Time Value of Money (TVM) Formulas
These formulas are used in the CedarSpring TVM software component. They are useful for complex TVM problems that involve present and future value amounts and also a series of equal payments. The payments can be made at the beginning or end of each period, and the compounding periods per year do not have to equal the payments per year. You may want to check the TVM Concepts section to see if the simpler formulas and detailed examples found there will better meet your needs.
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Payment
Definition.
Where:
- PMT = Payment
- PV = Present Value
- FV = Future Value
- ip = Interest Rate per period
- N = Number of periods
- k = 1 if payment is made at the end of the period; 1 + ip if made at the beginning of the period
Example: You are 65 years old and have saved $400,000 for retirement. You believe you will live 20 more years. You want to leave $100,000 to your family. You can invest at a nominal annual rate of 6% compounded monthly. What amount can you withdraw at the end of each month and still reach all your goals?
- PV = -400,000 (negative amount paid out (deposited))
- FV = 100,000 (positive amount withdrawn after 20 years)
- N = 240 (20 years x 12 months per year)
- ip = .005 (.06/12 annual rate / 12 months per year)
- k = 1 (payment at end of each month)
- PMT = ? (positive amount withdrawn monthly)
PMT = [-400000 + (-300000 / (3.3102 -1))] * -.005
PMT = [-400000 + -129858.8887] * -.005
PMT = 2,649.29
You can work through the example again with an online calculator that uses this formula.
Future Value
Definition.
This formula combines future value of a single amount and future value of an annuity.
Where:
- FV = Future Value
- PMT = Payment
- ip = Interest Rate per period
- N = Number of periods
- PV = Present Value
- k = 1 if payment is made at the end of the period; 1 + ip if made at the beginning of the period
Present Value
Definition.
This formula combines present value of a single amount and present value of an annuity.
Where:
- PV = Present Value
- PMT = Payment
- k = 1 if payment is made at the end of the period; 1 + ip if made at the beginning of the period
- FV = Future Value
- ip = Interest Rate per period
- N = Number of periods
Number of Periods
Definition.
There are straight-forward formulas for finding The Number of Periods (N) that require use of logarithms. However, the TVM Component was developed with the Java programming language which did not support natural logs or exponents for BigDecimal numbers at the time, so this alternate method was used instead. Values of all the known variables are substituted into the formula given below. Then different values for the remaining variable N are tried until the expression equals zero. The Newton-Raphson method is used to chose a series of values to try. This method converges on the answer with reasonably few attempts.
Where:
- PV = Present Value
- ip = Interest Rate per period
- N = Number of periods
- PMT = Payment
- k = 1 if payment is made at the end of the period; 1 + ip if made at the beginning of the period
- FV = Future Value
Interest Rate Per Year
Definition.The Interest Rate Per Year (IY) can be solved by first finding the nominal interest rate per payment period (ip). Then, if the number of compounding periods equals the payment periods per year, you can find the annual rate (IY) by multiplying ip times the number of payments per year. If they are not equal it becomes more complicated.
Start by substituting all known variables into the formula below. Then use the Newton-Raphson method (see the Number of Periods formula) to choose a series of values for ip until the expression equals zero.
You can use this rate of return formula when payments are not involved.
Where:
- PV = Present Value
- ip = Interest Rate per period
- N = Number of periods
- PMT = Payment
- k = 1 if payment is made at the end of the period; 1 + ip if made at the beginning of the period
- FV = Future Value
IY = ip x PY
Effective Annual Interest Rate
Definition.The known values for nominal Interest Rate Per Year (IY) and Number of Periods (N) are substituted into the formula below. Then different values for ie, the effective annual interest rate, are tried until the expression equals 0. Code in the TVM component uses the Newton-Raphson method with this formula to converge on the answer. This is a rearrangement of the effective rate formula on the interest page.
0 = (1 + IY)N -1 - ie
Where:
- IY = Nominal Annual Interest Rate
- N = Number of Periods
- ie = Effective Annual Interest Rate
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