17  Sinking Fund Loans

17.1 General Sinking Fund Loans

From the perspective of the lender, a sinking fund loan is a loan with a balloon payment and an interest rate of \(i\).

From the perspective of the borrower, a sinking fund loan has two sets of payments:

  • Loan payments are applied directly to the loan. They are used to pay the interest on the loan, and if the loan payments exceed the interest, then the excess is used to pay the loan down.
  • Sinking fund payments are used ot build up a separate fund. This fund accumulates at an effective interest rate \(j\) per unit time. Upon maturity of the loan, the accumulated value of the sinking fund is paid to the lender of the loan.

Upon maturity, the accumulated value of the loan minus the accumulated value of the loan payments is equal to the accumulated value of the sinking fund.

\[(1+i)^n\times B_0 - AV_n(\text{loan payments}, i) = AV_n(\text{sinking fund payments}, j)\]

The balance of the loan, \(B_t\), and the balance of the sinking fund, \(SFB_t\), at time \(t\) are:

\[\begin{align} B_t &= (1+i)^n \times B_0 - AV_t(\text{loan payments}, i) \\ SFB_t &= AV_t(\text{sinking fund payments},j)\end{align}\]

The net balance of the loan, \(N_t\), at time \(t\) is the loan balance minus the sinking fund balance:

\[N_t = B_t - SFB_t = (1+i)^t\times B_0 - AV_t(\text{loan payments}, i) = AV_t(\text{sinking fund payments}, j)\]

17.2 Sinking Fund Loans Assumptions

Most sinking fund loans have the folowing two characteristics:

  • The loan payment is equal to the interest on the loan.
  • The sinking fund payments are level.

17.2.1 Formula

An \(n\)-year loan is to be repaid at time \(n\) with the balance of a sinking fund such that:

  • The sinking fund payments are level and accumulate to the initial value of the loan at an interest rate of \(j\).
  • The periodic loan payments are equal to the interest on the loan.

Then, \(B_t = B_0\) for all \(t < n\), and the sinking fund payments equal to: \[SFP = \frac{B_0}{s_{\overline{n|}j}}\]

The sinking fund balance at time \(t\) is: \[SFB_t = SFP \times s_{\overline{n|}j}\] and the net balance is: \[N_t = B_0 - SFP \times s_{\overline{n|}j}\]

Example: A loan of \(\$40,000\) is to be repaid with annual payments using the sinking fund method over a period of \(20\) years. The annual effective interest rate on the loan is \(6\%\), and the annual effective interest rate on the sinking fund is \(4\%\). The annual payments made on the loan are equal to the interest due on the loan. The annual sinking fund payments accumulate for 20 years, at which time the loan is paid of.

  • Calculate the amount of each level sinking fund payment.
  • Calculate the loan balance at the end of 10 years.
  • Calculate the sinking fund balance at the end of 10 years.
  • Calculate the net amount of the loan at the end of 10 years.

Solution:

  • The amount of each sinking fund payment: \(SFP = \frac{40,000}{s_{\overline{20|}4\%}} = 1,343.27\)
  • The loan balance at the end of 10 years is \(B_{10} = B_0 = 40,000\)
  • The sinking fund balance at the end of 10 years is: \(SFB_{10} = SFP \times s_{\overline{10|}4\%} = 16,127.44\)
  • The net amount of the loan at the end of 10 years: \(40,000.00 - 16,127.44 = 23,872.56\)