What the finance calculator does
This finance calculator works like the time value of money (TVM) keys on a financial calculator such as the BA II Plus or the HP 12C. Five values describe a loan, an investment or a savings plan. Enter any four and it finds the fifth:
- N — the number of periods.
- I/Y — the interest rate per year.
- PV — the present value, the amount today.
- PMT — the payment made every period.
- FV — the future value, the amount after the last period.
The time value of money formula
All five values are tied together by one equation:
PV × (1 + i)N + PMT × (1 + i × B) × ((1 + i)N − 1) / i + FV = 0
- i is the rate per payment period: (1 + I/Y / C/Y)C/Y ÷ P/Y − 1. With yearly payments and yearly compounding it is simply I/Y.
- B is 1 when payments are made at the beginning of each period and 0 when they are made at the end.
- FV, PMT and PV are found directly from the equation, and N with a logarithm: N = ln((k − FV) / (PV + k)) / ln(1 + i), where k = PMT × (1 + i × B) / i.
- I/Y has no direct formula, so the calculator searches for the rate that makes the equation zero, as a financial calculator does.
Positive and negative amounts
The signs show which way the money moves. Money you receive is positive and money you pay out is negative. When you take a $20,000 loan, PV is +$20,000 and the payments are negative. When you put $20,000 into an account, PV is −$20,000 and the money you take out later is positive. If the result looks strange, check the signs first.
Example
You borrow $20,000 at 6% a year and pay $2,000 at the end of each year for 10 years. FV comes out at −$9,455.36: after the last payment you still owe $9,455.36. To pay the loan off in 10 years, set FV to 0 and solve for PMT: −$2,717.36 a year. With $2,000 a year, solving for N shows it takes 15.7 years.
Common TVM problems
| Problem | N | I/Y | PV | PMT | FV |
|---|---|---|---|---|---|
| Loan balance left | 10 | 6% | $20,000 | −$2,000 | −$9,455.36 |
| Mortgage payment* | 360 | 6.5% | $300,000 | −$1,896.20 | 0 |
| Retirement savings* | 360 | 7% | 0 | −$500 | $609,985.50 |
| Present value of a goal | 20 | 5% | −$37,688.95 | 0 | $100,000 |
| Rate that doubles money | 10 | 7.177% | −$10,000 | 0 | $20,000 |
| Months to pay off a card* | 52.0 | 22% | $10,000 | −$300 | 0 |
- The value in bold is the one solved for; the other four are the inputs.
- * Monthly: P/Y and C/Y are 12 and N is in months. The other rows use 1 and 1. Payments are made at the end of each period.
- The present value of the goal is negative because it is money you put in today to receive $100,000 later.
- For a single amount, the present value calculator and the future value calculator are quicker.
Settings: P/Y, C/Y and payment timing
- P/Y is the number of payments a year. With monthly payments, set it to 12 and give N in months.
- C/Y is how often the interest compounds each year. It is usually the same as P/Y. Canadian mortgages are an exception: they compound twice a year, so a $400,000 loan at 6% over 25 years costs $2,559.23 a month with C/Y 2, against $2,577.21 with C/Y 12.
- Beginning or end is when each payment is made. Loans are usually paid at the end of each period, rent and leases at the beginning. Saving $500 a month for 30 years at 7% gives $609,985.50 with payments at the end, and $613,543.75 at the beginning.
For everyday loans, the loan calculator and the mortgage calculator do the same math with an amortization schedule; to find the rate behind a payment, use the interest rate calculator.