Funding a Redemption Amount Different From Principal

Why the Redemption Amount Matters

A loan’s original principal and its contractual payment at maturity do not have to be equal. For example, a contract may require a maturity payment above or below the amount initially borrowed. If a separate sinking fund is intended to provide that payment, the fund must be designed around the amount actually due—not automatically around the original principal.

This tutorial considers a specific arrangement: the borrower receives principal $P$, pays interest each period on that fixed principal, and makes level deposits into a separate fund. At maturity, the fund is used to pay the contractual redemption amount $R$. Interest payments do not reduce $P$ in this arrangement, and the fund deposits are not loan-principal repayments.

Definition: The original principal $P$ is the amount borrowed at the start of the loan. The redemption amount $R$ is the lump sum contractually due at maturity. They may be equal, or $R$ may be greater or less than $P$. A sinking fund is a separate fund accumulated through deposits, here intended to provide $R$ at maturity.

Assume there are $n$ equal periods, a fixed effective loan interest rate $i_L$ per period, and a fixed effective sinking-fund yield $i_F$ per period. Unless stated otherwise, deposits are made at the end of each period, the fund earns interest at the end of each period, and there are no fees, taxes, defaults, or withdrawals. These assumptions make the two rates’ roles explicit: $i_L$ determines the loan-interest payments, while $i_F$ determines how quickly the fund accumulates.

Level Deposits to Accumulate the Redemption Amount

Let $D$ be the end-of-period deposit. The first deposit earns interest for $n-1$ periods, the second for $n-2$, and the final deposit earns no interest before maturity. Thus, at time $n$, the fund balance is the sum of all deposits accumulated to that date.

D\left((1+i_F)^{n-1}+(1+i_F)^{n-2}+\cdots+(1+i_F)+1\right) = D\,s_{\overline{n}|i_F}

The accumulated-value factor is $s_{\overline{n}|i_F}=\frac{(1+i_F)^n-1}{i_F}$ when $i_F\ne 0$, and $s_{\overline{n}|0}=n$. Equating the fund at maturity to the amount it must pay gives the required deposit.

Result: For level end-of-period deposits, no initial fund balance, and a constant effective fund yield $i_F$, the deposit required to accumulate redemption amount $R$ is $$ D=\frac{R}{s_{\overline{n}|i_F}}. $$ The formula applies when the fund is intended to provide the entire maturity payment $R$. It does not mean that the deposit is based only on $R-P$.

If the fund already contains $B_0$ at time 0, that balance accumulates to $B_0(1+i_F)^n$. The required level deposit is then $D=\frac{R-B_0(1+i_F)^n}{s_{\overline{n}|i_F}}$, provided the numerator is nonnegative. If the existing balance already exceeds $R$ at maturity, no positive level deposit is needed to reach the target; a separate withdrawal or surplus decision would have to be specified.

Borrower Outlay When Interest Is Paid Separately

Under the stated fixed-principal arrangement, the borrower pays $P i_L$ in loan interest each period. The borrower also contributes $D$ to the sinking fund each period. Therefore, the periodic outlay during the loan term is the sum of those two amounts. At maturity, the fund provides $R$ to the lender; do not add $R$ a second time to the borrower’s outlay if the deposits are being counted as the funding payments.

\text{periodic outlay}=P i_L+D \qquad\text{and}\qquad \text{total outlay over }n\text{ periods}=n(P i_L+D).

These totals describe the future interest payments and fund deposits only. They exclude the initial loan proceeds received by the borrower and any fees or other cash flows not specified. The formula is not appropriate if the loan contract instead reduces the interest-bearing balance over time or requires the borrower to pay the redemption amount directly in addition to the sinking-fund contributions.

Worked Applications

Worked Example: Redemption Above the Original Principal

A borrower receives $P=\$20{,}000$. The loan requires annual interest of $7\%$ on that fixed principal and a redemption payment of $R=\$20{,}500$ at the end of year 2. A separate fund earns $5\%$ effective annually. Find the equal end-of-year fund deposit and the borrower’s total outlay over the two years.

Method. Accumulate the full redemption amount in the fund, then add the fund deposit to the separately calculated annual loan interest. The main mistake avoided is targeting only the difference $R-P$.

Step 1: Find the fund accumulation factor. With $n=2$ and $i_F=0.05$, $s_{\overline{2}|0.05}=1+1.05=2.05$. Equivalently, $\frac{1.05^2-1}{0.05}=\frac{1.1025-1}{0.05}=2.05$.

Step 2: Find the annual deposit. $D=\frac{\$20{,}500}{2.05}=\$10{,}000$. The fund check is $\$10{,}000(1.05)+\$10{,}000=\$10{,}500+\$10{,}000=\$20{,}500$.

Step 3: Find interest and outlay. Annual loan interest is $P i_L=\$20{,}000(0.07)=\$1{,}400$. Annual borrower outlay is $\$1{,}400+\$10{,}000=\$11{,}400$, so the two-year total is $2(\$11{,}400)=\$22{,}800$.

Check. The fund reaches the exact redemption amount, and the total outlay consists of $\$2{,}800$ in loan interest plus $\$20{,}000$ in deposits, for $\$22{,}800$.

Worked Example: Redemption Below the Original Principal

A borrower receives $P=\$40{,}000$. The contract requires annual interest of $5\%$ on the fixed principal and a redemption payment of $R=\$33{,}100$ after 3 years. A separate sinking fund earns $10\%$ effective annually. Find the required equal year-end deposit and total borrower outlay.

Method. Use the contract’s actual redemption amount as the target. A redemption below $P$ changes the funding target; it does not change the stated basis for the annual interest payment.

Step 1: Find the fund accumulation factor. $s_{\overline{3}|0.10}=1+1.10+1.10^2=1+1.10+1.21=3.31$.

Step 2: Calculate the deposit. $D=\frac{\$33{,}100}{3.31}=\$10{,}000$. Check the accumulated value by deposit date: the first deposit accumulates to $\$10{,}000(1.10)^2=\$12{,}100$; the second to $\$10{,}000(1.10)=\$11{,}000$; and the third to $\$10{,}000$. Their sum is $\$33{,}100$.

Step 3: Calculate borrower outlay. Annual loan interest is $\$40{,}000(0.05)=\$2{,}000$. Annual outlay is $\$2{,}000+\$10{,}000=\$12{,}000$, and total outlay is $3(\$12{,}000)=\$36{,}000$.

Check. The total comprises $\$6{,}000$ of loan interest and $\$30{,}000$ of fund deposits. The fund pays the specified $\$33{,}100$ at maturity.

Worked Example: Including an Existing Fund Balance

A borrower receives $P=\$20{,}000$ and pays annual interest of $6\%$ on that fixed principal. The maturity redemption amount after 2 years is $R=\$20{,}810$. The separate fund already contains $B_0=\$4{,}000$ at time 0 and earns $5\%$ effective annually. Find the required level year-end deposit and the future borrower outlay over the two years.

Method. First accumulate the existing balance, then fund only the remainder of the redemption amount with level deposits. The existing balance is already in the fund, so it is not a new outlay during these two years.

Step 1: Accumulate the existing balance. $\$4{,}000(1.05)^2=\$4{,}000(1.1025)=\$4{,}410$.

Step 2: Find the amount that deposits must accumulate. $\$20{,}810-\$4{,}410=\$16{,}400$. Since $s_{\overline{2}|0.05}=2.05$, the level deposit is $D=\frac{\$16{,}400}{2.05}=\$8{,}000$.

Step 3: Check the maturity fund. The deposits accumulate to $\$8{,}000(1.05)+\$8{,}000=\$8{,}400+\$8{,}000=\$16{,}400$. Adding the existing balance’s accumulated value gives $\$16{,}400+\$4{,}410=\$20{,}810$.

Step 4: Calculate future outlay. Annual loan interest is $\$20{,}000(0.06)=\$1{,}200$. Annual outlay is $\$1{,}200+\$8{,}000=\$9{,}200$; over two years it is $2(\$9{,}200)=\$18{,}400$.

Check. The future outlay is $\$2{,}400$ in interest plus $\$16{,}000$ in new deposits. It excludes the original $\$4{,}000$ fund balance because that amount was already present at time 0.

Why the Deposit Formula Holds

The formula follows from the fund’s cash flows, not from the relationship between $R$ and $P$. Each end-of-period deposit contributes its own accumulated value at maturity. Under a constant effective fund rate, those values form a geometric series.

List the maturity values.
The deposit at time 1 is worth $D(1+i_F)^{n-1}$ at time $n$; the deposit at time 2 is worth $D(1+i_F)^{n-2}$; the final deposit is worth $D$.

Add the accumulated deposits.
The geometric-series sum is $D\frac{(1+i_F)^n-1}{i_F}=D s_{\overline{n}|i_F}$ when $i_F\ne0$. If $i_F=0$, the sum is $nD$.

Match the contractual target.
Because the fund must provide $R$, set $D s_{\overline{n}|i_F}=R$, giving $D=R/s_{\overline{n}|i_F}$. If an initial balance exists, add its accumulated value before setting the total equal to $R$.

The original principal does not appear in this derivation because the derivation concerns only the fund’s target. Principal does enter the outlay calculation through interest when the loan contract specifies that interest is paid on fixed principal $P$. The two calculations answer different questions and should be kept separate.

Common Exam Trap: Funding Only the Difference

For the first example, an incorrect calculation might treat $R-P=\$500$ as the amount the fund must accumulate and then calculate $\$500/2.05=\$243.90$ per year. That would accumulate only $\$500$, not the required $\$20{,}500$. The resulting fund would be short by $\$20{,}000$ at maturity.

The correction is to read the cash-flow arrangement carefully. Here the fund pays the entire maturity redemption, so the target is $R$, and the correct deposit is $\$20{,}500/2.05=\$10{,}000$. Funding only $R-P$ would be appropriate only if the problem separately specified another source or repayment arrangement for the remaining amount. Also avoid using $i_L$ in the fund accumulation factor unless the fund itself earns that rate.

Key takeaway: Identify the maturity payment the fund must actually provide, accumulate that amount at the fund’s yield, and calculate loan interest separately using the contract’s stated interest-bearing principal. The redemption amount and the original principal need not be the same.

Check Your Understanding

For each problem, assume no fees or taxes and use the stated rates and deposit timing. Report dollar amounts to the nearest cent.

A loan has fixed principal $\$12{,}000$, annual interest of $7\%$, and a redemption amount of $\$12{,}600$ after 2 years. A fund earns $5\%$ effective annually, with equal year-end deposits. Find the deposit and total borrower outlay. (Answer: deposit $\$6{,}146.34$; total outlay $\$13{,}972.68$)
A borrower receives $\$50{,}000$, pays $6\%$ annual interest on fixed principal, and must redeem the loan for $\$45{,}000$ after 3 years. The fund earns $10\%$ effective annually. Find the equal year-end deposit and total outlay. (Answer: deposit $\$13{,}595.17$; total outlay $\$49{,}785.50$)
A loan has fixed principal $\$60{,}000$, annual interest of $8\%$, and a redemption amount of $\$63{,}050$ after 3 years. The fund earns $5\%$ effective annually. Find the equal year-end deposit and the total borrower outlay. (Answer: deposit $\$20{,}000.00$; total outlay $\$74{,}400.00$)
A fund already holds $\$6{,}000$ at time 0 and earns $5\%$ effective annually. It must provide a redemption amount of $\$27{,}115$ in 2 years. The loan’s fixed principal is $\$26{,}000$, with annual interest of $4\%$. Find the equal year-end deposit and future total borrower outlay, excluding the pre-existing fund balance. (Answer: deposit $\$10{,}000.00$; future total outlay $\$22{,}080.00$)
A borrower receives $\$15{,}000$, pays annual interest of $6\%$ on fixed principal, and must accumulate a redemption amount of $\$16{,}500$ in 2 years. The fund earns $10\%$ effective annually, and equal deposits are made at the beginning of each year. Find each deposit and the total borrower outlay, including two annual interest payments. (Answer: deposit $\$7{,}142.86$; total outlay $\$16{,}085.71$)

This treatment is aligned with the SOA Exam FM (Financial Mathematics) syllabus. For any problem, first identify the contractual maturity amount, then confirm the fund’s rate and deposit timing, and finally add the separately specified loan-interest payments to the required deposits.

What You'll Learn