Sinking Funds Earning Less Than the Loan Rate

Why a Lower Fund Rate Matters

A sinking-fund loan keeps the loan principal outstanding during the term. The borrower pays the loan interest each period and makes separate deposits into a fund intended to accumulate to the principal due at maturity. If the fund earns less than the loan rate, each dollar deposited grows more slowly than it would at the loan rate. The borrower must therefore deposit more toward principal funding than under a direct-amortization loan, and the total periodic outlay is higher.

The comparison requires rates and periods to match. In this tutorial, both rates are effective rates per payment period, loan interest is paid at each period-end, and equal fund deposits are made at each period-end. The fund earns interest between deposits. Assume no fees, taxes, withdrawals, or default, and that the fund is available to repay the principal at the end of period $n$.

Definition: Let $L$ be the loan principal, $i$ the effective loan interest rate per period, $j$ the effective sinking-fund rate per period, and $n$ the number of periods. With end-of-period deposits, the level deposit $R$ that accumulates to $L$ is $R=L/s_{\overline{n}|}$ evaluated at fund rate $j$. The borrower's periodic outlay is the loan interest $iL$ plus the deposit $R$.

Deposit and Outlay Formulas

An end-of-period deposit earns interest for the periods remaining after it is made. Thus the first deposit earns for $n-1$ periods, the next for $n-2$, and the final deposit earns no interest before the fund is used. At rate $j$, the accumulated fund after $n$ deposits is $R s_{\overline{n}|}$. Setting this equal to the required principal gives the deposit.

R s_{\overline{n}|}=L \qquad\Longrightarrow\qquad R=\frac{L}{s_{\overline{n}|}} =\frac{Lj}{(1+j)^n-1} \quad (j>0).

When $j=0$, the fund earns no interest, so $n$ deposits must add directly to $L$: $R=L/n$. The total outlay per period is

C_{\text{sf}}=iL+R.

For comparison, a level-payment loan amortized directly over $n$ periods at rate $i$ has payment $L/a_{\overline{n}|}$, where the annuity-immediate is evaluated at $i$. These payments include both interest and principal repayment; they are not the same components as the sinking-fund interest and deposit.

Why the Sinking-Fund Outlay Is Higher

For a positive rate $i$, the accumulation factor for $n$ end-of-period payments satisfies $s_{\overline{n}|}=((1+i)^n-1)/i$. The direct-amortization payment can be written as loan interest plus a principal-funding amount:

\frac{L}{a_{\overline{n}|}\text{ at }i} =L\left(i+\frac{1}{s_{\overline{n}|}\text{ at }i}\right) =iL+\frac{L}{s_{\overline{n}|}\text{ at }i}.

To see the identity, $a_{\overline{n}|}=(1-v^n)/i$ and $s_{\overline{n}|}=((1+i)^n-1)/i$, both evaluated at $i$. Therefore $1/a_{\overline{n}|}=i(1+i)^n/((1+i)^n-1)=i+i/((1+i)^n-1)=i+1/s_{\overline{n}|}$. This is the direct-amortization payment expressed as an interest component and a principal-funding component.

For fixed $n$, $s_{\overline{n}|}$ increases as its rate increases: each earlier deposit earns at least as much, and the later deposits earn no less. Consequently, if $0\leq j<i$, then $s_{\overline{n}|}$ at $j$ is less than $s_{\overline{n}|}$ at $i$. Dividing the same positive principal $L$ by these factors shows that the sinking-fund deposit is greater than the direct-amortization principal-funding component. Both arrangements pay the same loan interest $iL$, so the sinking-fund outlay is greater.

Theorem: For $L>0$, $n\geq1$, and matched period rates $0\leq j<i$, a loan with equal end-of-period sinking-fund deposits has a greater periodic outlay than direct amortization at rate $i$. The difference is $$ C_{\text{sf}}-\frac{L}{a_{\overline{n}|}\text{ at }i} = \frac{L}{s_{\overline{n}|}\text{ at }j} - \frac{L}{s_{\overline{n}|}\text{ at }i} >0. $$

Worked Example: A Four-Period Loan With a Lower Fund Rate

A borrower takes a $10,000 loan for four years at an effective annual rate of 10%. The borrower pays interest annually and makes equal deposits at each year-end into a fund earning 5% effective annually. Find the annual deposit and total annual outlay, compare the outlay with the direct-amortization payment at 10%, and verify the fund target. Assume all amounts are in dollars and rates are annual.

Method. Accumulate the four deposits at the fund rate to determine the deposit. Add the annual loan interest to find outlay. For the comparison, express the direct payment as interest plus the principal-funding amount at the loan rate.

At 5%, the fund accumulation factor is

s_{\overline{4}|}\text{ at }5\% =\frac{(1.05)^4-1}{0.05} =\frac{1.21550625-1}{0.05} =4.310125.

The required annual deposit is

R=\frac{10{,}000}{4.310125} =2{,}320.118326\ldots \approx \$2{,}320.12.

Annual loan interest is $10{,}000(0.10)=\$1{,}000$, so the sinking-fund outlay is $1{,}000+2{,}320.118326\ldots=\$3{,}320.12$, rounded to cents.

At the loan rate, $s_{\overline{4}|}=(1.1^4-1)/0.10=(1.4641-1)/0.10=4.641$. The direct payment is

1{,}000+\frac{10{,}000}{4.641} =1{,}000+2{,}154.708037\ldots =\$3{,}154.71

to the nearest cent. The sinking-fund arrangement costs $3{,}320.118326\ldots-3{,}154.708037\ldots=\$165.41$ more per year, rounded to cents. As a check, the four deposits accumulate to $2{,}320.118326\ldots(4.310125)=\$10{,}000$, apart from the displayed rounding. The lower fund rate makes the required deposit larger than the direct loan's principal-funding component.

Common mistake avoided: Treating the sinking-fund deposit as the loan payment. The borrower pays both the $1,000 loan interest and the fund deposit; the deposit alone is not the total outlay.

Reading the Comparison Carefully

The inequality compares periodic outlays under specified assumptions. It does not say that the loan's interest rate changes when the fund earns a different rate. The loan interest remains $iL$, because the full principal $L$ is still outstanding in this model. The fund rate affects only how large the separate deposits must be to accumulate to $L$.

Quantity	Sinking-fund arrangement	Direct amortization
Loan interest paid each period	$iL$	Included in the level payment; interest portions can vary
Principal funding	$L/s_{\overline{n}\|}$ at fund rate $j$	$L/s_{\overline{n}\|}$ at loan rate $i$, as an equivalent decomposition
Total periodic outlay	Loan interest plus fund deposit	Level payment $L/a_{\overline{n}\|}$ at $i$

The direct-amortization column's principal-funding amount is a way to compare payments algebraically. It does not mean the borrower makes a separate fund deposit under direct amortization. In that arrangement, each payment is made to the lender and reduces the loan balance according to the amortization schedule.

Worked Example: A Fund With No Interest

A $12,000 loan is outstanding for three years at 8% effective annual interest. The borrower pays annual interest and makes equal year-end deposits to a fund that earns 0%. Find the annual outlay and compare it with direct amortization at 8%. Verify that the fund is sufficient at the end of year three.

Method. With a zero fund rate, the deposits accumulate without interest, so divide the principal equally among the three deposits. Then calculate the loan interest separately.

The deposit is $12{,}000/3=\$4{,}000$ per year. Annual loan interest is $12{,}000(0.08)=\$960$, so the periodic outlay is

C_{\text{sf}}=4{,}000+960=\$4{,}960.

For direct amortization, the loan-rate accumulation factor is

s_{\overline{3}|}\text{ at }8\% =\frac{(1.08)^3-1}{0.08} =\frac{1.259712-1}{0.08} =3.2464.

The direct payment is $960+12{,}000/3.2464=960+3{,}696.402169\ldots=\$4{,}656.40$, rounded to cents. Thus the sinking-fund outlay is $4{,}960-4{,}656.402169\ldots=\$303.60$ more per year, rounded to cents. The fund check is $3(4{,}000)=\$12{,}000$, exactly the principal due. This is consistent with the zero-yield assumption.

Common mistake avoided: Using the loan rate to accumulate the fund. The fund earns 0%, not 8%, so its deposits do not grow between dates.

Worked Example: Checking Each Fund Balance

A borrower has a $10,000 loan for two years at 12% effective annual interest. Equal deposits are made at each year-end into a fund earning 4% effective annually. Calculate the deposit, the total annual outlay, and the fund balance immediately after the final deposit. Compare with the direct-amortization payment at 12%.

Method. Use the two-period accumulation factor at 4% for the deposit. Then track the first deposit's interest and add the second deposit at the end of year two.

The fund accumulation factor is $s_{\overline{2}|}=1+1.04=2.04$. Therefore,

R=\frac{10{,}000}{2.04} =\$4{,}901.960784\ldots \approx \$4{,}901.96.

Annual loan interest is $10{,}000(0.12)=\$1{,}200$. The total annual outlay is $1{,}200+4{,}901.960784\ldots=\$6{,}101.96$, rounded to cents. To check the fund, the first deposit earns $4{,}901.960784\ldots(0.04)=\$196.078431\ldots$ during year one. Immediately before the second deposit the fund is $4{,}901.960784\ldots+196.078431\ldots=\$5{,}098.039216\ldots$. Adding the second deposit gives exactly $10,000, apart from displayed rounding.

At the loan rate, $s_{\overline{2}|}=1+1.12=2.12$. The direct payment is

1{,}200+\frac{10{,}000}{2.12} =1{,}200+4{,}716.981132\ldots =\$5{,}916.98.

The sinking-fund outlay exceeds this by $6{,}101.960784\ldots-5{,}916.981132\ldots=\$184.98$, rounded to cents. Both the balance check and the comparison use unrounded values before final rounding.

Common mistake avoided: Adding the second deposit before calculating the first deposit's earnings. The first deposit earns for one full period; the final deposit earns no interest before redemption.

Common Exam Traps

Comparing the deposit with the full direct payment. Compare the deposit with the direct payment's principal-funding component, not with the payment that also includes interest.
Using inconsistent rates. The factor $s_{\overline{n}|}$ for the deposit must use the fund rate $j$. The factor used to decompose the direct payment must use the loan rate $i$.
Ignoring payment timing. The formula $R=L/s_{\overline{n}|}$ assumes deposits at period-end. Beginning-of-period deposits have a different accumulation pattern.
Rounding too early. Keep the deposit and accumulation calculations unrounded until the requested final monetary amount. Rounding each deposit to cents can leave a small residual in the fund.
Calling a fund deposit a principal repayment. In this model, the deposit goes into a separate fund. The loan principal remains outstanding until it is repaid from the fund at maturity.

The central comparison is straightforward once the cash flows are separated: the sinking-fund borrower pays $iL$ in loan interest plus a deposit based on $j$. If $j<i$, the fund accumulates less per dollar deposited, so the deposit and total periodic outlay exceed their direct-amortization counterparts.

Check Your Understanding

For each problem, assume annual periods, end-of-year fund deposits, and no fees or taxes. Round dollar answers to the nearest cent.

A $8,000 loan is outstanding for four years at 9% effective annual interest. The sinking fund earns 0%. Find the annual fund deposit and total annual outlay. (Answer: Deposit $2,000.00; outlay $2,720.00.)
A $15,608 loan is outstanding for three years at 8% effective annual interest. The fund earns 4%. Find the annual deposit and total annual outlay. (Answer: Deposit $5,000.00; outlay $6,248.64.)
A $10,000 loan is outstanding for two years at 12% effective annual interest. The fund earns 4%. Find the annual outlay and the direct-amortization payment at 12%; state which is larger. (Answer: Sinking-fund outlay $6,101.96; direct payment $5,916.98; sinking-fund outlay is larger.)
A $20,000 loan is outstanding for two years at 10% effective annual interest. The fund earns 5%. Find the annual deposit and total annual outlay. (Answer: Deposit $9,756.10; outlay $11,756.10.)
A $5,000 loan is outstanding for two years at 6% effective annual interest. The fund earns 2%. Find the sinking-fund outlay, the direct-amortization payment, and the difference. (Answer: Sinking-fund outlay $2,775.25; direct payment $2,727.18; sinking-fund outlay is $48.06 greater per year.)

Quantity	Sinking-fund arrangement	Direct amortization
Loan interest paid each period	\(iL\)	Included in the level payment; interest portions can vary
Principal funding	\(L/s_{\overline{n}\|}\) at fund rate \(j\)	\(L/s_{\overline{n}\|}\) at loan rate \(i\), as an equivalent decomposition
Total periodic outlay	Loan interest plus fund deposit	Level payment \(L/a_{\overline{n}\|}\) at \(i\)

What You'll Learn