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Tutorials 9 min read

Time Value of Money (TVM): Step-by-Step with BA II Plus

A worked time value of money tutorial for the BA II Plus: solve for FV, PV, PMT, N and I/Y with full keystrokes, plus annuity due and mismatched compounding.

The BA II Plus Calculator Team

The short answer

Every time value of money problem on the BA II Plus is the same five registers — N, I/Y, PV, PMT and FV — with four filled in and one computed. Set P/Y to the number of payments per year, enter money you pay out as negative and money you receive as positive, then press CPT and the key for the unknown. If the answer looks impossible, check that N is in periods rather than years and that PV and FV do not share the same sign.

Time value of money is one idea: a pound today is worth more than a pound next year, and the interest rate is the exchange rate between them. The BA II Plus turns that idea into five keys, and once you can see which of the five a question is really asking for, the keystrokes stop being the hard part.

This guide works through all five unknowns with real numbers, then covers the two situations that break people’s answers even when their keystrokes are right: annuities that pay at the start of the period, and interest that compounds more often than the payments arrive.

The one equation behind all five keys

The calculator is not running five separate formulas. It holds one equation and rearranges it:

The TVM identity
  1. PV + PMT × ä(N, i) + FV × (1 + i)^−N = 0

ä(N, i) is the present value of 1 per period for N periods at rate i. In BGN mode it is multiplied by (1 + i).

Read it as a statement about balance. Everything you put in and everything you take out, discounted to today, has to sum to zero. That is why the signs matter: if all three money terms were positive there would be nothing for the equation to balance against, and the calculator would return Error 5 rather than guess what you meant.

Two consequences follow, and they explain most confusion about this calculator:

  • PV and FV normally have opposite signs. You pay out now to receive later, or receive now to pay out later.
  • N and I/Y have to speak the same language. If N counts months, the rate the calculator uses has to be a monthly rate. P/Y and C/Y are what handle that conversion for you.

Before the first problem

Three settings, once. The complete setup walkthrough covers the reasoning; here is the short version.

Floating decimal, annual periods, clean registers
  1. 2ND
  2. FORMAT
  3. 9
  4. ENTER
  5. 2ND
  6. QUIT
  7. 2ND
  8. P/Y
  9. 1
  10. ENTER
  11. 2ND
  12. QUIT
  13. 2ND
  14. CLR TVM

I am using P/Y = 1 for the annual problems below and switching to 12 where payments are monthly. Set it deliberately every time; assuming it is right is how people lose marks.

Solving for FV: what a lump sum grows into

You invest £10,000 at 7% a year for 15 years. What is it worth at the end?

Future value of a single sum
  1. 2ND
  2. CLR TVM
  3. 15
  4. N
  5. 7
  6. I/Y
  7. 10000
  8. +|−
  9. PV
  10. 0
  11. PMT
  12. CPT
  13. FV

FV = 27,590.32. The £10,000 has nearly tripled, and the arithmetic is simply 10,000 × 1.07¹⁵ = 10,000 × 2.759032.

Notice the 0 PMT. There is no recurring payment in this problem, but if the previous problem had one the register still holds it, and the calculator will happily include a phantom annuity in your answer.

Solving for PV: what a future sum is worth today

Your child starts university in 18 years and the fees will be £25,000. How much do you need to set aside now at 7%?

Present value of a single sum
  1. 2ND
  2. CLR TVM
  3. 18
  4. N
  5. 7
  6. I/Y
  7. 0
  8. PMT
  9. 25000
  10. FV
  11. CPT
  12. PV

PV = −7,396.60. Negative because you have to hand the money over. Deposit £7,396.60 today, leave it for 18 years at 7%, and you have the £25,000.

Discounting is the same operation as compounding, run backwards, and the exponent doing the work is the reason distant cash flows matter so little. That £25,000 arriving in 18 years is worth 29.6% of its face value today. Push it out to 30 years and it is worth £3,284.18.

Solving for FV with a payment: the annuity

You save £500 at the end of every month for 25 years and earn 8% a year. How much is in the account?

Payments are monthly now, so P/Y goes to 12 and N counts months: 25 × 12 = 300.

Future value of a monthly savings plan
  1. 2ND
  2. P/Y
  3. 12
  4. ENTER
  5. 2ND
  6. QUIT
  7. 2ND
  8. CLR TVM
  9. 300
  10. N
  11. 8
  12. I/Y
  13. 0
  14. PV
  15. 500
  16. +|−
  17. PMT
  18. CPT
  19. FV

FV = 475,513.20. You contributed 300 × £500 = £150,000 of your own money. The other £325,513.20 is interest, and interest on interest, which is the whole reason anyone bothers starting early.

There is a shortcut for the N entry that saves a mental multiplication: type 25 then press 2ND xP/Y and the calculator multiplies by P/Y and leaves 300 on the display, ready for N.

Solving for N: how long will this take?

You owe £20,000 on a credit line at 9% and can pay £450 a month. When is it gone?

Number of payments to clear a balance
  1. 2ND
  2. CLR TVM
  3. 9
  4. I/Y
  5. 20000
  6. PV
  7. 450
  8. +|−
  9. PMT
  10. 0
  11. FV
  12. CPT
  13. N

N = 54.26 months — four and a half years. The calculator gives a fractional answer because the last payment does not need to be a full £450. In practice you make 54 payments of £450 and a 55th of about £118, which is what the fraction is telling you.

Solving for I/Y: what return am I actually getting?

Someone offers you a private investment: pay £18,500 today, receive £30,000 in eight years. Is that any good?

Implied annual return between two amounts
  1. 2ND
  2. P/Y
  3. 1
  4. ENTER
  5. 2ND
  6. QUIT
  7. 2ND
  8. CLR TVM
  9. 8
  10. N
  11. 18500
  12. +|−
  13. PV
  14. 0
  15. PMT
  16. 30000
  17. FV
  18. CPT
  19. I/Y

I/Y = 6.2291%. Now the offer is comparable to anything else quoted as an annual rate, which is the point of the exercise. Against a government bond at 4.5% it is generous; against a diversified equity portfolio and eight years of illiquidity it is thin.

CPT I/Y is the only TVM computation that iterates rather than solving directly, so it is the one that occasionally takes a moment or returns Error 7. It needs at least one sign change in the cash flows to have anything to converge on.

Solving for PMT: what does the goal cost per month?

You want £1,000,000 in 30 years and expect 7% a year. What do you have to put away each month?

Payment needed to hit a target
  1. 2ND
  2. P/Y
  3. 12
  4. ENTER
  5. 2ND
  6. QUIT
  7. 2ND
  8. CLR TVM
  9. 360
  10. N
  11. 7
  12. I/Y
  13. 0
  14. PV
  15. 1000000
  16. FV
  17. CPT
  18. PMT

PMT = −819.69. Under £820 a month, which surprises most people the first time they see it. Over 360 months you contribute £295,088 and the market contributes the other £704,912.

The uncomfortable part is what happens when you start later. Ask the same question with 25 years — 300 N CPT PMT — and the answer is £1,234.46. Five years of delay adds 51% to the monthly cost of the same goal, because the contributions you skipped were the ones with the longest to compound.

Run the calculation in the other direction and it becomes a retirement number: drawing £40,000 a year for 25 years at 5% needs a pot of £563,757.78 on the day you stop working, which is 25 N, 5 I/Y, 40000 PMT, 0 FV, CPT PV.

Ordinary annuity vs annuity due

A 20-year lease pays £40,000 a year. At a 5.5% discount rate, what is it worth?

Timing Mode Present value
End of each year END £478,015.30
Start of each year BGN £504,306.14
Same cash flows, same rate, different timing.

The gap is £26,290.84, and it is not a rounding artefact. Every payment arrives one year earlier, so every payment is discounted one period less. The relationship is exact:

Annuity due = ordinary annuity × (1 + i)
  1. 478,015.30 × 1.055 = 504,306.14

Useful as a sanity check when you are not sure which mode the calculator is in.

Rent, leases, insurance premiums and pension contributions are usually annuities due. Loans, bonds and mortgages are ordinary annuities. When a question says “payments begin immediately” or “the first payment is today”, that is BGN mode being described in words.

When compounding and payments disagree

P/Y and C/Y are separate registers for a reason. A savings account can credit interest monthly while you deposit once a year, and a Canadian mortgage compounds semiannually while it is paid monthly.

£10,000 at 6% nominal for 10 years, compounded monthly, no payments.

Annual periods, monthly compounding
  1. 2ND
  2. P/Y
  3. 1
  4. ENTER
  5. ↓
  6. 12
  7. ENTER
  8. 2ND
  9. QUIT
  10. 2ND
  11. CLR TVM
  12. 10
  13. N
  14. 6
  15. I/Y
  16. 10000
  17. +|−
  18. PV
  19. 0
  20. PMT
  21. CPT
  22. FV

Set P/Y first — it overwrites C/Y — then arrow down and set C/Y separately.

FV = 18,193.97, against 17,908.48 with annual compounding. The extra £285.49 comes from the same 6% being credited twelve times a year instead of once, which is exactly the difference between a 6% nominal rate and its 6.1678% effective equivalent.

Five mistakes that produce almost-right answers

  1. N in years with P/Y = 12. Thirty years is 360, not 30. This one is worth checking first every time, because the answer it produces looks like a number rather than an error.
  2. A stale register. 2ND CLR TVM costs one keystroke. A forgotten FV from the last question costs the whole answer.
  3. Both PV and FV positive. The calculator needs money moving in two directions. All-positive inputs give Error 5 when you ask for N or I/Y.
  4. The rate entered as a decimal. I/Y takes 6.5, not 0.065.
  5. BGN left on. It stays set until you change it, and it silently inflates every subsequent savings answer by one period of interest.

Every one of those is a two-second check. Running all five before you trust an answer takes less time than redoing the problem.

Next, the same registers applied to a real loan: amortization schedules, period by period, or move on to uneven cash flows with NPV on the cash flow worksheet.

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Do it without the keystrokes

The calculators used in this guide

Same arithmetic, same sign convention, with every substitution written out underneath the answer.

TVM Calculator

Solve for N, I/Y, PV, PMT or FV with P/Y, C/Y and BGN/END handling.

N · I/Y · PV · PMT · FV

Interest Conversion Calculator

Nominal ↔ effective rates for any compounding frequency, plus continuous.

2ND · ICONV

Amortization Calculator

Full payment schedule with BAL, PRN and INT for any P1–P2 range.

2ND · AMORT

FAQ

Frequently asked questions

What are the five TVM keys on the BA II Plus?

N is the number of compounding periods, I/Y is the nominal annual interest rate as a percentage, PV is the value today, PMT is the level payment each period, and FV is the value at the end. Enter any four and the calculator solves for the fifth.

How do I calculate present value on the BA II Plus?

Enter the number of periods into N, the rate into I/Y, the future amount into FV, zero into PMT if there is no recurring payment, then press CPT PV. The answer comes back with the opposite sign to FV, because paying money in today is what buys the amount you receive later.

Why is my TVM answer negative?

A negative answer is the calculator telling you the direction of the cash flow, not that you made a mistake. It reports money leaving your hands as negative, so a loan payment is negative while the loan balance you received is positive. The magnitude is what matters; the sign is a consistency check.

What is the difference between an ordinary annuity and an annuity due?

An ordinary annuity pays at the end of each period, which is END mode and the default. An annuity due pays at the start, which is BGN mode. Every payment in an annuity due earns one extra period of interest, so its value is the ordinary annuity multiplied by (1 + i).

Do I enter 6.5% as 6.5 or 0.065 in I/Y?

Enter 6.5. The I/Y register expects a percentage, so entering 0.065 prices the problem at less than a tenth of a percent and returns an answer that looks almost like no interest at all.

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