Prospective Valuation of the Loan Balance

What Prospective Valuation Measures

A loan balance on a payment date can be viewed as the amount needed at that date to account for the borrower’s remaining contractual payments. Prospective valuation finds that amount by discounting each unpaid scheduled installment back to the valuation date using the loan’s periodic interest rate. The date matters: after an installment is paid, it is no longer part of the balance; before it is paid, it remains due.

This approach is useful when the remaining payment stream is known and the question asks for the balance at a particular point in the schedule. It applies whether the remaining installments are level or differ in amount, provided their amounts and payment dates are specified.

Definition: The prospective outstanding balance at a payment date is the present value, at that date, of all installments scheduled after that date. Use the loan’s periodic rate for the interval between installments. For a level-payment loan with end-of-period installments, the balance immediately after payment $k$ is the present value of the $n-k$ installments still due.

Valuing the Remaining Payments

Let $i$ be the effective interest rate per payment period, $R$ the level installment, $n$ the total number of installments, and $k$ the number already paid. Suppose installments are paid at the end of each period. Immediately after installment $k$, the remaining $n-k$ installments are due one, two, and then successive periods later. Discounting each installment to the valuation date gives

B_k =R\left(v+v^2+\cdots+v^{\,n-k}\right) =R a_{\overline{n-k}|}, \qquad v=\frac{1}{1+i}.

Here $a_{\overline{m}|}$ is the present-value annuity-immediate factor for $m$ end-of-period payments at rate $i$. When $i\ne 0$, the factor can also be written as

a_{\overline{m}|} =\frac{1-v^m}{i} =\frac{1-(1+i)^{-m}}{i}.

For nonlevel payments, use the same prospective principle but value each payment individually. If $C_j$ is the amount due $j$ periods after the valuation date, the balance is the sum of its discounted values.

B=\sum_{j=1}^{m} C_jv^j.

These expressions assume a fixed periodic rate over the remaining term, payments at the stated period ends, and no additional amounts such as fees included in the balance. The rate must match the interval between payments. For example, a monthly payment stream requires a monthly rate, not an annual rate used without conversion.

Why the Prospective Formula Holds

The formula follows from valuing the same remaining cash flows at two dates. Consider a balance $B_k$ just after payment $k$, with $m=n-k$ installments of $R$ still due. At that date, the balance accumulates for $m$ periods to the date of the final installment. Each remaining installment also accumulates to that date. Equating the accumulated values gives

B_k(1+i)^m =R(1+i)^{m-1}+R(1+i)^{m-2}+\cdots+R.

Divide both sides by $(1+i)^m$. Each term on the right becomes one installment discounted from its due date to the valuation date:

B_k =R(1+i)^{-1}+R(1+i)^{-2}+\cdots+R(1+i)^{-m} =R a_{\overline{m}|}.

Thus, the annuity factor is not a separate loan rule: it is the sum of the discounted values of the remaining installments. The argument relies on valuing all cash flows at the same rate and respecting their dates.

Worked Example: Balance After the Second Annual Payment

A hypothetical loan has five annual end-of-year installments of $1,000. The effective annual interest rate is 10%, fixed for the loan. Find the balance immediately after the second installment has been paid. Ignore fees and assume the installments are paid exactly as scheduled.

Method. Use prospective valuation from the date immediately after payment 2. Of the five installments, three remain, due one, two, and three years from that date. The annual periodic rate is $i=0.10$, so $v=1/1.10=0.9090909\ldots$.

Discount the three installments separately:

\begin{aligned} B_2 &=1{,}000(v+v^2+v^3)\\ &=1{,}000(0.9090909+0.8264463+0.7513148)\\ &=1{,}000(2.4868520)\\ &=\$2{,}486.8520\ldots \end{aligned}

Using the annuity formula gives the same result:

B_2 =1{,}000\frac{1-(1.10)^{-3}}{0.10} =1{,}000(2.4868520\ldots) =\$2{,}486.85.

The balance is $2,486.85, rounded to the nearest cent. As a check, it is less than the undiscounted total of the remaining installments, $3,000, because each installment is due in the future. A common mistake is to count four installments by including the second payment again. The valuation is immediately after that payment, so only three installments remain.

Worked Example: Convert a Quoted Rate Before Valuing Monthly Payments

A hypothetical loan has monthly installments of $250. Immediately after the eighth monthly installment, four monthly installments remain. The loan rate is quoted as a nominal annual rate of 12% convertible monthly. Find the outstanding balance. Assume the quoted rate applies throughout the remaining term, installments are paid at month-end, and there are no fees.

Method. Convert the nominal annual quote to the rate per monthly payment period. A nominal annual rate of 12% convertible monthly means

i^{(12)}=0.12, \qquad i=\frac{0.12}{12}=0.01.

Thus the monthly rate is 1%. Because the balance is measured just after payment 8, the four payments are due at the ends of months 1 through 4 after the valuation date. The monthly discount factor is $v=1/1.01=0.9900990\ldots$.

\begin{aligned} B_8 &=250\left(\frac{1-(1.01)^{-4}}{0.01}\right)\\ &=250\left(\frac{1-0.9609803\ldots}{0.01}\right)\\ &=250(3.9019656\ldots)\\ &=\$975.4914\ldots \end{aligned}

The balance is $975.49, rounded to the nearest cent. This result is below the $1,000 total of the four unpaid installments, as expected when future payments are discounted. The exam trap is to use 12% as the monthly rate. The stated nominal rate is annual; the monthly rate is 12% divided by 12, or 1%.

Worked Example: Discount Unequal Remaining Installments

A hypothetical loan is valued immediately after a quarterly payment. The next three scheduled installments are $600 one quarter from the valuation date, $600 two quarters from the date, and $1,100 three quarters from the date. The effective interest rate is 4% per quarter, fixed over these periods. Find the balance, assuming no other amounts are due.

Method. Since the scheduled amounts differ, discount each one separately. The quarterly rate is $i=0.04$, and $v=1/1.04$. The required discount factors are

v=0.96153846,\qquad v^2=0.92455621,\qquad v^3=0.88899636.

The present values of the three installments are

\begin{aligned} 600v&=600(0.96153846)=\$576.92308,\\ 600v^2&=600(0.92455621)=\$554.73373,\\ 1{,}100v^3&=1{,}100(0.88899636)=\$977.89599. \end{aligned}

Add these values:

B =576.92308+554.73373+977.89599 =\$2{,}109.55280.

The balance is $2,109.55, rounded to the nearest cent. The calculation checks by comparing with the undiscounted total, $2,300: the present value is lower because all three installments are due in the future. Applying a level-payment annuity factor to the average installment would not be justified; each amount must be discounted according to its own due date.

Payment-Date Timing and Common Errors

The phrase “on a payment date” is not enough by itself to settle whether the installment due that day is included. A balance stated immediately after a payment excludes it. A balance immediately before that payment includes the amount due on that date, which is payable at the valuation time and therefore is not discounted by a full period. Read the timing description before counting installments or assigning discount exponents.

The annuity-immediate formula above is specifically for payments at the end of each period, with the first remaining payment one full period after the valuation date. If the facts instead place the first payment at the valuation time, its value is not $Rv$; it is $R$. Do not shift every payment by one period simply because a loan is described as having regular installments.

A reliable calculation can be organized as follows:

Fix the valuation instant.
Determine whether the scheduled payment on that date has already been made.

List the remaining cash flows.
Record each amount and its number of payment periods from the valuation date.

Match the rate to the period.
Convert the quoted rate if necessary so that $i$ is the rate for one interval between installments.

Discount and sum.
Use the annuity factor for level end-of-period installments, or discount each amount separately for a nonlevel stream.

The most frequent errors are counting a payment already made, using the annual quote as though it were the payment-period rate, and assigning an incorrect exponent to the first remaining installment. These errors change either the cash flows being valued or the date at which they are valued. A quick check is to confirm that every discount exponent equals the number of periods between the valuation date and that installment’s due date.

Key takeaway: At a payment date immediately after an installment, the prospective balance is the present value of the installments still unpaid. Use the periodic rate, count from the valuation date, and discount each payment by its actual number of periods to maturity.

Check Your Understanding

For each problem, identify the remaining payment dates before calculating. Assume no fees or additional amounts unless stated.

A loan is valued immediately after a year-end payment. Two annual installments of $500 remain, due one and two years later. The effective annual rate is 8%. Find the balance to the nearest cent. (Answer: $891.63)
Immediately after the third quarterly payment, five equal quarterly installments of $200 remain. The effective rate is 2% per quarter. Find the balance to the nearest cent. (Answer: $942.69)
A balance is measured immediately after a payment. The next three annual installments are $300, $400, and $500, due at the ends of years 1, 2, and 3. The effective annual rate is 5%. Find the balance to the nearest cent. (Answer: $1,080.44)
Immediately after a monthly payment, ten monthly installments of $150 remain. The rate is nominal annual 12% convertible monthly. Find the balance to the nearest cent. (Answer: $1,420.70)
Immediately after a quarterly payment, the next three installments are $400, $400, and $900, due one, two, and three quarters later. The effective rate is 3% per quarter. Find the balance to the nearest cent. (Answer: $1,589.02)

What You'll Learn