4.6 Sinking Funds

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:

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.

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

[latex]\boxed[/latex] Periodic Cost of Debt

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

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

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.

[latex]\begin FV & = & \$10,000,000 \\ I/Y & = & 4.5\% \\ P/Y & = & 2 \\ C/Y & = & 2 \\ t & = & 30 \mbox < years>\end[/latex]

Step 2: Calculate the sinking fund payment.

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

[latex]PMT=\$80,353.2748. \rightarrow \$80,353.28[/latex]

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

Step 3: Calculate the bond payment.

The bond payments are [latex]\$255,000[/latex].

Step 4: Calculate the periodic cost of the debt.

Step 5: Write as a statement.

The periodic cost of the debt is [latex]\$335,353.28[/latex]. This means that every six months the bank must pay out a total of [latex]\$335,353.28[/latex] because of the debt. Of this amount, [latex]\$255,000[/latex] goes to paying the bond payments and [latex]\$80,353.28[/latex] goes to the sinking fund to accumulate the [latex]\$10,000,000[/latex] 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 [latex]\$200,000[/latex] to raise money to build an expansion to its factory. The bonds have a coupon rate of [latex]3.9\%[/latex] compounded semi-annually and ten years to maturity. The company established a sinking fund earning [latex]2.7\%[/latex] 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.

a. Calculate the sinking fund payment.

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

b. Calculate the periodic expense of the debt.

Write as a statement.

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 [latex]\$10[/latex] million in bonds and has accumulated [latex]\$1[/latex] million in its sinking fund, the book value of the debt is [latex]\$9[/latex] million.

[latex]\boxed[/latex] Book Value

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:

To fill in a sinking fund schedule, you first need to have all of the details about the fund, including the loan amount ([latex]FV[/latex]), the sinking fund payment ([latex]PMT[/latex]), the number of payments ([latex]N[/latex]), 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
[latex]0[/latex] [latex]0[/latex] [latex]\text^1[/latex]
[latex]1[/latex] [latex]PMT^2[/latex] [latex]INT^3[/latex] [latex]INC^4[/latex] [latex]BAL^5[/latex] [latex]BV^6[/latex]
[latex]2[/latex] [latex]PMT^2[/latex] [latex]INT^3[/latex] [latex]INC^4[/latex] [latex]BAL^5[/latex] [latex]BV^6[/latex]
[latex]\vdots[/latex] [latex]\vdots[/latex] [latex]\vdots[/latex] [latex]\vdots[/latex] [latex]\vdots[/latex] [latex]\vdots[/latex]
[latex]N-1[/latex] [latex]PMT^2[/latex] [latex]INT^3[/latex] [latex]INC^4[/latex] [latex]BAL^5[/latex] [latex]BV^6[/latex]
[latex]N[/latex] [latex]PMT^2[/latex] [latex]INT^3[/latex] [latex]INC^4[/latex] [latex]BAL^5[/latex] [latex]BV^6[/latex]
Totals [latex]\text^8[/latex] [latex]\text^10[/latex] [latex]\text^9[/latex]

HOW TO

Fill In a Sinking Fund Schedule

Follow these steps to fill in a sinking fund schedule.

Step 1: In row [latex]0[/latex], the only entries are in the balance and book value columns. The initial balance is [latex]0[/latex] 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:

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:

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

Step 7: For each payment, repeat steps [latex]2[/latex] through [latex]6[/latex], including for the last row.

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

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

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

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 [latex]\$20,000[/latex] loan in two years. The company establishes a sinking fund earning [latex]4\%[/latex] 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

[latex]\begin FV & = & \$20,000 \\ I/Y & = & 4\% \\ P/Y & = & 2 \\ C/Y & = & 2 \\ t & = & 2 \mbox < years>\end[/latex]

Step 2: Calculate the sinking fund payment.

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

The sinking fund payment is [latex]\$4,852.48[/latex].

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 [latex]i=\frac=2\%[/latex].

Payment Number Payment Interest Increase Balance Book Value
[latex]0[/latex] [latex]\$0[/latex] [latex]\$20,000[/latex]
[latex]1[/latex] [latex]\$4,852.48[/latex] [latex]\;\;\;\;\;\;\;\;\;\;\$0\;\;\;\;\;\;\;\;\;\;[/latex][latex]\color<(0 \times 0.02)>[/latex] [latex]\$4,852.48[/latex][latex]\color[/latex] [latex]\$4,852.48[/latex][latex]\color[/latex] [latex]\$15,147.52[/latex][latex]\;\color[/latex]
[latex]2[/latex] [latex]\$4,852.48[/latex] [latex]\;\;\$97.05\;\;[/latex][latex]\color<(4582.48 \times 0.02)>[/latex] [latex]\$4,949.53[/latex][latex]\color[/latex] [latex]\$9,802.01[/latex][latex]\;\color[/latex] [latex]\$10,197.99[/latex][latex]\color[/latex]
[latex]3[/latex] [latex]\$4,852.48[/latex] [latex]\;\$196.04\;[/latex][latex]\color<(9802.01 \times 0.02)>[/latex] [latex]\$5,048.52[/latex][latex]\color[/latex] [latex]\$14,850.5[/latex][latex]\;\color[/latex] [latex]\$5,149.47[/latex][latex]\color[/latex]
[latex]4[/latex] [latex]\$4,852.48[/latex] [latex]\$297.01[/latex][latex]\color<(14,850.53 \times 0.02)>[/latex] [latex]\$5,149.49[/latex][latex]\color[/latex] [latex]\$20,000.02[/latex][latex]\color[/latex] [latex]-\$0.02[/latex][latex]\;\color[/latex]
Totals [latex]\$19,409.92[/latex][latex]\color<(4 \times 4852.48)>[/latex] [latex]\$590.10[/latex][latex]\color[/latex] [latex]\$20,000.02[/latex]

Things to Watch Out For

In the previous example, the final balance is slightly more than the required [latex]\$20,000[/latex] 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 [latex]\$20,000[/latex] 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 ([latex]N[/latex], [latex]PV[/latex], [latex]FV[/latex], [latex]PMT[/latex], [latex]I/Y[/latex], [latex]P/Y[/latex], [latex]C/Y[/latex]) . 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 [latex]PMT[/latex] (the sinking fund payment) is paid out to the fund, [latex]PMT[/latex] is negative. At the end of the term, [latex]FV[/latex] is received, so [latex]FV[/latex] is positive.
  3. Go to the amortization worksheet by pressing 2nd AMORT (the [latex]PV[/latex] button).
  4. To view the entries for a specific row of the schedule, set [latex]P_1[/latex] and [latex]P_2[/latex] to the row number. For example, to view the entries for row [latex]5[/latex], set [latex]P_1=5[/latex] and [latex]P_2=5[/latex].
    1. At the [latex]P_1[/latex] prompt, enter the row number and press ENTER.
    2. Press the down arrow.
    3. At the [latex]P_2[/latex] prompt, enter the row number and press ENTER.
    4. Press the down arrow.
    5. The [latex]BAL[/latex] entry is the balance entry for the corresponding row.
    6. Press the down arrow.
    7. The [latex]PRN[/latex] entry is the increase entry for the corresponding row.
    8. Press the down arrow.
    9. The [latex]INT[/latex] entry is the interest entry for the corresponding row.
    10. Press the down arrow the return to the [latex]P_1[/latex] screen.

    Key Takeaways

    On the amortization worksheet, [latex]BAL[/latex] is the balance entry, [latex]PRN[/latex] is the increase entry, and [latex]INT[/latex] 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:

    Make sure to re-enter [latex]PMT[/latex] 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 [latex]2[/latex] decimal places.

    Example 4.6.3

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

    Solution

    Step 1: Calculate the sinking fund deposit.

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

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

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

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

    Payment Number Payment Interest Increase Balance Book Value
    [latex]0[/latex] [latex]\$0[/latex] [latex]\$30,000[/latex]
    [latex]1[/latex] [latex]\$5,793.65[/latex] [latex]\$0[/latex] [latex]\$5,793.65[/latex] [latex]\$5,793.65[/latex] [latex]\$24,206.35[/latex]
    [latex]2[/latex] [latex]\$5,793.65[/latex] [latex]\$101.39[/latex] [latex]\$5,895.04[/latex] [latex]\$11,688.69[/latex] [latex]\$18,311.31[/latex]
    [latex]3[/latex] [latex]\$5,793.65[/latex] [latex]\$204.55[/latex] [latex]\$5,998.20[/latex] [latex]\$17,686.89[/latex] [latex]\$12,313.11[/latex]
    [latex]4[/latex] [latex]\$5,793.65[/latex] [latex]\$309.52[/latex] [latex]\$6,103.17[/latex] [latex]\$23,790.06[/latex] [latex]\$6,209.94[/latex]
    [latex]5[/latex] [latex]\$5,793.65[/latex] [latex]\$416.33[/latex] [latex]\$6,209.98[/latex] [latex]\$30,000.04[/latex] [latex]-\$0.04[/latex]
    Totals [latex]\$28,968.25[/latex] [latex]\$1,031.79[/latex] [latex]\$30,000.04[/latex]