Interest, Equivalence, and Compound Growth

Interest, Equivalence, and Compound Growth

$Learning Objectives$

Distinguish simple interest from compound interest and identify when each model applies.
Use single-payment compound factors to move cash flows between focal dates.
Explain economic equivalence at a stated effective periodic interest rate.
Match the interest-rate period to the cash-flow period before applying factors.
Solve for unknown present worth, future worth, rate, or number of periods.

$Interest$

Interest is the monetary charge for the use of money over time, or the return earned for postponing consumption and committing capital.

$Interest Rate$

The interest rate $i$ is interest per unit principal per stated period. An engineering-economy calculation is incomplete unless the rate period is identified.

$Simple Interest$

Simple interest is computed only on the original principal. Interest earned in prior periods does not itself earn interest.

Simple-Interest Future Worth

Future amount under simple interest for n periods.

F=P(1+in)

Variables

Symbol	Description	Unit
$F$	Future amount after n periods	-
$P$	Present principal at time zero	-
$i$	Simple interest rate per period	-
$n$	Number of interest periods	-

$Compound Interest$

Compound interest is computed on the principal plus accumulated interest, so each period's ending balance becomes the next period's interest-bearing balance.

Compound Amount Factor (F/P)

Moves one present amount forward n periods at effective periodic rate i.

F=P(1+i)^n

Variables

Symbol	Description	Unit
$F$	Equivalent future amount	-
$P$	Present amount	-
$i$	Effective interest rate per period	-
$n$	Number of periods	-

Single-Payment Present Worth Factor (P/F)

Discounts one future amount back n periods.

P=F(1+i)^{-n}

Variables

Symbol	Description	Unit
$P$	Equivalent present amount	-
$F$	Future amount	-
$i$	Effective interest rate per period	-
$n$	Number of periods	-

$Interactive Compound-Growth Explorer$

Change the principal, periodic rate, and number of periods to see compound growth and the inverse relationship between the $F/P$ and $P/F$ factors.

Present amount, PPHP

Interest rate per cash-flow period8.00%

Number of periods10

Future worth, F

₱215,892.50

F = P(1+i)^n using i = 8.00% per period.

Interest accumulated

₱115,892.50

Future worth minus original principal.

Reverse P/F check

₱100,000.00

Discounting the computed future amount returns the original present amount.

Growth by period

$Economic Equivalence$

Cash flows at different dates are economically equivalent when moving each to the same focal date at the stated interest rate gives the same value.

$The Focal-Date Principle$

Any date may be used as the focal date if every cash flow is moved consistently to that date using the same rate basis. Moving cash flows forward multiplies by $(1+i)^n$ ; moving them backward divides by the same factor. Correctly performed analyses give the same economic conclusion regardless of focal date.

Amount at source periodPHP

Effective rate per period8.00%

Source periodt = 2

Focal periodt = 8

Equivalent amount at focal date

₱1,586,874.32

Compounded across 6 periods.

Transformation

Use F/P

The rate and period count both use the same period basis.

Solve for the Compound Rate

Finds the effective periodic rate when P, F, and n are known.

i=\left(\frac{F}{P}\right)^{1/n}-1

Variables

Symbol	Description	Unit
$i$	Effective periodic interest rate	-
$F$	Known future amount	-
$P$	Known present amount	-
$n$	Number of periods	-

Solve for the Number of Periods

Finds n for a positive compound-growth ratio when the periodic rate is known.

n=\frac{\ln(F/P)}{\ln(1+i)}

Variables

Symbol	Description	Unit
$n$	Number of periods	-
$F/P$	Future-to-present amount ratio	-
$i$	Effective periodic interest rate	-

$Effective Periodic Rate Is the Working Rate$

The $i$ used in compound factors must be effective for one cash-flow period. A quoted nominal annual rate cannot be inserted directly into a monthly factor unless the quote itself is an effective monthly rate.

$Interpolation and Rounded Factor Tables$

Published factor tables are rounded approximations. Calculator or software evaluation of the exact factor is preferable when precision matters. If interpolation is required by an exam method, state that it is an approximation and keep adequate intermediate precision.

$Single-Payment Equivalence Workflow$

STEP-BY-STEP

Draw the known and unknown cash flows on a timeline.
Convert the quoted rate to an effective rate per cash-flow period.
Choose a focal date.
Count the exact number of compounding periods between each cash flow and the focal date.
Apply $F/P$ when moving forward and $P/F$ when moving backward.
Combine equivalent amounts algebraically with signs intact.
Check whether the magnitude and direction of change are reasonable.

$Reasonableness Checks$

For $i>0$ , a positive amount moved forward should increase.
For $i>0$ , a positive amount discounted backward should decrease.
$F/P$ and $P/F$ for the same $i$ and $n$ must multiply to $1$ .
The number of periods must be consistent with the stated rate period.
Do not round the interest factor before the final result unless instructed.

Key Takeaways

Simple interest does not compound; compound interest earns interest on accumulated interest.
Single-payment equivalence uses $F=P(1+i)^n$ and its inverse.
Economic equivalence is always relative to a stated rate and timing basis.
Any focal date works when all cash flows are moved consistently.
The working rate must be effective for the same period represented by the cash-flow timeline.
Inverse factor and magnitude checks catch many time-value errors before they propagate.