N · I/Y · PV · PMT · FV
TVM Calculator
Solve any time value of money problem for N, I/Y, PV, PMT or FV — with separate payment and compounding frequencies, begin or end timing, and the whole formula written out underneath. The same equation the BA II Plus solves, with nothing hidden.
- Solves all five registers
- P/Y and C/Y handled separately
- BGN and END timing
- Every formula shown
Background
What the time value of money actually says
A pound today is worth more than a pound next year, because today’s pound can be invested and next year’s cannot. Everything in finance that involves waiting — a mortgage, a bond, a pension, a lease, a capital project — is an application of that one sentence. The time value of money framework turns it into arithmetic by pinning every cash flow to a date and moving it forward or backward at an assumed rate.
The BA II Plus reduces the whole subject to five registers and one equation. Once four of the five are known the fifth is fixed, which is why the calculator has a CPT key rather than five different functions. The equation it satisfies is:
PV + PMT × ä(N, i) + FV × (1 + i)−N = 0
where ä(N, i) = (1 − (1 + i)−N) ÷ i, multiplied by (1 + i) in BGN mode, and i is the rate for one payment period rather than the annual rate you type in.
Read left to right, it says: the value you start with, plus the present value of every payment, plus the present value of whatever is left at the end, must net to nothing. If it does not net to nothing, the deal has not been described consistently — which is almost always a sign problem rather than a maths problem.
The five registers
What each key means, and what a sensible value looks like
| Key | Meaning | Watch out for |
|---|---|---|
| N | The total number of payment periods, not the number of years. | 30 years of monthly payments is 360, not 30. |
| I/Y | The nominal annual rate, entered as a percentage. | Enter 6.5, not 0.065, and never the monthly rate. |
| PV | The value at time zero — the loan advanced or the lump sum invested. | Positive when the money comes to you. |
| PMT | The equal amount paid or received each period. | Must be constant; uneven flows belong in the NPV calculator. |
| FV | Whatever is left at the end — a balloon payment or a target balance. | Zero for a fully amortising loan, which is easy to leave stale. |
Two settings sit behind those five and change what they mean. P/Y is payments per year and C/Y is compounding periods per year; together they turn I/Y into the rate for one payment period. The payment timing switch — END or BGN — decides whether each payment lands at the close of a period or its start.
The formulas
How each variable is calculated
Four of the five have a closed-form solution: rearrange the balance equation and read the answer off. The rate does not, which is why it is the one that takes the calculator a moment.
- Present value
- PV = − [ PMT × ä + FV × (1 + i)−N ]
- Discount the payment stream and the terminal value back to today, then flip the sign so the equation nets to zero.
- Future value
- FV = − (PV + PMT × ä) × (1 + i)N
- Collapse the opening value and the payments into a single present value, then compound the whole thing forward N periods.
- Payment
- PMT = − (PV + FV × (1 + i)−N) ÷ ä
- The amount that has to be spread over N periods, divided by the annuity factor. This is the mortgage payment formula.
- Number of periods
- N = − ln[ (−PV − k) ÷ (FV − k) ] ÷ ln(1 + i), k = PMT ÷ i
- Taking logs of both sides isolates the exponent. The answer is often fractional, which means the final payment is smaller than the rest.
- Interest rate
- solve PV + PMT × ä(N, i) + FV × (1 + i)−N = 0 for i
- No algebraic rearrangement exists once PMT is non-zero, because i appears both inside the annuity factor and in the discount term. It is found by iteration — the same reason the hardware pauses for a fraction of a second on CPT I/Y.
Worked example
A £320,000 mortgage over 30 years at 6.5%
This is the default loaded into the calculator above, so you can follow along and change one number at a time to see what moves.
-
Translate the problem into registers
N = 360 · I/Y = 6.5 · PV = 320,000 · FV = 0 · P/Y = C/Y = 12 · END
Thirty years of monthly payments is 360 periods. The loan is money coming to you, so PV is positive.
-
Find the monthly rate
i = 6.5 ÷ 12 ÷ 100
i = 0.541667% per month
-
Find the annuity factor
ä = (1 − 1.00541667−360) ÷ 0.00541667
ä = 158.210820
-
Divide the loan by the factor
PMT = − 320,000 ÷ 158.210820
PMT = −2,022.62 per month
Negative because it leaves your account. Over 360 payments that is 728,142 in total, of which 408,142 is interest — more than the house.
On the hardware the whole thing is six entries: 2ND FV · 360 N · 6.5 I/Y · 320000 PV · 0 FV · CPT PMT. Change I/Y to 7 and press CPT PMT again — the payment rises by 106.35 a month, which is what half a percentage point costs on a loan this size.
Troubleshooting
Five mistakes that account for most wrong answers
-
Entering years instead of periods
N counts payment periods. Multiply the years by P/Y before you enter it — 25 years of quarterly payments is 100.
-
Giving every register the same sign
The equation cannot balance if all the money flows the same way. One side of the deal has to be negative, and which side is arbitrary as long as you are consistent.
-
A stale FV or PMT from the previous problem
The hardware keeps every register until you clear it, so a leftover FV silently changes the answer. Press 2ND FV first, every time.
-
Leaving P/Y at its last value
P/Y survives a TVM clear and even a battery change. If a monthly answer looks twelve times too big, this is why.
-
Using END for an annuity due
Rent, leases and most insurance premiums are paid up front. Switching to BGN multiplies the annuity factor by (1 + i), which on a long term is not a rounding difference.
Where to go next
Beyond a level payment
TVM assumes every payment is identical. The moment that stops being true, or the moment you want to see inside a single payment, a different worksheet is the right tool.
- Uneven cash flows belong in the NPV calculator or the IRR calculator, which take up to 24 groups with their own frequencies.
- To split each payment into interest and principal, use the amortization calculator — the same loan, period by period.
- To compare a rate that compounds monthly with one that compounds daily, convert both with the interest conversion calculator.
- For coupon bonds with a settlement date and a day-count convention, use the bond calculator.
- To practise the keystrokes exactly as they appear in the exam room, run the full BA II Plus emulator, or read the study guides.
FAQ
TVM questions, answered
The seven things people ask most about the time value of money keys.
What does TVM stand for on the BA II Plus?
TVM means time value of money — the idea that a sum available today is worth more than the same sum later, because today’s money can earn a return. On the BA II Plus it refers to the five keys along the top row: N, I/Y, PV, PMT and FV. Enter any four and the calculator solves the fifth.
Why do I have to enter one value as a negative number?
Because the TVM equation only balances when the cash flows point in opposite directions. Money coming to you is positive, money leaving you is negative. A £320,000 mortgage is money coming to you, so PV is positive and the monthly payment comes back negative. If you enter both the loan and the payment as positive numbers, no rate or term can satisfy the equation and the solver reports an error.
What is the difference between P/Y and C/Y?
P/Y is how many payments happen per year and C/Y is how many times the interest compounds per year. They are usually the same — a monthly mortgage is 12 and 12. Canadian mortgages are the classic exception, quoting a rate that compounds semi-annually while payments are monthly, which is P/Y 12 with C/Y 2. When they differ, the rate per payment period becomes (1 + I/Y ÷ C/Y)C/Y ÷ P/Y − 1 rather than a simple division.
When should I use BGN instead of END?
Use BGN when the payment happens at the start of each period — rent, lease payments, insurance premiums and most annuities due. Use END for anything that pays in arrears, which covers nearly every loan and mortgage. Switching to BGN multiplies the annuity factor by (1 + i), so the same cash flows are worth one period more of interest.
Why does my answer differ from the back of the textbook by a few pence?
Almost always rounding. Textbook solutions often use a rate rounded to four decimal places or annuity factors from a table, while this solver and the hardware both carry full internal precision. A second cause is the number of periods: 30 years of monthly payments is N = 360, not 30. Check that N counts periods and I/Y is the annual rate before assuming an error.
How does the calculator solve for I/Y?
There is no closed-form solution for the rate once a payment is involved, so it is found numerically. This solver runs Newton–Raphson from several starting rates and falls back to a bracketed bisection over the range −99.99% to 10,000% per period, which is the same belt-and-braces approach that stops the hardware returning Error 5 on awkward inputs. The iteration count is reported with the result.
Can I clear the registers between problems?
Press the Reset button and every field returns to its starting value, which is the equivalent of 2ND FV (Clear TVM) on the hardware. On the device itself, forgetting to clear is the single most common source of wrong answers, because a leftover FV or a stale P/Y silently changes the result.
Keep going
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