Skip to content
Get started
Get startedLog in

Time value of money for the CFA® Program Level I: formulas and worked examples

Discount future cash flows to find present value and compound today's money to find future value. Formulas, worked examples and a BA II Plus workflow for annuities, perpetuities and uneven cash flows.

CFA® Program Level I guideBy High Performer CertsPublished 8 min read

Key facts

  • Present value discounts cash flows to today; future value compounds them to a specified later date.
  • The effective interest rate and the number of periods must use the same time unit.
  • An ordinary annuity pays at period ends; an annuity due pays at period beginnings.
  • A level perpetuity starting one period from now has present value equal to its payment divided by a constant positive discount rate.
  • Uneven cash flows must be discounted separately before their present values are added.
  • Effective annual rates include the effect of compounding within the year.

Discount future cash flows to find present value; compound today's money to find future value. Match the rate to the period and distinguish single amounts, equal payments and uneven cash flows. (NYU Stern, time value of money (secondary source))

This guide supports study for the CFA® Program Level I, run by CFA Institute. (CFA Institute, Level I exam) Our questions complement CFA Institute's own curriculum; they do not replace it. The course page explains the wider course. Examples use illustrative amounts and constant rates, exclude fees and tax, and round monetary answers to pennies.

What do the time value of money symbols mean?

PVPV is value today, FVFV is value at the specified future date, rr is the effective rate per period, and nn counts periods. PMTPMT is an equal periodic payment; CFtCF_t is the cash flow at time tt. Time zero means now. A payment there needs no discounting. (NYU Stern, time value of money (secondary source))

Write the rate's time unit beside it. Four years means four annual periods or 48 monthly periods. Use the matching effective rate. The valuation formulas use positive receipt amounts; calculator entries also distinguish receipts from payments using signs.

How do you calculate present and future value?

A single amount compounds forward and discounts back using reciprocal factors. (NYU Stern, time value of money (secondary source))

FV=PV(1+r)nPV=FV(1+r)nFV = PV(1+r)^n \qquad PV = \frac{FV}{(1+r)^n}

Suppose £6,400 earns 5.5% per year for four years, with interest reinvested. Substitute 0.055 for the rate in the formula:

FV=6,400(1.055)4≈7,928.48FV = 6{,}400(1.055)^4 \approx 7{,}928.48

The balance is £7,928.48, including £1,528.48 of interest. Discounting the unrounded balance for four periods at 5.5% returns £6,400. Keep full precision between steps; round the final answer.

How do ordinary annuities and annuities due differ?

An ordinary annuity has equal payments at period ends; an annuity due has them at period beginnings. For an ordinary annuity, present value is one period before the first payment and future value is at the last payment. These formulas assume a non-zero rate. (NYU Stern, time value of money (secondary source))

PVordinary=PMT1−(1+r)−nrPV_{\text{ordinary}} = PMT\frac{1-(1+r)^{-n}}{r}FVordinary=PMT(1+r)n−1rFV_{\text{ordinary}} = PMT\frac{(1+r)^n-1}{r}

Take five annual payments of £1,350 at 4.5%. Their total undiscounted amount is £6,750. Substitution gives:

PV=1,3501−(1.045)−50.045≈5,926.47PV = 1{,}350\frac{1-(1.045)^{-5}}{0.045} \approx 5{,}926.47FV=1,350(1.045)5−10.045≈7,385.46FV = 1{,}350\frac{(1.045)^5-1}{0.045} \approx 7{,}385.46

Move each payment one period earlier to obtain an annuity due. At the same valuation dates, multiply each ordinary-annuity value by 1+r1+r. Each payment has one fewer period of discounting or one more period of compounding. (NYU Stern, time value of money (secondary source)) In this annual example, the factor is 1.045: present value becomes £6,193.16 and future value becomes £7,717.80.

The same five payments, with different timing
Payment patternPayment datesPV dateFV date
OrdinaryYears 1, 2, 3, 4, 5Year 0Year 5
DueYears 0, 1, 2, 3, 4Year 0Year 5

At a zero rate, simply add the payments: both values are £6,750 in this example. The displayed fractions would divide by zero, so do not substitute zero directly into them.

How do you value a perpetuity?

A level perpetuity pays the same amount indefinitely, starting one period from now. At a constant positive discount rate its present value is the payment divided by that rate. (NYU Stern, time value of money (secondary source))

PV=PMTr,r>0PV = \frac{PMT}{r},\quad r>0

For £840 annually starting next year at 6%, the calculation is £840 ÷ 0.06 = £14,000. A useful check is that 6% of £14,000 equals the £840 annual payment. There is no finite final payment date.

How do you discount uneven cash flows?

Discount each payment separately, then add values at the common date. (NYU Stern, time value of money (secondary source)) Include any initial payment at time zero to obtain net present value (NPV). (Texas Instruments, NPV definition and calculation)

PVfuture flows=∑t=1nCFt(1+r)tNPV=CF0+PVfuture flowsPV_{\text{future flows}} = \sum_{t=1}^{n}\frac{CF_t}{(1+r)^t}\qquad NPV = CF_0 + PV_{\text{future flows}}

Assume receipts of £1,250, £1,700 and £2,200 at the ends of years one, two and three, discounted at 7%:

PV=1,2501.07+1,7001.072+2,2001.073≈4,448.93PV = \frac{1{,}250}{1.07} + \frac{1{,}700}{1.07^2} + \frac{2{,}200}{1.07^3} \approx 4{,}448.93

If the initial outlay is £4,200, enter it as negative: NPV = −£4,200 + £4,448.93 = £248.93. The £4,448.93 figure values the future receipts; £248.93 also accounts for today's cost. Replacing these receipts with their average changes the valuation.

How do you convert a nominal rate to an effective annual rate?

For a nominal annual rate jj compounded mm times annually, the effective annual rate (EAR) includes interest on interest. (NYU Stern, time value of money (secondary source))

EAR=(1+jm)m−1EAR = \left(1+\frac{j}{m}\right)^m-1

A nominal 7.2% compounded monthly means 0.6% each month. Over a year:

EAR=(1+0.072/12)12−1≈7.4424%EAR = (1+0.072/12)^{12}-1 \approx 7.4424\%

To recover a monthly effective rate from an EAR, take the twelfth root of one plus EAR and subtract one. Using the unrounded EAR above returns 0.6%. Dividing an effective annual rate by twelve does not reverse compounding; dividing the stated nominal rate by twelve is appropriate here because its quotation specifies monthly compounding.

What is a reliable BA II Plus calculator workflow?

Use the TVM keys for a single amount or equal regular payments, and the cash-flow worksheet for unequal amounts. (Texas Instruments, TVM keys) (Texas Instruments, cash-flow worksheet) The following setup reproduces the annual examples above.

  1. Clear TVM memory: press 2nd → CLR TVM. This clears N, I/Y, PV, PMT and FV; it does not change P/Y or C/Y. (Texas Instruments, P/Y and C/Y)
  2. Set the frequency: open 2nd → P/Y, enter 1 → ENTER, move down to C/Y and confirm it is also 1, then 2nd → QUIT. (Texas Instruments, P/Y and C/Y)
  3. Check payment timing: open 2nd → BGN. If necessary, use 2nd → SET to select END, then quit. Select BGN for the annuity-due example and restore END afterwards. (Texas Instruments, signs and payment timing)
  4. Enter all known values: for the single amount use N = 4, I/Y = 5.5, PV = −6400 and PMT = 0. Press CPT → FV to obtain £7,928.48. Enter interest as a percentage number here, not the decimal 0.055 used in the formula. (Texas Instruments, future-value inputs)
  5. Reconcile the signs: money paid out is negative; money received is positive. After clearing again, use END, N = 5, I/Y = 4.5, PMT = 1350, FV = 0, then CPT → PV. The result is −£5,926.47: the outlay balancing the positive receipts. (Texas Instruments, signs and payment timing)

For the uneven example, open CF, then 2nd → CLR WORK. Enter CF0 = −4200, C01 = 1250, C02 = 1700 and C03 = 2200, pressing ENTER for each. Use the down arrow to move through the fields and set each frequency, F01–F03, to 1. Open NPV, enter I = 7 → ENTER, move down to NPV and press CPT: £248.93. With CF0 set to zero instead, the result is £4,448.93. (Texas Instruments, cash-flow entries) (Texas Instruments, NPV definition and calculation)

Check the answer against your timeline: our single future balance exceeds its starting amount, ordinary-annuity PV is below total receipts, and annuity-due PV exceeds the ordinary value. Unexpected comparisons suggest a timing or input error.

Frequently asked questions

What is the difference between present value and future value?

Present value expresses future cash flows in today's money. Future value compounds money to a specified later date. Use a rate and period count measured in the same time unit.

When should I use an annuity-due calculation?

Use it when equal regular payments occur at the beginning of each period. At the same valuation dates, multiply the ordinary-annuity present or future value by one plus the periodic rate.

What is the present value formula for a level perpetuity?

Divide the periodic payment by the constant positive periodic discount rate, with the first payment one period from now. For £840 annually at 6%, present value is £14,000.

Can I divide an effective annual rate by twelve for monthly calculations?

No. Take the twelfth root of one plus the effective annual rate, then subtract one. A nominal annual rate quoted with monthly compounding is divided by twelve to obtain its monthly rate.

Why does the BA II Plus return a negative present value?

The calculator distinguishes cash paid out from cash received. Positive future receipts produce a negative present value representing the balancing initial outlay. The sign alone does not indicate an error.

Sources

  1. Aswath Damodaran, NYU Stern School of Business (secondary source): A Primer on the Time Value of Money. Accessed 29 October 2026.
  2. Texas Instruments: Solution 11254: Time Value of Money Functions (TVM) on the BA II PLUS and BA II PLUS PROFESSIONAL. Accessed 29 October 2026.
  3. Texas Instruments: Solution 11234: Computing Future Value on a BA II PLUS or a BA II PLUS PROFESSIONAL. Accessed 29 October 2026.
  4. Texas Instruments: Solution 11262: Different P/Y and C/Y Settings on the BA II PLUS PROFESSIONAL and BA II PLUS. Accessed 29 October 2026.
  5. Texas Instruments: Solution 21590: Unexpected Answer Using the BA II PLUS and BA II PLUS PROFESSIONAL. Accessed 29 October 2026.
  6. Texas Instruments: BA II Plus Guidebook: Cash Flow Worksheet. Accessed 29 October 2026.
  7. Texas Instruments: BA II Plus Guidebook: Cash Flow Worksheet Variables. Accessed 29 October 2026.
  8. Texas Instruments: Solution 11241: Computing Net Present Value (NPV) and Internal Rate of Return (IRR) Using the BA II PLUS and the BA II PLUS PROFESSIONAL. Accessed 29 October 2026.
  9. CFA Institute: CFA Program Level I exam: format and topic weights. Accessed 26 October 2026.

High Performer Certs is an independent study platform. It is not affiliated with, endorsed by, or sponsored by CFA Institute or the Global Association of Risk Professionals (GARP). We are not a CFA Institute Prep Provider and not a GARP Exam Preparation Provider. CFA Institute does not endorse, promote, review or warrant the accuracy or quality of our products and services. GARP does not endorse, promote, review or warrant the accuracy of our products or services. All questions and explanations are our own original material and contain no CFA Institute or GARP exam, curriculum or practice questions. CFA®, Chartered Financial Analyst®, CIPM® and GIPS® are registered trademarks owned by CFA Institute. FRM® and GARP® are trademarks owned by the Global Association of Risk Professionals.

Cookie settings

Choose which optional tools we may use. We store your choice in your browser for 12 months (local storage, “hp.consent”). Details: privacy policy.

Strictly necessaryAlways on

Remembering this choice and, in the app, keeping you signed in. No tracking.

Statistics without cookiesAlways on

Our own visit counter and Vercel Web Analytics. Nothing is stored on your device, no IP addresses are kept and you cannot be recognised on another day.

Heatmaps and session recordings that show how visitors use our public pages. Form fields and personal details are masked. Sets the cookies _clck (1 year) and _clsk (1 day). Microsoft processes the data as an independent controller and may transfer it to the USA. Never used in the app or on sign-in pages.