Sinking Funds Earning More Than the Loan Rate

Why a Higher Fund Rate Matters

In a sinking-fund loan, the borrower pays loan interest as it comes due and makes separate deposits into a fund intended to accumulate the principal. If the fund earns more than the loan rate, each deposit grows faster than it would at the loan rate. As a result, less needs to be deposited each period to accumulate the principal, and the borrower’s total periodic outlay can be lower than the level payment on a directly amortized loan.

The comparison depends on keeping two rates and two cash flows distinct. The loan rate determines the loan interest payment. The fund rate determines how deposits accumulate. The fund’s higher yield does not change the interest owed on the loan.

Definition: Suppose a borrower receives principal $L$, pays loan interest at an effective rate $i$ each period, and repays the principal at the end of period $n$. The borrower makes equal deposits $R$ into a separate fund at the end of each period, and the fund earns an effective rate $j$ per period. Assume $n$ is a positive integer, the rates apply to the same period, there are no fees or taxes, and the fund earns its stated rate throughout. The fund is intended to accumulate exactly $L$ by the end of period $n$.

Required Deposit and Total Outlay

With end-of-period deposits, the first deposit earns interest for $n-1$ periods, the second for $n-2$ periods, and the last deposit earns no interest before the target date. Thus the accumulated fund at time $n$ is $R s_{\overline{n}|\,j}$, where $s_{\overline{n}|\,j}$ is the accumulated value of an annuity-immediate with $n$ payments of 1 at rate $j$. Equating that value to $L$ gives the required deposit.

R s_{\overline{n}|\,j}=L \qquad\Longrightarrow\qquad R=\frac{L}{s_{\overline{n}|\,j}}

For $j\ne 0$, the accumulated-value factor is $s_{\overline{n}|\,j}=((1+j)^n-1)/j$, so $R=Lj/((1+j)^n-1)$. At $j=0$, each deposit simply remains unchanged, so the fund at time $n$ is $nR$ and $R=L/n$. This zero-rate case must be stated separately if using the expression with $j$ in its denominator.

s_{\overline{n}|\,j}= \begin{cases} \dfrac{(1+j)^n-1}{j}, & j\ne 0,\\[6pt] n, & j=0, \end{cases} \qquad R= \begin{cases} \dfrac{Lj}{(1+j)^n-1}, & j\ne 0,\\[6pt] \dfrac{L}{n}, & j=0. \end{cases}

Each period, the loan interest is $Li$, so the borrower’s total periodic outlay—interest plus the separate fund deposit—is $Li+R$. The principal is not being repaid directly through these deposits: the loan remains outstanding until redemption, while the fund accumulates separately.

Key result: When $j>i\ge 0$ and $n\ge 2$, the sinking-fund deposit is smaller than the deposit needed at the loan rate, and the total outlay $Li+R$ is less than the level payment on a directly amortized loan with principal $L$, rate $i$, and term $n$. For $n=1$, there is no time for any deposit to earn interest, so the two total payments are equal.

Why the Outlay Is Lower

The reason is the growth of the deposit stream, together with the standard level-payment formula for a directly amortized loan. The following comparison assumes nonnegative rates and matching payment periods. It also identifies why the one-period case is an exception to a strict inequality.

Compare fund accumulation factors.
For a nonnegative rate $r$, $s_{\overline{n}|\,r}=\sum_{k=0}^{n-1}(1+r)^k$. If $j>i\ge 0$, then each term with $k\ge 1$ is larger at $j$ than at $i$. Therefore, when $n\ge 2$, $s_{\overline{n}|\,j}>s_{\overline{n}|\,i}$.

Compare deposits.
Both factors are positive, so $L/s_{\overline{n}|\,j}<L/s_{\overline{n}|\,i}$. A higher fund rate reduces the deposit required to accumulate the same principal.

Express the direct-amortization payment.
For $i>0$, the level payment is $L/a_{\overline{n}|\,i}$. The annuity factors satisfy $s_{\overline{n}|\,i}=(1+i)^n a_{\overline{n}|\,i}$, and $(1+i)^n-1=i s_{\overline{n}|\,i}$. Consequently, $L/a_{\overline{n}|\,i}=Li+L/s_{\overline{n}|\,i}$.

Compare total outlays.
The sinking-fund outlay is $Li+L/s_{\overline{n}|\,j}$. Since $L/s_{\overline{n}|\,j}<L/s_{\overline{n}|\,i}$, it is less than the direct payment above. If $i=0$, the direct payment is $L/n=L/s_{\overline{n}|\,0}$, and the same comparison holds for $j>0$ and $n\ge 2$.

For $n=1$, $s_{\overline{1}|\,j}=1$ for every fund rate: the only deposit is made at the redemption date and earns no interest. The deposit is $L$, and the total outlay $Li+L$ equals the single direct-amortization payment. For longer terms, the higher fund rate has periods in which to increase the value of earlier deposits.

Worked Comparisons

Worked Example: Three-Year Loan With a Higher Fund Yield

A borrower receives $L=\$10{,}000$ for three years. Loan interest is paid annually at an effective rate of $i=5\%$, and equal fund deposits are made at each year-end. The fund earns an effective annual rate of $j=10\%$. Find the required deposit, total annual outlay, direct-amortization payment at 5%, and the annual outlay reduction.

Method. Accumulate the end-of-year deposits at the fund rate, add loan interest at the loan rate, and compare that total with the direct level payment.

1. Find the fund accumulation factor. The three deposits earn interest for two, one, and zero years, respectively:

s_{\overline{3}|\,0.10} =1.10^2+1.10+1 =1.21+1.10+1 =3.31.

2. Calculate the deposit. The fund must accumulate $\$10{,}000$:

R=\frac{10{,}000}{3.31} =\$3{,}021.1480\ldots \approx \$3{,}021.15.

3. Calculate the sinking-fund outlay. Annual loan interest is $10{,}000(0.05)=\$500$. Therefore, the annual outlay is $\$500+\$3{,}021.1480\ldots=\$3{,}521.1480\ldots$, or $\$3{,}521.15$.

4. Calculate the direct payment. At 5%, the accumulated-value factor for three deposits is

s_{\overline{3}|\,0.05} =1.05^2+1.05+1 =1.1025+1.05+1 =3.1525.

The direct payment can be written as $Li+L/s_{\overline{3}|\,i}$. Thus it is $\$500+\$10{,}000/3.1525=\$500+\$3{,}172.0856\ldots=$ $\$3{,}672.09$.

5. Compare and check. The annual reduction in outlay is $\$3{,}672.0856\ldots-\$3{,}521.1480\ldots=\$150.9376\ldots$, or $\$150.94$. The fund check is $3.31(\$3{,}021.1480\ldots)=\$10{,}000$, as required. Money is rounded to the nearest cent only at the end.

Common mistake avoided: The direct payment is not just the amount $L/s_{\overline{3}|\,i}$. That term is the principal-funding component in the comparison; the $\$500$ annual loan interest must also be included.

Worked Example: A Two-Year Loan With a Larger Rate Difference

A borrower takes a $\$5{,}000$ loan for two years. The loan rate is 5% effective per year, and a fund earns 15% effective per year. Deposits are made at each year-end. Calculate the fund deposit and total outlay per year, then compare with direct amortization.

1. Accumulate the deposits. The first deposit earns one year of interest and the second earns none:

s_{\overline{2}|\,0.15}=1.15+1=2.15.

2. Find the deposit and total outlay.

R=\frac{\$5{,}000}{2.15} =\frac{\$100{,}000}{43} =\$2{,}325.581395\ldots \approx \$2{,}325.58.

Loan interest each year is $0.05(\$5{,}000)=\$250$, so total annual outlay is $\$250+\$2{,}325.581395\ldots=$ $\$2{,}575.58$.

3. Compare with direct amortization. At the loan rate, $s_{\overline{2}|\,0.05}=1.05+1=2.05$. The direct payment is $\$250+\$5{,}000/2.05=\$250+\$2{,}439.024390\ldots=$ $\$2{,}689.02$. The sinking-fund arrangement reduces the annual outlay by $\$2{,}689.024390\ldots-\$2{,}575.581395\ldots=\$113.442995\ldots$, or $\$113.44$.

4. Check the fund at redemption. Just before the second deposit, the first deposit has grown to $1.15R$. Adding the second deposit gives $1.15R+R=2.15R=\$5{,}000$. Money is rounded to the nearest cent only for reported amounts.

Common mistake avoided: The 15% fund rate is not the loan rate. The interest due is $\$250$, based on 5% of the original loan principal, not 15% of that principal.

Worked Example: Zero Loan Interest but Positive Fund Interest

A hypothetical loan of $\$6{,}000$ requires a single principal repayment after three years. The loan rate is 0%, while the fund earns 8% effective annually. Equal deposits are made at each year-end. Find the deposit and compare the total outlay with direct amortization at 0%.

1. Calculate the accumulation factor.

s_{\overline{3}|\,0.08} =1.08^2+1.08+1 =1.1664+1.08+1 =3.2464.

2. Calculate the fund deposit and outlay.

R=\frac{\$6{,}000}{3.2464} =\$1{,}848.201085\ldots \approx \$1{,}848.20.

The loan interest is zero, so the total annual outlay is $\$1{,}848.20$. Direct amortization at 0% requires three equal payments of $\$6{,}000/3=\$2{,}000$, so the annual outlay is reduced by approximately $\$151.80$.

3. Check the accumulated fund. The three deposits accumulate to $R(1.1664+1.08+1)=3.2464R=\$6{,}000$. The calculation remains valid even though the loan rate is zero; the fund rate is positive and is applied only to the fund.

Common mistake avoided: A zero loan rate does not imply a zero fund rate. Keep the two rates separate, and use the appropriate zero-rate formula if the fund rate itself is zero.

Common Mistakes and Practical Checks

Applying the fund rate to the loan. Loan interest is $Li$, not $Lj$. The fund rate affects deposit accumulation only.
Using the wrong deposit timing. The formulas here assume end-of-period deposits. A beginning-of-period deposit earns one extra period of interest, so it does not have the same accumulation factor.
Ignoring the zero-rate case. The expression $Lj/((1+j)^n-1)$ has a zero denominator at $j=0$. In that case use $R=L/n$.
Comparing only the deposit with the direct payment. The sinking-fund total outlay includes both the deposit and loan interest. The direct payment also covers both principal and interest.
Assuming the advantage is strict for every term. With only one period, the sole fund deposit earns no interest. A higher fund rate therefore does not reduce the required deposit or total outlay in that case.
Mixing rate periods. The loan and fund rates in the formulas must both correspond to the same payment period. A rate quoted on a different basis must be converted before comparing them.

Key takeaway: For matching periods, end-of-period deposits, and $j>i\ge 0$, a sinking fund accumulates more per dollar deposited than it would at the loan rate. For a term of at least two periods, that reduces the required deposit and makes the combined interest-plus-deposit outlay lower than the corresponding direct-amortization payment. This comparison is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, assume the stated rates are effective per payment period, deposits are made at the end of each period, and there are no fees or taxes. Round money to the nearest cent.

A $\$8{,}000$ loan runs for two years at a 4% annual loan rate. A separate fund earns 9% annually. Find the level fund deposit, total annual outlay, and direct-amortization payment. (Answer: Deposit $\$3{,}827.75$; sinking-fund outlay $\$4{,}147.75$; direct payment $\$4{,}241.57$.)
A $\$9{,}000$ loan has no interest and is repaid after three years using equal annual fund deposits. The fund earns 5% annually. Find the deposit, total annual outlay, and direct-amortization payment. (Answer: Deposit and outlay $\$2{,}854.88$; direct payment $\$3{,}000.00$.)
A $\$6{,}000$ loan runs for two years at 6% annually, and a separate fund earns 12% annually. Find the deposit, total annual outlay, direct payment, and annual outlay reduction. (Answer: Deposit $\$2{,}830.19$; sinking-fund outlay $\$3{,}190.19$; direct payment $\$3{,}272.62$; reduction $\$82.43$.)
A $\$2{,}000$ loan is due after one year. The loan rate is 4%, and the fund rate is 10%. Find the required fund deposit and total outlay, and explain whether the higher fund rate lowers the outlay. (Answer: Deposit $\$2{,}000.00$; total outlay $\$2{,}080.00$; no reduction relative to direct amortization, because the only deposit earns no interest before redemption.)

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