Interest, Equivalence, and Compound Growth

Interest, Equivalence, and Compound Growth

$Learning Objectives$

Distinguish simple from compound interest and identify the growth model assumed.
Read and annotate a dated single-payment cash-flow timeline.
Convert single present and future amounts using consistent $F/P$ and $P/F$ factors.
Match nominal and effective rate conventions to period counts and cash-flow dates.
Move multiple signed payments to one focal date and verify equivalent amounts.
Solve for an unknown compound rate or period count while checking domain limits.

$Interest$

Interest is the monetary charge for the use of money over time or the return earned for deferring the use of capital. The resulting amount depends on both the agreed rate and the time interval.

$Interest Rate$

The interest rate $i$ is interest per unit principal per stated period. In the compound formulas below, $i$ means the effective rate per compounding/cash-flow period. A number such as 8% without its period is insufficient data.

$Simple Interest$

Simple interest is computed only on the original principal $P$ : accumulated interest does not itself earn additional interest. Here $n$ is the number of equal interest periods and $i$ the simple rate for one such period.

Simple-Interest Future Worth

Non-compounding growth for n simple-interest periods.

F=P(1+in)

Variables

Symbol	Description	Unit
$F$	Future amount, in the same currency as P	-
$P$	Original principal at time zero	-
$i$	Simple rate per stated period	-
$n$	Number of matching periods	-

$Compound Interest$

In a compound account the balance after each period becomes the principal for the next period. At a constant effective rate, $P$ , $P(1+i)$ , $P(1+i)^2$ , and so on describe successive end-of-period amounts. All examples in this section use discrete compounding unless stated otherwise.

Compound Amount Factor (F/P)

Forward equivalence of one amount over n equal periods.

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 from P to F	-

$Simple Versus Compound: Same Initial Principal$

For an analytical example with $P=\text{PHP }100{,}000$ , $i=20\%$ per year, and years $n=0,1,2,3,4$ , simple growth gives PHP 100,000, 120,000, 140,000, 160,000, and 180,000. Compound growth gives PHP 100,000, 120,000, 144,000, 172,800, and 207,360. The two models agree after one year, then diverge because only the compound model earns interest on interest.

$Growth Diagram Interpretation$

The labeled end balances follow $F=P(1+0.20n)$ and $F=P(1.20)^n$ . Axis values are in thousands of PHP; marks at whole years identify exact payment dates. The graph is model-derived, not measured account data.

Single-Payment Present Worth Factor (P/F)

Discount a single future amount n periods backward.

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

Variables

Symbol	Description	Unit
$P$	Equivalent amount at the earlier focal date	-
$F$	Amount at the later date	-
$i$	Effective rate for one period	-
$n$	Number of periods between the dates	-

$Read the Single-Payment Timeline$

Suppose PHP 100,000 at time zero is equivalent to PHP 121,000 at the end of year 2 at $i=10\%$ effective per year. This is one amount at each date, not a uniform annual series. A two-year forward shift multiplies by $(1.10)^2$ , and a two-year backward shift divides by the same factor.

$Factor-Placement Check$

Multiplying $100{,}000(1.10)^2$ gives 121,000 PHP; discounting $121{,}000/(1.10)^2$ gives 100,000 PHP. The arrows explain movement of value, not an additional cash payment.

$Interactive Compound-Growth Explorer$

Change the principal, periodic rate, and number of periods to see compounding and verify that the $F/P$ and $P/F$ factors are reciprocals for the same $i$ and $n$ .

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

$Economic Equivalence$

Amounts at different times are equivalent at a specified interest rate if converting each amount to the same focal date produces the same result. Equivalence is conditional on the rate and the timing assumptions.

$The Focal-Date Principle$

One may choose time zero, year three, or another common date. Move each signed cash flow forward by multiplying by the appropriate compound factor, or backward by dividing by it; then sum amounts at the focal date. Using a different focal date does not change which properly comparable alternative is preferred, provided the same assumptions apply throughout.

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

Implied effective periodic rate for positive P and F and nonzero n.

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

Variables

Symbol	Description	Unit
$i$	Effective rate per period	-
$F$	Positive future amount	-
$P$	Positive present amount	-
$n$	Positive number of periods	-

Solve for the Number of Periods

Logarithmic inverse of compound growth when i differs from zero.

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

Variables

Symbol	Description	Unit
$n$	Number of compounding periods	-
$F/P$	Positive future-to-present ratio	-
$i$	Effective rate with i greater than minus one and i not equal to zero	-

$Rate and Period Edge Cases$

The factor $(1+i)^n$ requires $1+i>0$ for ordinary real-valued logarithmic inversion. If $i=0$ , then $F=P$ for any $n$ and the logarithmic formula for $n$ is undefined. If $i>0$ , a positive single amount grows moving forward; if $-1<i<0$ , it shrinks. The common growth examples assume $P>0$ , $F>0$ , and $n>0$ .

$Nominal Quotes Versus Effective Rates$

A nominal annual quote $j$ compounded $m$ times per year implies an effective rate per compounding interval of $j/m$ , when the quote uses that convention. The corresponding effective annual rate is $(1+j/m)^m-1$ . For $j=12\%$ nominal per year compounded monthly, $m=12$ : the working monthly rate is $1\%$ , not $12\%$ , and the effective annual rate is approximately $12.6825\%$ . This is a mathematical illustration, not a market quote.

$Effective Periodic Rate Is the Working Rate$

The compounding-rate period and number of factors must match the model's time interval. For annual effective rates use years, for monthly effective rates use months. Twelve months correspond to $n=12$ monthly factors, not $n=1$ with the nominal annual percentage.

$Interpolation and Rounded Factor Tables$

Table factors are rounded approximations. Retain calculator precision through intermediate steps, and explain any interpolation if a table method is required.

$Single-Payment Equivalence Workflow$

STEP-BY-STEP

Draw a timeline with the dates of every known and unknown amount.
Identify whether the rate is simple, effective, nominal compounded, or another stated convention.
Convert the quoted rate to the compounding period represented on the timeline.
Choose a focal date and count the exact number of intervals to each cash flow.
Apply $F/P$ when moving forward or $P/F$ when moving backward, keeping signs.
Sum all amounts at one date; keep unrounded factors until the final result.
Check rate-period consistency and whether growth or discounting has the expected direction.

$Reasonableness Checks$

For $i>0$ , a positive amount carried forward should increase.
For $i>0$ , discounting a positive amount backward should decrease it.
$F/P$ times $P/F$ for identical $i$ and $n$ must equal one.
The number of intervals must match the effective-rate period.
For $i=0$ do not divide by $\ln(1+i)$ .

Key Takeaways

Simple interest grows linearly in $n$ ; compound interest earns interest on accumulated interest.
$F=P(1+i)^n$ and $P=F(1+i)^{-n}$ are reciprocal movements between dated amounts.
Time value depends on a specified rate, compounding convention, and focal date.
Monthly amounts and monthly rates require month counts; nominal annual quotes are not automatic annual effective rates.
Rate and period inverse formulas have domain and zero-rate exceptions.
Rate-period, magnitude, sign, and inverse-factor checks expose common errors.

$References$

Kneifel and Webb, Life Cycle Costing Manual for the Federal Energy Management Program, NIST Handbook 135e2022 (2022). Consulted for the general principle of a common focal date and consistent discounting; U.S. federal guidance, not a Philippine mandate. Numerical illustrations here are independently computed classroom examples.