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CFA Level I Exam · The Time Value of Money in Finance

Uneven Cash Flows and Loan Amortization Explained

Updated 7 October 2026 · Fact-checked

Uneven cash flows are valued by discounting each payment separately and adding the results. Level payments on a loan use the annuity formula for the payment. An amortization schedule then splits each payment into interest, which is the periodic rate times the opening balance, and principal, which is the rest. The balance falls by the principal.

Understand Uneven Cash Flow Series and Loan Amortization

A series of uneven cash flows is a set of payments that differ in size. The annuity formula does not apply, because it needs equal payments. Instead you treat every cash flow as its own single-sum problem and add the results.

To find the present value, discount each cash flow back to time 0 using the number of periods it sits from today, then sum. To find the future value, compound each cash flow forward to the chosen end date, then sum. Both rest on one idea: money at different dates can only be added after you move it to the same date.

A loan amortization problem is the reverse use of an annuity. The lender gives you an amount today, and you repay it in equal payments. The payment is the amount that makes the present value of all payments equal the loan. Each payment covers the interest on the balance first, and whatever is left reduces the principal.

Because the balance falls each period, interest falls and principal rises, while the payment stays the same. The outstanding balance at any date equals the present value of the payments still to come, discounted at the loan rate. That gives you two ways to find a balance, and they should agree.

Key formulas to remember

Present value of uneven cash flows
PV = Σ [CFt ÷ (1 + r)^t], for t = 1 to N
Use the periodic rate r. Each cash flow gets its own t. A cash flow at time 0 is not discounted.
Future value of uneven cash flows
FV at time N = Σ [CFt × (1 + r)^(N − t)]
A cash flow received at time N is not compounded. Alternatively, FV = PV × (1 + r)^N.
Periodic rate
r = stated annual rate ÷ m, and N = years × m
m is payments per year. Match the rate and the number of periods to the payment frequency.
Level loan payment
PMT = Loan ÷ [(1 − (1 + r)^−N) ÷ r]
Valid for equal payments at the end of each period (ordinary annuity).
Interest portion of a payment
Interest(t) = r × Balance(t − 1)
Use the balance at the start of the period.
Principal portion and new balance
Principal(t) = PMT − Interest(t); Balance(t) = Balance(t − 1) − Principal(t)
The final balance should be zero apart from rounding.
Balance as PV of remaining payments
Balance after k payments = PMT × [(1 − (1 + r)^−(N − k)) ÷ r]
A shortcut that skips the schedule.
Total interest over the loan
Total interest = (PMT × N) − Loan
Holds for a fully amortizing loan with equal payments.

How to solve Uneven Cash Flow Series and Loan Amortization questions

Use this order for any uneven cash flow or loan question. It keeps the timing and rate matched.

  1. 1Draw a timeline and mark each cash flow with its period number. Note whether payments come at the end or start of each period.
  2. 2Decide the target date: time 0 for present value, the last date for future value, or another date the question names.
  3. 3Set the periodic rate and number of periods to match the payment frequency. For monthly payments use annual rate ÷ 12 and years × 12.
  4. 4If the cash flows are unequal, discount or compound each one separately and sum. If they are equal, use the annuity formula or the TVM keys.
  5. 5For a loan, solve for the payment first: PV = loan, N, I/Y, FV = 0, then compute PMT.
  6. 6For a schedule, compute interest = rate × opening balance, principal = PMT − interest, and new balance = old balance − principal.
  7. 7For a balance far into the loan, use the present value of the remaining payments instead of building every row.
  8. 8Check reasonableness: interest falls each period, principal rises, and total payments exceed the loan.

Quickest way: Calculator shortcuts for uneven flows and amortization

When to use it: Use this when you have 90 seconds per question and the numbers are not simple.

  1. Uneven cash flows on the TI BA II Plus: first press CF, then 2nd CLR WORK to clear old entries. Then enter CF0 and press ENTER, then the down arrow; enter each cash flow and its frequency (F01, F02...) the same way. Press NPV, enter I (the periodic rate as a percent), ENTER, down arrow, CPT.
  2. Gaps in the timeline: enter 0 as a cash flow for each empty period so the timing stays right.
  3. For a future value of uneven flows, compute the NPV first, then multiply by (1 + r)^N, or enter PV, I/Y and N and compute FV.
  4. Level loan payment: enter N, I/Y, PV as the loan (negative or positive, but then PMT takes the opposite sign), FV = 0, CPT PMT. Set P/Y to 1 on the TI so I/Y is the periodic rate.
  5. Balance after k payments: enter N = remaining payments, keep I/Y and PMT, FV = 0, CPT PV. This is the outstanding balance.
  6. On the BA II Plus you can also use 2nd AMORT: enter P1 and P2, then scroll to see BAL, PRN and INT for that range. On the HP 12C, first enter the loan terms (n, i, PV and PMT). Then enter the number of payments to amortize and press f AMORT to see interest, x≷y for principal, and RCL PV for the remaining balance.
  7. Eliminate options: the payment times N must exceed the loan, and the balance must fall below the loan after every payment.

Common mistakes in Uneven Cash Flow Series and Loan Amortization

  • Discounting a cash flow by the wrong number of periods

    You count years from the wrong starting point, or forget that a cash flow at time 0 is not discounted.

    Fix: Write the period number under each cash flow on a timeline before you calculate. Use that number as the exponent.

  • Using the annuity formula for unequal payments

    Annuity keys are fast and familiar, so you apply them out of habit.

    Fix: Use the annuity formula only when every payment is the same. Otherwise discount each flow, or use the CF and NPV keys.

  • Mixing an annual rate with monthly payments

    The question gives an annual rate and a monthly mortgage, and you enter both as given.

    Fix: Divide the stated annual rate by 12 and multiply the years by 12, or set the calculator payments per year consistently.

  • Calculating interest on the original loan every period

    You treat the loan like simple interest on the full amount.

    Fix: Interest = periodic rate × balance at the start of that period. The balance shrinks after each payment.

  • Confusing the payment with the principal repaid

    The payment looks like the amount that reduces the debt.

    Fix: Principal = payment − interest. In early periods most of the payment is interest.

  • Forgetting to clear the CF worksheet or sign errors on the TVM keys

    Old entries stay in memory, and PV and PMT need opposite signs.

    Fix: Clear the worksheet before each problem, and enter the loan and the payment with opposite signs. An answer with the wrong sign is still the right size.

Worked examples

Example 1

An investment pays €10,000 at the end of year 1, €15,000 at the end of year 2 and €20,000 at the end of year 3. The discount rate is 8% a year. The present value is closest to: A) €37,996 B) €41,667 C) €45,000

Show the solution
  1. Year 1: 10,000 ÷ 1.08 = 9,259.26.
  2. Year 2: 15,000 ÷ 1.08² = 15,000 ÷ 1.1664 = 12,860.08.
  3. Year 3: 20,000 ÷ 1.08³ = 20,000 ÷ 1.259712 = 15,876.64.
  4. Sum: 9,259.26 + 12,860.08 + 15,876.64 = 37,995.98, about €37,996.
  5. Check: C is the undiscounted total (45,000). B discounts the total by only one year (45,000 ÷ 1.08 = 41,667). Both ignore the different timing of each payment.

Answer: A) €37,996

Example 2

A borrower takes a $100,000 loan at 8% a year, repaid in five equal annual payments at the end of each year. The payment is about $25,046. How much principal is repaid in the second payment? A) $17,046 B) $18,409 C) $25,046

Show the solution
  1. Annuity factor = [1 − 1.08^−5] ÷ 0.08 = 3.99271. Payment = 100,000 ÷ 3.99271 = 25,045.65 (calculator: N = 5, I/Y = 8, PV = −100,000, FV = 0, CPT PMT).
  2. Year 1 interest = 0.08 × 100,000 = 8,000. Principal = 25,045.65 − 8,000 = 17,045.65.
  3. Balance after year 1 = 100,000 − 17,045.65 = 82,954.35.
  4. Year 2 interest = 0.08 × 82,954.35 = 6,636.35.
  5. Year 2 principal = 25,045.65 − 6,636.35 = 18,409.30, about $18,409.
  6. Check: principal rises each year, so it must exceed $17,046 and be less than the whole payment. That removes A and C.

Answer: B) $18,409

Exam tips

  • Questions are standalone and have three options. Often one option is the undiscounted sum and another discounts everything by a single period. Spot these traps first.
  • Always check that the rate and the number of periods match the payment frequency. This is the most common way marks are lost on mortgage-style items.
  • Use the balance-as-PV-of-remaining-payments shortcut when a question asks for the balance after many payments. Building the full schedule takes too long.
  • Use sense checks to eliminate options: early payments are mostly interest, the balance falls every period, and total payments exceed the loan.
  • Practise the CF and NPV keys until you can enter a five-flow series, including zero periods, in well under a minute.

Practice questions from The Time Value of Money in Finance

Uneven Cash Flow Series and Loan Amortization in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Uneven Cash Flow Series and Loan Amortization: frequently asked questions

How do I find the present value of uneven cash flows?

Discount each cash flow to time 0 using PV = CF ÷ (1 + r)^t, where t is its period number. Then add the results. On the TI BA II Plus you can enter the flows in the CF worksheet and press NPV.

How do I find the future value of an unequal cash flow series?

Compound each cash flow to the end date using CF × (1 + r)^(N − t), then add them. A quicker way is to find the present value and multiply it by (1 + r)^N. Both give the same answer.

How is a loan amortization schedule built?

Calculate the level payment first. For each period, interest equals the periodic rate times the opening balance, principal equals the payment minus interest, and the new balance is the old balance minus the principal. Repeat until the balance reaches zero.

What is the outstanding balance on a loan?

It is the present value of the payments that remain, discounted at the loan's periodic rate. It also equals the original loan minus all principal repaid so far. Use whichever method is faster for the question.