Chapter 3 · Van Horne, 13th Ed.

The Time Value of Money

Quick MBA study notes — read in 10 minutes

1The Big Idea

Would you rather have $1,000 today or $1,000 in 10 years? Obviously today — because money today can be invested and put to work earning interest. This single idea — that money has a time value — underlies nearly every financial decision you'll ever make.

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Key Concept The interest rate is the tool that lets us move cash flows — forward or backward — to a single point in time so we can fairly compare them.

Analogy: comparing $1,000 today to $2,000 in 10 years is like comparing prices in two different currencies — you first have to convert them to the same "currency" (a single point in time) before you can judge which is worth more.

2Simple vs. Compound Interest

Simple interest is earned only on the original principal. Compound interest is earned on the principal plus all previously accumulated interest — "interest on interest."

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Key Concept Compounding is why small, early investments snowball. $2,000/year invested at age 21 grows to ~$773,000 by 65 at 8% — nearly double what the same amount invested starting at 31 would yield.

3Future Value & Present Value — The Core Formulas

Concept	Formula
Future value (single amount)	FVₙ = P₀(1+i)ⁿ
Present value (single amount)	PV₀ = FVₙ / (1+i)ⁿ

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Key Concept Future value asks "what will this money grow to?" Present value asks "what is a future amount worth today?" They're mirror images of the same idea.

Always start with a time line — draw out each cash flow at the period it occurs. It's the single most useful habit for avoiding mistakes in these problems.

4Annuities — Ordinary vs. Due

An annuity is a series of equal payments over time. The timing of those payments changes the formula:

Type	Cash flow timing	Valued as of
Ordinary annuity	End of each period	One period before the first cash flow
Annuity due	Beginning of each period	The first cash flow itself

PVAₙ = R × [(1 − 1/(1+i)ⁿ) / i]

Example

$1,000 ordinary annuity for 3 years at 8% → PVIFA = 2.577 → PV = $2,577

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Key Concept An annuity due is always worth slightly more than an equivalent ordinary annuity — you get each payment one period earlier, so it has more time to grow (or less time discounted away).

5Mixed (Uneven) Cash Flows

When cash flows aren't equal each period, there's no shortcut formula — you simply discount or compound each flow individually and sum the results. If you spot an annuity pattern hidden inside a mixed stream, use the annuity formula for that portion to save time.

6Solving for the Unknown Rate or Payment

Time value formulas can be rearranged to solve for whatever's missing — rate, payment, or number of periods — as long as you know the other three variables.

Example

Need $9,500 in 8 years by saving $1,000/year? Solve FVIFA = 9.5 → rate is just under 5%.

7Compounding Frequency

FVₙ = PV₀ (1 + i/m)^(mn)

Example — $100 for 3 years at 8% nominal

Compounding	Future Value
Annual	$125.97
Semiannual	$126.53
Quarterly	$126.82

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Key Concept The more frequently interest compounds, the higher the future value — and the lower the present value of a future amount. As compounding frequency approaches infinity, you get continuous compounding.

8Amortization Schedules

A loan repaid in equal installments splits each payment between interest (on the remaining balance) and principal.

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Key Concept Early payments are mostly interest; later payments are mostly principal — the interest portion shrinks each period because it's charged only on the shrinking balance.

9Formula Cheat-Sheet

Concept	Formula
Future value	FVₙ = P₀(1+i)ⁿ
Present value	PV₀ = FVₙ/(1+i)ⁿ
FV of ordinary annuity	FVAₙ = R[((1+i)ⁿ−1)/i]
PV of ordinary annuity	PVAₙ = R[(1−1/(1+i)ⁿ)/i]
Compounding m times/year	FVₙ = PV₀(1+i/m)^(mn)

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A dollar today is worth more than a dollar tomorrow — the interest rate is simply the exchange rate that lets you compare money across time.