Planning Basics: The Time Value of Money

"Money is better than poverty, if only for financial reasons." - Woody Allen

It approaches hyperbole to state that the most fundamental concept in finance and hence of financial planning is that money has a “time value.” Money in hand today is worth more than money that is expected to be received in the future. “A bird in the hand…” and all that. Why is that so? Well, a dollar that you receive today can be invested such that you will have more than a dollar at some future time. If only our Congressmen knew this, but I digress.


First an example to show what we mean. Let’s say you have 100 dollars of money today and you invest it for one year and earning 5 percent interest. It will be worth 105 dollars after one year. Therefore, 100 dollars paid now or 105 dollars paid exactly one year from now both have the same value to the recipient assuming 5 percent interest.

But the method is robust enough that it also allows the valuation of a likely stream of income in the future, in such a way that the annual incomes are “discounted” and then added together, thus providing a lump-sum "present value" of the entire income stream.

All of the standard calculations for time value of money derive from a basic algebraic expression for the present value of a future sum, discounted to the present by an amount equal to the time value of money. For example, the future value FV to be received in one year is discounted (at the rate of interest r) to give the present value PV thusly : PV = FV − r•PV = FV/(1+r).

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

Present Value

The current worth of a future sum of money or stream of cash flows given a specified rate of return. Future cash flows are discounted at the discount rate, and the higher the discount rate, the lower the present value of the future cash flows. Determining the appropriate discount rate is the key to properly valuing future cash flows, whether they be earnings or obligations.

Present Value of an Annuity.

An annuity is a series of equal payments or receipts that occur at evenly spaced intervals. Leases and rental payments are examples. The payments or receipts occur at the end of each period for an ordinary annuity while they occur at the beginning of each period for an annuity due.

Present Value of a Perpetuity

An annuity with an infinite and constant stream of identical cash flows

Future Value

Future Value is the value of an asset or cash at a specified date in the future that is equivalent in value to a specified sum today.

Future Value of an Annuity

The future value of a stream of payments (annuity), assuming the payments are invested at a given rate of interest.

Basic Equations, Identities and Combining Equations

Several basic equations that represent the equalities listed above. The solutions may be found using these formulas or, since we are in the twenty-first century, a financial calculator or a spreadsheet using Microsoft Excel for example. The formulas are programmed into most financial calculators and several spreadsheet functions (such as PV, FV, RATE, NPER, and PMT).

The equations are identities. For any of the equations below, the formula may be rearranged to determine one of the other unknowns. (Note: 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 “force” the solution 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.

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. I can't stress this enough.

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.

Finding the Formulae

Here is where I would naturally present all the formula regarding each of the concepts listed above and explain what each of the abbreviations ( i for interest rate, PV for present value, etc) meant. But I am not going to waste the pixels. You can find this in any finance textbook or in numerous sources on the Web. Instead I am going to give you a few links where you can find all of the relevant formulas and even examples.

From Investopedia

From StudyFinance.com

From GetObjects.com

I know some critics who have claimed that "this is all there is to" financial planning. I don't agree but even some practicioners have claimed as much. This knowledge IS however extremely valuable and study of it and its application is invaluable. That is why I have started the New Year with this post.

Good reading.