Time Value of Money

 
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The time value of money is based on the premise that an investor prefers to receive a payment of a fixed amount of money today, rather than an equal amount in the future, all else being equal. In particular, if one received the payment today, one can then earn interest on the money until that specified future date.

All of the standard calculations are based on the most basic formula, the present value of a future sum, "discounted" to the present. For example, a sum of FV to be received in one year is discounted (at the appropriate rate of r) to give a sum of PV at present.

Some standard calculations based on the time value of money are:

  • Present Value (PV) of an amount that will be received in the future.
  • Present Value of a Annuity (PVA) is the present value of a stream of (equally-sized) future payments, such as a mortgage.
  • Present Value of a Perpetuity is the value of a regular stream of payments that lasts "forever", or at least indefinitely.
  • Future Value (FV) of an amount invested (such as in a deposit account) now at a given rate of interest.
  • Future Value of an Annuity (FVA) is the future value of a stream of payments (annuity), assuming the payments are invested at a given rate of interest.

Calculations

There are several basic equations that represent the equalities listed above. The solutions may be found using (in most cases) the formulas, a financial calculator or a spreadsheet. The formulas are programmed into most financial calculators and several spreadsheet functions (such as PV, FV, RATE, NPER, and PMT).

For any of the equations below, the formulae may also be rearranged to determine one of the other unknowns. In the case of the standard annuity formula, however, there is no closed-form algebraic solution for the interest rate (although financial calculators and spreadsheet programs can readily determine solutions through rapid trial and error algorithms).

These equations are frequently combined for particular uses. For example, bonds can be readily priced using these equations. A typical coupon bond is composed of two types of payments: a stream of coupon payments similar to an annuity, and a lump-sum return of capital at the end of the bond's maturity - that is, a future payment. The two formulas can be combined to determine the present value of the bond.

An important note is that the interest rate i is the interest rate for the relevant period. For an annuity that makes one payment per year, i will be the annual interest rate. For an income or payment stream with a different payment schedule, the interest rate must be converted into the relevant periodic interest rate, For example, a monthly rate for a mortgage with monthly payments requires that the interest rate be divided by 12 (see the example below). See compound interest for details on converting between different periodic interest rates.

The rate of return in the calculations can be either the variable solved for, or a predefined variable that measures a discount rate, interest, inflation, rate of return, cost of equity, cost of debt or any number of other analogous concepts. The choice of the appropriate rate is critical to the exercise, and the use of an incorrect discount rate will make the results meaningless.

For calculations involving annuities, you must decide whether the payments are made at the end of each period (known as an ordinary annuity), or at the beginning of each period (known as an annuity due). If you are using a financial calculator or a spreadsheet, you can usually set it for either calculation. The following formulas are for an ordinary annuity. If you want the answer for the Present Value of an annuity due simply multiply the PV of an ordinary annuity by (1 + i).

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