Lesson 2 of 6 6–9% of the exam

You are reading the 2026 curriculum (exams up to November 2026). Sitting in February 2027 or later? Go to the 2027 Quantitative Methods guide.

By Daniil Tarakanov Updated Oct 16, 20262026 curriculum

Annuities and level-payment loans

A level-payment loan (a mortgage, a car loan) is an annuity: the same payment every period, part interest and part principal, until the balance is zero. Its payment is A = r × PV ÷ [1 − (1 + r)⁻ᵗ], using the periodic rate and the number of payments.

6 min cram, 24 min full read. Based on the 2026 CFA® Program Level I curriculum, Quantitative Methods, Learning Module 2. This is the 2026 version of the lesson. Sitting in February 2027 or later? Read the updated 2027 version.

Tailor this to you: take the self-assessment

Before this: Pricing bonds: zero-coupon, coupon and perpetual

On the exam

Calculate and interpret the present value (PV) of fixed-income and equity instruments based on expected future cash flows

Calculate and interpret the present value and payment of level-payment (amortizing) instruments. Expect a monthly mortgage payment, the present value of an annuity, and a question on how the interest and principal shares of a payment change.

How deeply to learn each part

Know the idea: Understand it in general. It frames other topics, but you will not calculate with it.
Use it: Understand it well enough to solve a question with it. Recognize when it applies.
Memorize: Learn it by heart. The exam will not give it to you.

You know this one. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still do.

You tested strong here. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still know it.

What is an annuity worth?

An annuity pays the same amount every period for a fixed number of periods. In an ordinary annuity the first payment arrives at the end of the first period. Discount each payment and add, or use the closed form.

Present value of an ordinary annuity Memorize

PV = A \times \frac{1 - (1 + r)^{-t}}{r}

Worked example Ten years of income

An income of USD 5,000 at the end of each year for 10 years, discounted at 7%.

Discount factor: 1.07⁻¹⁰ = 0.5083.
Annuity factor: (1 − 0.5083) ÷ 0.07 = 7.0236.
Multiply: 5,000 × 7.0236 = USD 35,117.91.

On your calculator

2NDCLR TVM with P/Y set to 1
10N7I/Y5000PMT
CPTPV −35,117.91

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · I/Y interest rate per year (per period when P/Y is 1) · PMT payment · CPT compute · PV present value

fCLEAR FIN
10n7i5000PMT
PV −35,117.91

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · i interest rate per period · PMT payment · PV present value

Both calculators normally work in end-of-period mode. Only change the mode when a question says payments come at the start of each period, and change it back afterwards.

How do you find a loan payment?

A level-payment instrument pays the same amount every period, so it is an annuity. A fully repaid amortizing loan , such as a mortgage or a car loan, is the lender’s side of the same picture: the loan amount is the present value, and the payment is the unknown. Turn the formula over.

Level payment on a loan Memorize

A = \frac{r \times PV}{1 - (1 + r)^{-t}}

Use the periodic rate and the number of payments. A monthly loan needs a monthly rate (annual rate ÷ 12) and months (years × 12).

Worked example A 25-year mortgage

A GBP 250,000 mortgage over 25 years at 4.8% a year, paid monthly.

Periodic rate: 4.8% ÷ 12 = 0.4% a month.
Number of payments: 25 × 12 = 300.
Payment: 0.004 × 250,000 ÷ (1 − 1.004⁻³⁰⁰) = GBP 1,432.49 a month.

Total paid: 300 × 1,432.4924 (the unrounded payment) = GBP 429,747.72, so GBP 179,747.72 more than the loan is interest.

On your calculator

2NDCLR TVM with P/Y set to 1, so I/Y is the monthly rate
300N0.4I/Y250000PV
CPTPMT −1,432.49, and FV stays 0 after the clear

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · I/Y interest rate per year (per period when P/Y is 1) · PV present value · CPT compute · PMT payment

fCLEAR FIN
300n0.4i250000PV
PMT −1,432.49

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · i interest rate per period · PV present value · PMT payment

Rearrange itOrdinary annuity: solve for value or payment

The question asks for

An investment pays 5,000.00 at the end of each period for 10 periods. The discount rate is 7.00% per period. Its present value is closest to:

Rate per period, r%Number of payments, tPayment each period, A

P V = A \frac{1 - ( 1 + r ) ^{- t}}{r}

5,000.00 × (1 − 1.07^−10) ÷ 0.07 = 35,117.91

Present value, PV35,117.91

On your calculator

2NDCLR TVMwith P/Y set to 1
10N7I/Y5000PMT
CPTPVnegative: it is the money you pay

fCLEAR FIN
10n7i5000PMT
PVnegative: it is the money you pay

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · i interest rate per period · PMT payment · PV present value

InteractiveA level-payment loan: same payment, shifting mix

Try this 0 of 3

Move the year slider to the first year where principal beats interest
Switch to one payment a year and compare total interest
Cut the total interest on the worked loan to less than half

Loan amount (GBP)250,000

Annual rate4.8%

Term (years)25 years

Balance after yearyear 1

Payment a month1,432.49

Total interest179,747.72

Balance after year 1244,694.38

Interest
Principal repaid
Balance

First month: interest1,000.00

First month: principal432.49

Year 1: interest share69.1%

Show working

Periodic rate 4.8% ÷ 12 = 0.4%; payments 25 × 12 = 300.

A = r × PV ÷ [1 − (1 + r)^−t] = 0.004 × 250,000 ÷ [1 − 1.004⁻³⁰⁰] = 1,432.49 a month

First payment: interest 250,000 × 0.004 = 1,000.00; principal 1,432.49 − 1,000.00 = 432.49.

Balance after 12 payments = 250,000 × 1.004¹² − 1,432.49 × [1.004¹² − 1] ÷ 0.004 = 244,694.38

Total paid 300 × 1432.4924 (the unrounded payment) = 429,747.72; minus the 250,000 borrowed leaves 179,747.72 of interest.

On your calculator

2NDCLR TVMwith P/Y set to 1
300N0.4I/Y250000PV0FV
CPTPMT−1,432.49
12NCPTFV−244,694.38, the balance after 12 payments

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · I/Y interest rate per year (per period when P/Y is 1) · PV present value · FV future value · CPT compute · PMT payment

fCLEAR FIN
300n0.4i250000PV0FV
PMT−1,432.49
12nFV−244,694.38, the balance after 12 payments

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · i interest rate per period · PV present value · FV future value · PMT payment

Every payment is the same size, but the mix shifts from mostly interest to mostly principal.

How a loan can be repaid

Loan cash-flow patternseach is a different stack of payments, priced by discounting

Interest only Know the idea interest every period, the whole principal at the end
Fully amortizing Memorize equal payments that repay interest and principal, balance reaches zero
- Level-payment loan priced as an annuity: mortgage, car loan
Partly amortizing (balloon) Know the idea smaller payments, then a large final payment

How does each payment split between interest and principal?

Each period, interest is the rate times the balance at the start of the period. The rest of the fixed payment repays principal and lowers the balance, so next period’s interest is a little smaller.

Interest in a payment Use it

\text{Interest} = r \times \text{opening balance}, \quad \text{Principal} = A - \text{Interest}

Worked example The first payment and the balance after a year

Continuing the GBP 250,000 mortgage with a payment of GBP 1,432.49.

Interest in payment 1: 250,000 × 0.004 = GBP 1,000.00.
Principal in payment 1: 1,432.49 − 1,000.00 = GBP 432.49.
After 12 payments, the balance is the loan’s future value minus the payments’ future value: GBP 244,694.38.

Step 2: the split

250000×0.004= 1,000.00 interest
+|-+1432.49= 432.49 principal

+|- change sign

250000ENTER0.004× 1,000.00 interest
CHS1432.49+ 432.49 principal

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Step 3: the balance, straight after finding the payment

12N I/Y, PV and PMT are still in memory
CPTFV −244,694.38, the balance still owed

N number of periods · CPT compute · FV future value

12n i, PV and PMT are still in memory
FV −244,694.38, the balance still owed

n number of periods · FV future value

The mix moves steadily from interest to principal while the payment never changes.

Payment	Interest	Principal	Balance after
1	1,000.00	432.49	249,567.51
12	980.59	451.91	244,694.38
150	648.52	783.97	161,345.61
300	5.71	1,426.79	0.00

Exam tip ”What happens to the principal share?” It rises. “What happens to the payment?” Nothing, it is level.

Go deeper Annuity due, interest-only and balloon loans

An annuity due pays at the start of each period, so every payment is discounted one period less. Its value is the ordinary-annuity value times (1 + r). The USD 5,000 income stream above would be worth 35,117.91 × 1.07 = 37,576.16 if each payment came at the start of the year.

Annuity due, by multiplying

35117.91×1.07= 37,576.16

35117.91ENTER1.07× 37,576.16

ENTER enter (on the HP, puts the number on the stack)

Two other loan patterns appear in Fixed Income. An interest-only loan pays interest each period and the whole principal at the end, so it is priced like a bond. A balloon loan amortizes only part of the principal and ends with a large final payment.

Go deeper Why prepayment matters for mortgage pools

A level-payment mortgage is a known stack of cash flows only if the borrower never repays early. Borrowers can repay ahead of schedule, especially when rates fall and they can refinance. That risk shortens the investor’s cash flows at the worst time and is why mortgage-backed securities need their own treatment in Fixed Income.

Memorize this

MemorizeAnnuity formulas

Want	Formula
Present value	$A \times [1 - (1 + r)^{-t}] / r$
Payment	$r \times PV \,/\, [1 - (1 + r)^{-t}]$
Interest in a payment	$r \times$ opening balance
Principal in a payment	$A -$ interest

Memory hook The payment formula is the PV formula turned over: PV in, A out.

Know it cold Rate and number of payments must match the payment period: monthly loan, monthly rate, monthly count. With a fully amortizing loan, FV = 0.

How a loan can be repaid

Loan cash-flow patternseach is a different stack of payments, priced by discounting

Interest only Know the idea interest every period, the whole principal at the end
Fully amortizing Memorize equal payments that repay interest and principal, balance reaches zero
- Level-payment loan priced as an annuity: mortgage, car loan
Partly amortizing (balloon) Know the idea smaller payments, then a large final payment

Present value of an ordinary annuity

PV = A \times \dfrac{1 - (1 + r)^{-t}}{r}

Quick example

EUR 2,000 at the end of each year for 6 years, discounted at 5%. 2,000 × (1 − 1.05⁻⁶) ÷ 0.05 = EUR 10,151.38.

On your calculator

2NDCLR TVM with P/Y set to 1
6N5I/Y2000PMT
CPTPV −10,151.38

fCLEAR FIN
6n5i2000PMT
PV −10,151.38

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · i interest rate per period · PMT payment · PV present value

Level payment on a loan

A = \dfrac{r \times PV}{1 - (1 + r)^{-t}}

Quick example

Borrow 400,000 over 30 years at 5.4% a year, paid monthly. r = 0.45% a month, t = 360. A = 2,246.12 a month.

On your calculator

2NDCLR TVM
360N0.45I/Y400000PV
CPTPMT −2,246.12

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · I/Y interest rate per year (per period when P/Y is 1) · PV present value · CPT compute · PMT payment

fCLEAR FIN
360n0.45i400000PV
PMT −2,246.12

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · i interest rate per period · PV present value · PMT payment

Interest in a payment

\text{Interest} = r \times \text{opening balance}, \quad \text{Principal} = A - \text{Interest}

Quick example

Balance 250,000 at 0.4% a month, payment 1,432.49. Interest 1,000.00, principal 432.49.

On your calculator

250000×0.004= 1,000.00 interest
+|-+1432.49= 432.49 principal

+|- change sign

250000ENTER0.004× 1,000.00 interest
CHS1432.49+ 432.49 principal

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

How it is asked

Monthly mortgage payment

30-year USD 400,000 loan at 5.4%, monthly. Payment?

N = 360, I/Y = 0.45, PV = 400,000, FV = 0, compute PMT: 2,246.12. Interest only (1,800) is the trap.
Present value of level payments

EUR 2,000 a year for 6 years at 5%. Value today?

N = 6, I/Y = 5, PMT = 2,000, compute PV: 10,151.38. The undiscounted total (12,000) is the trap.
Interest and principal over time

In a level-payment loan, what happens to the principal share of each payment?

It increases, because interest is charged on a shrinking balance.

Exam traps

Using the annual rate and annual periods, then dividing the payment by 12. Convert rate and count to months first.
Forgetting FV = 0 on a fully amortizing loan. A leftover FV from an earlier problem corrupts the answer.
Treating the payment as interest only (balance × rate ÷ 12). A level payment also repays principal.

Checkpoint

3 exam-style questions. About 90 seconds each.

1A borrower takes a 30-year USD 400,000 mortgage at 5.4% a year, paid monthly. The monthly payment is closest to:
2In a level-payment loan, as time passes, the share of each payment that repays principal most likely:
3An investment pays EUR 2,000 at the end of each year for 6 years. At a 5% discount rate its present value is closest to:

Key takeaways

Level-payment loans are annuities.
Use the periodic rate and the number of payments.
Payment fixed, interest share falls, principal share rises.

Present value of an ordinary annuity

PV = A \times \dfrac{1 - (1 + r)^{-t}}{r}

Quick example

EUR 2,000 at the end of each year for 6 years, discounted at 5%. 2,000 × (1 − 1.05⁻⁶) ÷ 0.05 = EUR 10,151.38.

On your calculator

2NDCLR TVM with P/Y set to 1
6N5I/Y2000PMT
CPTPV −10,151.38

fCLEAR FIN
6n5i2000PMT
PV −10,151.38

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · i interest rate per period · PMT payment · PV present value

Level payment on a loan

A = \dfrac{r \times PV}{1 - (1 + r)^{-t}}

Quick example

Borrow 400,000 over 30 years at 5.4% a year, paid monthly. r = 0.45% a month, t = 360. A = 2,246.12 a month.

On your calculator

2NDCLR TVM
360N0.45I/Y400000PV
CPTPMT −2,246.12

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · I/Y interest rate per year (per period when P/Y is 1) · PV present value · CPT compute · PMT payment

fCLEAR FIN
360n0.45i400000PV
PMT −2,246.12

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · i interest rate per period · PV present value · PMT payment

Interest in a payment

\text{Interest} = r \times \text{opening balance}, \quad \text{Principal} = A - \text{Interest}

Quick example

Balance 250,000 at 0.4% a month, payment 1,432.49. Interest 1,000.00, principal 432.49.

On your calculator

250000×0.004= 1,000.00 interest
+|-+1432.49= 432.49 principal

+|- change sign

250000ENTER0.004× 1,000.00 interest
CHS1432.49+ 432.49 principal

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Next lesson Valuing a stock from its dividends

Questions about this lesson?

Where this comes back

Fixed-Income Cash Flows and Types Fixed Income
Mortgage-Backed Securities: RMBS, CMOs and CMBS Fixed Income

Questions candidates ask

How do you calculate a monthly mortgage payment?

Convert to monthly terms: rate ÷ 12 and years × 12. Then A = r × PV ÷ [1 − (1 + r)⁻ᵗ], or on the calculator enter N, I/Y and PV with FV = 0 and compute PMT. A 250,000 loan over 25 years at 4.8% costs 1,432.49 a month.

What is the difference between an annuity and a perpetuity?

An annuity pays a fixed amount each period for a set number of periods, so it ends. A perpetuity pays for ever. The annuity formula turns into the perpetuity formula, PMT ÷ r, as the number of periods grows without limit.

How much of a loan payment is interest?

The periodic rate times the balance at the start of the period. On a 250,000 loan at 0.4% a month, the first payment includes 1,000 of interest. The rest repays principal, and the interest share falls every month.

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