The Prospective Balance Principle

Why the Balance Date Matters

An outstanding loan balance is not just a number attached to a loan: it is a value at a particular date. To calculate it prospectively, identify the installments still owed and value those payments at the balance date. The timing convention matters. A balance immediately before an installment is due includes that installment; a balance immediately after it has been paid does not.

The calculation also depends on the rate used to value the payments. For a contractual outstanding balance, use the loan’s interest basis, expressed per payment period. The discounting period must match the time from the balance date to each payment—not the time from the start of the loan.

Definition: The prospective balance at a focal date is the present value, at that date, of all installments that remain unpaid. Under a level periodic-rate model, if unpaid payments $C_j$ are due $j$ payment periods after the loan begins, and the focal date is $k$ periods after the loan begins, then $$ B_k=\sum_{j\in J_k}\frac{C_j}{(1+i)^{j-k}}, $$ where $J_k$ contains the payment dates for installments unpaid at the focal date, and $i$ is the effective interest rate per payment period. A payment due exactly at the focal date belongs in $J_k$ if the balance is measured immediately before payment, but not if it is measured immediately after payment.

Set the Focal Date and the Rate First

A focal date is the date at which the balance is being measured. Before calculating, state whether that date is immediately before or immediately after any scheduled payment. Then list the unpaid installments and count the periods from the focal date to each one. For example, a payment due one period after the focal date is discounted once, regardless of how many periods have passed since the loan began.

Use the interest rate in the same units as the payment periods. If the contract gives an effective annual rate $i$ and payments are $m$ times per year, the effective rate per payment period is $(1+i)^{1/m}-1$. If it gives a nominal annual rate $i^{(m)}$, convertible $m$ times per year, the periodic rate is $i^{(m)}/m$. These conversions assume the stated annual rate applies consistently and that the payment periods are equal. They do not justify using a rate on a different basis, such as a market yield, when the question asks for the contractual outstanding balance.

Once the periodic rate is known, discount each unpaid installment by the number of payment periods between the focal date and that installment. Do not discount payments already made, and do not include a payment that has just been made in a balance immediately after that payment.

Worked Example: Balance After the First Annual Payment

A loan requires four annual payments of $1,000, each made at the end of a year. The effective annual loan rate is 25%. Find the outstanding balance immediately after the first payment. State the method and check the result at the next payment date.

Method. Use the prospective principle: discount only the unpaid installments to the date immediately after payment 1. The rate per annual payment period is $i=0.25$. Because payment 1 has just been made, the unpaid payments are due at the ends of years 2, 3, and 4. Their distances from the focal date at year 1 are one, two, and three periods.

B_1=\frac{1{,}000}{1.25}+\frac{1{,}000}{1.25^2}+\frac{1{,}000}{1.25^3}

Evaluate each term:

\frac{1{,}000}{1.25}=800,\qquad \frac{1{,}000}{1.5625}=640,\qquad \frac{1{,}000}{1.953125}=512.

Therefore $B_1=800+640+512=1{,}952$. The outstanding balance is $1,952.00.

Check. Accumulate this balance one period, to immediately before payment 2: $1{,}952(1.25)=2{,}440$. At that date the value is the $1,000 payment due then plus the value of payments 3 and 4: $1{,}000+1{,}000/1.25+1{,}000/1.25^2=1{,}000+800+640=2{,}440$. The values agree.

Mistake avoided. Discounting the three remaining payments by two, three, and four periods would count from the loan’s start rather than from the balance date, understating the balance. The focal date is year 1, so the correct exponents are one, two, and three.

Why Discounting the Unpaid Payments Works

The prospective balance is the amount at the focal date that is financially equivalent, under the loan’s interest assumptions, to the remaining payment obligation. Each installment has a value at that date: discount it for the number of periods from the focal date until it is due. Adding those values gives the value of the full unpaid stream.

The justification follows from the accumulation rule. If an amount $X$ at the focal date accumulates for $n$ periods at effective rate $i$, its value at the later date is $X(1+i)^n$. Therefore the amount at the focal date equivalent to a payment $C$ due $n$ periods later must satisfy $X(1+i)^n=C$, so $X=C/(1+i)^n$. Applying this to each unpaid installment and adding the values gives the prospective balance formula:

B_k=\sum_{j\in J_k}\frac{C_j}{(1+i)^{j-k}}.

This argument assumes a consistent effective rate per payment period, payments made on the stated dates, and no additional fees or charges included in the balance. If the interest rate changes, payment dates are irregular, or a separate payoff charge applies, the simple single-rate formula needs adjustment. Do not treat those situations as though they were governed by one constant periodic rate.

Convert the Rate to the Payment Period

A quoted annual rate is not automatically the rate for a monthly or quarterly period. Convert it before discounting. For a nominal annual rate convertible monthly, divide by 12 to obtain the monthly rate. For an effective annual rate and monthly payments, take the twelfth root of $1+i$ and subtract 1. The conversion affects every term in the balance calculation.

Worked Example: Monthly Payments and a Nominal Annual Rate

A loan has monthly payments of $101. The contract rate is a nominal annual rate of 12% convertible monthly. Find the balance immediately after the payment at the end of month 2 if the remaining payments are due at the ends of months 3, 4, and 5. Use the contractual rate and check the balance immediately before the next payment.

Method. Convert the nominal rate to the monthly rate: $i=0.12/12=0.01$, or 1% per month. At the focal date, month 2 immediately after its payment, the remaining payment dates are one, two, and three months away.

B_2=\frac{101}{1.01}+\frac{101}{1.01^2}+\frac{101}{1.01^3}.

The terms are $101/1.01=100$, $101/1.0201\approx99.009901$, and $101/1.030301\approx98.029605$. Thus $B_2\approx297.039506$, which is $297.04 to the nearest cent.

Check. Accumulating one month to immediately before payment 3 gives $297.039506(1.01)\approx300.009901$. At that date the value is the $101 payment due then plus the value of the month 4 and month 5 payments at that same date: $101+101/1.01+101/1.01^2\approx101+100+99.009901=300.009901$. The check agrees.

Mistake avoided. Using 12% as the monthly rate would apply the annual nominal rate twelve times in each year. The monthly periodic rate is 1%, not 12%.

Payments Due on the Balance Date

A payment due exactly on the focal date creates a frequent timing ambiguity. If the balance is immediately before payment, that payment is still owed and is included at its full amount: it is due at the focal date, so it is not discounted. If the balance is immediately after payment, it has been settled and is excluded. The other unpaid payments are discounted from that date as usual.

Worked Example: Balance Immediately Before a Scheduled Payment

A loan has three remaining semiannual payments of $105, due at the ends of periods 3, 4, and 5. The nominal annual rate is 10% convertible semiannually. Find the balance immediately before the period 3 payment, and compare it with the balance immediately after that payment. Round money to the nearest cent.

Method. The semiannual rate is $0.10/2=0.05$, or 5% per half-year. Immediately before period 3, the payment due at period 3 is still unpaid and is included without discounting. The payments at periods 4 and 5 are one and two periods away.

B_{3^-}=105+\frac{105}{1.05}+\frac{105}{1.05^2}.

The terms are $105$, $105/1.05=100$, and $105/1.1025\approx95.238095$. Therefore $B_{3^-}\approx300.238095$, or $300.24.

Immediately after the period 3 payment, that $105 installment is no longer unpaid. The balance is $B_{3^+}=105/1.05+105/1.05^2=100+95.238095\approx195.238095$, or $195.24. The before-payment balance equals the payment plus the after-payment balance: $105+195.238095=300.238095$.

Check. The difference between the two balances is exactly the payment made at period 3. This confirms that the only change between the two focal-date conventions is whether that installment is still owed.

Mistake avoided. Omitting the installment due at period 3 from a balance immediately before payment would give the after-payment balance instead. Conversely, including it in a balance immediately after payment would count an installment that has already been paid.

Common Traps and a Reliable Procedure

The method is simple once the dates and rate are fixed, but small timing errors can change the answer materially. In particular, a balance at a later date cannot generally be checked by merely accumulating the earlier balance across intervening payment dates. Payments made during that interval must also be accounted for. A direct prospective calculation at the later date avoids that shortcut.

Mark the focal date.
Write whether the balance is just before or just after any installment on that date.

Identify unpaid payments.
Include a payment due at the focal date only for a just-before-payment balance.

Match the rate to the period.
Convert the stated rate to an effective rate per payment period before calculating.

Discount from the focal date.
Use the number of payment periods between the focal date and each unpaid payment.

Check at the next payment date.
Accumulating the balance to immediately before the next payment should match that payment plus the value, at that date, of the installments still unpaid afterward.

The prospective balance is useful because it gives a direct valuation of the remaining obligation without needing to reconstruct every earlier allocation of payments. It is also a clear way to test whether a proposed balance uses the right rate and focal date. This material is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Key takeaway: A prospective balance is the value, at a stated focal date, of every installment still unpaid. Decide whether a payment on that date is included, convert the contract rate to the payment period, and discount each remaining payment from the focal date—not from the loan’s origination.

Check Your Understanding

For each problem, identify the payment periods remaining from the stated balance date and use the rate on the payment-period basis.

A loan has annual payments of $420. The effective annual rate is 5%. Find the balance immediately after the first payment if two payments remain, due one and two years later. (Answer: $780.95)
Two payments of $600 are due at the ends of periods 3 and 4. The effective rate is 20% per period. Find the balance immediately before the period 3 payment. (Answer: $1,100.00)
Monthly payments of $101 remain at the ends of months 5, 6, and 7. The nominal annual rate is 12% convertible monthly. Find the balance immediately after the payment at the end of month 4. Round to the nearest cent. (Answer: $297.04)
A loan has payments of $300 at the end of quarter 3 and $600 at the end of quarter 4. Its nominal annual rate is 8% convertible quarterly. Find the balance immediately after the payment at the end of quarter 2. Round to the nearest cent. (Answer: $870.82)
A loan has two remaining semiannual payments of $500, due one and two half-years after the balance date. The effective annual rate is 21%. Find the balance at the stated date. (Answer: $867.77)

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