# 4.6 Sinking Funds

## Learning Objectives

• Construct a sinking fund schedule

## Formula & Symbol Hub

#### Symbols Used

• $BAL$ = Principal balance
• $BV$ = Book value
• $FV$ = Face value of bond
• $PMT$ = Annuity payment amount
• $N$ = Number of annuity payments
• $I/Y$ = Nominal interest rate
• $P/Y$ = Number of payments per year or payment frequency
• $C/Y$ = Number of compounds per year or compounding frequency

#### Formulas Used

• ##### Formula 4.1 – Bond Payment

$\text{Payment}=\mbox{Face Value} \times \mbox{periodic bond rate}$

• ##### Formula 4.2 – Purchase Price on an Interest Payment Date

$\text{Purchase Price}=\mbox{Present Value of Face Value}+\mbox{Present Value of Bond Payments}$

• ##### Formula 4.3 – Premium (Bonds)

$\text{Premium}=\mbox{Purchase Price}-\mbox{Face Value}$

• ##### Formula 4.4 – Discount (Bonds)

$\text{Discount}=\mbox{Face Value}-\mbox{Purchase Price}$

• ##### Formula 4.5 – Quoted Price (Bonds)

$\mbox{Flat Price}-\mbox{Accrued Interest}$

• ##### Formula 4.6 – Quoted Price as a Percentage of Face Value

$\frac{\mbox{Quoted Price}}{\mbox{Face Value}} \times 100\%$

• ##### Formula 4.7 – Periodic Cost of Debt

$\mbox{Sinking Fund Payment}+\mbox{Periodic Interest Payment}$

• ##### Formula 4.8 – Book Value

$\mbox{Loan Amount}-\mbox{Balance}$

## Introduction

A sinking fund is a special account into which an investor, either an individual or a business, makes annuity payments so that sufficient funds are on hand by a specified date to meet a future savings goal or debt obligation. In its simplest terms, a sinking fund is a financial savings place. As the definition indicates, a sinking fund has is used for one of two main purposes:

• Capital Savings. When your goal is to acquire some form of a capital asset by the end of the fund, you have a capital savings sinking fund. What is a capital asset? It is any tangible property that is not easily converted into cash. For example, saving up to buy a home, car, warehouse, or even new production machinery are all capital savings.
• Debt Retirement. When your goal is to pay off some form of debt by the end of the fund, you have a debt retirement sinking fund. Perhaps as a consumer, if you were able to get a $0\%$ interest plan with no payments for one year, you might want to make monthly payments into your own savings account so that you would have the needed funds to pay off your purchase when it comes due. Businesses usually set up sinking funds for the retirement of stocks, bonds, and debentures.

## Sinking Funds and Debt Retirement

Whether the sinking fund is for capital savings or debt retirement, the mathematical calculations and procedures are identical. However, this section will focus on using sinking funds for debt retirement. Now, why discuss sinking funds in the chapter about bonds? Many bonds carry a sinking fund provision. Once the bond has been issued, the company must start regular contributions to a sinking fund because large sums of money have been borrowed over a long time frame, and investors need assurance that the bond issuer will be able to repay its debt upon bond maturity.

To provide further assurance to bondholders, the sinking fund is typically managed by a neutral third party rather than the bond-issuing company. This third-party company ensures the integrity of the fund, working toward the debt retirement in a systematic manner according to the provisions of the sinking fund. Investors much prefer bonds or debentures that are backed by sinking funds and third-party management because they are less likely to default.

Sinking funds are an alternative way to pay off a loan or debt. A sinking fund is used to accumulate the principal only owed on a debt so that the principal of the debt can be repaid in its entirety on the maturity date. For example, sinking funds are used to accumulate the face value of bonds so that money is available to pay the face value at maturity. Sinking funds are NOT used to pay the interest due on the debt. For example, sinking funds are not used to pay the periodic bond payments.

In the case of bonds or debentures, sinking funds are most commonly set up as ordinary simple annuities that match the timing of the bond interest payments. Thus, when a bond issuer makes a bond interest payment to its bondholders, it also makes an annuity payment to its sinking fund. In other applications, any type of annuity is possible, whether ordinary or due, general or simple.

When a sinking fund is used to retire a debt, there are two interest rates associated with the debt.

• Interest rate for the loan or debt. Because sinking funds are used to pay off loans, bonds, or other debts, there is an interest rate for calculating the amount of interest charged on the loan. For example, if the sinking fund is used to accumulate the face value of a bond, the interest rate on the debt is the coupon rate of the bond. This interest rate is only for calculating the interest due on the loan or debt, and does not have anything to do with the money deposited into the sinking fund.
• Interest rate for the sinking fund. Because a sinking fund is an interest-earning account, there is an interest rate for calculating the interest earned by the money in the sinking fund.

When a sinking fund is established to retire a debt, there are two different periodic costs or expenses made in relation to the debt.

• Periodic interest payments on the debt. The interest rate for the loan or debt is used to calculate the periodic interest payment. For example, if a sinking fund is used to accumulate the face value of a bond, the bond issuer still has to make the periodic bond payments to the bond holders. These periodic interest payments are not directly related to the sinking fund, but are a cost associated with the debt.
• Periodic payments made to the sinking fund.When a periodic interest payment is made on the debt, a payment is made into the sinking fund with the goal of accumulating the loan amount. The interest rate for the sinking fund is used to calculate the periodic annuity payment for the sinking fund. For example, if a sinking fund is used to accumulate the face value of a bond, the bond issuer makes a deposit into the sinking fund at the same time they make the bond payment. Because these periodic sinking fund payments are made as a result of a debt, they are also costs associated with the debt.

The periodic cost of a debt retired with a sinking fund is the total amount paid each payment interval as a result of the debt.

### $\boxed{4.7}$ Periodic Cost of Debt

$\Large\begin{eqnarray*}{\color{red}{\mbox{Periodic Cost of the Debt}}}&=&{\color{blue}{\mbox{Periodic Interest Payment}}}\\&+&{\color{green}{\mbox{Periodic Sinking Fund Payment}}}\end{eqnarray*}$

The payments made into a sinking fund form an annuity, and are calculated the same as any other annuity payment with the future value of the sinking fund set to the loan amount and using the interest rate associated with the sinking fund. Because the goal of the sinking fund is to accumulate at least the required amount, sinking fund payments are always rounded UP to the next cent. Consequently, all of the payments made to a sinking fund, including the last payment, are the same.

### Example 4.6.1

A bank issued a $\10,000,000$ face value bond carrying a $5.1\%$ coupon and $30$ years until maturity. The bank set up a sinking earning $4.5\%$ to accumulate the face value of the bond.

1. Calculate the sinking fund payment.
2. Calculate the periodic expense of the debt.
Solution

Step 1: The given information for the sinking fund is

Because no other information is given, the frequency of the payments (for both the bond and the sinking fund) and the compounding frequencies (for the coupon rate and the sinking fund rate) are assumed to be semi-annual.

$\begin{eqnarray*} FV & = & \10,000,000 \\ I/Y & = & 4.5\% \\ P/Y & = & 2 \\ C/Y & = & 2 \\ t & = & 30 \mbox{ years} \end{eqnarray*}$

Step 2: Calculate the sinking fund payment.

 PMT Setting END $N$ $2 \times 30=60$ $PV$ $0$ $FV$ $10,000,000$ $PMT$ $?$ $I/Y$ $4.5$ $P/Y$ $2$ $C/Y$ $2$

$PMT=\80,353.2748... \rightarrow \80,353.28$

The sinking fund payment is $\80,353.28$. Remember, sinking fund payments always get rounded UP to the next cent.

Step 3: Calculate the bond payment.

$\begin{eqnarray*} \mbox{Bond Payment}& = & FV \times \frac{\mbox{coupon rate}}{2} \\ & = & 10,000,000\times \frac{0.051}{2}\\ & = & \255,000\end{eqnarray*}$

The bond payments are $\255,000$.

Step 4: Calculate the periodic cost of the debt.

$\begin{eqnarray*} \mbox{Periodic Cost of the Debt} & = & \mbox{Periodic Interest Payment}+\mbox{Periodic Sinking Fund Payment}\\ & = & 255,000+80,353.28 \\ & = & \335,353.28\end{eqnarray*}$

Step 5: Write as a statement.

The periodic cost of the debt is $\335,353.28$. This means that every six months the bank must pay out a total of $\335,353.28$ because of the debt. Of this amount, $\255,000$ goes to paying the bond payments and $\80,353.28$ goes to the sinking fund to accumulate the $\10,000,000$ face value of the bonds.

### Key Takeaways

The goal of a sinking fund is to accumulate the loan amount so that the loan amount can be paid off in one lump-sum payment at the end of the term. So, the loan amount becomes the future value of the sinking fund.

Sinking fund payments always get rounded UP to the next cent. This ensures that the final amount in the sinking fund will be at or over the loan amount. Rounding the payment up guarantees that the balance in the sinking fund at the end of the term will always be at or over the loan amount.

### Try It

1) A company issued bonds worth $\200,000$ to raise money to build an expansion to its factory. The bonds have a coupon rate of $3.9\%$ compounded semi-annually and ten years to maturity. The company established a sinking fund earning $2.7\%$ compounded semi-annually to accumulate the face value of the bonds.

1. Calculate the sinking fund payment.
2. Calculate the periodic expense of the debt.
Solution

a. Calculate the sinking fund payment.

 PMT Setting END $N$ $2 \times 10=20$ $PV$ $0$ $FV$ $200,000$ $PMT$ $?$ $I/Y$ $2.7$ $P/Y$ $2$ $C/Y$ $2$

$PMT=\8,777.63$

b. Calculate the periodic expense of the debt.

$\begin{eqnarray*} \mbox{Bond Payment} & = & FV \times \frac{\mbox{coupon rate}}{2} \\ & = & 200,000\times \frac{0.039}{2}\\ & = & \3,900\end{eqnarray*}$

Write as a statement.

$\begin{eqnarray*} \mbox{Periodic Cost of the Debt} & = & \mbox{Periodic Interest Payment}+\mbox{Periodic Sinking Fund Payment}\\ & = & 3,900+8,777.63 \\ & = & \12,677.63\end{eqnarray*}$

## Sinking Fund Schedules

When a company takes out a loan or issues bonds, these are debts to the company. Through a sinking fund the company saves up money to extinguish that debt. The book value of the debt is the difference between the principal amount owing on the debt (i.e. the loan amount or face value of the bond) and the accumulated balance in the sinking fund at any point in time. For example, if the company issued $\10$ million in bonds and has accumulated $\1$ million in its sinking fund, the book value of the debt is $\9$ million.

### $\boxed{4.8}$ Book Value

$\Large{\color{red}{\mbox{Book Value}}}={\color{blue}{\mbox{Principal}}}-{\color{green}{\mbox{Fund Balance}}}$

A sinking fund schedule is a table that records the sinking fund contribution, the interest earned by the fund, the increase in the fund, the accumulated balance for every payment, and the current book value of the debt. A sinking fund schedule is very similar to an amortization schedule except that the balance increases instead of decreases and the interest is earned instead of being paid.

A sinking fund schedule has six columns:

• Payment Number. There is a row for every payment into the sinking fund.
• Payment. The sinking fund payment ($PMT$). Because the sinking fund payment is rounded up, all of the payments are the same, including the last payment.
• Interest. The interest earned by the fund at the end of each payment interval.
• Increase. The total amount added to the fund with each payment interval.
• Balance. The current amount accumulated in the fund for each payment interval.
• Book Value. The book value of the debt.

To fill in a sinking fund schedule, you first need to have all of the details about the fund, including the loan amount ($FV$), the sinking fund payment ($PMT$), the number of payments ($N$), and the sinking fund’s interest rate. If any of these quantities are missing, calculate out the missing value before completing the sinking fund schedule.

 Payment Number Payment Interest Increase Balance Book Value $0$ $0$ $\text{Loan Amount}^1$ $1$ $PMT^2$ $INT^3$ $INC^4$ $BAL^5$ $BV^6$ $2$ $PMT^2$ $INT^3$ $INC^4$ $BAL^5$ $BV^6$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $N-1$ $PMT^2$ $INT^3$ $INC^4$ $BAL^5$ $BV^6$ $N$ $PMT^2$ $INT^3$ $INC^4$ $BAL^5$ $BV^6$ Totals $\text{Total Payments}^8$ $\text{Total Interest}^10$ $\text{Total Increase}^9$

### HOW TO

#### Fill In a Sinking Fund Schedule

Follow these steps to fill in a sinking fund schedule.

Step 1: In row $0$, the only entries are in the balance and book value columns. The initial balance is $0$ and the initial book value is the loan amount (the future value of the sinking fund).

Step 2: Each entry in the payment column is the sinking fund payment. If you have to calculate out the payment, remember to round the payment up to the next cent. All of the payments in this column are the same, including the last payment.

Step 3: Calculate the interest. The interest is the balance from the previous row times the periodic interest rate:

$\mbox{Interest}=\mbox{Balance from Previous Row} \times i$

Note: this calculation uses the periodic sinking fund rate, not the periodic interest rate associated with the loan.

Step 4: Calculate the increase. The increase is the sum of the payment and the interest:

$\mbox{Increase}=PMT+\mbox{Interest}$.

Step 5: Calculate the new balance. The balance is the sum of the balance in the previous row and the increase:

$\mbox{Balance}=\mbox{Balance from Previous Row}+\mbox{Increase}$

Step 6: Calculate the new book value. The book value is the difference between the book value from the previous row and the increase:

$\mbox{Book Value}=\mbox{Book Value from Previous Row}-\mbox{Increase}$.

Step 7: For each payment, repeat steps $2$ through $6$, including for the last row.

Step 8: The total payments is the sum of the payment column:

$\mbox{Total Payments}=N \times PMT$.

Step 9: The total increase is the sum of the increase column, and is the last balance entry:

$\mbox{Total Increase}=\mbox{Balance in Last Row}$.

Step 10: The total interest is the sum of the interest column, and equals the difference between the other two column totals:

$\mbox{Total Interest}=\mbox{Total Payments}-\mbox{Total Increase}$.

## Paths to Success

The manual calculation of the interest entry above is based on the assumption that the payment frequency and the compounding frequency are equal. If the payment frequency and the compounding frequency are not equal, an interest conversion would be required to convert the interest rate to the equivalent rate with the compounding frequency equal to the payment frequency. However, if you use the TI BAII Plus’s built-in amortization worksheet (described below), no interest conversion is required.

As you fill in the schedule, round the entries to two decimal places.

The sinking fund schedule presented here assumes the payments are made at the end of the payment interval. That is, the sinking fund schedule presented above is for an ordinary annuity. If the sinking fund is an annuity due (payments at the beginning), the calculations are the same except for the interest column, where the interest is based on both the balance from the previous row and the payment.

### Example 4.6.2

A company has to repay a $\20,000$ loan in two years. The company establishes a sinking fund earning $4\%$ compounded semi-annually and makes end-of-six-month payments into the fund to accumulate the loan amount. Construct the sinking fund schedule.

Solution

Step 1: The given information is

$\begin{eqnarray*} FV & = & \20,000 \\ I/Y & = & 4\% \\ P/Y & = & 2 \\ C/Y & = & 2 \\ t & = & 2 \mbox{ years} \end{eqnarray*}$

Step 2: Calculate the sinking fund payment.

 PMT Setting END $N$ $2 \times 2=4$ $PV$ $0$ $FV$ $20,000$ $PMT$ $?$ $I/Y$ $4$ $P/Y$ $2$ $C/Y$ $2$

$PMT=\4,852.48$

The sinking fund payment is $\4,852.48$.

Step 3: Complete the sinking fund schedule.

Because the payment frequency and the compounding frequency are equal, no interest conversion is required. The calculations for each entry are shown in blue. The periodic interest rate is $i=\frac{4\%}{2}=2\%$.

 Payment Number Payment Interest Increase Balance Book Value $0$ $\0$ $\20,000$ $1$ $\4,852.48$ $\;\;\;\;\;\;\;\;\;\;\0\;\;\;\;\;\;\;\;\;\;$$\color{blue}{(0 \times 0.02)}$ $\4,852.48$$\color{blue}{(4,852.48+0)}$ $\4,852.48$$\color{blue}{(0+4,852.48)}$ $\15,147.52$$\;\color{blue}{(20,000-4,582.48)}$ $2$ $\4,852.48$ $\;\;\97.05\;\;$$\color{blue}{(4582.48 \times 0.02)}$ $\4,949.53$$\color{blue}{(4852.48+97.05)}$ $\9,802.01$$\;\color{blue}{(4852.48+4949.53)}$ $\10,197.99$$\color{blue}{(15,147.52-4949.53)}$ $3$ $\4,852.48$ $\;\196.04\;$$\color{blue}{(9802.01 \times 0.02)}$ $\5,048.52$$\color{blue}{(4852.48+196.04)}$ $\14,850.5$$\;\color{blue}{(9802.01+5048.52)}$ $\5,149.47$$\color{blue}{(10,197.99-5048.52)}$ $4$ $\4,852.48$ $\297.01$$\color{blue}{(14,850.53 \times 0.02)}$ $\5,149.49$$\color{blue}{(4852.48+297.01)}$ $\20,000.02$$\color{blue}{(14,850.53+5149.49)}$ $-\0.02$$\;\color{blue}{(5149.47-5149.49)}$ Totals $\19,409.92$$\color{blue}{(4 \times 4852.48)}$ $\590.10$$\color{blue}{(20,000.02-19,409.92)}$ $\20,000.02$

### Things to Watch Out For

In the previous example, the final balance is slightly more than the required $\20,000$ because the sinking fund payment was rounded up to the next cent. By rounding the payment up, we ensure that the sinking fund has at least $\20,000$ at the end of the term.

Although the calculations in a sinking fund schedule are relatively straightforward, the manual calculations are time-consuming, especially when the schedule has a lot of rows. The amortization worksheet on a financial calculator, such as the TI BAII Plus, can be used to quickly calculate the entries for each row of the schedule.

## Using the TI BAII Plus Calculator to Construct a Sinking Fund Schedule

To use the amortization worksheet to complete a sinking fund schedule:

1. Solve for any unknown quantities about the sinking fund. You need to know all of the information about the sinking fund first before you can use the amortization worksheet.
2. Enter all the value of all seven time value of money variables into the calculator ($N$, $PV$, $FV$, $PMT$, $I/Y$, $P/Y$, $C/Y$). If you calculated the payment in the first step, you must re-enter it rounded up to the next cent and with the correct cash flow sign. Make sure the payment setting is set to END, and obey the cash flow sign convention. Because $PMT$ (the sinking fund payment) is paid out to the fund, $PMT$ is negative. At the end of the term, $FV$ is received, so $FV$ is positive.
3. Go to the amortization worksheet by pressing 2nd AMORT (the $PV$ button).
4. To view the entries for a specific row of the schedule, set $P_1$ and $P_2$ to the row number. For example, to view the entries for row $5$, set $P_1=5$ and $P_2=5$.
1. At the $P_1$ prompt, enter the row number and press ENTER.
2. Press the down arrow.
3. At the $P_2$ prompt, enter the row number and press ENTER.
4. Press the down arrow.
5. The $BAL$ entry is the balance entry for the corresponding row.
6. Press the down arrow.
7. The $PRN$ entry is the increase entry for the corresponding row.
8. Press the down arrow.
9. The $INT$ entry is the interest entry for the corresponding row.
10. Press the down arrow the return to the $P_1$ screen.
5. Repeat the previous step with a different row number to view the entries for a different row.

### Key Takeaways

On the amortization worksheet, $BAL$ is the balance entry, $PRN$ is the increase entry, and $INT$ is the interest entry.

You cannot get the entries for the last column, the book value, from the amortization worksheet on the calculator. This entry will still need to be calculated manually. You can find the book value for any row by subtracting the balance for the row from the loan amount:

$\mbox{Book Value} =\mbox{Principal}-\mbox{Fund Balance}$

Make sure to re-enter $PMT$ rounded up to the next cent before using the amortization worksheet. Otherwise, you will not get the correct entries for the sinking fund schedule.

As you read the entries off of the amortization worksheet on the calculator and put them in the schedule, round the entries to $2$ decimal places.

### Example 4.6.3

A company set up a sinking fund to accumulated the $\30,000$ they need to repay a loan. The sinking fund earns $3.5\%$ compounded semi-annually. The company made semi-annual deposits into the sinking fund for $2.5$ years. Construct the sinking fund schedule.

Solution

Step 1: Calculate the sinking fund deposit.

 PMT Setting END $N$ $2 \times 2.5=5$ $PV$ $0$ $FV$ $30,000$ $PMT$ $?$ $I/Y$ $3.5$ $P/Y$ $2$ $C/Y$ $2$

$PMT=\5,793.65$

Step 2: Enter the information into the time value of money buttons on the calculator.

 PMT Setting END $N$ $5$ $PV$ $0$ $FV$ $30,000$ $PMT$ $-5,793.65$ $I/Y$ $3$ $P/Y$ $2$ $C/Y$ $2$

Step 4: Complete the sinking fund schedule using the amortization worksheet on the calculator.

 Payment Number Payment Interest Increase Balance Book Value $0$ $\0$ $\30,000$ $1$ $\5,793.65$ $\0$ $\5,793.65$ $\5,793.65$ $\24,206.35$ $2$ $\5,793.65$ $\101.39$ $\5,895.04$ $\11,688.69$ $\18,311.31$ $3$ $\5,793.65$ $\204.55$ $\5,998.20$ $\17,686.89$ $\12,313.11$ $4$ $\5,793.65$ $\309.52$ $\6,103.17$ $\23,790.06$ $\6,209.94$ $5$ $\5,793.65$ $\416.33$ $\6,209.98$ $\30,000.04$ $-\0.04$ Totals $\28,968.25$ $\1,031.79$ $\30,000.04$
• Row 1: In the amortization worksheet, set $P_1=1$ and $P_2=1$. The entry for the last column (the book value) is $30,000-5,793.65$.
• Row 2: In the amortization worksheet, set $P_1=2$ and $P_2=2$. The entry for the last column (the book value) is $30,000-11,688.69$.
• Row 3: In the amortization worksheet, set $P_1=3$ and $P_2=3$. The entry for the last column (the book value) is $30,000-17,686.89$.
• Row 4: In the amortization worksheet, set $P_1=4$ and $P_2=4$. The entry for the last column (the book value) is $30,000-23,790.06$.
• Row 5: In the amortization worksheet, set P$1=5$ and P$2=5$. The entry for the last column (the book value) is $30,000-30,000.04$.
• Totals Row:
• The increase total is the last balance entry ($\30,000.04$).
• The payments total is the sum of the payments: $5 \times 5,793.65=28,968.25$.
• The interest total is the difference in the other two column totals: $30,000.04-28,968.25=1,031.79$.

### Try It

2) A $\10,000$ loan was repaid using a sinking fund that was earning $4.5\%$ compounded semi-annually. Deposits were made every six months into the fund for three years to accumulate the loan amount. Construct the sinking fund schedule.

Solution
 PMT Setting END $N$ $2 \times 3=6$ $PV$ $0$ $FV$ $10,000$ $PMT$ $?$ $I/Y$ $4.5$ $P/Y$ $2$ $C/Y$ $2$

$PMT=\1,575.35$

 Payment Number Payment Interest Increase Balance Book Value $0$ $\0$ $\10,000$ $1$ $\1,575.35$ $\0$ $\1,575.35$ $\1,575.35$ $\8,424.65$ $2$ $\1,575.35$ $\35.45$ $\1,610.80$ $\3,186.15$ $\6,813.85$ $3$ $\1,575.35$ $\71.69$ $\1,647.04$ $\4,833.18$ $\5,166.82$ $4$ $\1,575.35$ $\108.75$ $\1,684.10$ $\6,517.28$ $\3,482.72$ $5$ $\1,575.35$ $\146.64$ $\1,721.99$ $\8,239.27$ $\1,760.73$ $6$ $\1,575.35$ $\185.38$ $\1,760.73$ $\10,000$ $\0$ Totals $\9,452.10$ $\547.90$ $\10,000$

## Other Sinking Fund Calculations

Similar to working with loans, the amortization worksheet on the financial calculator can be applied to sinking funds to find a partial sinking fund schedule, to find the total interest or the total increase for a series of payments, to find the balance in the fund after any payment, or to find the book value after any payment.

### Example 4.6.4

A company took out a $\25,000$ loan and establishes a sinking fund earning $2.7\%$ compounded semi-annually to accumulate the loan amount. The company makes semi-annual payments into the fund for ten years. Construct a partial sinking fund schedule showing the details of payment #$7$, the last two payments and the totals.

Solution

Step 1: Calculate the sinking fund deposit.

 PMT Setting END $N$ $2 \times 10=20$ $PV$ $0$ $FV$ $25,000$ $PMT$ $?$ $I/Y$ $2.7$ $P/Y$ $2$ $C/Y$ $2$

$PMT=\1,097.21$

Step 2: Enter the information into the time value of money buttons on the calculator.

 PMT Setting END $N$ $20$ $PV$ $0$ $FV$ $25,000$ $PMT$ $-1,097.21$ $I/Y$ $2.7$ $P/Y$ $2$ $C/Y$ $2$

Step 3: Complete the partial sinking fund schedule using the amortization worksheet on the calculator.

 Payment Number Payment Interest Increase Balance Book Value $7$ $\1,097.21$ $\91.93$ $\1,189.14$ $\7,988.62$ $\17,011.38$ $19$ $\1,097.21$ $\299.54$ $\1,396.75$ $\23,584.57$ $\1,415.43$ $20$ $\1,097.21$ $\318.39$ $\1,415.60$ $\25,000.17$ $-\0.17$ Totals $\21,944.20$ $\3,055.97$ $\25,000.17$
• Row 7: In the amortization worksheet, set $P_1=7$ and $P_2=7$.
• Row 19: In the amortization worksheet, set $P_1=19$ and $P_2=19$.
• Row 20: In the amortization worksheet, set $P_1=20$ and $P_2=20$.
• Book Value for each Row:$\mbox{Book Value}=25,000-\mbox{Balance}$
• Totals Row:
• The increase total is the last balance entry ($\25,000.17$).
• The payments total is the sum of the payments: $20 \times 1,097.21=21,944.20$.
• The interest total is the difference in the other two column totals: $25,000.17-21,944.20=3,055.97$.

## Using the TI BAII Plus Calculator for Other Sinking Fund Calculations

To use the amortization worksheet to find the total interest, the total increase or the balance for a sinking fund:

1. Solve for any unknown quantities about the sinking fund. You need to know all of the information about the sinking fund first before you can use the amortization worksheet.
2. Enter all the value of all seven time value of money variables into the calculator ($N$, $PV$, $FV$, $PMT$, $I/Y$, $P/Y$, $C/Y$). If you calculated the payment in the first step, you must re-enter it rounded up to the next cent and with the correct cash flow sign. Make sure the payment setting is set to END, and obey the cash flow sign convention. Because $PMT$ (the sinking fund payment) is paid out to the fund, $PMT$ is negative. At the end of the term, $FV$ is received, so $FV$ is positive.
3. Go to the amortization worksheet by pressing 2nd AMORT (the $PV$ button).
4. To find the balance in the fund after a payment, set $P_1$ and $P_2$ equal to the payment number. The $BAL$ entry is the balance after payment number $P_2$. Note that the $BAL$ entry is only tied to the value of $P_2$ and does not depend on the value of $P_1$.
5. To view the total increase or the total interest for a series of payments, set $P_1$ to the first payment number of the series of payments and set $P_2$ to the last payment number of the series of payments. For example, to view the total increase or the total interest for payments four to seven, set $P_1=4$ and $P_2=7$. In the outputs from the amortization worksheet:
• The $PRN$ entry is the total increase from payment number $P_1$ to payment number $P_2$.
• The $INT$ entry is the total interest from payment number $P_1$ to payment number $P_2$.

### Key Takeaways

The $PRN$ entry on the amortization worksheet is the sum of the increase entries in the sinking fund schedule starting at payment number $P_1$ and ending at payment number $P_2$. For example, if $P_1=4$ and $P_2=7$. the $PRN$ entry tells you the sum of the increase column in the sinking fund schedule starting with payment number $4$ and ending with payment number $7$.

The $INT$ entry on the amortization worksheet is the sum of the interest entries in the sinking fund schedule starting at payment number $P_1$ and ending at payment number $P_2$. For example, if $P_1=4$ and $P_2=7$, the $INT$ entry tells you the sum of the interest column in the sinking fund schedule starting with payment number $4$ and ending with payment number $7$.

The calculator thinks in terms of payment numbers, not years. That is, $P_1$ and $P_2$ must be payment numbers.

### Example 4.6.5

A company issued $\500,000$ worth of bonds with $15$ years to maturity. The company set up a sinking fund earning $4.7\%$ compounded semi-annually to accumulate the face value of the bonds and made semi-annual payments into the sinking fund.

1. What is the sinking fund payment?
2. What was the balance in the fund after payment $18$?
3. By how much did the sinking fund increase with payment $9$?
4. How much interest was paid in year $10$?
5. What was the book value in the fund at the end of year $8$?
Solution

Step 1: Calculate the payment.

 PMT Setting END $N$ $12 \times 15=30$ $PV$ $0$ $FV$ $500,000$ $PMT$ $?$ $I/Y$ $4.7$ $P/Y$ $2$ $C/Y$ $2$

$PMT=\11,663.61$

Step 2: Calculate the balance for payment $18$.

To find the balance for payment $18$, set $P_1=18$ and $P_2=18$.

 PMT Setting END $N$ $30$ $PV$ $0$ $FV$ $500,000$ $PMT$ $-11,663.61$ $I/Y$ $4.7$ $P/Y$ $2$ $C/Y$ $2$ $P_1$ $18$ $P_2$ $18$

$BAL=\257,632.82$

After $18$ payments, the balance in the fund is $\257,632.82$.

Step 3: Calculate the increase for payment $9$.

To find the interest for payment $9$, set $P_1=9$ and $P_2=9$.

 PMT Setting END $N$ $30$ $PV$ $0$ $FV$ $500,000$ $PMT$ $-11,663.61$ $I/Y$ $4.7$ $P/Y$ $2$ $C/Y$ $2$ $P_1$ $9$ $P_2$ $9$

$PRN=\14,045.45$

The increase for payment $9$ is $\14,045.45$.

Step 4: Calculate the interest paid for year ten.

To find the interest paid in year two, set $P_1$ to first payment number of year ten and $P_2$ to last payment of year ten. There are $2$ payments a year, so the last payment made in year two is $20$ ($2\times 10$). The first payment made in year ten is $19$ ($2\times 9+1$). So, to find the interest paid in year ten, set $P_1=19$ and $P_2=20$.

 PMT Setting END $N$ $30$ $PV$ $0$ $FV$ $500,000$ $PMT$ $-11,663.61$ $I/Y$ $4.7$ $P/Y$ $2$ $C/Y$ $2$ $P_1$ $10$ $P_2$ $20$

$INT=\12,525.11$

The interest paid in year ten is $\12,525.11$.

Step 5: Calculate the book value after eight years.

To find the book value, calculate the balance after eight years and then subtract the balance from $\500,000$. To find the balance after eight years, enter the payment number that corresponds to the last payment made in year eight. There are $2$ payments a year, so the last payment made in year eight is $16$ ($8\times 2$). So, to find the balance after eight years, set $P_1=16$ and $P_2=16$.

 PMT Setting END $N$ $30$ $PV$ $0$ $FV$ $500,000$ $PMT$ $-11,663.61$ $I/Y$ $4.7$ $P/Y$ $2$ $C/Y$ $2$ $P_1$ $16$ $P_2$ $16$

$BAL=\223,407.95$

$\begin{eqnarray*}\mbox{Book Value} & = & 500,000-223,407.95 \\ & = & \276,592.05 \end{eqnarray*}$

Step 6: Write as a statement.

After eight years, the book value in the fund is $\276,592.05$.

### Things to Watch Out For

A common mistake occurs in translating years into payment numbers. You often need to find the total interest or the total increase in the sinking fund for a particular year. To do this, you need to set $P_1$ equal to the number of the first payment that occurs in that year and $P_2$ equal to the number of the last payment that occurs in that year.

For example, suppose you have monthly payments and you want to know the total interest in the fourth year. In error, you might calculate that the fourth year begins with payment $36$ and ends with payment $48$, and so enter $P_1=36$ and $P_2=48$. But the $36^{th}$ payment is actually the last payment of the third year. The first payment to occur in year four is the $37^{th}$. So, if you wanted to find the total interest in year $4$, $P_1=37$ and $P_2=48$.

When you need to find the first payment number and last payment number for a particular year, there are two methods you can use to calculate the correct payment numbers.

1. Calculate the payment number at the end of the year in question by multiplying $P/Y$ by the year number: $P/Y\times\text{year number}$. To find the payment number at the start of the year, subtract the payment frequency less one ($P/Y - 1$) from the last payment number of the year. For example, suppose the you have monthly payments and want to find the interest paid for year $4$. The last payment in year $4$ is $12\times 4=48$. The first payment in year $4$ is $48-(12-1)=37$.
2. Calculate the payment number at the end of the year in question by multiplying $P/Y$ by the year number: $P/Y\times\text{year number}$. To find the payment number for the first payment in the year, multiply $P/Y$ by the previous year number and then add one to it: $P/Y\times (year-1)+1$. For example, suppose the you have monthly payments and want to find the interest paid for year $4$. The first payment in year $4$ is $12\times 3+1=37$. The last payment of the fourth year remains at payment $48$.

### Try It

3) A company sold $\100,000$ worth of bonds and set up a sinking fund earning $4\%$ compounded semi-annually to retire the bonds in nine years. The company made semi-annual deposits into the sinking fund.

1. What is the sinking fund payment?
2. How much interest did the sinking fund earn with the $10$th payment?
3. By how much did the sinking fund increase with the $7$th payment?
4. What is the balance on the loan after eight years?
5. How much interest did the sinking fund earn in year three?
6. By how much did the sinking fund increase in year six?
7. What is the book value in the fund after year five?
Solution

a. Calculate the payment.

 PMT Setting END $N$ $2 \times 9=18$ $PV$ $0$ $FV$ $100,000$ $PMT$ $?$ $I/Y$ $4$ $P/Y$ $2$ $C/Y$ $2$

$PMT=\4,670.22$

b. Calculate the interest for the $10^{th}$ payment.

 PMT Setting END $N$ $18$ $PV$ $0$ $FV$ $100,000$ $PMT$ $-4,670.22$ $I/Y$ $4$ $P/Y$ $2$ $C/Y$ $2$ $P_1$ $10$ $P_2$ $10$

$INT=\911.13$

c. Calculate the increase for the $7^{th}$ payment.

 PMT Setting END $N$ $18$ $PV$ $0$ $FV$ $100,000$ $PMT$ $-4,670.22$ $I/Y$ $4$ $P/Y$ $2$ $C/Y$ $2$ $P_1$ $7$ $P_2$ $7$

$PRN=\5,259.43$

d. Calculate the balance at the end of year $8$.

 PMT Setting END $N$ $18$ $PV$ $0$ $FV$ $100,000$ $PMT$ $-4,670.22$ $I/Y$ $4$ $P/Y$ $2$ $C/Y$ $2$ $P_1$ $16$ $P_2$ $16$

$BAL=\87,049.56$

e. Calculate the interest for year $3$.

 PMT Setting END $N$ $18$ $PV$ $0$ $FV$ $100,000$ $PMT$ $-4,670.22$ $I/Y$ $4$ $P/Y$ $2$ $C/Y$ $2$ $P_1$ $5$ $P_2$ $6$

$INT=\871.06$

f. Calculate the increase for year $6$.

 PMT Setting END $N$ $18$ $PV$ $0$ $FV$ $100,000$ $PMT$ $-4,670.22$ $I/Y$ $4$ $P/Y$ $2$ $C/Y$ $2$ $P_1$ $11$ $P_2$ $12$

$PRN=\11,499.80$

g. Calculate the book value after year $5$.

 PMT Setting END $N$ $18$ $PV$ $0$ $FV$ $100,000$ $PMT$ $-4,670.22$ $I/Y$ $4$ $P/Y$ $2$ $C/Y$ $2$ $P_1$ $10$ $P_2$ $10$

$BAL=\51,137.61$

$\begin{eqnarray*}\mbox{Book Value} & = & 100,000-51,137.61 \\ & = & \48,862.39 \end{eqnarray*}$

## Section 4.6 Exercises

1. A company set up a sinking fund earning $6.2\%$ effective to accumulate $\10,000$ to repay a loan. The company makes quarterly payments into the sinking fund for two years. Construct the sinking fund schedule for the loan.
Solution
 Payment Number Payment Interest Increase Balance Book Value $0$ $\0$ $\10,000$ $1$ $\1,185.21$ $\0$ $\1,185.21$ $\1,185.21$ $\8,814.79$ $2$ $\1,185.21$ $\17.96$ $\1,203.17$ $\2,388.38$ $\7,611.62$ $3$ $\1,185.21$ $\36.19$ $\1,221.40$ $\3,609.78$ $\6,390.22$ $4$ $\1,185.21$ $\54.70$ $\1,239.91$ $\4,849.68$ $\5,150.32$ $5$ $\1,185.21$ $\73.48$ $\1,258.69$ $\6,108.38$ $\3,891.62$ $6$ $\1,185.21$ $\92.55$ $\1,277.76$ $\7,386.14$ $\2,613.86$ $7$ $\1,185.21$ $\111.92$ $\1,297.13$ $\8,683.27$ $\1,316.73$ $8$ $\1,185.21$ $\131.57$ $\1,316.78$ $\10,000.05$ $-\0.05$ Totals $\9,481.68$ $\518.37$ $\10,000.05$
2. A company issued $\5,000,000$ worth of bonds and set up a sinking fund earning $7\%$ compounded semi-annually to retire the bonds in six years. The company made semi-annual payments into the fund. Construct a sinking fund schedule showing the details of third year, the last two payments and the totals.
Solution
 Payment Number Payment Interest Increase Balance Book Value $5$ $\342,419.75$ $\50,514.79$ $\392,934.54$ $\1,836,214.23$ $\3,163,785.77$ $6$ $\342,419.75$ $\64,267.50$ $\406,687.25$ $\2,242,901.47$ $\3,757,098.53$ $11$ $\342,419.75$ $\140,597.13$ $\483,016.88$ $\4,500,077.59$ $\49,922.41$ $12$ $\342,419.75$ $\157,502.72$ $\499,922.47$ $\5,000,000.05$ $-\0.05$ Totals $\4,109,037$ $\890,963.05$ $\5,000,000.05$
3. A company borrowed $\800,000$ to expand their factory. They establish a sinking fund earning $5\%$ compounded quarterly to accumulate the loan amount in $8$ years. The company made monthly payments into the sinking fund. Construct a sinking fund schedule showing the details of payment $37$, the last two payments and the totals.
Solution
 Payment Number Payment Interest Increase Balance Book Value $37$ $\6,800.52$ $\1,093.21$ $\7,893.73$ $\271,355.37$ $\528,644.63$ $95$ $\6,800.52$ $\3,236.08$ $\10,036.60$ $\789,922.04$ $\10,077.96$ $96$ $\6,800.52$ $\3,277.72$ $\10,078.24$ $\800,000.29$ $-\0.29$ Totals $\652,849.92$ $\890,963.05$ $\5,000,000.05$
4. A $\500,000$ bond with a $6.5\%$ coupon is redeemable in ten years. The bond comes with a sinking fund that requires deposits timed to match the bond payments. The sinking fund earns $4\%$ compounded semi-annually.
1. What is the sinking fund payment?
2. What is the periodic expense of the debt?
3. What is the balance in the fund after two years?
4. How much interest did the fund earn with the $17$th payment?
5. By how much did the fund increase with the $8$th payment?
6. How much interest did the fund earn in year nine?
7. By how much did the fund increase in year six?
8. What is the book value at the end of year seven?
Solution

a. $\20,578.40$; b. $\36,828.40$; c. $\84,816.10$; d. $\7,671.33$; e. $\23,638.11$; f. $\15,907.66$; g. $\50,671.61$; h. $\171,281.91$

5. A $\125,000$ face value bond is issued with $15$ years to maturity and a coupon rate of $5\%$. A sinking fund is established to accumulate the face value of the bond. The sinking fund earns $7.4\%$ compounded semi-annual and the payments are time to match the bond payments.
1. What is the sinking fund payment?
2. What is the periodic expense of the debt?
3. What is the balance in the fund half way through the term?
4. How much interest did the fund earn with the $20$th payment?
5. By how much did the fund increase with the $10$th payment?
6. How much interest did the fund earn in year seven?
7. By how much did the fund increase in year three?
8. What is the book value at the end of year twelve?
Solution

a. $\2,342.79$; b. $\5,467.79$; c. $\45,878.92$; d. $\2,329.50$; e. $\3,248.94$; f. $\2,694.64$; g. $\5,518.73$; h. $\36,885.21$

6. A company borrowed $\750,000$ at $3.8\%$ compounded quarterly which required them to make quarterly interest payments. The company set up a sinking fund earning $4.9\%$ compounded quarterly to retire the debt in eight years. The company makes quarterly payments into the sinking fund at the same time they make the quarterly interest payments on the loan.
1. What is the sinking fund payment?
2. What is the periodic expense of the debt?
3. What is the balance in the fund at the end of year five?
4. How much interest did the fund earn with the $13$th payment?
5. By how much did the fund increase with the $25$th payment?
6. How much interest did the fund earn in year four?
7. By how much did the fund increase in year six?
8. What is the book value at the end of year two?
Solution

a. $\19,284.57$; b. $\33,534.57$; c. $\434,053.97$; d. $\3,033.85$; e. $\25,829.55$; f. $\13,789.23$; g. $\100,229.96$; h. $\588,944.27$