Time Value of Money: Annuities, Perpetuities, and Compounding

Comprehensive vocabulary flashcards covering the Time Value of Money lecture slides, including compounding frequencies, APR, EAR, perpetuities, and ordinary annuities versus annuities due.

Compounding

The process of determining the future value of an investment made today by making the amount bigger through earning interest on interest.

Discounting

The process of determining the present value of cash to be received at a future date by making the amount smaller.

Time Value of Money (Core Drivers)

The principle that cash received sooner is more valuable due to three primary reasons: inflation deteriorating purchasing power, lost opportunity cost of waiting to invest, and the risk that payment might not be received.

Valuation Building Block of Finance

The principle that the financial value of any asset today is the summation of its future cash flows discounted to the present day at a rate reflecting the risk associated with those cash flows.

Simple Interest

Interest paid exclusively on the original principal sum.

Compound Interest

Interest paid on both the principal sum and on all prior interest earned.

Compounding Period Consistency Rule

The requirement that interest rate ($r$) and time ($t$) must be expressed in the exact same terms/units (e.g., monthly interest rate with the number of months) when applying the future value formula $FV = PV \times (1 + r)^t$.

Annual Percentage Rate (APR)

The nominal, stated interest rate quoted on a loan or investment per year (e.g., $10.25\%$ per year).

Effective Annual Rate (EAR)

The actual annual rate paid or received taking into account the compounding frequency within the year, calculated as $EAR = \left[1 + \left(\frac{APR}{m}\right)\right]^m - 1$, where $m$ is the number of compounding periods per year.

Perpetuity

A perpetual stream of cash flows that continues forever, which can be constant (flat) or grow at a constant rate.

Present Value of a Growing Perpetuity Formula

$PV_0 = \frac{C_1}{r - g}$, where $C_1$ is the payment in period 1, $r$ is the discount rate, $g$ is the constant growth rate, and $r > g$.

Present Value of a Flat Perpetuity Formula

$PV_0 = \frac{C_1}{r}$, derived from the perpetuity formula when the growth rate $g = 0$.

Annuity

A finite series of equal payments that occur on a regular, fixed schedule.

Ordinary Annuity

An annuity where the payments occur at the end of each period.

Annuity Due

An annuity where payments occur at the beginning of each period, making its value greater than an otherwise equivalent ordinary annuity.

Present Value of an Annuity Due Relationship

The present value obtained by multiplying the present value of an otherwise identical ordinary annuity by $(1 + r)$ to account for the additional period of time value of money.

Present Value of an Ordinary Annuity Formula

$PV = C \left[\frac{1 - \frac{1}{(1 + r)^t}}{r}\right]$, which simplifies the sum of the individual discounted cash flows over $t$ periods at discount rate $r$.

Growing Annuity

A finite stream of cash flows occurring on a regular schedule where the payments increase at a constant rate over time.