Time Value of Money Formulas

Why This Matters

The time value of money (TVM) is the foundational principle behind virtually everything in corporate finance: valuing stocks and bonds, making capital budgeting decisions, and structuring loan payments. When you're tested on TVM, you're really being tested on your ability to translate cash flows across time, recognizing that a dollar today isn't worth the same as a dollar tomorrow.

Don't just memorize these formulas. Understand what problem each one solves and when to apply it. Exam questions rarely ask you to simply plug numbers into an equation. They present scenarios where you must first identify which formula applies, then execute the calculation. Master the logic of discounting (moving future values backward), compounding (moving present values forward), and annuitizing (handling streams of payments), and you can tackle any TVM problem thrown at you.

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Single Cash Flow Formulas

These formulas handle the simplest TVM scenario: one lump sum moving through time. Compounding grows value forward; discounting shrinks value backward. Master these first since they're the building blocks for everything else.

Present Value (PV)

$PV = \frac{FV}{(1 + r)^n}$

This calculates what a future lump sum is worth today at discount rate $r$ over $n$ periods. Discounting reverses the effect of compound interest, answering: "What would I pay now for money I'll receive later?"

• Core application: valuing any single future cash flow, from a zero-coupon bond's face value to a lawsuit settlement paid years from now

Future Value (FV)

$FV = PV \times (1 + r)^n$

This projects what today's money grows to after $n$ compounding periods at rate $r$. Compounding captures the snowball effect: you earn interest on your interest each period.

• Core application: projecting investment growth, calculating how much a deposit today becomes at retirement

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Annuity Formulas

Annuities involve equal payments at regular intervals: loan payments, lease payments, retirement withdrawals. The formulas below aggregate multiple cash flows into a single value using a shortcut rather than discounting each payment individually.

Present Value of an Annuity (PVA)

$PVA = PMT \times \frac{1 - (1 + r)^{-n}}{r}$

This finds today's value of $n$ equal payments of $PMT$ discounted at rate $r$. The piece $\frac{1 - (1 + r)^{-n}}{r}$ is called the annuity discount factor. It represents the sum of all individual discount factors collapsed into one expression.

• Core application: pricing bonds (coupon stream), calculating maximum loan amounts, valuing pension payouts

Future Value of an Annuity (FVA)

$FVA = PMT \times \frac{(1 + r)^n - 1}{r}$

This calculates what a stream of equal deposits grows to at the end of $n$ periods. Compounding stacks each payment at a different horizon: the first payment compounds $n - 1$ times, the second $n - 2$ times, and the last payment doesn't compound at all.

• Core application: projecting 401(k) balances, calculating total savings from regular contributions

Present Value of a Perpetuity

$PV = \frac{PMT}{r}$

This values an infinite stream of equal payments. It's the simplest TVM formula. It works because as $n \to \infty$, the term $(1 + r)^{-n} \to 0$ in the PVA formula, and the annuity discount factor simplifies to just $\frac{1}{r}$.

• Core application: valuing preferred stock dividends, endowments, and any cash flow assumed to continue forever

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Interest Rate Conversions

Nominal rates can be misleading when compounding frequency varies. These formulas let you compare apples to apples by converting everything to a true annual yield.

Effective Annual Rate (EAR)

$EAR = \left(1 + \frac{i}{m}\right)^m - 1$

This converts a nominal (stated) annual rate $i$ compounded $m$ times per year into its true annual equivalent. More frequent compounding produces a higher EAR because interest-on-interest accumulates faster within the year.

• Core application: comparing credit cards (daily compounding) to mortgages (monthly compounding) to bonds (semiannual compounding)

APR vs. EAR

APR (Annual Percentage Rate) is the nominal rate lenders are required to quote. It simply divides the periodic rate by the number of periods, ignoring intra-year compounding. EAR captures that compounding and shows you the true cost.

For example, a 12% APR compounded monthly means a monthly rate of $\frac{0.12}{12} = 0.01$. The EAR is:

$EAR = (1 + 0.01)^{12} - 1 = 0.1268 = 12.68\%$

You're actually paying 12.68%, not 12%. Exams love testing whether students recognize this distinction.

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Investment Performance Metrics

These formulas evaluate whether an investment creates value. They combine discounting with decision rules to answer: "Is this worth doing?"

Compound Annual Growth Rate (CAGR)

$CAGR = \left(\frac{\text{Ending Value}}{\text{Beginning Value}}\right)^{1/n} - 1$

This calculates the smoothed annual return that would produce the observed growth over $n$ years. CAGR ignores volatility along the way, so it's useful for comparing investments with different holding periods on an equal footing.

• Core application: benchmarking portfolio performance, comparing growth rates across different time horizons

Net Present Value (NPV)

$NPV = \sum_{t=1}^{n} \frac{CF_t}{(1 + r)^t} - C_0$

NPV sums all discounted future cash flows ($CF_t$) and subtracts the initial investment ($C_0$).

• Decision rule: accept projects with NPV > 0. A positive NPV means the project earns more than the required return and creates shareholder value.
• Core application: capital budgeting, project evaluation, M&A valuation. NPV is considered the gold standard for investment decisions.

Internal Rate of Return (IRR)

IRR is the discount rate $r$ that makes $NPV = 0$. You typically solve for it iteratively or with a financial calculator since there's no clean algebraic solution.

• Decision rule: accept projects with IRR > cost of capital. The project's return exceeds what investors require.
• Limitation: IRR can give misleading results with non-conventional cash flows (where signs flip more than once) or when comparing mutually exclusive projects of different sizes. Always cross-check with NPV.

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Quick Reference Table

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Self-Check Questions

1. You're offered $\$10{,}000$ five years from now or a lump sum today. Which formula determines the minimum you'd accept today, and what variables do you need?

2. What mathematical relationship connects PVA and the perpetuity formula? Under what condition does PVA simplify to the perpetuity formula?

3. A bank offers 6% APR compounded monthly; another offers 6.1% compounded annually. Which gives the higher effective return, and how do you prove it?

4. An investment has a positive NPV but an IRR below the company's cost of capital. Is this possible? Explain what might cause this apparent contradiction.

5. You're saving $\$500$/month for 30 years at 7% annual return. Which formula calculates your ending balance, and why would using the PVA formula give you a meaningless answer here?