Time Value of Money (TVM) FormulasThese 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.
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?
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.
Number of Periods
Interest Rate Per YearDefinition.
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.
IY = ip x PY
Effective Annual Interest RateDefinition.
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
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