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Exam FM · Sinking Funds and Principal Splits · Tutorial 285 of 998

Separating Loan Interest From Principal Funding

Learn to calculate each period’s loan interest separately from the sinking-fund deposit that builds toward repayment of principal.

Intermediate 11 min read

What You'll Learn

  • Distinguish interest owed on an outstanding loan from money deposited into a sinking fund.
  • Identify the assumptions that make the loan-interest amount level from period to period.
  • Calculate a level end-of-period fund deposit using the fund’s yield and payment timing.
  • Check that the fund accumulates to the principal due at maturity.
  • Avoid using the fund yield to calculate the lender’s interest charge.

Why Separate the Two Amounts?

A sinking-fund loan keeps its principal outstanding during the loan term while the borrower builds a separate fund to repay that principal at maturity. The borrower therefore has two different obligations to manage in each period: interest owed to the lender and a deposit toward future principal repayment. These amounts are calculated using different information.

The lender’s interest charge is based on the loan balance and the loan’s interest rate. The sinking-fund deposit is based on the amount the fund must accumulate, the fund’s yield, and the deposit timing. In particular, depositing money into a separate fund does not, by itself, reduce the principal owed to the lender.

Definition: In a sinking-fund loan with principal \(P\), the loan interest for a period is the contractual interest owed to the lender for that period. A sinking-fund deposit is money set aside in a separate fund to accumulate toward repayment of principal. Unless the loan contract says otherwise, the deposit is not a payment that reduces the loan balance.

The Two Calculations Use Different Rates

Assume the loan has level principal \(P\), interest is payable at the end of each period, and the principal remains outstanding until maturity. Let \(i_L\) be the effective loan interest rate for one payment period. The interest owed in each period is \(P i_L\). This amount stays level because both the principal and the loan rate stay level.

Now suppose the borrower makes \(n\) equal deposits at the end of each period into a separate fund earning an effective rate \(i_F\) per period. The deposit is chosen so that the accumulated fund equals \(P\) at the end of period \(n\). If \(R\) is the deposit each period, then

$$ R=\frac{P}{s_{\overline{n}|i_F}} \qquad\text{where}\qquad s_{\overline{n}|i_F}=\frac{(1+i_F)^n-1}{i_F}. $$

The rate \(i_L\) is used for the loan-interest calculation; the rate \(i_F\) is used for the fund-deposit calculation. Both rates must be expressed per payment period before applying these formulas. If \(i_F=0\), the fund earns no interest, so \(s_{\overline{n}|0}=n\) and the level deposit is \(P/n\).

Why the Deposit Formula Works

Each end-of-period deposit earns fund interest only after it has been made. The first deposit earns for \(n-1\) periods, the next for \(n-2\) periods, and the last deposit earns no interest before the maturity date. Thus, at the end of period \(n\), the accumulated fund is

$$ R(1+i_F)^{n-1}+R(1+i_F)^{n-2}+\cdots+R(1+i_F)+R =R\sum_{r=0}^{n-1}(1+i_F)^r =R s_{\overline{n}|i_F}. $$

The sum is a geometric accumulation series. Setting the fund balance equal to the principal required at maturity gives \(R s_{\overline{n}|i_F}=P\), and dividing by the accumulation factor gives the stated deposit formula. This derivation depends on equal end-of-period deposits, a constant fund rate, no withdrawals, and a target fund balance of \(P\). It does not apply unchanged if deposits are made at the beginning of periods, the fund rate varies, or the amount due at maturity differs from \(P\).

1
Find the loan-interest rate for one period.
Use the contractual loan rate, converted to the same period as the interest payments.
2
Calculate interest owed.
For a level-principal, interest-only loan, multiply the outstanding principal by the per-period loan rate.
3
Find the fund deposit separately.
Use the fund yield, number of deposits, and deposit timing to accumulate the required principal.
4
Check the two results against their purposes.
Interest is due to the lender for the period; deposits and fund earnings accumulate in the separate fund.

Worked Examples

Worked Example: Quarterly Interest and a Two-Deposit Fund

A borrower receives a principal of $20,400 under a hypothetical loan. The loan charges interest at an effective 3% per quarter, with interest payable at the end of each quarter and the full principal due after two quarters. The borrower makes equal deposits into a separate fund at the end of each quarter; the fund earns an effective 2% per quarter. Find the interest owed each quarter and the required deposit per quarter.

Method. Calculate the lender’s interest using the loan rate, then calculate the fund deposit using the fund rate. The principal remains $20,400 throughout the two quarters.

Interest owed each quarter:

$$ 20{,}400(0.03)=612. $$

The borrower owes $612 of interest each quarter. This calculation does not use the fund’s 2% yield.

Fund deposit: There are two end-of-quarter deposits, and the fund rate per quarter is 0.02. The accumulation factor is

$$ s_{\overline{2}|0.02} =\frac{(1.02)^2-1}{0.02} =\frac{1.0404-1}{0.02} =\frac{0.0404}{0.02} =2.02. $$

Therefore, the unrounded deposit is

$$ R=\frac{20{,}400}{2.02}=10{,}099.00990099\ldots $$

Rounded to the nearest cent, the borrower deposits $10,099.01 per quarter. The check using the unrounded deposit is \(10{,}099.00990099\ldots(1.02)+10{,}099.00990099\ldots=20{,}400\). If both deposits are made in cents, they accumulate to $20,400.0002, which is $20,400.00 when rounded to the nearest cent.

Common mistake avoided: using the 2% fund yield to calculate the lender’s interest would give \(20{,}400(0.02)=408\), but that is not the loan’s contractual rate. The correct lender interest is $612 per quarter; the 2% rate determines how much must be deposited into the fund.

Worked Example: A Five-Period Loan With Different Annual Rates

A hypothetical $120,000 loan has level principal outstanding for five years, with annual interest payable at each year-end and principal due at the end of year five. The loan rate is 7% effective annually. A separate fund earns 4% effective annually and receives equal deposits at each year-end. Find the annual interest owed and the annual fund deposit.

Interest calculation: The principal stays at $120,000, so each year’s interest is

$$ 120{,}000(0.07)=8{,}400. $$

Thus, interest owed is $8,400 at each year-end.

Deposit calculation: There are five end-of-year deposits, and the fund yield is 4% per year. First calculate the accumulation factor:

$$ s_{\overline{5}|0.04} =\frac{(1.04)^5-1}{0.04} =\frac{1.2166529024-1}{0.04} =5.41632256. $$

The unrounded annual deposit is

$$ R=\frac{120{,}000}{5.41632256} =22{,}155.2536\ldots $$

Rounded to the nearest cent, the deposit is $22,155.25 per year. Using the unrounded deposit, the fund accumulates to \(22{,}155.2536\ldots(5.41632256)=120{,}000\). The cent-rounded deposit accumulates to about $119,999.98, a small rounding shortfall; the exact level deposit is the amount used in the formula.

The two annual amounts have different roles: $8,400 is interest owed under the loan contract, while $22,155.25 is deposited into the fund toward the future principal repayment. The fund yield of 4% does not replace the loan rate of 7% in the interest calculation.

Worked Example: Checking Every Period of a Three-Year Fund

A hypothetical borrower owes $50,000 of principal for three years. The loan is interest-only during its term, charges an effective 6% annually, and requires principal repayment at the end of year three. Equal deposits are made at each year-end into a separate fund earning an effective 5% annually. Determine the loan interest and deposit each year, then verify the fund target.

Interest each year:

$$ 50{,}000(0.06)=3{,}000. $$

So the loan interest is $3,000 in each of years one, two, and three.

Fund deposit: For three end-of-year deposits at 5%, the accumulation factor is

$$ s_{\overline{3}|0.05} =\frac{(1.05)^3-1}{0.05} =\frac{1.157625-1}{0.05} =3.1525. $$

The exact level deposit is

$$ R=\frac{50{,}000}{3.1525} =15{,}860.4282\ldots $$

Rounded to the nearest cent, the deposit is $15,860.43 each year. The accumulation check using the exact deposit is \(15{,}860.4282\ldots(3.1525)=50{,}000\). If the rounded cent amount is used for all three deposits, the fund accumulates to approximately $50,000.01, so a schedule using strictly cent-level cash flows may require a small final adjustment.

The interest does not decline as the fund grows: the loan principal remains $50,000 until maturity. Fund growth changes the amount available for principal repayment, not the balance on which the lender charges interest under these assumptions.

Timing, Assumptions, and Limits

The calculations above require a particular loan structure. The principal remains constant, interest is payable rather than added to the balance, and principal is repaid at maturity. If the borrower repays principal during the term, subsequent loan interest is calculated on the reduced outstanding balance, not on the original principal. If interest is unpaid and capitalizes, the balance may also change. In either case, the simple level-interest calculation \(P i_L\) no longer describes every period.

Deposit timing also matters. With deposits at the beginning rather than the end of each period, every deposit earns one additional period of fund interest. For \(n\) equal beginning-of-period deposits, the fund accumulation at the end of period \(n\) is \(R(1+i_F)s_{\overline{n}|i_F}\); the required deposit is correspondingly smaller than for end-of-period deposits. Do not use an end-of-period accumulation factor when the deposits are made at the beginning.

When loan and fund rates are quoted with different time units, convert each separately to an effective rate per payment period before calculating. For example, if interest is payable quarterly, the loan rate used for each interest charge must be the effective loan rate per quarter, and the fund rate used to accumulate quarterly deposits must also be the effective fund yield per quarter. The two converted rates need not be equal.

Common Exam Traps

  • Applying the fund yield to the loan balance. This confuses the fund’s investment return with the contractual rate charged by the lender. Calculate loan interest from the loan rate.
  • Treating the deposit as a principal payment to the lender. In a sinking-fund arrangement, the loan principal normally stays outstanding while assets accumulate separately in the fund.
  • Using the loan rate to find the required fund deposit. The fund deposit depends on the fund yield, not automatically on the loan rate.
  • Mixing rate periods. A quarterly interest charge requires a quarterly loan rate; do not multiply the principal by an annual rate for each quarter.
  • Ignoring deposit timing or rounding. Beginning-of-period and end-of-period deposits accumulate differently, and rounded deposits can leave a small final surplus or shortfall.
Key takeaway: For a level-principal, interest-only sinking-fund loan, calculate the period’s interest as \(P i_L\), and calculate the separate level end-of-period deposit as \(P/s_{\overline{n}|i_F}\). The first uses the loan rate; the second uses the fund yield and deposit timing.

Check Your Understanding

For each problem, keep the lender’s interest calculation separate from the fund-deposit calculation. Unless stated otherwise, deposits are equal, made at the end of each period, and the fund begins with a zero balance.

  1. A hypothetical $30,000 loan remains outstanding for three months. Interest is payable monthly at an effective 1% per month. A separate fund earns an effective 0.5% per month and receives three end-of-month deposits to accumulate $30,000. Find the monthly interest and deposit. (Answer: $300 interest per month; $9,950.17 deposited per month.)
  2. A hypothetical $120,000 interest-only loan lasts five years, with annual interest at an effective 7% and principal due at maturity. A separate fund earns an effective 4% annually and receives five end-of-year deposits. Find both annual amounts. (Answer: $8,400 interest per year; $22,155.25 deposited per year.)
  3. A hypothetical $40,000 principal remains outstanding for four quarters. The loan charges effective interest of 2.5% per quarter; a separate fund earns 1.5% per quarter and receives four end-of-quarter deposits. Find the quarterly interest and deposit. (Answer: $1,000 interest per quarter; $9,777.79 deposited per quarter.)
  4. A hypothetical $18,000 principal is due after six quarters. Loan interest is payable at an effective 2% per quarter. A separate fund earns an effective 1% per quarter and receives six equal deposits at the beginning of each quarter. Find the interest per quarter and the beginning-of-quarter deposit. (Answer: $360 interest per quarter; $2,896.90 deposited per quarter.)

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