Section 1Economic Factors and Business Information
Time Value of Money
45 min read
· Lesson 1 of 4
Time Value of Money
A dollar today is worth more than a dollar tomorrow — not because of inflation alone, but because today's dollar can be invested to earn a return. Every investment valuation on the Series 66 traces back to this single idea. This chapter builds the toolkit: future value, present value, NPV and IRR, the Rule of 72, and how to think about real (inflation-adjusted) returns.
Time value of money (TVM) is the foundational concept behind every investment valuation on the Series 66. A dollar today is worth more than a dollar tomorrow because today's dollar can be invested to earn returns. That single relationship drives bond pricing, equity valuation, retirement planning, and the accept/reject decisions an adviser makes on a client's behalf.
The exam will not ask you to crunch complex TVM calculations by hand. It will ask you to reason about relationships — when interest rates rise, what happens to present value? When compounding gets more frequent, what happens to future value? Get the relationships locked in and the rest follows.
Opportunity cost: the engine behind TVM
Three forces make today's dollar more valuable than tomorrow's. In rough order of importance for the Series 66:
1. Investable return
Today's dollar can earn interest, dividends, or capital gains between now and then.
2. Inflation
A dollar tomorrow buys less than a dollar today when prices rise.
3. Risk & uncertainty
A future payment might not arrive — promised cash is not the same as cash in hand.
The Series 66 sometimes calls the first one opportunity cost: by waiting, you forgo what that dollar could have earned elsewhere. Even in a zero-inflation, zero-risk world, the investable return alone is enough to make money sooner more valuable than money later.
The Core Relationship to Memorize: Interest rates
and present value move in opposite directions.
When rates rise → PV falls. When rates fall → PV rises.
This applies to bonds (bond prices fall when rates rise),
annuities (the present value of future payments drops),
and all TVM calculations. If a Series 66 question describes
rising interest rates and asks about PV, the answer is always
lower.
Concept Check
Interest rates in the economy unexpectedly rise. All else equal, what happens to the present value of a fixed future cash flow?
Present value moves inversely with the discount rate. A fixed future cash flow becomes worth less today when the discount rate rises, because each future dollar is divided by a larger (1+r) raised to n. The cash flow itself doesn't change in nominal terms, but its today-equivalent is lower. This relationship is the foundation of bond pricing — when rates rise, bond prices fall — and applies equally to any single-payment PV calculation. Timing of the cash flow magnifies the effect but does not reverse it.
Concept Check
A client says, 'I'd rather have $1,000 right now than $1,000 a year from now.' From a time-value-of-money perspective, what is the primary reason this preference is rational?
The core time-value principle is opportunity cost: a dollar received today can be invested at the prevailing rate of return and grow to more than a dollar by next year. If the risk-free rate is 4%, $1,000 today becomes $1,040 in one year, making today's $1,000 worth $1,040 in future terms. Inflation and counterparty risk reinforce the preference but are secondary on the Series 66 — even in a zero-inflation, zero-default world, the investable rate of return alone justifies preferring money sooner.
Section 2 of 5~9 min · 3 concept checks
Future value & compounding
Future value (FV)
Future value calculates what a present sum will be worth at a specified point in the future, given a rate of return:
FV = PV × (1 + r)n
PV = present value (the amount today)
r = interest rate per period
n = number of compounding periods
Worked example: $10,000 invested at 6% for 10 years grows to $10,000 × (1.06)10 = $17,908. Notice the growth is non-linear: in year 1 you earn $600 of interest; in year 10 you earn $1,014, because you're earning interest on the interest from prior years.
Simple vs. compound interest
The Series 66 expects you to distinguish between simple and compound interest. The gap is small over short periods and dramatic over long ones.
Simple interest
Interest is earned only on the original principal.
FV = PV × (1 + r × n)
$10,000 at 6% for 30 yrs = $28,000
Compound interest
Interest is earned on principal and all accumulated interest.
FV = PV × (1 + r)n
$10,000 at 6% for 30 yrs = $57,435
Same principal, same rate, same time horizon — but compound interest more than doubles the simple-interest result over 30 years. This gap is the engine of long-term wealth building, which is why retirement-planning chapters lean heavily on compounding intuition.
Compounding frequency
More frequent compounding always produces a higher future value, all else equal — but the marginal gain shrinks fast. Here's $10,000 at 6% for 10 years across six compounding frequencies:
The gap between annual and continuous compounding on $10,000 over 10 years is just $313 — small in dollars, but the directional concept is tested.
Effective annual rate (EAR) vs. stated/APR
When a rate is quoted as an annual percentage (APR) but compounds more than once per year, the rate you actually earn — the effective annual rate (EAR) — is higher than the stated rate.
EAR = (1 + APR ÷ n)n − 1
where n is the number of compounding periods per year. A 6% APR compounded:
Annually → EAR = 6.00%
Semi-annually → EAR = 6.09%
Monthly → EAR = 6.17%
Daily → EAR = 6.18%
EAR matters when comparing investments quoted on different compounding bases. A bank offering "6.10% APR compounded daily" actually pays slightly less than one offering "6.15% APR compounded annually" — the second has the higher EAR.
Concept Check
An investor deposits $5,000 today at 6% interest. Holding all other variables constant, which change would result in the highest future value at the end of 10 years?
Future value is most sensitive to the rate of return. On $5,000 over 10 years, a one-percentage-point increase from 6% to 7% raises FV by roughly $880 — far more than adding one more year of growth at 6% (about $540) or switching from annual to monthly compounding (about $145). More frequent compounding does help, but with sharply diminishing marginal effect: monthly to daily adds only about $13. The exam pairs these levers as distractors; the rate increase wins decisively.
Concept Check
Which of the following compounding frequencies produces the highest future value, all other inputs being equal?
More frequent compounding always produces a higher future value, with continuous compounding (theoretical infinite frequency) representing the upper limit. The progression is annual < semi-annual < quarterly < monthly < daily < continuous. However, marginal gains shrink quickly: the gap between daily and continuous on $10,000 at 6% for 10 years is just $1. The Series 66 wants you to know the directional rule, not the dollar differences. Continuous compounding wins any ranking question.
Concept Check
Two CDs are advertised with the same stated annual rate of 5%. Bank A compounds annually; Bank B compounds monthly. Which statement is most accurate?
When the same stated rate is compounded more frequently, the effective annual rate (EAR) is higher. Bank B's EAR is (1 + 0.05/12)^12 − 1 = 5.12%, while Bank A's EAR equals the stated 5%. The depositor at Bank B earns about $1.20 more per $1,000 deposited over one year. The distinction between stated/APR and effective rates is exam-testable: when comparing investments quoted on different compounding bases, EAR is the apples-to-apples figure.
Section 3 of 5~8 min · 3 concept checks
Present value & discount rate
Present value (PV)
Present value is the reverse of future value — it discounts a future cash flow back to today's equivalent:
PV = FV ÷ (1 + r)n
The same formula rearranged. What changes is the question: instead of "what will today's money grow to?" you're asking "what is a promised future amount worth today, given the rate I could otherwise earn?"
Worked example: A client is promised $10,000 in 5 years. With a 6% discount rate, that promise is worth $10,000 ÷ (1.06)5 = $7,473 today. If you could pay less than $7,473 to acquire that promise, you'd earn more than 6%; if you'd have to pay more, you'd earn less.
What goes into the required rate of return
The discount rate you apply to a future cash flow isn't arbitrary — it reflects what you'd require to take on that particular investment. It has two building blocks:
Risk-free rate
Risk premium
~3-month T-bill
Compensation for uncertainty
Risk-free rate — what you could earn with near-zero default risk (the 3-month Treasury bill is the standard proxy). Compensates you purely for the passage of time.
Risk premium — the additional return demanded for accepting uncertainty about the actual cash flows. Equities demand a larger premium than investment-grade corporate bonds, which demand more than Treasuries.
Required return = risk-free rate + risk premium. As either piece moves, the required return moves with it — which is why rising interest rates pressure equity valuations across the board.
Picking the right discount rate
The discount rate should match the risk of the cash flow you're discounting. Two practical implications:
Riskier cash flows → higher discount rate → lower PV. A speculative startup's projected earnings get discounted at a much higher rate than a Treasury coupon, which is exactly why those earnings are worth less today than a Treasury promise of the same nominal amount.
Don't mix rates within one analysis. Discounting some cash flows at the risk-free rate and others at an equity hurdle rate inside the same NPV calculation produces nonsense.
For exam purposes, you'll be given the required rate of return (sometimes called the hurdle rate). You don't have to construct it — but you should recognize what it represents.
Holding period return & annualization
Holding period return (HPR) is the total return earned over the time you held an investment, expressed as a percentage of the starting value:
HPR = (Ending value − Beginning value + Income) ÷ Beginning value
To compare investments held for different lengths of time, HPR has to be annualized. A 21% return earned over 3 years is not 7% per year on a compound basis — it's 6.56% per year, because (1.0656)3 ≈ 1.21. Multiplying or dividing by years is the rough cut; compounding is the right answer.
This concept comes back hard in Chapter 12 (portfolio performance) where you'll distinguish time-weighted from dollar-weighted returns.
Concept Check
A client is promised $20,000 in exactly five years from a private investment. If the appropriate discount rate is 7%, the present value of this promise is closest to:
Present value = $20,000 / (1.07)^5 = $20,000 / 1.4026 = $14,260. The promise is worth $14,260 today. Two distractor reasoning errors: $18,692 multiplies by (1 − 0.07×5), the simple-interest analog; $21,400 grows the amount forward at 7% rather than discounting back. The correct mental check is direction — discounting back must produce a smaller number than the future amount, which rules out $21,400 and $28,051 immediately.
Concept Check
An adviser is evaluating two potential investments for a client: a high-grade corporate bond and a speculative private equity stake. Which statement about the required rate of return is most accurate?
The required rate of return equals the risk-free rate plus a risk premium specific to the investment. Riskier cash flows demand larger premiums, so a speculative private equity stake is discounted at a higher rate than an investment-grade bond. The result is a lower PV per dollar of expected cash flow for the riskier investment — which is exactly why those cash flows must be larger to justify the investment. Using one blended rate across investments with very different risk profiles produces meaningless comparisons.
Concept Check
An investor's holding period return on a 3-year investment was 21%. What is the closest approximation of the compound annualized return?
Annualizing a multi-year HPR is a compounding calculation, not a simple division. The correct formula is (1 + HPR)^(1/years) − 1 = (1.21)^(1/3) − 1 = 1.0656 − 1 = 6.56%. Dividing 21% by 3 to get 7% is the most common error: it treats the return linearly and overstates the per-year compound rate. The Series 66 tests this directly because annualization mistakes are common in performance reporting — Module 3's portfolio-performance chapter revisits the concept with time-weighted vs. dollar-weighted measures.
Section 4 of 5~12 min · 4 concept checks
NPV & IRR decisions
Net present value (NPV)
NPV extends the present-value idea to a stream of cash flows. You discount every future cash flow back to today, sum the present values, and subtract the initial investment:
NPV = Σ [CFt ÷ (1 + r)t] − Initial investment
The decision rule is the heart of it:
NPV > 0 — the investment adds value beyond the required rate of return. Accept.
NPV < 0 — the investment destroys value relative to the required rate. Reject.
NPV = 0 — the investment earns exactly the required rate. Indifferent.
NPV is measured in dollars, which makes it directly comparable across projects of any size. It is the academic and CFA-curriculum preferred metric whenever NPV and IRR disagree (more on that below).
Internal rate of return (IRR)
IRR is the discount rate that makes the NPV of an investment exactly zero. Think of it as the investment's own annualized rate of return, given its cash flows.
The decision rule mirrors NPV:
IRR > required rate of return — accept the investment.
IRR < required rate of return — reject.
IRR is useful for comparing investments with different cash-flow patterns on a return basis. But it carries a hidden assumption that exam questions probe: IRR assumes you can reinvest interim cash flows at the IRR itself, which is often unrealistic — especially when the IRR is very high. NPV's reinvestment assumption (at the required rate / discount rate) is more conservative and more realistic.
Worked example · should the client invest?
Scenario. A client is considering a $50,000 investment that returns $15,000 per year for 4 years. The client's required rate of return is 8%. Should they accept or reject?
1
Discount each year's cash flow to present value
Year 1: $15,000 ÷ 1.081 = $13,889
Year 2: $15,000 ÷ 1.082 = $12,860
Year 3: $15,000 ÷ 1.083 = $11,907
Year 4: $15,000 ÷ 1.084 = $11,025
2
Sum the present values
$13,889 + $12,860 + $11,907 + $11,025 = $49,681
3
Subtract the initial investment
$49,681 − $50,000 = NPV = −$319
Decision
NPV is negative, so the project's return is below the client's 8% required rate. Reject. Note this doesn't mean the client would "lose $319" — the cash flows are positive; they just don't compensate for the time value of capital at the required rate.
TVM calculator
How much will today's money grow to? Set the inputs below.
Future value
$18,194
$10,000 grows to $18,194 over 10 years · 1.82× multiplier
FV = $10,000 × (1 + 0.06/12)^(12×10)
NPV vs. IRR — side by side
NPV
Preferred
Measures
Dollar value added at the required rate
Decision rule
Accept if NPV > 0
Reinvestment assumption
Cash flows reinvested at the required rate (realistic)
Mutually exclusive projects
Reliable — always pick the higher NPV
IRR
Measures
Annualized rate of return implied by the cash flows
Decision rule
Accept if IRR > required rate
Reinvestment assumption
Cash flows reinvested at the IRR itself (often unrealistic)
Mutually exclusive projects
Can give wrong ranking when project sizes differ
Exam: when NPV and IRR disagree on which of two mutually exclusive projects to accept, the answer is NPV. The phrase "theoretically superior" or "academically preferred" in a question is your signal to pick NPV.
The Series 66 won't ask you to calculate complex TVM problems by hand. Instead, expect conceptual questions: "If interest rates rise, what happens to the present value of a future cash flow?" (It decreases.) Focus on the relationships, not the arithmetic.
Concept Check
An investment has a net present value (NPV) of −$5,000. This means:
A negative NPV means the present value of expected cash flows is less than the cost of the investment when discounted at the required rate — so the investment does not meet the investor's hurdle. It does NOT mean a $5,000 nominal loss. The future cash flows may still be positive in absolute dollars; they're simply not large enough to compensate for the time value of capital at the required rate. Decision rule: NPV > 0 accept, NPV < 0 reject. This negative-NPV-as-nominal-loss interpretation is one of the most common Series 66 traps.
Concept Check
An investment has an IRR of 11% and the client's required rate of return is 9%. The adviser should:
When IRR exceeds the required rate of return (the hurdle rate), the investment is acceptable because it earns a return above what the investor demands. This is mathematically equivalent to having a positive NPV at the required rate. High IRR does not flag excess risk — risk is captured in the required rate itself, which the adviser sets to reflect the investment's risk profile. Both NPV > 0 and IRR > required rate point the same direction for a single project; they only conflict when ranking mutually exclusive projects of different sizes.
Concept Check
An adviser is comparing two mutually exclusive projects. Project A has an NPV of $50,000 and an IRR of 22%. Project B has an NPV of $80,000 and an IRR of 18%. Both clear the client's hurdle rate. Which project should the adviser recommend?
When NPV and IRR rankings conflict on mutually exclusive projects, NPV is the correct decision rule. NPV measures the actual dollar value added to the client; IRR can mislead because of its unrealistic reinvestment assumption (that interim cash flows compound at the IRR itself). Project B adds $30,000 more in present-value terms to the client's wealth. The exam phrase 'theoretically superior' or 'academically preferred' is your signal to choose the NPV-based answer. Splitting between mutually exclusive projects is a non-option — they're mutually exclusive by definition.
Concept Check
A client is offered an investment requiring $100,000 today, with projected returns of $30,000 per year for 4 years. The client's required rate of return is 10%. Without calculating exact NPV, the adviser should:
The shortcut here is the Rule of 72: at 10%, money roughly doubles every 7.2 years. Four years of $30,000 cash flows discounted at 10% sum to about $95,096 — below the $100,000 cost. NPV is approximately −$4,904, so reject. The trap is treating $120,000 in nominal future cash flows as equivalent to $100,000 today: that ignores the time value of money entirely. Total nominal flows must exceed the initial investment by enough to also cover the required rate of return.
Section 5 of 5~7 min · 3 concept checks
Real-world TVM
The Rule of 72
A quick mental-math shortcut. To estimate the years it takes for an investment to double, divide 72 by the annual rate of return.
Years to double ≈ 72 ÷ rate
Years to double
12.0
Common values to memorize: at 4% → 18 years · at 6% → 12 years · at 8% → 9 years · at 12% → 6 years. The rule is approximate — it gets less accurate at very high or very low rates — but it's exact enough for Series 66 questions.
Real vs. nominal returns
Every return quote is one of two flavors:
Nominal return — the headline rate, not adjusted for inflation. "Your portfolio returned 7% last year" is a nominal figure.
Real return — the nominal return adjusted for inflation. It tells you what your purchasing power actually did.
The Fisher relationship ties the two together. The exam-friendly approximation:
Real return ≈ Nominal return − Inflation rate
So a portfolio returning 7% nominally during a year of 3% inflation delivered about 4% in real purchasing-power terms. The exact Fisher equation — (1 + nominal) = (1 + real) × (1 + inflation) — is mathematically tighter, but the subtraction shortcut is what Series 66 questions use.
The takeaway: a high nominal return in a high-inflation environment can be a real loss. Retirement-planning conversations should always run on real returns.
Inflation & purchasing power
The same PV math that prices a future cash flow tells you what your purchasing power will look like in retirement. At 3% annual inflation, $1 today buys what $0.55 will buy in 20 years:
$1 ÷ (1.03)20 = $0.55
The intuition flips for the client's spending: a $60,000 lifestyle today costs roughly $108,367 in nominal dollars 20 years from now — that's $60,000 × (1.03)20. Building a retirement plan in today's dollars and ignoring inflation is the most common amateur mistake.
This connects directly to the retirement-planning chapter in Module 3, where TVM math becomes the engine of every projection.
Annuities & perpetuities — a quick primer
Two specialized TVM patterns come up repeatedly elsewhere in the course. Recognize them so the math doesn't feel like a surprise:
Annuity — a stream of equal payments for a finite period. A pension that pays $40,000 per year for 25 years is an annuity. The PV of an annuity is the sum of the PVs of each payment; that's the engine behind retirement-income calculations.
Perpetuity — a stream of equal payments that goes on forever. Preferred stock is the classic example: a fixed dividend with no maturity. Its present value collapses to a simple formula:
PV = Annual payment ÷ Required return
A perpetuity paying $5 a year, valued at an 8% required return, is worth $5 ÷ 0.08 = $62.50. This formula resurfaces in Module 2's equity-valuation chapter when we price preferred stock.
Concept Check
A client invests $100,000 at 6% annual return. Using the Rule of 72, approximately how long will it take to double the investment to $200,000?
The Rule of 72 estimates doubling time as 72 divided by the annual rate of return: 72 / 6 = 12 years. The rule is an approximation — the exact doubling time at 6% compounded annually is 11.9 years — but it's accurate enough for Series 66 questions. The reverse formula works too: a client who wants to double their money in 9 years needs roughly 72 / 9 = 8% per year. The rule applies only to doubling; tripling has its own approximation (about 114) but it isn't tested.
Concept Check
A client's portfolio returned 8% nominally during a year in which inflation was 3.5%. The client's approximate real return for the year was:
The exam-friendly Fisher approximation: real return ≈ nominal return − inflation. So 8.0% − 3.5% = 4.5%. The real return is what the client's portfolio earned in purchasing-power terms — what their dollars can actually buy. The exact Fisher formula, (1 + nominal) = (1 + real)(1 + inflation), gives a slightly tighter 4.35%, but the subtraction shortcut is the answer expected on the Series 66. Adding inflation to nominal (11.5%) reverses the relationship; choosing 8.0% ignores inflation entirely.
Concept Check
A client wants to live on the equivalent of $80,000 (today's dollars) annually when they retire in 20 years. If inflation averages 3% per year, approximately how much in nominal dollars will the client need in the first year of retirement to maintain that purchasing power?
Inflation grows the nominal cost of a fixed lifestyle: $80,000 × (1.03)^20 = $80,000 × 1.806 = $144,489. The client needs roughly $144,489 in year-one-of-retirement dollars to buy what $80,000 buys today. Ignoring inflation ($80,000) is the most common amateur retirement-planning error. The $104,000 distractor adds simple 3% × 20 = 60% growth — incorrect because inflation compounds. $240,000 triples the original — directionally right but excessive at 3% inflation. The compound calculation is the only correct approach.
SummaryCram aid & consolidated traps
Chapter summary
Exam essentials · everything in this chapter, distilled
The core relationship
Rate ↑ ⇒ PV ↓. Rate ↓ ⇒ PV ↑.
FV formula
FV = PV × (1+r)n
PV formula
PV = FV ÷ (1+r)n
NPV rule
NPV > 0 → accept. NPV < 0 → reject.
IRR rule
IRR > required rate → accept.
Rule of 72
Years to double ≈ 72 ÷ rate
Real return
≈ Nominal − Inflation
Perpetuity PV
Annual payment ÷ required return
Common traps to expect on the exam
"NPV of −$5,000 means you lose $5,000." No — it means the project's return falls short of the required rate by $5,000 worth of present value. The cash flows can still be positive in absolute terms.
IRR rankings flip when project sizes differ. When forced to pick between two mutually exclusive projects, the higher-IRR project may have lower NPV. NPV wins that tiebreaker.
"More frequent compounding always wins." True, but the marginal gain shrinks fast. Don't be tricked into picking the difference as a big number — annual vs. continuous on $10K over 10 years is just $313.
Nominal vs. real returns get mixed up. A 6% nominal return in a 4% inflation year is only a 2% real return. Long-horizon planning needs real-return inputs.
Rule of 72 is for doubling, not for any other multiple. Tripling has its own rule (~114), but you won't be tested on that — only doubling.