Splitting Principal Between Amortization and a Sinking Fund

Two Routes for Repaying One Loan

A borrower can repay different portions of a loan in different ways. One portion may be reduced through scheduled installments that cover both interest and principal. Another portion may remain outstanding until maturity, while the borrower makes deposits to a separate fund intended to accumulate enough to repay it. To calculate the borrower’s outlay correctly, keep the loan account and the fund account distinct.

The model below assumes a loan principal $P$, a fixed term of $n$ periods, and end-of-period payments. The amortized portion is $P_A$; the portion assigned to the sinking fund is $P_F$, so $P=P_A+P_F$. The loan earns an effective rate $i$ per period, while the separate fund earns an effective rate $j$ per period. If payments are annual, these are annual effective rates. The fund is intended to accumulate exactly $P_F$, with no initial fund balance, fees, taxes, or withdrawals before maturity.

Definition: A split-principal loan divides the original principal into an amortized tranche $P_A$ and a sinking-fund tranche $P_F$. The amortized tranche is repaid through level end-of-period installments at rate $i$. Interest on the sinking-fund tranche, $iP_F$, is paid each period, while level deposits accumulate separately at rate $j$ to repay $P_F$ at maturity.

This definition specifies an important feature of the arrangement: a deposit into the separate fund is not a payment to the lender. The lender’s loan balance falls as the amortized tranche is repaid, but the sinking-fund tranche remains outstanding until maturity. At that date, the accumulated fund is used to meet the remaining principal obligation.

Payment and Deposit Formulas

For the amortized tranche, let $R$ be the level installment paid at the end of each period. The present value of $n$ such payments at the loan rate $i$ must equal $P_A$. Using the annuity-immediate notation, this gives $R=P_A/a_{\overline{n}|}$, where $a_{\overline{n}|}$ is evaluated at $i$.

For the sinking-fund tranche, let $D$ be the level end-of-period deposit. The deposits accumulate at fund rate $j$, so their value at the end of period $n$ is $D s_{\overline{n}|}$, evaluated at $j$. Setting that value equal to $P_F$ gives $D=P_F/s_{\overline{n}|}$.

R=\frac{P_A}{a_{\overline{n}|\,i}} \qquad\text{and}\qquad D=\frac{P_F}{s_{\overline{n}|\,j}}

The notation makes the different roles of the rates explicit: $i$ values the installments that repay the amortized loan tranche, and $j$ accumulates the fund deposits. The sinking-fund portion also incurs loan interest $iP_F$ per period. Therefore, the borrower’s regular total outlay is

C=R+iP_F+D =\frac{P_A}{a_{\overline{n}|\,i}}+iP_F+ \frac{P_F}{s_{\overline{n}|\,j}}.

This formula assumes the same payment dates and period length for loan interest, installments, and fund deposits. If rates change, payment frequencies differ, the fund has an initial balance, or the target at maturity is not $P_F$, the account values must be adjusted to reflect those conditions rather than applying this formula unchanged.

Why the Formulas Hold

The installment formula follows from a present-value equation. Each end-of-period installment $R$ is discounted to time 0 at the loan rate $i$. The sum of those discounted payments is $R a_{\overline{n}|\,i}$. Because those payments repay the amortized tranche, their present value must equal $P_A$, which yields $R=P_A/a_{\overline{n}|\,i}$.

The deposit formula follows from accumulating each deposit to maturity. The first deposit earns interest for $n-1$ periods, the second for $n-2$, and the final deposit earns no interest before the end-of-period valuation date. Thus the accumulated value is

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

Since that fund must provide $P_F$ at maturity, $D s_{\overline{n}|\,j}=P_F$, and hence $D=P_F/s_{\overline{n}|\,j}$. The total-outlay formula then adds the three cash amounts the borrower must provide each period: the installment on the amortized tranche, interest on the sinking-fund tranche, and the separate fund deposit.

Worked Example: Calculating the Regular Outlay

A borrower takes a $100,000 loan for two years. The borrower will amortize $42,000 through two level annual installments and fund the remaining $58,000 with end-of-year deposits. The loan rate is 10% effective annually, and the fund earns 5% effective annually. Find the annual installment, fund deposit, interest paid on the sinking-fund tranche, and total annual outlay. Assume the fund starts at zero and is used to repay exactly $58,000 at the end of year 2.

Method. Value the amortized tranche as a two-payment annuity-immediate at the loan rate. Accumulate the two fund deposits at the fund rate. A common mistake is to use the loan rate for both accounts even though the fund earns a different rate.

Step 1: Find the amortized installment. Here $P_A=42{,}000$, $n=2$, and $i=0.10$. The annuity factor is

a_{\overline{2}|\,0.10} =\frac{1}{1.10}+\frac{1}{1.10^2} =0.9090909+0.8264463 =1.7355372.

Therefore $R=42{,}000/1.7355372$, or $24,200. To check, the present value of the installments is $24{,}200/1.10+24{,}200/1.10^2=22{,}000+20{,}000=42{,}000$.

Step 2: Find the fund deposit. The two end-of-year deposits accumulate at 5%. Thus $s_{\overline{2}|\,0.05}=1+1.05=2.05$, and

D=\frac{58{,}000}{2.05} =28{,}292.6829\ldots

The annual deposit is $28,292.68. Its maturity value is $28{,}292.6829\ldots\times2.05=58{,}000$.

Step 3: Add the interest on the sinking-fund tranche. The loan still has $58,000 of sinking-fund principal outstanding during each year, so annual interest on that tranche is $0.10(58{,}000)=$5{,}800$.

Step 4: Calculate the total outlay. The borrower pays the amortized installment, sinking-tranche interest, and fund deposit:

C=24{,}200+5{,}800+28{,}292.6829\ldots =58{,}292.6829\ldots

Rounded to the nearest cent, the annual outlay is $58,292.68. The installment’s present value is $42,000, and the fund accumulates to $58,000, so the two principal portions are both accounted for.

Reading the Two Accounts Over Time

A schedule helps distinguish the principal reduction in the loan account from the growth in the fund. On the amortized tranche, each installment consists of interest on the current loan balance and principal repayment. On the sinking-fund tranche, the loan balance stays level at $P_F$ during the term; the fund balance grows because of interest and deposits.

Worked Example: Checking the Tranche Balances at Maturity

A $70,000 loan has a two-year term. The borrower amortizes $21,000 through level annual installments and uses a sinking fund for the remaining $49,000. The loan rate is 10% effective annually, and the fund earns 5% effective annually. Payments and deposits are made at each year-end. Find the amortized-tranche schedule, the fund balance at each year-end, and the regular annual outlay.

Method. First derive the installment from the amortized tranche, then apply the loan balance recurrence. Separately accumulate the fund. This avoids treating the fund balance as though it were a payment that has already reduced the loan.

Step 1: Find the installment. As in the two-period 10% annuity calculation, $a_{\overline{2}|\,0.10}=1.7355372$. Therefore

R=\frac{21{,}000}{1.7355372}=$12{,}100.

Step 2: Track the amortized tranche. In year 1, interest is $0.10(21{,}000)=$2{,}100$. The installment repays $12{,}100-2{,}100=$10{,}000$ of principal, leaving a balance of $11,000. In year 2, interest is $0.10(11{,}000)=$1,100$. The $12,100 installment then repays $12{,}100-1{,}100=$11{,}000$, leaving a zero balance.

Step 3: Track the fund. The fund must accumulate $49,000 in two years. Since $s_{\overline{2}|\,0.05}=2.05$,

D=\frac{49{,}000}{2.05} =$23{,}902.4390\ldots.

After year 1, the fund balance is $23,902.4390. At the end of year 2, the first deposit has grown to $23{,}902.4390(1.05)=$25{,}097.5610$, and the second deposit adds $23,902.4390. Their sum is $49,000.

Step 4: Combine the cash flows. Interest on the $49,000 sinking-fund tranche is $0.10(49{,}000)=$4{,}900$ each year. The regular outlay is therefore

C=12{,}100+4{,}900+23{,}902.4390\ldots =$40{,}902.4390\ldots.

The annual outlay is $40,902.44, rounded to the nearest cent. At maturity, the amortized tranche has a zero balance and the fund has accumulated exactly $49,000, matching the principal left outstanding on the sinking-fund tranche.

Solving for the Principal Split

The same structure can be used in reverse when the total outlay is given and the borrower’s principal split is unknown. Write $P_A=xP$ and $P_F=(1-x)P$, where $x$ is the fraction assigned to amortization. Substituting into the outlay equation gives

C=xP\left(\frac{1}{a_{\overline{n}|\,i}}\right) +(1-x)P\left(i+\frac{1}{s_{\overline{n}|\,j}}\right).

This is a linear equation in $x$. Its solution must satisfy $0\leq x\leq1$, because neither principal portion can be negative. The endpoint $x=1$ represents a fully amortized loan; $x=0$ represents principal fully funded through the sinking fund, with loan interest paid separately. If a proposed outlay leads to $x$ outside this interval, it is not achievable under the stated model.

Worked Example: Finding the Amortized Fraction from the Outlay

A $100,000 loan has a two-year term and a 10% effective annual loan rate. A separate fund earns 0% effective annually, and the borrower makes equal end-of-year deposits. The stated total annual outlay is $58,571.43. Find the fraction of principal assigned to amortization and check the resulting components.

Method. Express each component of the outlay as a function of the unknown amortized fraction $x$, then solve the resulting linear equation. The fund earns zero, so its two deposits simply add to the target principal.

Step 1: Express the installment per dollar of amortized principal. For two periods at 10%, the installment factor is

\frac{1}{a_{\overline{2}|\,0.10}} =\frac{0.10(1.10)^2}{(1.10)^2-1} =0.5761904762\ldots.

Thus the installment on $100{,}000x$ is $57{,}619.04762x$ dollars.

Step 2: Express the sinking-fund costs. The sinking-fund principal is $100{,}000(1-x)$. Its annual loan interest is $10{,}000(1-x)$. With a zero fund yield and two end-of-year deposits, each deposit is $100{,}000(1-x)/2=50{,}000(1-x)$. Therefore

$$ C=57{,}619.04762x+10{,}000(1-x)+50{,}000(1-x) =60{,}000-2{,}380.95238x. $$

Step 3: Solve and check. Set this expression equal to the stated outlay:

58{,}571.42857=60{,}000-2{,}380.95238x, \qquad x=\frac{1{,}428.57143}{2{,}380.95238}=0.60.

Therefore, $60,000 is amortized and $40,000 is assigned to the fund. The installment is $0.5761904762(60{,}000)=$34{,}571.43$; sinking-tranche interest is $0.10(40{,}000)=$4{,}000$; and each fund deposit is $40{,}000/2=$20{,}000$. Their sum is $58,571.43, confirming the split.

Common Mistakes and Model Checks

The most common error is subtracting the fund deposit from the loan balance when calculating loan interest. A deposit is placed in a separate account; it does not, by itself, reduce the amount owed to the lender. For the $70,000 loan in the schedule example, the year-1 deposit is about $23,902.44. Incorrectly subtracting it from the $70,000 loan principal would produce interest of about $0.10(70{,}000-23{,}902.44)=$4{,}609.76$. That is not the contractual interest. The correct year-1 loan interest is $0.10(70{,}000)=$7{,}000$, comprising $2,100 on the amortized tranche and $4,900 on the sinking-fund tranche.

Another error is omitting the separate interest payment on $P_F$, or counting that interest twice by adding it again to the amortized installment. The installment $R$ covers interest and principal only on $P_A$. The term $iP_F$ covers interest on the outstanding sinking-fund principal, while $D$ is a deposit into the fund, not an interest payment to the lender.

Before accepting a result, check the two maturity conditions: the amortized-tranche balance should be zero after the final installment, and the accumulated sinking fund should equal $P_F$. Also check that $P_A+P_F=P$ and that the total outlay includes exactly one installment, one sinking-tranche interest payment, and one fund deposit per period.

Check Your Understanding

Assume end-of-period payments, no initial fund balance, and no fees or taxes unless a question says otherwise. Round dollar amounts to the nearest cent.

A $40,000 loan has a two-year term. The amortized tranche is $16,000, the fund tranche is $24,000, and both the loan and fund rates are 5% effective annually. Find the annual installment, annual fund deposit, and total annual outlay. (Answer: $8,604.88 installment; $11,707.32 deposit; $21,512.20 total outlay.)
A $80,000 loan runs for two years. The borrower amortizes $30,000 and funds the rest. The loan rate is 10% effective annually, and the fund earns 0% effective annually. Find the annual outlay. (Answer: $47,285.71.)
A $120,000 two-year loan has $60,000 amortized and $60,000 assigned to a sinking fund. The loan rate is 10% and the fund rate is 5%, both effective annually. Find the annual installment, fund deposit, and total outlay. (Answer: $34,571.43 installment; $29,268.29 deposit; $69,839.72 total outlay.)
A $100,000 loan runs for two years at a 10% effective annual loan rate. The fund earns 0% effective annually. If the annual total outlay is $58,571.43, what percentage of the principal is amortized? (Answer: 60% is amortized and 40% is assigned to the fund.)

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