6.5 Miscellaneous Application Problems

In this section, you will learn to:

Find the outstanding balance of a loan before the loan term ends.
Solve financial situations that involve multiple stages of savings and/or annuities.
Find the fair market value of a bond.
Construct an amortization schedule for a loan.

We already developed the formulas needed for most finance problems. In this section, we put those tools together to solve practical applications.

This section matters because these are exactly the kinds of calculations people make in real life. If you refinance or sell a house, you need the outstanding loan balance. If you compare investments, you need bond valuation. If you take out a mortgage or car loan, you are living inside an amortization schedule whether you realize it or not.

Many finance problems look intimidating at first because they involve several steps. Usually the key is to identify what kind of calculation each stage uses: compound interest, present value, future value, or annuity payments.

OUTSTANDING BALANCE ON A LOAN

One very common finance problem is finding the balance still owed at some point during the life of a loan. For example, suppose someone buys a house with a 30-year mortgage but sells the house years before the mortgage ends. At the time of the sale, that borrower must pay off the lender, so the borrower needs to know the remaining balance.

Since many long-term loans are paid off early, this is an extremely practical calculation.

To find the outstanding balance of a loan at a specified time, we find the present value $P$ of all future payments that have not yet been made. In this setting:

$t$ represents the time still remaining on the loan.
$nt$ represents the number of future payments still remaining.

This is critical in refinancing and home sales. The bank does not care how much you originally borrowed at that moment—it cares about the present value of the payments you still owe.

Example 6.5.1

Mr. Jackson bought his house in 1995 and financed the loan for 30 years at an interest rate of \$7.8\%. His monthly payment was \$1260. In 2015, Mr. Jackson decides to pay off the loan. Find the balance of the loan he still owes.

Example 6.5.1 Solution

Notice that the original amount of the loan is not given. We do not need it.

The original loan was for 30 years. Since 20 years have passed, there are 10 years remaining.

Because the payments are monthly,

$$ 12(10)=120 $$

payments remain.

From the lender's point of view, Mr. Jackson still owes 120 monthly payments of \$1260. Since he wants to pay the whole loan off now, we find the present value of those remaining payments.

Using the present value formula for an annuity:

$$ P\left(1+\frac{0.078}{12}\right)^{120}= \frac{1260\left[\left(1+\frac{0.078}{12}\right)^{120}-1\right]}{0.078/12} $$ $$ P(2.17597)=227957.85 $$ $$ P=\$104,761.48 $$

So the balance Mr. Jackson still owes is \$104,761.48.

Outstanding Balance Formula

If a loan has payment $m$ dollars made $n$ times per year at interest rate $r$, and there are $t$ years still remaining on the loan, then the outstanding balance $P$ is given by:

$$ P\left(1+\frac{r}{n}\right)^{nt}= \frac{m\left[\left(1+\frac{r}{n}\right)^{nt}-1\right]}{r/n} $$

Here, $t$ is not the original term of the loan. It is the amount of time still remaining.

If the problem does not directly tell you how much time is left on the loan, calculate it first:

$$ t=\text{original term of loan}-\text{time already passed} $$

There are other ways to find the outstanding balance, but this is usually the simplest one.

One alternate method is to use an amortization schedule, which we will discuss later in this section. An amortization schedule shows each payment, the interest portion, the principal portion, and the remaining balance after each payment. It is tedious to do by hand, but spreadsheets make it manageable.

Another possible method is to calculate $A-B$, where:

$A$ = the original loan amount accumulated to the date when we want the outstanding balance.
$B$ = the accumulated value of all payments that have been made by that date.

This method is mathematically valid, but it requires more work because it uses a compound interest calculation, an annuity calculation, and then a subtraction. The formula above is much faster.

Try It Now 6.5.1

A borrower still has 5 years left on a loan with monthly payments of \$850. If the interest rate is \$6\% compounded monthly, find the outstanding balance.

Try It Now 6.5.1 Solution

There are $12(5)=60$ payments remaining.

$$ P\left(1+\frac{0.06}{12}\right)^{60}= \frac{850\left[\left(1+\frac{0.06}{12}\right)^{60}-1\right]}{0.06/12} $$

Solving gives:

$$ P\approx \$43,969.65 $$

PROBLEMS INVOLVING MULTIPLE STAGES OF SAVINGS AND/OR ANNUITIES

Now consider situations like these:

Suppose a baby, Aisha, is born and her grandparents invest \$8000 in a college fund. The money remains invested for 18 years until Aisha enters college, and then it is withdrawn in equal semiannual payments over the 4 years that Aisha expects to need to finish college. The college investment fund earns \$5\% interest compounded semiannually. How much money can Aisha withdraw from the account every six months while she is in college?
Aisha graduates college and starts a job. She saves \$1000 each quarter in a retirement savings account. Suppose that Aisha saves for 30 years and then retires. At retirement she wants to withdraw money as an annuity that pays a constant amount every month for 25 years. During the savings phase, the retirement account earns \$6\% interest compounded quarterly. During the annuity payout phase, the retirement account earns \$4.8\% interest compounded monthly. Calculate Aisha's monthly retirement annuity payout.

These problems may look complicated, but they can be broken into smaller parts. Often there is a savings stage followed by an annuity stage. The accumulated value from the first stage often becomes the present value for the second stage.

In multi-stage finance problems, the output of one stage becomes the input of the next stage. That is the big idea. Finish one stage carefully, then carry that value into the next calculation.

Example 6.5.2

Suppose a baby, Aisha, is born and her grandparents invest \$8000 in a college fund. The money remains invested for 18 years until Aisha enters college; then it is withdrawn in equal semiannual payments over the 4 years that Aisha expects to attend college. The college investment fund earns \$5\% interest compounded semiannually. How much can Aisha withdraw from the account every six months while she is in college?

Example 6.5.2 Solution

Part 1: Accumulation of college savings

Find the accumulated value at the end of 18 years of \$8000 invested at \$5\% compounded semiannually.

$$ A=8000\left(1+\frac{0.05}{2}\right)^{2\cdot18} =8000(1.025)^{36} =8000(2.432535) $$ $$ A=\$19,460.28 $$

Part 2: Semiannual annuity payout for college expenses

Now find the semiannual payout for 4 years using the accumulated savings from Part 1 at the same interest rate.

The amount

$$ A=\$19,460.28 $$

from Part 1 is the accumulated value at the end of the savings period. That amount becomes the present value

$$ P=\$19,460.28 $$

for the annuity calculation.

$$ 19460.28\left(1+\frac{0.05}{2}\right)^{2\cdot4}= \frac{m\left[\left(1+\frac{0.05}{2}\right)^{2\cdot4}-1\right]}{0.05/2} $$ $$ 23710.46=m(8.73612) $$ $$ m=\$2,714.07 $$

Aisha will be able to withdraw \$2714.07 every six months for college expenses.

This kind of two-stage problem is how college funds really work. First the money grows. Then the same account turns into a source of repeated withdrawals.

Example 6.5.3

Aisha graduates college and starts a job. She saves \$1000 each quarter, depositing it into a retirement savings account. Suppose that Aisha saves for 30 years and then retires. At retirement she wants to withdraw money as an annuity that pays a constant amount every month for 25 years. During the savings phase, the retirement account earns \$6\% interest compounded quarterly. During the annuity payout phase, the retirement account earns \$4.8\% interest compounded monthly. Calculate Aisha's monthly retirement annuity payout.

Example 6.5.3 Solution

Part 1: Accumulation of retirement savings

Find the accumulated value at the end of 30 years of \$1000 deposited at the end of each quarter into a retirement savings account earning \$6\% interest compounded quarterly.

$$ A=\frac{1000\left[\left(1+\frac{0.06}{4}\right)^{4\cdot30}-1\right]}{0.06/4} $$ $$ A=\$331,288.19 $$

Part 2: Monthly retirement annuity payout

Now find the monthly annuity payments for 25 years using the accumulated savings from Part 1 at an interest rate of \$4.8\% compounded monthly.

The amount

$$ A=\$331,288.19 $$

at the end of the savings phase becomes the present value

$$ P=\$331,288.19 $$

for the retirement payout phase.

$$ 331288.19\left(1+\frac{0.048}{12}\right)^{12\cdot25}= \frac{m\left[\left(1+\frac{0.048}{12}\right)^{12\cdot25}-1\right]}{0.048/12} $$ $$ 1097285.90=m(578.04483) $$ $$ m=\$1,898.27 $$

Aisha will have a monthly retirement annuity income of \$1898.27 when she retires.

Try It Now 6.5.2

An account contains \$5000 and earns \$4\% compounded annually for 10 years. After that, the money is withdrawn in equal annual payments over 5 years at the same interest rate. What is the annual withdrawal?

Try It Now 6.5.2 Solution

First grow the account:

$$ A=5000(1.04)^{10}\approx \$7,401.22 $$

Now use that as the present value for a 5-year annuity:

$$ 7401.22(1.04)^5=\frac{m[(1.04)^5-1]}{0.04} $$

So,

$$ m\approx \$1,663.70 $$

FAIR MARKET VALUE OF A BOND

Whenever a business—or the U.S. government—needs to raise money, it may do so by selling bonds. A bond is a certificate of promise that states the terms of the agreement. Usually a bond has a face value of \$1000 and a specified maturity date.

The bondholder pays the face value to buy the bond. In return, the bondholder is promised two things:

The return of the face value at maturity.
A fixed interest payment every six months.

As market interest rates change, the price of the bond changes too. Bonds are bought and sold at their fair market value.

If market interest rates go up, the bond becomes less attractive compared with newer investments, so its value drops. In that case, the bond trades at a discount.

If market interest rates go down, the bond becomes more attractive, so its value rises. In that case, the bond trades at a premium.

Bond valuation matters to investors because it tells them what a bond is actually worth today, not just what is printed on the certificate. The bond's price depends on how its promised payments compare with current market rates.

Example 6.5.4

The Orange Computer Company needs to raise money to expand. It issues a 10-year \$1000 bond that pays \$30 every six months. If the current market interest rate is 7%, what is the fair market value of the bond?

Example 6.5.4 Solution

The bond promises two things:

\$1000 at the end of 10 years
\$30 every six months for 10 years

So we find two present values and add them.

Let $P_1$ be the present value of the face amount.

$$ P_1\left(1+\frac{0.07}{2}\right)^{20}=1000 $$

Since the interest is paid twice a year,

$$ nt=2(10)=20 $$

Then:

$$ P_1(1.9898)=1000 $$ $$ P_1=\$502.56 $$

Now let $P_2$ be the present value of the \$30 semiannual payments.

$$ P_2\left(1+\frac{0.07}{2}\right)^{20}= \frac{30\left[\left(1+\frac{0.07}{2}\right)^{20}-1\right]}{0.07/2} $$ $$ P_2(1.9898)=848.39 $$ $$ P_2=\$426.37 $$

Present value of the lump-sum \$1000 = \$502.56
Present value of the \$30 semiannual payments = \$426.37

Therefore, the fair market value of the bond is:

$$ P=P_1+P_2=502.56+426.37=\$928.93 $$

Because the market interest rate of \$7\% is higher than the bond's implied interest rate of \$6\%, the bond is selling at a discount.

Example 6.5.5

A state issues a 15-year \$1000 bond that pays \$25 every six months. If the current market interest rate is \$4\%, what is the fair market value of the bond?

Example 6.5.5 Solution

The bond promises:

\$1000 at the end of 15 years
\$25 every six months for 15 years

So again, we find the present value of the face amount and the present value of the coupon payments, then add them.

Here,

$$ nt=2(15)=30 $$

Let $P_1$ be the present value of the lump sum \$1000.

$$ P_1\left(1+\frac{0.04}{2}\right)^{30}=1000 $$ $$ P_1=\$552.07 $$

Let $P_2$ be the present value of the \$25 semiannual payments.

$$ P_2\left(1+\frac{0.04}{2}\right)^{30}= \frac{25\left[\left(1+\frac{0.04}{2}\right)^{30}-1\right]}{0.04/2} $$ $$ P_2(1.8114)=1014.20 $$ $$ P_2=\$559.90 $$

Present value of the lump sum \$1000 = \$552.07
Present value of the \$25 semiannual payments = \$559.90

Therefore, the fair market value of the bond is:

$$ P=P_1+P_2=552.07+559.90=\$1111.97 $$

Because the market interest rate of \$4\% is lower than the bond's implied interest rate of \$5\%, the bond is selling at a premium.

Bond Valuation Summary

To find the fair market value of a bond:

Find the present value $P_1$ of the face amount $A$ payable at maturity:

$$ A=P_1\left(1+\frac{r}{n}\right)^{nt} $$

Find the present value $P_2$ of the semiannual payments of amount $m$:

$$ P_2\left(1+\frac{r}{n}\right)^{nt}= \frac{m\left[\left(1+\frac{r}{n}\right)^{nt}-1\right]}{r/n} $$

Add the two present values:

$$ P=P_1+P_2 $$

Try It Now 6.5.3

A 10-year \$1000 bond pays \$40 every six months. If the market rate is \$10\%, will the bond sell at a premium or a discount?

Try It Now 6.5.3 Solution

The bond pays \$80 per year on a \$1000 face value, so its implied rate is \$8\%.

Since the market rate is \$10\%, the bond's return is lower than the market rate.

So the bond sells at a discount.

AMORTIZATION SCHEDULE FOR A LOAN

An amortization schedule is a table that lists all payments on a loan, separates each payment into the portion that pays interest and the portion that repays principal, and shows the outstanding balance after each payment.

Amortization schedules are used in essentially every mortgage, car loan, and installment loan. They show why early loan payments are interest-heavy and why the principal shrinks more slowly at the beginning.

Example 6.5.6

An amount of \$500 is borrowed for 6 months at a rate of \$12\%. Make an amortization schedule showing the monthly payment, the monthly interest on the outstanding balance, the portion of the payment contributing toward reducing the debt, and the outstanding balance.

Example 6.5.6 Solution

The monthly payment is \$86.27.

For the first month, the outstanding balance is \$500, so the monthly interest is:

$$ (\text{outstanding balance})(\text{monthly interest rate})=(500)\left(\frac{0.12}{12}\right)=\$5 $$

This means that, in the first month, out of the \$86.27 payment, \$5 goes toward interest and the remaining \$81.27 goes toward the balance. The new balance is:

$$ 500-81.27=418.73 $$

In the second month, the outstanding balance is \$418.73, so the monthly interest is:

$$ (418.73)\left(\frac{0.12}{12}\right)=\$4.19 $$

Out of the \$86.27 payment, \$4.19 goes toward interest and the remaining \$82.08 goes toward the balance. The new balance is:

$$ 418.73-82.08=336.65 $$

The process continues as shown below.

Payment #	Payment	Interest	Debt Payment	Balance
1	\$86.27	\$5.00	\$81.27	\$418.73
2	\$86.27	\$4.19	\$82.08	\$336.65
3	\$86.27	\$3.37	\$82.90	\$253.75
4	\$86.27	\$2.54	\$83.73	\$170.02
5	\$86.27	\$1.70	\$84.57	\$85.45
6	\$86.27	\$0.85	\$85.42	\$0.03

The last balance of \$0.03 is caused by rounding.

Notice what happens over time: the interest part gets smaller and the debt-payment part gets larger. That happens because interest is computed on the remaining balance, and the remaining balance keeps shrinking.

How to Build an Amortization Schedule

For each payment:

Compute the interest on the current balance.
Subtract that interest from the payment to get the amount applied to principal.
Subtract the principal payment from the old balance to get the new balance.

An amortization schedule is often long and tedious to calculate by hand. For example, a 30-year mortgage with monthly payments would require:

$$ (12)(30)=360 $$

rows in the table.

A 5-year car loan with monthly payments would require:

$$ 12(5)=60 $$

rows.

Spreadsheet software is very useful for these repetitive calculations.

Most of the other applications in this section's problem set are reasonably straightforward, but they require careful interpretation. Also remember: there is often more than one valid way to solve a finance problem.

Try It Now 6.5.4

A borrower owes \$1000 on a loan with monthly interest rate \$1\%. If the monthly payment is \$90, how much of the first payment goes to principal?

Try It Now 6.5.4 Solution

Monthly interest:

$$ 1000(0.01)=\$10 $$

Principal paid:

$$ 90-10=\$80 $$

So \$80 of the first payment reduces the debt.

SECTION 6.5 PROBLEM SET: MISCELLANEOUS APPLICATION PROBLEMS

Problem Set 6.5

For problems 1-4, assume a \$200{,}000 house loan is amortized over 30 years at an interest rate of \$5.4\%.

Problem 1. Find the monthly payment.	Problem 2. Find the balance owed after 20 years.
Problem 3. Find the balance of the loan after 100 payments.	Problem 4. Find the monthly payment if the original loan were amortized over 15 years.

Problem 5. Mr. Patel wants to pay off his car loan. The monthly payment for his car is \$365, and he has 16 payments left. If the loan was financed at \$6.5\%, how much does he owe?

Problem 6. An amount of \$2000 is borrowed for a year at a rate of \$7\%. Make an amortization schedule showing the monthly payment, the monthly interest on the outstanding balance, the portion of the payment going toward reducing the debt, and the balance.

Problem 7. Fourteen months after Dan bought his new car he lost his job. His car was repossessed by his lender after he made only 14 monthly payments of \$376 each. If the loan was financed over a 4-year period at an interest rate of 6.3%, how much did the car cost the lender? In other words, how much did Dan still owe on the car?	Problem 8. You have a choice of either receiving \$5,000 at the end of each year for the next 5 years or receiving \$3000 per year for the next 10 years. If the current interest rate is 9%, which is better?
Problem 9. Mr. Smith is planning to retire in 25 years and would like to have \$250,000 then. What monthly payment made at the end of each month to an account that pays 6.5% will achieve his objective?	Problem 10. Assume Mr. Smith has reached retirement and has \$250,000 in an account which is earning 6.5%. He would now like to make equal monthly withdrawals for the next 15 years to completely deplete this account. Find the withdrawal payment.
Problem 11. Mrs. Garcia is planning to retire in 20 years. She starts to save for retirement by depositing \$2000 each quarter into a retirement investment account that earns 6% interest compounded quarterly. Find the accumulated value of her retirement savings at the end of 20 years.	Problem 12. Assume Mrs. Garcia has reached retirement and has accumulated the amount found in question 11 in a retirement savings account. She would now like to make equal monthly withdrawals for the next 15 years to completely deplete this account. Find the withdrawal payment. Assume the account now pays 5.4% compounded monthly.

Problem 13. A ten-year \$1,000 bond pays \$35 every six months. If the current interest rate is 8.2%, find the fair market value of the bond. Hint: You must do the following. Find the present value of \$1000. Find the present value of the \$35 payments. The fair market value of the bond = a + b	Problem 14. Find the fair market value of the ten-year \$1,000 bond that pays \$35 every six months, if the current interest rate has dropped to 6%. Hint: You must do the following. Find the present value of \$1000. Find the present value of the \$35 payments. The fair market value of the bond = a + b
Problem 15. A twenty-year \$1,000 bond pays \$30 every six months. If the current interest rate is 4.2%, find the fair market value of the bond. Hint: You must do the following. Find the present value of \$1000. Find the present value of the \$30 payments. The fair market value of the bond = a + b	Problem 16. Find the fair market value of the twenty-year \$1,000 bond that pays \$30 every six months, if the current interest rate has increased to 7.5%.

Problem 17. Mr. and Mrs. Nguyen deposit \$10,000 into a college investment account when their new baby grandchild is born. The account earns 6.25% interest compounded quarterly.

a. When their grandchild reaches the age of 18, what is the accumulated value of the college investment account?

b. The Nguyens' grandchild has just reached the age of 18 and started college. If she is to withdraw the money in the college savings account in equal monthly payments over the next 4 years, how much money will be withdrawn each month?

Problem 18. Mr. Singh is 38 and plans to retire at age 65. He opens a retirement savings account.

a. Mr. Singh wants to save enough money to accumulate \$500,000 by the time he retires. The retirement investment account pays 7% interest compounded monthly. How much does he need to deposit each month to achieve this goal?

b. Mr. Singh has now reached age 65 and retires. How much money can he withdraw each month for 25 years if the retirement investment account now pays \$5.2\% interest, compounded monthly?