Payment Holidays and Capitalized Interest

What Changes During a Payment Holiday?

A payment holiday suspends scheduled installments for an agreed period. It does not necessarily suspend interest. The loan contract determines what happens to interest during the holiday: it may accrue as a separate amount, or it may be added to principal. Those treatments can produce different balances and different installments when repayment resumes.

The calculations require a clear timeline. Let $B$ be the loan principal immediately before the holiday begins, $i$ the effective rate per loan period, and $h$ the number of complete periods in the holiday. Assume that no installments or other payments are made during those $h$ periods. Unless stated otherwise, resumed installments are level, paid at the end of each period, and the first is due one period after the holiday ends.

Definition: Under a separate-interest arrangement in this tutorial, accrued interest is recorded separately from principal, earns no interest itself during the holiday, and is due as a lump sum when payments resume. The original principal remains $B$. Under capitalization, each period's interest is added to the principal, so later interest is charged on the increased balance.

The separate-interest definition is an explicit model, not a claim about every loan contract. A contract may require accrued interest to earn interest, or may add it to principal at specified dates. Follow the contract's stated rule. If the accrued interest itself earns interest, the amount compounds and is not the non-interest-bearing separate amount modeled here.

Accumulating Interest and Recasting Payments

For a constant effective rate $i$ per period, capitalization over $h$ complete periods gives a balance of $B(1+i)^h$. Under the separate-interest model, interest is calculated on the unchanged principal in each period, so the separate amount is $Bih$. Once payments resume, a level installment can be found by equating the balance to the present value of the installments.

\begin{aligned} \text{Capitalized balance at resumption} &= B(1+i)^h,\\ \text{Separate interest due at resumption} &= Bih,\\ \text{Level installment for }n\text{ periods} &= \frac{\text{amount being amortized}}{a_{\overline{n}|}},\\ a_{\overline{n}|} &= \frac{1-(1+i)^{-n}}{i}. \end{aligned}

The phrase “amount being amortized” matters. When interest is capitalized, the entire balance $B(1+i)^h$ is repaid through the installments. When separate interest is due as a lump sum at resumption, the installments amortize $B$, and the borrower also pays the separate interest. If the separate interest is instead spread across the resumed installments without interest, its share is added to each installment.

Set the period and date.
Identify the effective rate per period, the number of holiday periods, and when the first resumed installment is due.

Apply the contract's interest treatment.
Compound the balance if interest is capitalized; calculate the separate amount under the stated non-capitalizing rule if interest is kept separate.

Specify what the installments repay.
Use the principal or capitalized balance as appropriate, and account separately for any amount due at resumption.

Value the resumed installments.
Use an annuity-immediate factor for end-of-period payments and the number of installments in the revised term.

Why the Balance and Payment Formulas Hold

The capitalized-balance formula follows by applying the one-period accumulation rule repeatedly. Starting with $B$, after one holiday period the balance is $B(1+i)$. After the next period it is $B(1+i)(1+i)=B(1+i)^2$. Repeating this argument $h$ times gives $B(1+i)^h$.

For the separate-interest model, the principal remains $B$ throughout the holiday. Each of the $h$ periods therefore adds $Bi$ of interest, with no interest charged on earlier accrued interest. The total is $hBi$. This is simple accumulation on the original principal, not compound accumulation.

To justify the installment formula, suppose $n$ equal payments $R$ are made at the ends of periods, with the first payment one period after resumption. Their values at the resumption date are $Rv, Rv^2,\ldots,Rv^n$, where $v=(1+i)^{-1}$. Thus their present value is $R(v+v^2+\cdots+v^n)=Ra_{\overline{n}|}$. Setting that value equal to the balance being repaid gives $R=\text{balance}/a_{\overline{n}|}$.

This valuation assumes the balance and installment schedule use the same rate period. For example, a monthly installment schedule requires a rate per month and a term stated in months. An annual rate cannot be inserted directly as a monthly rate unless it is converted consistently.

Worked Example: Capitalizing Interest During a Monthly Holiday

A loan has a balance of $10,000 immediately before a two-month payment holiday. The effective monthly interest rate is 1%, and interest is capitalized monthly. After the holiday, the borrower will make 10 equal monthly payments, with the first due one month later. Find the balance at resumption and the installment. The method is to accumulate the balance through the holiday, then amortize it over the resumed term; this avoids treating the missed installments as if they were payments.

Step 1: Accumulate for two months. The balance after the first month is $10{,}000(1.01)=\$10{,}100.00$. After the second month it is $10{,}100(1.01)=\$10{,}201.00$. Equivalently, $10{,}000(1.01)^2=\$10{,}201.00$.

Step 2: Calculate the 10-payment factor.

a_{\overline{10}|} =\frac{1-(1.01)^{-10}}{0.01} \approx 9.4713045.

Step 3: Find the installment. The 10 payments must have a value of $10,201 at resumption, so $R=10{,}201/9.4713045\approx\$1{,}076.9372$. The level installment is $1,076.94, rounded to the nearest cent.

Check. Using the unrounded installment, its value at the resumption date is approximately $\$1{,}076.9372(9.4713045)=\$10{,}201.00$, the accumulated balance. Actual payment rounding may require a small adjustment to the final payment.

Worked Example: Separate Interest Due at Resumption

A $8,000 loan has a one-year payment holiday. The effective annual rate is 6%. The contract keeps holiday interest separate, with no interest earned on accrued interest, and requires that amount to be paid as a lump sum when installments resume. The $8,000 principal is then repaid in four equal annual installments, with the first installment one year after resumption. Find the separate amount, the installment, and the total amount due on the resumption date.

Step 1: Calculate separate interest. There is one holiday period, so the interest due is $8{,}000(0.06)(1)=\$480.00$. The principal remains $8,000.

Step 2: Value the four installments. At the 6% effective annual rate,

a_{\overline{4}|} =\frac{1-(1.06)^{-4}}{0.06} \approx 3.4651057.

Step 3: Calculate the installment. The installments amortize only the $8,000 principal: $R=8{,}000/3.4651057\approx\$2{,}308.7319$. The installment is $2,308.73, rounded to the nearest cent.

Step 4: State the amount due at resumption. The separate $480.00 interest is due then, in addition to the first installment if that installment is due on the resumption date. Under the timeline in this problem, the first installment is due one year later, so the immediate amount due at resumption is the $480.00 lump sum.

Check. The present value at resumption of the four unrounded installments is approximately $\$2{,}308.7319(3.4651057)=\$8{,}000.00$. Adding the separately due $480.00 does not change the principal amortized by those installments.

Worked Example: Comparing Separate and Capitalized Interest

A borrower owes $10,000 immediately before a one-year holiday. The effective annual rate is 10%. Compare two hypothetical contract terms, each followed by two equal annual installments, with the first installment one year after the holiday. Under the first term, interest is capitalized. Under the second, interest is kept separate without interest on accrued interest and is due as a lump sum at resumption. Find the resumption balance or separate amount, the installment under each term, and the total of the two installments.

Capitalized-interest term. The resumption balance is $10{,}000(1.10)=\$11{,}000.00$. The two-payment factor is $a_{\overline{2}|}=1+(1.10)^{-1}=1+10/11=21/11$. Therefore, $R=11{,}000/(21/11)=121{,}000/21\approx\$5{,}761.90$. The total of the two installments is approximately $2(5{,}761.9048)=\$11{,}523.81$.

Separate-interest term. The separate amount is $10{,}000(0.10)(1)=\$1{,}000.00$, while principal remains $10,000. Using the same two-payment factor, $R=10{,}000/(21/11)=110{,}000/21\approx\$5{,}238.10$. The total of the two installments is approximately $2(5{,}238.0952)=\$10{,}476.19$. The separate $1,000.00 is also due at resumption, so total cash paid across the lump sum and installments is approximately $11,476.19.

Check and interpretation. Each installment stream has the correct value at resumption: the capitalized stream is worth $11,000, and the separate-term installment stream is worth $10,000, with the separate $1,000 due independently. The total cash amounts differ because the two terms use different interest treatment during the holiday.

Common Mistakes and Contract Details

A payment holiday changes the timing of payments, but the precise financial effect depends on the agreement. A calculation is incomplete until it states whether interest is capitalized, how any separate amount is handled, and when resumed payments are due.

Using simple interest for a capitalized balance. For two periods, a 10% capitalized rate gives $B(1.10)^2$, not $B(1+0.10(2))$. The latter is simple accumulation with no interest on prior interest.
Adding separate interest to principal without authorization. If the contract keeps interest separate, the resumed installments may amortize only the original principal. Adding the interest to that amount models capitalization instead.
Forgetting when the first resumed installment occurs. Annuity-immediate payments begin one period after the valuation date. A payment due immediately at resumption has a different present-value factor and must be treated separately.
Mixing annual and monthly units. The rate, holiday length, and installment term must use matching periods. For monthly payments, use the effective monthly rate and count the remaining term in months.
Counting suspended installments as principal payments. No installment is made during the holiday in these models. The balance changes only according to the specified interest treatment.

If separate interest is spread equally across $n$ resumed payments without earning interest, each payment includes an additional $H/n$, where $H$ is the separate amount. This is different from capitalizing $H$ and amortizing it with interest. The contract's allocation rule determines which calculation applies.

Key takeaway: First determine the holiday balance under the contract's interest rule. Then identify exactly which amount the resumed installments repay and value those installments at the matching periodic rate. Separate accrued interest and capitalized interest are not interchangeable.

Check Your Understanding

For each problem, use the stated holiday-interest treatment and keep the rate period consistent with the payment schedule.

A $6,000 balance has a three-month holiday at an effective monthly rate of 2%, with interest capitalized monthly. Find the balance at resumption. (Answer: $\$6{,}000(1.02)^3=\$6{,}367.25$.)
A $9,000 balance has a one-year holiday at an effective annual rate of 10%. Interest is kept separately, earns no interest, and is due at resumption. The principal is repaid in two equal annual installments, the first one year later. Find the separate amount and installment. (Answer: separate amount $\$900.00$; installment $9{,}000/(21/11)=\$4{,}714.29$.)
A $15,000 balance has a two-period holiday at an effective rate of 3% per period, with interest capitalized each period. Find the balance at resumption. (Answer: $\$15{,}000(1.03)^2=\$15{,}913.50$.)
A $20,000 balance has a two-year holiday at an effective annual rate of 5%. Compare the resumption amounts under capitalization and under separate, non-interest-bearing accrual. (Answer: capitalized balance $\$20{,}000(1.05)^2=\$22{,}050.00$; separate interest $\$2{,}000.00$, with principal still $\$20{,}000.00$.)
A $12,000 balance has a three-month holiday at an effective monthly rate of 1%, with monthly capitalization. The borrower then makes six equal monthly payments, the first one month after resumption. Give the resumed balance and the formula for the installment. (Answer: balance $\$12{,}000(1.01)^3=\$12{,}363.61$; installment $12{,}000(1.01)^3/a_{\overline{6}|}$, with $a_{\overline{6}|}=[1-(1.01)^{-6}]/0.01$.)

What You'll Learn