Time Value of Money Concepts

Assumptions Inherent in TVOM Calculations

There are several important assumptions underlying the TVOM equations and it would not be prudent to perform calculations without being fully aware of their influence:

  • Money is always invested and always productive so that returns can be reinvested at a rate equal to i.
    (This assumption is illustrated and discussed in the example problem #23.)

  • The yield curve is flat so that short term interest rates are equivalent to long term interest rates.

  • Time periods are all of equal length.

  • Payments are all equal and either all inflows or all outflows.

  • The interest rate is constant throughout the term.

  • Annuities are simple, certain, discrete and ordinary.

This last assumption requires further explanation:

  • Annuity-Certain - one with a fixed number of payments and the assurance that the payments will be made (i.e., they are not contingent on any event that cannot be entirely foretold)

  • Discrete Annuity - one with equal intervals between successive payment dates

  • Simple annuity - one with payments and interest conversions on the same date

  • Ordinary annuity - one in which all the payments are made at the end of the period (But see annuity-due for an alternative arrangement.)

Dealing with Restrictions Imposed by the Assumptions

Often you'll find in real-life that problems don't fit neatly into formulas and assumptions that have been derived from theory.

Fortunately, there are 'work-arounds' for many of the restrictions imposed by the assumptions above.

A very versatile approach to overcoming the restrictions of overburdening assumptions is to borrow from John Locke's epistemology, one of the basic tenets of which is that complex things are built from combinations of simpler things.

In other words, break the problem up into smaller pieces.

Uneven payments? Changing interest rates? Unequal time periods?

All of these issues violate the basic assumptions mentioned above. However, they can all be handled quite easily.

Remember, any annuity can be broken up into a series of individual single sums. Likewise, a single sum with a large term can be broken down into a series of smaller single sums with shorter terms. In either case, the 'aggregate' present or future value is simply the summation of all the individual pieces.

An example of how this can be done in practice is illustrated in Example Problem #12.

A Flat Yield Curve Will Not Persist

Another key assumption that can cause practical problems is that the yield curve is flat. In other words, there is no difference between long and short term interest rates.

This is almost never the case.

In choosing the right interest rate, the time horizon exerts a powerful influence. At a minimum, the interest rate should always be adjusted for the time to maturity.

Other factors that may need to be considered are: credit risk, inflation, taxes, options or unusual contractual terms, the nature and type of investment, alternative investments, and anticipated economic activity.

The need for precision is also a significant factor to consider when deciding on a interest rate.

See Interest Rates for more information on choosing an appropriate value for i.

Definitions

The terminology used with TVOM calculations is not precise. For example, in practice the term interest rate, discount rate, yield, and rate of return are often used interchangeably.

In an effort to avoid equivocation, here are some basic definitions that may be helpful:

  • Money that is loaned earns money for the lender and such money is called interest.

  • The amount of money which is owed and upon which the interest is earned is called principal.

  • The rate of interest in a given interval is numerically equal to the interest earned in one interval on a unit of principal per unit of time.

  • The term is the interval extending from beginning of the first compounding period to the end of the last compounding period.

  • Compound interest is interest charged on interest.

  • The nominal rate of interest is the stated annual rate of interest not taking into account compounding.

  • The effective rate of interest is the actual annual rate of interest taking into account the effect of compounding.

  • An annuity is a series of periodic payments.

  • The accumulated value of an annuity is the total accumulated values of all payments and interest as of the end of the annuity term.

  • A coupon bond is a bond that makes periodic (usually semi-annual) interest payments.

  • A discount bond (a.k.a. zero-coupon bond) is a bond that does not make periodic interest payments. Instead it is sold at a 'discount' and the difference between its face value and the price paid is the equivalent of a single interest payment made at maturity. T-Bills are an example of a discount bond.

The Interest Rates page provides a more detailed discussion on the distinction among the various types of interest rates.