline \end{array}\;[/latex] button.

[latex]N=241.4[/latex], so it will take 241.4 months to pay off the mortgage with payments of $1,000. To convert this to years, we can use a conversion factor.

[latex]241.4 \text{ months} \times \frac{1 \text{ year}}{12 \text{ months}} \approx 20.12[/latex] years.

Return to Section 4.4

If a mortgage at a 6% interest rate has payments of $1,000 a month, how much will the loan balance be 10 years from the end of the loan?

[latex]N =10 \times 12[/latex] because the monthly payments will be made for 10 years.

[latex]I\% = 6[/latex] because the interest rate is 6%.

[latex]PV[/latex] is what you want to know.

[latex]PMT=-1000[/latex] because you can afford payments of $1,000.

[latex]FV = 0[/latex] because after you’ve made all payments, you owe no money on the car.

[latex]P/Y=12[/latex] because you are making 12 payments per year.

[latex]C/Y=12[/latex] because [latex]P/Y[/latex] and [latex]C/Y[/latex] are always the same number.

Plug all values into the TVM Solver, and click the [latex]\begin{array}{|c|}\hline \text{Solve for PV}\\ \hline \end{array}\;[/latex] button.

[latex]PV=90073.45[/latex], so 10 years from the end of the loan, there will be a balance of $90,073.45.

Return to Section 4.4

A couple purchases a home with a $180,000 mortgage at 4% for 30 years with monthly payments. What will the remaining balance on their mortgage be after 5 years?

First we will calculate the monthly payment amount.

[latex]N =30 \times 12 = 360[/latex] because they will make monthly payments for 30 years.

[latex]I\% = 4[/latex] because the interest rate is 4%.

[latex]PV=180000[/latex] because they will owe $180,000 when they take out the mortgage loan.

[latex]PMT[/latex] is what we want to know.

[latex]FV = 0[/latex] because after they’ve made all payments, they owe no money on the home.

[latex]P/Y=12[/latex] because they are making 12 payments per year.

[latex]C/Y=12[/latex] because [latex]P/Y[/latex] and [latex]C/Y[/latex] are always the same number.

Plug all values into the TVM Solver, and click the [latex]\begin{array}{|c|}\hline \text{Solve for PMT}\\ \hline \end{array}\;[/latex] button.

The couple can afford payments of $859.35.

Now that we know the monthly payments, we can determine the remaining balance. We want the remaining balance after 5 years, when 25 years will be remaining on the loan, so we calculate the loan balance that will be paid off with the monthly payments over those 25 years.

[latex]N =25 \times 12[/latex] because we want to know the balance on the mortgage for the last 25 years.

[latex]I\% = 4[/latex] because the interest rate is 4%.

[latex]PV[/latex] is what we want to know.

[latex]PMT=-859.35[/latex] because the couple can afford payments of $859.35.

[latex]FV = 0[/latex] because after they’ve made all payments, they owe no money on the home.

[latex]P/Y=12[/latex] because they are making 12 payments per year.

[latex]C/Y=12[/latex] because [latex]P/Y[/latex] and [latex]C/Y[/latex] are always the same number.

Plug all values into the TVM Solver, and click the [latex]\begin{array}{|c|}\hline \text{Solve for PV}\\ \hline \end{array}\;[/latex] button.

The loan balance after 5 years, with 25 years remaining on the loan, will be $162,805.99.

Return to Section 4.4

Ula just got a new job. She had $70,000 in a 401(k) at her old job that she rolled over into an IRA. She plans to continue contributing $5,000 a year into her IRA. If her account earns 6% compounded annually, how much will she have in her IRA in 10 years?

The challenge with this problem is that our savings annuity formula assumed that our account balance started at zero. While we could try to build a new formula to handle this situation, we can more easily envision it as two different money streams, as if the original $70,000 is invested in a different account than future contributions.

In the first stream of money, Ula has $70,000 earning 6% interest for 10 years, which is a basic compound interest problem:

[latex]P = $70,000, r = 0.06, t = 10, k = 1[/latex]

Using the compound interest formula to solve for the ending balance:

[latex]A = $125,359.34[/latex]

In the second stream of money, Ula is depositing $5,000 a year, a savings annuity problem:

[latex]N =10[/latex] because annual deposits are made into the account for 10 years.

[latex]I\% = 6[/latex] because the interest rate is 6%.

[latex]PV=0[/latex] because the account has no money in it when it is first opened.

[latex]PMT=-5000[/latex] because Ula is depositing $5,000 per year.

[latex]FV[/latex] is what we want to know.

[latex]P/Y=1[/latex] because Ula is making 1 payment per year.

[latex]C/Y=1[/latex] because [latex]P/Y[/latex] and [latex]C/Y[/latex] are always the same number.

Five years ago a couple purchased a home for $260,000, making a 20% down payment and financing the rest with a 30-year adjustable rate mortgage fixed at 3% for the first 5 years. Now that the fixed rate period is up, the couple is facing a higher adjustable rate. They now plan to refinance into a fixed rate 30-year mortgage at 4%. What will their new monthly payments be? Assume there are no costs associated with the refinance.