Calculating Total Periodic Borrower Outlay

What Counts as Periodic Borrower Outlay?

A sinking-fund loan separates two uses of the borrower’s periodic cash: interest is paid to the lender, while deposits are made to a fund that will accumulate to the principal due at maturity. The borrower’s periodic outlay is the sum of those payments. It is not just the interest, and it is not generally the principal divided by the number of periods.

This calculation assumes a level-principal loan: the original principal remains outstanding until maturity, and the borrower pays its interest each period. A separate fund earns interest and receives equal deposits at the end of each period. At maturity, the fund is used to pay the principal. The formulas below do not include fees, taxes, withdrawals from the fund, or a final fund surplus or shortfall.

Definition: Let $P$ be the loan principal, $n$ the number of payment periods, $i_L$ the effective loan rate per period, and $j_F$ the effective sinking-fund rate per period. If $R$ is the level end-of-period fund deposit, then the periodic borrower outlay is $O=P i_L+R$. The loan interest payment is $P i_L$; the deposit $R$ is chosen so that the fund accumulates to $P$ at the end of period $n$.

The loan and fund rates must be expressed over the same period used for the payments. For example, with quarterly payments, both rates in the outlay calculation must be effective quarterly rates. If a nominal annual rate $i^{(4)}$ is convertible quarterly, its effective rate per quarter is $i^{(4)}/4$. If a rate is given as an effective annual rate, convert it to the effective rate for the payment interval before using it.

Finding the Deposit Before Adding the Interest

For end-of-period deposits, the first deposit earns interest for $n-1$ periods, the second for $n-2$ periods, and the final deposit earns no interest before maturity. Thus, the fund balance at maturity is a geometric accumulation of the deposits. Using $s_{\overline{n}|\,j_F}$ for the accumulated value of an annuity-immediate of 1 per period at rate $j_F$, the required deposit satisfies $R s_{\overline{n}|\,j_F}=P$.

R=\frac{P}{s_{\overline{n}|\,j_F}} = \begin{cases} \displaystyle P\frac{j_F}{(1+j_F)^n-1}, & j_F\ne 0,\\[6pt] \displaystyle \frac{P}{n}, & j_F=0. \end{cases} \qquad O=P i_L+R

The zero-rate case follows because $n$ deposits of $R$ accumulate to $nR$, so $nR=P$. Once $R$ is found, add it to the loan interest for one period. This gives a level periodic outlay under the stated assumptions: the loan interest is constant because the outstanding principal stays at $P$, and the fund deposit is level by design.

Why the accumulation formula holds: The deposits accumulate to $R(1+j_F)^{n-1}+R(1+j_F)^{n-2}+\cdots+R$. Factoring out $R$, this is a geometric sum with $n$ terms and common ratio $1+j_F$. Its value is $R\big((1+j_F)^n-1\big)/j_F=R s_{\overline{n}|\,j_F}$ when $j_F\ne 0$. Setting this maturity balance equal to $P$ gives the required deposit. The last deposit is not multiplied by an interest factor because it is made at the maturity date.

Worked Outlay Calculations

Worked Example: Annual Interest and Annual Fund Deposits

A borrower receives $P=\$120{,}000$ on a level-principal loan. The effective annual loan rate is 6%, and the fund earns an effective annual rate of 4%. The borrower makes equal deposits at each year-end for 10 years. Find the annual outlay. Ignore fees and taxes.

Method. First accumulate the 10 end-of-year deposits to the principal due, then add one year’s loan interest. The rates are already effective annual rates, matching the annual periods.

The loan interest each year is

P i_L=\$120{,}000(0.06)=\$7{,}200.

The fund accumulation factor is

s_{\overline{10}|\,0.04} =\frac{(1.04)^{10}-1}{0.04} =\frac{1.480244284-1}{0.04} =12.0061071.

Therefore the annual deposit is

R=\frac{\$120{,}000}{12.0061071}=\$9{,}994.91.

The annual outlay is $\$7{,}200+\$9{,}994.91=\$17{,}194.91$. Rounded to the nearest cent, the borrower pays $\$17{,}194.91$ each year. As a check, the unrounded deposit multiplied by the accumulation factor equals $\$120{,}000$, the amount required at maturity. A common mistake is to treat the $\$7{,}200$ interest payment as if it also funded principal; it does not, so the fund deposit must be added.

Worked Example: Converting Nominal Rates to the Payment Period

A hypothetical borrower takes a $\$100{,}000$ level-principal loan for two years. The loan rate is 10% nominal per year convertible semiannually, and the fund rate is 20% nominal per year convertible semiannually. Equal deposits are made at each half-year end. Find the half-year outlay.

Method. Convert each nominal rate to an effective half-year rate, use four periods, and then add the half-year loan interest to the required deposit. The two nominal annual rates are not themselves the rates to substitute into the annuity formula.

The period rates are

i_L=\frac{0.10}{2}=0.05, \qquad j_F=\frac{0.20}{2}=0.10, \qquad n=2(2)=4.

The loan interest each half-year is $\$100{,}000(0.05)=\$5{,}000$. The fund accumulation factor and deposit are

s_{\overline{4}|\,0.10} =\frac{(1.10)^4-1}{0.10} =\frac{1.4641-1}{0.10} =4.641, \qquad R=\frac{\$100{,}000}{4.641}=\$21{,}547.08.

Thus the half-year outlay is $\$5{,}000+\$21{,}547.08=\$26{,}547.08$, rounded to the nearest cent. The four deposits accumulate to $\$100{,}000$ at the end of the fourth half-year. The outlay is a half-year amount; multiplying or dividing it to report an annual amount would require a separate, clearly stated annual-total calculation.

Worked Example: Finding the Principal Supported by an Outlay Limit

A borrower can budget at most $\$11{,}050$ at each year-end. A two-year level-principal loan charges an effective annual rate of 5%, and a separate fund earns an effective annual rate of 10%. Deposits are made at each year-end and the fund must equal the loan principal at maturity. Find the largest principal consistent with the outlay limit.

Method. Express both parts of the outlay per dollar of principal. Then solve the resulting linear equation for $P$. This avoids guessing a principal and repeatedly recalculating the deposit.

For two annual deposits, the accumulation factor is

s_{\overline{2}|\,0.10} =\frac{(1.10)^2-1}{0.10} =2.1.

The fund deposit per dollar of principal is $1/2.1=10/21$, and the annual loan interest per dollar is $0.05=1/20$. Therefore,

O=P\left(\frac{1}{20}+\frac{10}{21}\right) =P\left(\frac{221}{420}\right).

Set the outlay equal to the maximum budget and solve:

\$11{,}050=P\frac{221}{420}, \qquad P=\$11{,}050\frac{420}{221}=\$21{,}000.

Check: annual interest is $\$21{,}000(0.05)=\$1{,}050$, and the required fund deposit is $\$21{,}000/2.1=\$10{,}000$. Their sum is exactly $\$11{,}050$. The maturity fund balance is $\$10{,}000(2.1)=\$21{,}000$, so the principal is fully funded.

Timing, Interpretation, and Common Traps

The outlay formula is a cash-flow statement for one payment period. It combines a payment to the lender and a deposit into the fund, even though those amounts go to different destinations. The fund deposit is still part of the borrower’s cash outlay: the borrower must have the money available and transfer it to the fund.

The deposit formula above specifically assumes end-of-period deposits. If deposits are made at the beginning of each period, they earn interest for one additional period compared with end-of-period deposits, so the required deposit is different. Do not use the end-of-period accumulation factor without checking the stated timing. Likewise, if payment periods and rate-conversion periods do not match, first convert the rates to the actual payment interval.

A frequent error is to calculate $P/n$ as the deposit even when the fund earns interest. That amount is correct only when $j_F=0$. Another error is to add the full principal $P$ to every period’s outlay. Under this model, principal is accumulated through the periodic fund deposits and paid from the fund at maturity; it is not an additional periodic payment to the lender. If the question instead says the borrower pays principal directly in installments, it describes a different cash-flow structure.

Match the periods.
Identify the number of payment periods and convert the loan and fund rates to effective rates per payment period.

Fund the maturity principal.
Use the stated deposit timing to find the level deposit that accumulates to $P$.

Add the period’s loan interest.
For an unchanged principal balance, calculate $P i_L$, then add the required fund deposit.

Check both cash flows.
Confirm that interest uses the loan rate and that the fund deposits accumulate to the amount due at maturity.

This separation is useful whenever a question asks what the borrower must pay each period rather than only how much interest the loan charges. It also provides a direct way to test a budget: calculate the interest component and the principal-funding component independently, then compare their sum with the allowed outlay. Keep the destination and timing of each amount explicit; those details determine which formula applies.

Check Your Understanding

For each problem, assume there are no fees, taxes, or fund withdrawals, and that the stated deposits are made at the end of each period.

A $\$60{,}000$ level-principal loan has a 4% effective annual rate, and its fund earns 0%. The loan matures in three years with equal annual deposits. Find the annual outlay. (Answer: $\$22{,}400$ per year.)
A $\$80{,}000$ loan has a 7% effective annual rate. A fund earns 5% effective annually and receives equal deposits for two years. Find the annual outlay, rounded to the nearest cent. (Answer: $\$44{,}624.39$ per year.)
A borrower owes $\$50{,}000$ on a level-principal loan for one year, with quarterly interest payments. The loan rate is 12% nominal per year convertible quarterly, and the fund rate is 8% nominal per year convertible quarterly. Find the quarterly outlay, rounded to the nearest cent. (Answer: $\$13{,}631.19$ per quarter.)
A two-year loan charges 5% effective annual interest, and a fund earns 10% effective annually. The borrower can pay at most $\$11{,}050$ at each year-end, including both loan interest and the fund deposit. Find the maximum principal. (Answer: $\$21{,}000$.)

What You'll Learn