BJU · Engineering Economics · Lecture 3

The Time Value of Money

Why one dollar today is not one dollar tomorrow — and how engineers price the gap.

Instructor
Dr. Zhijiang Chen
Session
Friday · 2026-06-05
Room
SY 109
Duration
110 minutes
BJU · ENGINEERING ECONOMICSLecture 3 — Time Value of Money

Learning objectives

What you will be able to do by 16:30 today

  1. Translate a verbal cash-flow problem into the canonical (P, F, A, i, n) notation and a cash-flow diagram.
  2. Compute future and present values under simple, discrete-compound, and continuously-compounded interest.
  3. Convert fluently between nominal annual rate, periodic rate, and Effective Annual Rate (EAR) for any compounding frequency.
  4. Justify which compounding model is appropriate for a given financial instrument — and recognise when the choice changes the answer materially.
Park, Contemporary Engineering Economics, 6e — ch. 32
BJU · ENGINEERING ECONOMICSLecture 3 — Time Value of Money

Agenda · 110 minutes

How we will spend the session

I.
Motivation — would you rather have $1,000 today or $1,100 next year?
10 min
II.
The cash-flow language: P, F, A, i, n and the diagram
10 min
III.
Simple interest vs. compound interest
15 min
IV.
Single-payment factors: F/P and P/F — with worked examples
20 min
—
Stretch break
10 min
V.
Nominal vs. periodic vs. effective annual rate (EAR)
15 min
VI.
Continuous compounding — the ert limit
10 min
VII.
In-class discussion — credit cards, payday loans, & APR vs APY
15 min
VIII.
Synthesis, pitfalls, and a preview of Lecture 4
5 min
Dr. Zhijiang Chen — Engineering Economics 2026 Spring3
BJU · ENGINEERING ECONOMICSPart I — Motivation
i.

Why a dollar moves through time.

Three forces — earning power, inflation, and risk — make time itself a price.

Dr. Zhijiang Chen — Engineering Economics 2026 Spring4
BJU · ENGINEERING ECONOMICSPart I — Motivation

Opening question

Would you rather have $1,000 today, or $1,100 next year?

Show of hands. Before anyone justifies their answer with a number, notice that the question itself is incomplete — it cannot be answered without three pieces of context:

  • Earning power. What else could $1,000 do for you in twelve months?
  • Purchasing power. Will $1,100 next year buy what $1,000 buys today?
  • Risk. How confident are you the $1,100 will actually arrive?

Engineering economics is the discipline of pricing those three forces explicitly so we can compare cash that does not arrive at the same time.

Dr. Zhijiang Chen — Engineering Economics 2026 Spring5
BJU · ENGINEERING ECONOMICSPart I — Motivation

Decomposition

The interest rate, dissected

The single number i we use throughout this course is in fact a bundle:

$$i \;\approx\; \underbrace{r_f}_{\text{real risk-free}} \;+\; \underbrace{\pi}_{\text{inflation premium}} \;+\; \underbrace{\rho}_{\text{risk premium}}$$
This is the Fisher decomposition in its additive approximation; we will tighten it in Lecture 11. For now, treat i as the price the market charges to move one dollar one year forward.

Two lenses on the same rate

  • Lender's lens. i is the rent collected for parting with capital.
  • Borrower's lens. i is the price paid for spending money you have not yet earned.

An engineering project's lens

  • i is the opportunity cost — the return forgone by not deploying the same capital in the next-best alternative.
  • That alternative is often called the Minimum Attractive Rate of Return (MARR), to which we return in Lecture 5.
Park, ch. 3 §3.1 · Newnan, ch. 4 §4.26
BJU · ENGINEERING ECONOMICSPart II — Notation
ii.

A vocabulary engineers can compute with.

P, F, A, i, n — five letters that carry the rest of the course.

Dr. Zhijiang Chen — Engineering Economics 2026 Spring7
BJU · ENGINEERING ECONOMICSPart II — Notation

The five symbols

Cash-flow notation and the cash-flow diagram

SymbolMeaningUnit
PPresent worth — a single sum located now, at n = 0$ at time 0
FFuture worth — a single sum located at the end of period n$ at time n
AAnnuity — a uniform amount occurring at the end of each period$ per period
iInterest rate per perioddecimal / period
nNumber of interest periodsperiods

Conventions: end-of-period cash flow, arrows up for receipts, arrows down for disbursements, time on the horizontal axis.

0 1 2 3 … n P F n periods, interest rate i per period

Single-payment diagram: invest P today, receive F after n periods.

Dr. Zhijiang Chen — Engineering Economics 2026 Spring8
BJU · ENGINEERING ECONOMICSPart III — Simple vs. compound
iii.

Two ways interest accrues.

One is a straight line. The other is a curve — and almost everything in modern finance is the curve.

Dr. Zhijiang Chen — Engineering Economics 2026 Spring9
BJU · ENGINEERING ECONOMICSPart III — Simple vs. compound

Definition

Simple interest — the linear case

Interest is charged only on the original principal. Each period adds the same dollar amount; nothing is ever charged on previously accrued interest.

$$ I \;=\; P \cdot i \cdot n \qquad\text{and}\qquad F \;=\; P\,(1 + i \cdot n) $$
I is total interest accrued, P is principal, i is the periodic rate, n is the number of periods.

Where you'll actually see it

  • Short-term Treasury bills quoted on a "discount" basis.
  • Some commercial paper and money-market quotations.
  • Statutory interest on overdue invoices in many jurisdictions.
  • Convenience approximations (back-of-envelope checks).
Park, ch. 3 §3.2.110
BJU · ENGINEERING ECONOMICSPart III — Simple vs. compound

Definition

Compound interest — the exponential case

Each period, interest is computed on the current balance — principal plus all previously accrued interest. Interest earns interest.

$$ F \;=\; P\,(1 + i)^{n} $$
The factor $(1+i)^{n}$ is the single-payment compound-amount factor, written $(F/P,\,i,\,n)$ in the textbook tables.

The same $1,000 at 6 %, compounded annually

YearOpening balanceInterestClosing balance
11,000.0060.001,060.00
21,060.0063.601,123.60
31,123.6067.421,191.02
41,191.0271.461,262.48
51,262.4875.751,338.23

After five years the compound balance exceeds the simple balance by $38.23 — a 12.7 % bigger pile of interest from a single keyword change.

Park, ch. 3 §3.2.211
BJU · ENGINEERING ECONOMICSPart III — Simple vs. compound

Side by side

The growth curves diverge — and the gap never closes

0 10 20 30 40 50 years (n) $1k $5k $10k $15k $20k balance F simple compound at year 40, $10.3k vs $3.4k

$1,000 invested at 6 % per year, simple vs. compound, over 50 years.

What the picture tells us

  • For small n, the two curves are nearly indistinguishable. Many short-horizon decisions are insensitive to the choice.
  • For large n, the gap explodes — this is the famed exponential behaviour Einstein never actually called the eighth wonder of the world (but the quote sticks because the math is correct).
  • The doubling time under compounding is approximated by the Rule of 72: $n \approx 72/i\%$. At 6 %, money doubles roughly every 12 years.

From this point in the course, every formula assumes compound interest unless stated otherwise.

Dr. Zhijiang Chen — Engineering Economics 2026 Spring12
BJU · ENGINEERING ECONOMICSPart IV — Single-payment factors
iv.

Moving one dollar through time, in either direction.

The compound-amount factor pushes a sum forward; its reciprocal — the present-worth factor — pulls a sum back.

Dr. Zhijiang Chen — Engineering Economics 2026 Spring13
BJU · ENGINEERING ECONOMICSPart IV — Single-payment factors

The F / P factor

Future Value of a single present sum

$$ F \;=\; P\,(1 + i)^{n} \;=\; P \cdot (F/P,\, i,\, n) $$
Read the factor aloud as "F given P, at rate i, over n periods."

Reading the textbook table

Tables in the back of Park give $(F/P, i, n)$ for standard rates. To find $F$ for $P = \$1{,}000$, $i = 6\%$, $n = 5$:

  1. Open the 6 % table.
  2. Find the row $n = 5$.
  3. Read the F/P column: 1.3382.
  4. Multiply: $F = 1{,}000 \cdot 1.3382 = \$1{,}338.20$.

In practice we just type =FV(0.06,5,0,-1000) into Excel or call (1.06)**5 in Python — but on the midterm you'll see tables, so know the layout.

Park, ch. 3 §3.3.1 · Newnan, ch. 4 §4.514
BJU · ENGINEERING ECONOMICSPart IV — Single-payment factors

The P / F factor

Present Value of a single future sum

$$ P \;=\; \dfrac{F}{(1 + i)^{n}} \;=\; F \cdot (P/F,\, i,\, n) $$
$(P/F, i, n) = (1+i)^{-n}$ is called the discount factor; it always lies in $(0, 1]$ for $i \ge 0$.

Discounting answers the inverse question: "What single sum today would I need, set aside at rate i, to grow to F by year n?"

Every NPV calculation in the rest of this course is built from $(P/F, i, n)$ applied to each year's cash flow.

Behaviour as i and n increase

  • For fixed F and n, higher i ⇒ lower P. Future cash is cheaper to promise.
  • For fixed F and i, larger n ⇒ lower P. Distant cash is cheaper still.
Park, ch. 3 §3.3.215
BJU · ENGINEERING ECONOMICSPause
—

Ten-minute stretch.

When we come back: compounding more often than once a year, and what "12.99 % APR" really costs you.

Dr. Zhijiang Chen — Engineering Economics 2026 Spring16
BJU · ENGINEERING ECONOMICSPart V — Nominal vs. effective
v.

The rate the bank quotes is not the rate you pay.

Compounding frequency turns a single nominal rate into a family of periodic and effective rates.

Dr. Zhijiang Chen — Engineering Economics 2026 Spring17
BJU · ENGINEERING ECONOMICSPart V — Nominal vs. effective

Compounding more than once a year

Three rates that live on the same loan

When interest compounds m times per year, the rate you quote, the rate the bank applies, and the rate you actually pay are three different numbers.

A loan disclosed as 12 % APR compounded monthly is in fact a 12.683 % EAR. The difference is real money.

Park, ch. 3 §3.4 · TILA Reg-Z §1026.1418
BJU · ENGINEERING ECONOMICSPart V — Nominal vs. effective

Derivation

Where the EAR formula comes from

Start with $\$1$ invested for one year at nominal rate r, compounded m times per year. Each period multiplies the balance by $\left(1 + r/m\right)$:

$$ F_{1\text{ year}} \;=\; 1 \cdot \left(1 + \tfrac{r}{m}\right)^{m} $$

The effective annual rate $i_a$ is, by definition, the rate that produces the same one-year balance compounded only once:

$$ 1 + i_a \;=\; \left(1 + \tfrac{r}{m}\right)^{m} $$
Subtract 1 from both sides to recover the formula on the previous slide. The EAR is what you should quote when comparing two loans whose compounding frequencies differ.
Dr. Zhijiang Chen — Engineering Economics 2026 Spring19
BJU · ENGINEERING ECONOMICSPart V — Nominal vs. effective

Worked comparison

Two mortgage offers, side by side

BankNominal APR rCompounding mPeriodic iEAR ia
Alpha5.95 %Monthly (12)0.4958 %/mo6.116 %
Beta 6.00 %Semi-annual (2)3.000 %/half-yr6.090 %
Gamma5.92 %Daily (365)0.01622 %/day6.097 %

Despite Alpha having the lowest nominal rate, Beta is the cheapest offering when compounded properly. Compounding frequency reshuffles the ranking.

A closed-form to keep handy

For any nominal $r$ and compounding $m$:

$$ i_a = (1 + r/m)^{m} - 1 $$

For $r = 5.95\%, m = 12$: $\;(1 + 0.0595/12)^{12} - 1 = 0.06116$.

Dr. Zhijiang Chen — Engineering Economics 2026 Spring20
BJU · ENGINEERING ECONOMICSPart VI — Continuous compounding
vi.

What happens when we compound every instant.

The limit as $m \to \infty$ is finite, elegant, and closes the chapter.

Dr. Zhijiang Chen — Engineering Economics 2026 Spring21
BJU · ENGINEERING ECONOMICSPart VI — Continuous compounding

The continuous limit

From discrete (1+r/m)m to er

Take the EAR formula and let the number of compounding periods per year grow without bound. From calculus:

$$ \lim_{m \to \infty} \left(1 + \tfrac{r}{m}\right)^{m} \;=\; e^{r} $$
This is one of the definitions of Euler's number, and the gateway from discrete to continuous time finance.

So under continuous compounding at nominal rate r:

$$ F \;=\; P\,e^{r n} \qquad P \;=\; F\,e^{-r n} \qquad i_a \;=\; e^{r} - 1 $$
Time n is now measured in years (matching the units of r) and need not be an integer.
Park, ch. 3 §3.5 · Hull, Options, Futures & Other Derivatives, ch. 422
BJU · ENGINEERING ECONOMICSPart VI — Continuous compounding

Comparison and worked example

How much does compounding frequency really cost?

Setup. Nominal rate $r = 12\%$ per year, principal $P = \$10{,}000$, horizon $n = 1$ year.

CompoundingmPeriodic rateEAR iaF after 1 yr
Annual 1 12.00000 %12.0000 %11,200.00
Semi-annual 2 6.00000 % 12.3600 %11,236.00
Quarterly 4 3.00000 % 12.5509 %11,255.09
Monthly 12 1.00000 % 12.6825 %11,268.25
Weekly 52 0.23077 % 12.7341 %11,273.41
Daily 365 0.03288 % 12.7475 %11,274.75
Continuous∞—12.7497 %11,274.97

The asymptote between daily and continuous is only $\$0.22$ on $\$10{,}000$ — negligible in practice, conceptually fundamental.

Dr. Zhijiang Chen — Engineering Economics 2026 Spring23
BJU · ENGINEERING ECONOMICSPart VI — Continuous compounding

Diagnostic

Five mistakes that account for most of the homework points lost on this material

  1. Using the nominal rate where the periodic rate belongs.

    If you compound monthly, you must use $r/12$, not $r$, as the per-period rate. Plugging $r$ directly into $(1+i)^n$ with $n$ in months over-charges interest by a factor that grows with n.

  2. Mismatching i's period and n's period.

    If n is in months, i must be a monthly rate. The single most common silent error.

  3. Comparing two offers by APR alone.

    Always convert to EAR first when compounding frequencies differ.

  4. Forgetting that simple-interest formulas exist.

    A homework problem that explicitly says "simple interest" is not a typo; do not silently switch to compound.

  5. Reporting too many decimal places.

    Cash answers belong to the cent ($\pm 0.005$); rate answers to the basis point ($\pm 0.0001$). More digits implies precision you do not have.

Dr. Zhijiang Chen — Engineering Economics 2026 Spring24
BJU · ENGINEERING ECONOMICSPart VII — In-class discussion
In-class discussion · 15 minutes

Why does my credit-card statement say "19.99 % APR" when I'm actually paying more than 22 %?

"Find your most recent credit-card statement — or pull up one online. Locate the disclosed APR, the cycle length, and any 'cash advance' rate. Convert the APR to an EAR, and decide: is the lender being honest, legal, or both?"

Dr. Zhijiang Chen — Engineering Economics 2026 Spring25
BJU · ENGINEERING ECONOMICSPart VIII — Synthesis

Where we have been

One picture, four formulas, one habit

The four formulas to memorise

SettingForward (F)Backward (P)
Simple interest$F = P(1 + in)$$P = F/(1 + in)$
Discrete compound$F = P(1 + i)^{n}$$P = F\,(1+i)^{-n}$
Continuous compound$F = P\,e^{rn}$$P = F\,e^{-rn}$
EAR conversion$i_a = (1 + r/m)^{m} - 1$ · or · $i_a = e^{r} - 1$
Today's content compresses to: money has a price that depends on time, that price compounds, and a quote is not a cost until you convert to an effective rate.
Dr. Zhijiang Chen — Engineering Economics 2026 Spring26
BJU · ENGINEERING ECONOMICSAssignment

Problem set 3 — due Wednesday 2026-06-10, 23:59

Practice and apply

Required

  1. Park 3.7, 3.12, 3.19, 3.24 — straight drill on the four formulas.
  2. Statement exercise. Take one bill or statement from your own life (utilities, tuition, phone plan, subscription) and decompose any financing terms it contains into APR, periodic rate, and EAR. One paragraph of commentary on whether the disclosure is informative.
  3. Modelling exercise. Compare two student-loan offers (data on Canvas): which has the lower EAR, and by how much over a 10-year horizon?

Optional · extension

  • Park 3.41 — continuous-compounding mortgage problem.
  • Read Hull §4.1–4.3 for a derivatives perspective on continuous compounding.
Dr. Zhijiang Chen — Engineering Economics 2026 Spring27
BJU · ENGINEERING ECONOMICSComing next

Lecture 4 · Saturday 2026-06-06 · SY 109

Cash Flow Analysis & Equivalence

Today we moved a single sum through time. Next session we move streams — uniform annuities, arithmetic gradients, geometric gradients — and prove that any cash-flow pattern can be reduced to an equivalent single sum.

Bring

  • Today's cash-flow notation, fluent.
  • A financial calculator or laptop with Excel / Sheets / Python.
  • The first three problems of Park ch. 4 attempted (not necessarily solved).
Dr. Zhijiang Chen — Engineering Economics 2026 Spring28
BJU · ENGINEERING ECONOMICSReferences

For further reading

Sources and recommended texts

Primary text

  • Park, C. S. Contemporary Engineering Economics, 6th ed. Pearson, 2019. Chapter 3.
  • Newnan, D. G.; Eschenbach, T. G.; Lavelle, J. P. Engineering Economic Analysis, 14th ed. Oxford, 2020. Chapter 4.

Quant-finance complement

  • Hull, J. C. Options, Futures & Other Derivatives, 11th ed. Pearson, 2021. Chapter 4 §4.1–4.3 on continuous compounding.

Regulatory background

  • U.S. CFPB. Truth in Lending Act (Regulation Z), 12 CFR §1026 — APR disclosure requirements.
Dr. Zhijiang Chen — Engineering Economics 2026 Spring29