Balloon Payments and Partially Amortizing Loans

Introduction

A loan does not have to be fully repaid through its scheduled installments. A borrower and lender may agree to smaller level payments during the loan term, with the remaining principal due as a balloon payment at maturity. This arrangement changes the size of the periodic payment, but it does not change the basic valuation principle: the loan amount must equal the value of all payments made under the agreement.

The central calculation is to determine how much principal remains after the scheduled installments. Alternatively, if the balloon amount is specified, the same relationship determines the level installment. The key is to keep the rate, payment interval, number of payments, and balloon date aligned.

Definition: A partially amortizing loan is a loan for which scheduled payments reduce the principal but do not fully repay it by the stated maturity date. For a loan of principal $L$, with $n$ equal payments $R$ at the ends of successive periods and a fixed effective rate $i$ per period, let $B_n$ denote the balance immediately after the $n$th scheduled payment. If $B_n>0$, that balance is the balloon amount due at maturity. If the scheduled installment is also due on that date, the total maturity-date cash payment is $R+B_n$.

Valuing the Installments and Balloon

Assume that the loan is issued at time 0, payments are made at the ends of periods 1 through $n$, interest is credited at the same fixed effective rate $i$ in each period, and there are no fees or other cash flows. The balloon balance is measured just after the scheduled installment at time $n$. These assumptions match the ordinary-annuity timing used for end-of-period loan payments.

The loan amount equals the present value of the $n$ installments plus the balloon due at time $n$. With $v=1/(1+i)$, the relationship is:

L=R a_{\overline{n}|}+B_n v^n

Here $a_{\overline{n}|}$ is the present value of $n$ unit payments at the ends of the periods. Solving for the installment when the balloon is specified gives:

R=\frac{L-B_n v^n}{a_{\overline{n}|}}

For a fixed installment, the balance can instead be found by accumulating all cash flows to time $n$:

B_n=L(1+i)^n-Rs_{\overline{n}|}

The factor $s_{\overline{n}|}$ accumulates $n$ unit payments made at the ends of periods to time $n$. These present-value and accumulated-value approaches are equivalent: one values every cash flow at the loan date; the other values every cash flow at maturity.

Why the Balance Formula Holds

At time $n$, the original loan principal has accumulated to $L(1+i)^n$. Each scheduled payment has reduced that accumulated amount. The first payment, made at time 1, accumulates for $n-1$ periods; the second accumulates for $n-2$ periods; and the final payment is already at time $n$. Thus the accumulated value of the installments at time $n$ is:

R(1+i)^{n-1}+R(1+i)^{n-2}+\cdots+R(1+i)+R =R\frac{(1+i)^n-1}{i} =Rs_{\overline{n}|}

Subtracting this accumulated value from the accumulated original principal leaves the balance immediately after the final scheduled installment:

B_n=L(1+i)^n-Rs_{\overline{n}|}

This derivation also explains why simply subtracting $nR$ from the accumulated principal is incorrect: earlier payments earn interest after they are made, so they must be accumulated to the same valuation date as the loan.

Worked Example: Find the Balloon After Three Annual Payments

A hypothetical borrower receives a loan of $100,000. The loan charges an effective annual rate of 10%, and the borrower makes three end-of-year payments of $25,000. Find the balloon balance immediately after the third payment, and state the total cash payment due on that date if the third installment is also due.

Method. Accumulate the original principal and each installment to the end of year 3. The annual payment period matches the effective annual rate, so $i=0.10$ and $n=3$.

The accumulated original principal is:

100{,}000(1.10)^3=100{,}000(1.331)=133{,}100

The accumulated value of the three payments is $25{,}000s_{\overline{3}|}$, where:

s_{\overline{3}|}=\frac{(1.10)^3-1}{0.10} =\frac{1.331-1}{0.10} =3.31

Therefore, the installments have an accumulated value of $25{,}000(3.31)=82{,}750$. The remaining balance is:

B_3=133{,}100-82{,}750=50{,}350

The balloon is $50,350.00. Because the third scheduled installment of $25,000 is also due at maturity, the total cash payment on that date is $25{,}000+50{,}350=$ $75,350.00.

Check. The present value of the three installments is $25{,}000a_{\overline{3}|}$, with $a_{\overline{3}|}=2.48685199$, or about $62,171.30. The present value of the balloon is $50{,}350/(1.10)^3$, or about $37,828.70. Together they equal $100,000.00, as required.

Choosing the Installment for a Specified Balloon

A contract may specify both the maturity date and the balloon amount, leaving the level installment to be determined. Use the present-value equation when it is convenient to discount the balloon to time 0. Use the accumulated-value equation when it is convenient to carry all amounts to time $n$. Either method requires the balloon to be valued at the agreed maturity date, not at the date of the last earlier installment.

Worked Example: Find the Installment for a Target Balloon

A hypothetical $100,000 loan has an effective annual rate of 10% and a two-year term. The borrower will make two equal end-of-year installments, followed immediately after the second installment by a separate balloon of $50,000. Find the installment.

Method. Work at the end of year 2. The original loan accumulates to $100{,}000(1.10)^2=$$121,000. The two installments accumulate to $Rs_{\overline{2}|}$, where:

s_{\overline{2}|}=\frac{(1.10)^2-1}{0.10} =\frac{1.21-1}{0.10} =2.10

At maturity, the accumulated loan must cover the accumulated installments and the $50,000 balloon:

$$ 121{,}000=2.10R+50{,}000 $$

Rearrange and solve:

R=\frac{121{,}000-50{,}000}{2.10} =\frac{71{,}000}{2.10} =33{,}809.5238\ldots

Rounded to the nearest cent, each installment is $33,809.52. The maturity-date cash flow is the final installment plus the balloon, or $33,809.52 + $50,000.00 = $83,809.52.

Check. Before rounding, the accumulated installments are $2.10(33{,}809.5238\ldots)=$$71,000. Added to the $50,000 balloon, this is $121,000, the accumulated value of the original loan.

Payment Periods and Rate Consistency

The rate $i$ in these formulas is the effective rate per payment period. The exponent $n$ counts payment periods, not necessarily years. For example, if payments are quarterly and the stated rate is a nominal annual rate $i^{(4)}$ convertible quarterly, the quarterly rate is $i^{(4)}/4$. If instead the stated rate is an effective annual rate, the equivalent quarterly rate is $(1+i)^{1/4}-1$. Identify the rate convention before substituting into a loan formula.

The following example uses a nominal annual rate convertible at the same frequency as the payments.

Worked Example: Quarterly Payments and a Balloon

A hypothetical business borrows $150,000 at a nominal annual rate of 8% convertible quarterly. It makes eight end-of-quarter payments of $15,000 over two years. Find the balance after the eighth payment and the total cash due at that date if the eighth installment is also due.

Method. There are four quarters per year, so the effective quarterly rate is $i=0.08/4=0.02$. The term is $n=8$ quarters. Accumulate all cash flows to the end of quarter 8.

The loan accumulated to that date is:

150{,}000(1.02)^8 =150{,}000(1.171659381002) =175{,}748.90715

The accumulation factor for the eight payments is:

s_{\overline{8}|} =\frac{(1.02)^8-1}{0.02} =\frac{1.171659381002-1}{0.02} =8.5829690501

The accumulated value of the payments is $15{,}000(8.5829690501)=128{,}744.53575$. Therefore:

B_8=175{,}748.90715-128{,}744.53575 =47{,}004.37140

To the nearest cent, the balloon is $47,004.37. The total maturity-date cash payment, including the eighth installment, is $15,000.00 + $47,004.37 = $62,004.37.

Check. The eight installments accumulated to $128,744.53575, and the remaining balance is $47,004.37140. Their sum is $175,748.90715, which matches the accumulated loan principal.

Interpreting the Result and Avoiding Common Errors

A positive $B_n$ means the installments do not fully repay the loan by maturity. For a specified loan amount and term, an installment equal to $L/a_{\overline{n}|}$ would fully amortize the loan, leaving no balloon under these assumptions. A smaller installment leaves a positive balance; a larger installment produces a zero or negative result from the formula, indicating that the assumed payments would repay more than the original loan balance by that date.

A balloon is a balance, not automatically the entire final-date cash flow. If the contract requires both a regular installment and repayment of the balance on the same date, both amounts are due. If the contract instead describes the balloon as the single final payment inclusive of the regular installment, the cash-flow definition must be adjusted: the specified final payment is not an additional amount on top of that installment.

A common mistake is to treat every installment as though it were made at maturity. For the two-year loan in the preceding example, using $121{,}000=2R+50{,}000$ would give $R=$$35,500. This is wrong because the first installment was paid one year earlier and accumulates for one year. The correct maturity-date equation is $121{,}000=1.10R+R+50{,}000=2.10R+50{,}000$, giving $R=$$33,809.52. The general factor $s_{\overline{n}|}$ accounts for the different payment dates.

When checking a calculation, use the same valuation date for every amount. A present-value check compares the loan at time 0 with discounted installments and balloon; an accumulated-value check compares accumulated principal with accumulated installments and the balloon at time $n$. Mixing a payment at one date with a balance at another without adjusting for interest is a timing error.

Key takeaway: For end-of-period level installments at effective rate $i$ per period, the balloon after $n$ installments is $B_n=L(1+i)^n-Rs_{\overline{n}|}$. If the balloon is specified, solve the same equation for $R$. Confirm the payment timing, rate period, balloon date, and whether the final installment is additional to the balloon.

Check Your Understanding

For each problem, assume payments are made at the ends of periods and the stated balloon is due immediately after the last scheduled installment. Round dollar answers to the nearest cent where needed.

A $60,000 loan charges an effective annual rate of 5%. The borrower makes three annual installments of $15,000. Find the balance after the third installment. (Answer: $22,170.00)
A $90,000 loan earns an effective annual rate of 10%. There are two equal annual installments and a $40,000 balloon after the second installment. Find each installment. (Answer: $32,809.52)
A $50,000 loan charges a nominal annual rate of 8% convertible quarterly. The borrower makes four quarterly payments of $8,000. Find the balloon after the fourth payment. (Answer: $21,148.74)
A $120,000 loan has an effective annual rate of 6% and three annual installments of $30,000. Find the balloon after the third installment and the total cash due on that date if the third installment is also payable. (Answer: balloon $47,413.92; total cash due $77,413.92)
A $100,000 loan charges an effective annual rate of 10%. It has a three-year term and a target balloon of $25,000 after the third end-of-year installment. Find the level annual installment. (Answer: $32,658.61)

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