Time Value of Money — The Only Time Travel Allowed on the CFA

"Money now is like a reliable friend. Money later might ghost you." — your future fiduciary self

You just finished Ethics, where we learned to put clients' interests first, disclose everything that matters, and avoid fuzzy math and shady promises. Good — because TVM (Time Value of Money) is financial ethics in numerical form: you owe your client the honest present value of future cash flows, not a wishful back-of-the-napkin number that makes your pitch deck look pretty.

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What is the Time Value of Money (TVM)? (Short answer, long impact)

TVM is the principle that a dollar today is worth more than a dollar tomorrow because of the potential to earn returns (interest), risk, inflation, and opportunity cost. This is the foundation for valuing bonds, loans, investments, pensions, and basically everything that involves cash over time.

Think of it like this: would you rather have $100 now or $100 in one year? Unless you are a gourmet hoarder of crisp paper, you take the $100 today and invest it, buy snacks, or both.

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The core formulas (bring snacks; it gets friendlier fast)

Present Value (PV) and Future Value (FV) are two sides of the TVM coin.

• Future value of a single sum:

FV = PV * (1 + r)^n

• Present value of a single sum:

PV = FV / (1 + r)^n

Where:

• r = interest rate per period (as a decimal)
• n = number of periods

If payments repeat (annuity):

• Future value of an ordinary annuity (payments at end of period):

FV_ann = PMT * [((1 + r)^n - 1) / r]

• Present value of an ordinary annuity:

PV_ann = PMT * [1 - (1 + r)^-n] / r

• Perpetuity (infinite annuity) present value:

PV_perp = PMT / r

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Important flavors of interest rates

• Nominal (stated) rate: annual rate not accounting for compounding within the year (e.g., 12% compounded monthly).
• Effective annual rate (EAR): actual annual rate after intra-year compounding.

Conversion:

EAR = (1 + r_nom/m)^m - 1

Where m = compounding periods per year.

Why care? Because a 12% nominal rate compounded monthly gives you more actual return than 12% compounded annually. Your clients care. So should you.

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Timelines — your secret weapon (and sanity saver)

Always sketch a timeline: it's the difference between guesswork and precision. Mark period 0 (today), then 1, 2, ... n. Place cash inflows as positive, outflows as negative. For annuities, mark uniform PMTs.

Example timeline (ordinary annuity of $100 for 3 years):

0 1 2 3
|----|----|----|
-100 -100 -100 (if payments at ends)

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Quick examples (so it actually sticks)

1. Single sum: You want $1,000 in 5 years. If you can earn 6% annually, how much do you need to invest today?

PV = 1000 / (1.06)^5 = 1000 / 1.3382256 = $747.26

So putting down $747.26 today gets you to $1,000 in 5 years. If you tell a client otherwise, you're not only wrong — you're violating the fiduciary spirit we celebrated in Ethics.

2. Ordinary annuity: A client will receive $200 at the end of each year for 4 years. Discount rate = 5%. Present value?

PV = 200 * [1 - (1.05)^-4] / 0.05 = 200 * 3.54595 = $709.19

3. Perpetuity: Preferred stock pays $5 forever; required return = 8%. Value = 5 / 0.08 = $62.50.

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Calculator tips (BA II Plus / TI style)

Most of the CFA quantitative questions assume you know how to use a financial calculator.

• Keys: N (periods), I/Y (rate), PV, PMT, FV.
• Example: PV of $1,000 in 5 years at 6%: enter N=5, I/Y=6, PMT=0, FV=1000, then compute PV.

If you rely on spreadsheets, know the functions: =PV(rate,nper,pmt,fv) and =FV(rate,nper,pmt,pv).

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Common pitfalls (aka traps that make graders cry)

• Mixing nominal and effective rates. If compounding isn't annual, convert.
• Confusing annuity due vs ordinary annuity. (Annuity due payments are at period-start — treat them like ordinary annuity multiplied by (1 + r)).
• Forgetting sign convention: PV negative if cash outflow; keep consistent.
• Using the wrong n (years vs periods).

Table: Quick comparison

Concept	Use when	Formula snippet
Single sum PV	One future lump sum	PV = FV/(1+r)^n
Ordinary annuity PV	Payments at period end	PV = PMT*[1-(1+r)^-n]/r
Annuity due PV	Payments at period start	PV_ann_due = PV_ann * (1+r)
Perpetuity PV	Infinite equal payments	PV = PMT/r

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Ethics tie-in (because you did Ethics, remember?)

When advising clients about the value of future cash flows — whether a bond, a pension, or a projected return — you are not allowed to fudge the discount rate to make returns look prettier. Using inappropriate assumptions is an ethical issue. Be transparent: state the discount rate, justify it (market rates, required return), and show sensitivity (what if rate is 1% higher?). That’s not just good math; it's professional integrity.

If you wouldn’t sign your name to it, don’t present it to a client. TVM is where quantitative correctness meets ethical responsibility.

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Quick practice (try it now — brain gains!)

Problem: You will receive $500 annually at the end of each year for 6 years. Discount rate = 4%. What is the PV? (Show steps).

Solution sketch:

1. Recognize ordinary annuity: PMT = 500, r = 0.04, n = 6.
2. Use formula: PV = 500 * [1 - (1.04)^-6] / 0.04.
3. Compute (1.04)^-6 = 0.790314, so bracket = 1 - 0.790314 = 0.209686 / 0.04 = 5.24215.
4. PV = 500 * 5.24215 = $2,621.08.

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Final punchline and takeaways

• TVM is the DNA of valuation: everything that happens in finance is just PVs and FVs dressed up in different clothes.
• Always use correct rate-period alignment, timeline sketches, and consistent signs.
• Be explicit about assumptions — this is both good practice and ethical duty (hi again, Ethics!).

Leave this ready-to-use mantra on repeat: A number without a discount rate is a story without context. Now go forth, discount like a scholar, and never let a client’s future get undervalued.

Version checklist: practice a few calculator problems, draw timelines, and when in doubt, convert nominal to effective.

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Need more? Ask for 5 timed practice problems with step-by-step BA II Plus keystrokes (I’ll make them viciously educational).