Integrated Sinking-Fund Loan Problem Workshop

Introduction

A sinking-fund loan requires two accounts to be tracked at once. The borrower pays interest on the outstanding loan, while separately making deposits into a fund intended to provide money for principal redemption. If the loan rate and fund yield differ, the fund’s accumulated value cannot be inferred from the loan balance. Existing fund money, deposits made between regular dates, and changing yields make that distinction especially important.

The method in this workshop is to establish the amount due at maturity, accumulate each fund contribution from its own deposit date, and compare the resulting fund value with the redemption amount. For a loan with both amortized principal and a sinking fund, first identify the amount of principal assigned to each method. The sinking-fund deposit is an investment contribution, not a repayment to the lender.

Definition: A sinking fund is an account in which deposits accumulate to meet a specified future payment, often the principal or redemption amount of a loan. A fund surplus at maturity is the fund value minus the amount it must pay; a fund shortfall is the amount by which the required payment exceeds the fund value.

A Consistent Valuation Framework

Assume all rates and dates are known, and that fund deposits are credited to the account on their stated dates. A deposit made after interest is credited on a date does not earn that date’s interest; it earns interest only in subsequent periods. Unless stated otherwise, loan interest is paid separately from the fund, and no fund withdrawal occurs before maturity.

Let $R$ be the redemption amount due at time $T$, and let $F(T)$ be the accumulated value of the fund immediately before redemption. For an initial balance $B_0$, deposits $D_j$ at times $t_j$, and annual effective fund rate $i_F$, with fractional-year accumulation explicitly modeled by $(1+i_F)^s$, the value is:

F(T)=B_0(1+i_F)^T+\sum_j D_j(1+i_F)^{T-t_j}.

This single-rate expression applies only when the fund earns the same effective annual rate throughout the accumulation period and the stated fractional-year convention is appropriate. If the rate changes, use the product of the accumulation factors for the periods each amount remains in the fund. In either case, compare the accumulated fund with the redemption amount:

\text{surplus}=F(T)-R,\qquad \text{shortfall}=R-F(T).

A positive surplus means the fund exceeds the required payment; a positive shortfall means additional money is needed. Do not subtract deposits from the loan principal unless the contract actually applies them as principal repayments.

Why Each Deposit Accumulates Separately

A fund is valued by carrying each amount forward from the date it enters the account. For example, if an account begins with $B_0$, earns an effective rate $i_F$ for one period, and then receives a deposit $D_1$, its value immediately after that deposit is $B_0(1+i_F)+D_1$. After another period, the value is:

\bigl(B_0(1+i_F)+D_1\bigr)(1+i_F) =B_0(1+i_F)^2+D_1(1+i_F).

The initial balance earns two periods of interest, while the later deposit earns one. Repeating this argument gives one accumulated term for each deposit, with an exponent equal to the number of periods from its deposit date to the valuation date. This also explains why an existing balance must be accumulated from time zero rather than treated as a deposit at a later date.

Set the maturity target.
State whether the target is principal, a contractual redemption amount, or the remaining principal after any amortization.

Map the fund timeline.
List the opening balance, each deposit date, and the interest factors applying after each amount enters the account.

Accumulate and compare.
Find the fund value at maturity, then subtract the target in the appropriate order to report a surplus or shortfall.

Track loan cash flows separately.
Calculate coupon interest on the outstanding loan principal and identify any principal repaid before maturity.

Worked Problems

Worked Example: Finding an Irregular Final Deposit

A hypothetical company borrows $100,000 for four years at an effective annual loan interest rate of 5%. Interest is paid annually, and the contractual redemption amount at the end of year 4 is $105,000. A separate sinking fund earns an effective annual rate of 3%; fractional-year accumulation follows $(1.03)^s$. The fund starts with $12,000 at time 0, receives $10,000 at time 1 and $15,000 at time 2.5, each after any interest credited on that date, and receives a final deposit $X$ at time 3.5. Find $X$ so that the fund exactly meets redemption.

Method. Accumulate each existing amount to time 4. The opening balance earns four years, the time-1 deposit earns three years, and the time-2.5 deposit earns 1.5 years. The final deposit earns half a year.

F(4)=12{,}000(1.03)^4+10{,}000(1.03)^3 +15{,}000(1.03)^{1.5}+X(1.03)^{0.5}.

The known maturity values are:

12{,}000(1.03)^4=13{,}506.10572,\qquad 10{,}000(1.03)^3=10{,}927.27000, $$ $$ 15{,}000(1.03)^{1.5} =15{,}000(1.0453358312\ldots) =15{,}680.03747\ldots.

Their unrounded sum is $40,113.41319 approximately. Therefore the final deposit must provide $105,000 minus this amount at maturity:

X=\frac{105{,}000-40{,}113.41319\ldots}{(1.03)^{0.5}} =\frac{64{,}886.58681\ldots}{1.0148891565\ldots} =63{,}934.6537\ldots.

Rounded to the nearest cent, $X=\$63{,}934.65$. Using the rounded deposit, the fund accumulates to approximately $105,000.00 to the nearest cent. The annual loan-interest payment is $0.05(100{,}000)=\$5{,}000$ each year; those coupon payments are separate and do not reduce the $105,000 redemption target.

Check and trap avoided. The deposit at time 2.5 earns 1.5 years, not two full years, and the time-3.5 deposit earns only half a year. Applying a full year of interest to either would overstate the fund and understate the required final deposit.

Worked Example: Changing Fund Yields and a Maturity Shortfall

A hypothetical borrower has a $90,000 loan with a 5.5% effective annual interest rate. The loan pays annual interest and has a contractual redemption amount of $94,500 at the end of year 3. A separate fund starts with $8,000. It earns 2% in year 1, 3% in year 2, and 4% in year 3. After year-1 interest is credited, the borrower deposits $12,000; after year-2 interest is credited, the borrower deposits $20,000. Find the fund value and the additional amount needed at maturity.

Method. Use the actual sequence of annual fund rates. The opening balance experiences all three rates. The first deposit experiences only the year-2 and year-3 rates, and the second deposit experiences only the year-3 rate.

$$ F(3)=8{,}000(1.02)(1.03)(1.04) +12{,}000(1.03)(1.04)+20{,}000(1.04). $$

For the opening balance, $(1.02)(1.03)=1.0506$, and $1.0506(1.04)=1.092624$, so its maturity value is $8{,}000(1.092624)=\$8{,}740.992$. The first deposit accumulates by $1.03(1.04)=1.0712$, giving $12{,}000(1.0712)=\$12{,}854.40$. The second deposit grows by 4%, giving $20{,}000(1.04)=\$20{,}800$. Thus:

F(3)=8{,}740.992+12{,}854.40+20{,}800 =\$42{,}395.392.

The fund is short of the redemption amount by $94{,}500-42{,}395.392=\$52{,}104.608$, or $52,104.61 to the nearest cent. The annual loan-interest payment is $0.055(90{,}000)=\$4{,}950$, paid separately; total coupon interest over three years is $14,850. It is not part of the fund shortfall calculation.

Check and trap avoided. The fund value is below redemption, so the difference is a shortfall, not a surplus. Applying the final 4% rate to the time-1 deposit for two years would incorrectly give it an extra period of growth.

Worked Example: Combining Amortized Principal and a Sinking Fund

A hypothetical $120,000 loan has a four-year term and a 6% effective annual interest rate. The borrower repays $7,500 of principal at the end of each year, leaving $90,000 to be redeemed at the end of year 4. A separate fund earns 4% annually, starts with $6,000 at time 0, and receives deposits of $16,000, $18,000, $20,000, and $21,000 at the ends of years 1 through 4, respectively, after that year’s fund interest is credited. Find the maturity shortfall and the annual loan-interest payments.

Fund calculation. The fund target is the $90,000 remaining after the four scheduled principal repayments. The opening balance earns four years; the four deposits earn three, two, one, and zero years, respectively.

F(4)=6{,}000(1.04)^4+16{,}000(1.04)^3 +18{,}000(1.04)^2+20{,}000(1.04)+21{,}000.

The factors are $(1.04)^2=1.0816$, $(1.04)^3=1.124864$, and $(1.04)^4=1.16985856$. The individual maturity values are therefore:

6{,}000(1.16985856)=7{,}019.15136,\quad 16{,}000(1.124864)=17{,}997.824, $$ $$ 18{,}000(1.0816)=19{,}468.80,\quad 20{,}000(1.04)=20{,}800,\quad 21{,}000=21{,}000.

Adding these amounts gives $F(4)=\$86{,}285.77536$. The shortfall is $90{,}000-86{,}285.77536=\$3{,}714.22464$, or $3,714.22 to the nearest cent.

Loan-interest calculation. Interest is based on the principal outstanding during each year. The balances at the start of years 1 through 4 are $120,000, $112,500, $105,000, and $97,500. The corresponding payments are:

0.06(120{,}000)=7{,}200,\quad 0.06(112{,}500)=6{,}750,\quad 0.06(105{,}000)=6{,}300,\quad 0.06(97{,}500)=5{,}850.

Check and trap avoided. The $7,500 annual principal repayments reduce the loan balance; the sinking-fund deposits do not. Counting deposits as additional principal repayments would reduce the stated $90,000 maturity target incorrectly. The final $21,000 deposit is made at maturity after fund interest is credited, so it earns no interest in this calculation.

Common Mistakes in Integrated Problems

The arithmetic is usually manageable once each amount is assigned to the correct account and date. Errors tend to arise when loan and fund cash flows are blended, or when a date convention is overlooked.

Using the loan rate for the fund. The loan rate determines interest owed to the lender; the fund rate determines how deposits grow. They may be different.
Giving a deposit the wrong number of periods. A deposit made after interest is credited does not earn interest for that period. Count only the future periods or factors.
Ignoring an existing fund balance. Accumulate it from its actual starting date, just as for any other fund amount.
Applying one rate across changing yields. When rates vary, multiply the applicable factors in chronological order for each amount.
Confusing a fund deposit with a principal repayment. A deposit builds a separate asset. It does not reduce the debt unless the loan contract says it is paid to the lender.
Reversing the maturity comparison. Compare fund value with the stated redemption target, not automatically with original principal. A redemption premium or prior amortization changes the target.

For a final check, verify that every deposit has the right accumulation duration, that the fund target matches the principal still due, and that loan interest is calculated from the loan balance rather than the fund balance. This account-by-account approach is useful for integrated loan questions aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, keep loan cash flows and fund accumulation separate. Round money to the nearest cent.

A fund starts with $4,000 and earns an effective annual rate of 3% for three years. Deposits of $5,000 and $6,000 are made at the ends of years 1 and 2, after interest is credited. Find the value immediately after the year-3 interest credit. (Answer: $15,855.41)
A fund earns 2% annually. It starts with $7,000, receives $10,000 at the end of year 1 and $12,000 at the end of year 3, and must contain $50,000 at the end of year 4. All deposits occur after interest is credited on their dates. Find the deposit required at the end of year 2. (Answer: $18,810.93)
A $60,000 loan charges 5% effective annual interest, paid separately each year for three years. A zero-balance sinking fund earns 3% annually and receives equal deposits at the ends of years 1, 2, and 3, after interest is credited. Find the annual deposit needed to accumulate $60,000 at maturity and the total loan interest paid. (Answer: Annual deposit $19,411.82; total loan interest $9,000.00)
A loan’s contractual redemption amount is $30,000 at the end of year 3. Its separate fund starts with $10,000, earns 2% in year 1, 3% in year 2, and 4% in year 3, and receives deposits of $5,000 after year-1 interest and $8,000 after year-2 interest. Find the maturity fund value and state the surplus or shortfall. (Answer: Fund value $24,602.24; shortfall $5,397.76)

What You'll Learn