Why $1 today is not the same as $1 tomorrow — the single most important quantitative idea in this course.
Instructor
Dr. Zhijiang Chen
Session
No. 03 of 16
Date
5 June 2026
Room
SY109
Duration
110 minutes
Format
Lecture + Computation
Bring a calculator (or your phone). Today every slide has a number on it.
Lecture IIIAgenda02 / 24
Today's Plan
Four formulas. One question.
Why is $1 today worth more than $1 in five years — and how much more, exactly?
§
Topic
Minutes
I.
Why money has time value
15
II.
Future value & present value (single sum)
25
III.
Simple vs compound interest
15
IV.
Nominal, effective & continuous rates
20
—
Discussion: what's your personal discount rate?
10
V.
Worked problems — software examples
20
HW3 brief, questions
5
Part — One
I
Why money has time value — and why this matters for every software decision.
"A bird in the hand is worth two in the bush." — Aesop, ~600 BC.
§ IWhy04 / 24
Three reasons a future dollar is worth less
Three forces, one conclusion.
Force
Why it makes future money worth less today
Opportunity
Money today can be invested and grow. $1 invested at 8% becomes $1.47 in 5 years.
Risk
Future money is uncertain. The promised dollar in 5 years may not arrive at all.
Inflation
The future dollar buys less. 3% inflation cuts purchasing power by ~14% over 5 years.
All three sum into a single rate — the discount rate — that we will use to translate any future cash flow into today's terms.
In software, the same logic applies to engineering effort: an hour today is more valuable than an hour next quarter, because it can be invested into work that compounds.
Part — Two
II
Future value & present value — the two core formulas of engineering economics.
§ IIFuture value06 / 24
Single sum, compound interest
Future Value (FV).
FV = PV × (1 + i)n
Where:
PV = present value (today's dollar amount) ·
i = interest rate per period ·
n = number of periods.
Example: $100 today, invested at 8% annually, for 5 years:
FV = 100 × (1.08)5 = 100 × 1.4693 = $146.93
Future value is what you have at the end. Compounding is the engine: each year's interest itself earns interest.
§ IIPresent value07 / 24
The reverse direction
Present Value (PV).
PV = FV ÷ (1 + i)n
Example: a vendor will pay you $100,000 in 5 years. At a 10% opportunity cost of capital, what is that worth to you today?
Translation: a guaranteed $100K in five years is economically equivalent to $62,092 today — at this firm's cost of capital.
PV is the lens that lets us compare cash flows at different points in time on a common ruler. It is the foundation of NPV, which arrives in Lecture 5.
§ IISoftware example08 / 24
A real software-decision example
License now, or license later?
A vendor offers two pricing terms for the same software:
Option
Pay now
Pay in 3 years
PV @ 8%
A. Upfront
$50,000
—
$50,000
B. Deferred
—
$60,000
$47,629
B has the higher nominal price but the lower present value. Provided you're confident in the firm's solvency three years out, B is the better deal — at 8%.
Notice how much the discount rate matters. At 4%, B's PV becomes $53,340 and A wins instead. The discount rate is a policy choice — and decisions hinge on it.
Part — Three
III
Simple vs Compound — and why nobody uses simple any more.
§ IIISimple vs Compound10 / 24
A small choice, a large difference
Two ways to earn interest.
Simple interest
FV = PV × (1 + i × n)
Interest is only on the original principal. The principal stays the same; interest is constant per period.
Example: $1,000 at 8%, 20 years → $2,600.
Compound interest
FV = PV × (1 + i)n
Interest itself earns interest. The base grows each period.
Example: $1,000 at 8%, 20 years → $4,661.
Almost every real financial product uses compounding. Engineering-economics formulas assume compounding unless stated otherwise.
Part — Four
IV
Nominal · Effective · Continuous — three names for the same idea, calibrated differently.
§ IVRate conventions12 / 24
Read the quote carefully
The 18% APR that wasn't.
Convention
Formula
At 18% nominal, 12 periods
Nominal annual rate (r)
quoted "headline" rate, ignores compounding frequency
18.00%
Effective annual rate (i)
(1 + r/m)m − 1
(1+0.015)12−1 = 19.56%
Continuous compounding
er − 1
e0.18−1 = 19.72%
The same "18%" hides three different actual rates — credit cards, mortgages, and bonds use different conventions and the differences compound.
When reading a contract or financial model, always identify which rate convention is being used. The wrong assumption silently inflates or deflates every NPV calculation that follows.
§ IVContinuous compounding13 / 24
When periods → 0
Continuous compounding.
FV = PV × er·n · PV = FV × e−r·n
Used widely in derivatives pricing, treasury bond math, and academic finance. For engineering economics decisions we usually stay with annual or monthly compounding — but you should be able to translate.
Example: $1 at 10% continuously compounded for 1 year:
FV = 1 × e0.10 = $1.1052 (vs $1.10 simple annual)
The "spread" between annual and continuous compounding feels small per period but matters for long horizons and high rates. We will revisit when we discuss risk and option-style decisions.
Discussion10 minutes14 / 24
Introspection in pairs
%
What is your personal discount rate?
I will offer you $100 today, or $X in one year. What value of X just barely tips you to wait?
Solve for your implied annual discount rate: r = X/100 − 1.
Now repeat for $1,000 today vs $X in one year. Does your rate change? Why?
Compare with your partner. Are you more or less patient than the firm's discount rate (typically 8–15%)?
Patient people are economically valuable. So are impatient ones — they tend to act faster on opportunity.
Part — Five
V
Worked problems — software-decision examples using PV and FV.
§ VExample 116 / 24
Discounting a TCO worksheet
Yesterday's worksheet — discounted.
A team estimates the following annual TCO cash outflows for an internal AI tool. Find the PV of total cost at a 10% discount rate.
Year
Cash outflow
Discount factor
PV
0 (build)
$80,000
1.0000
$80,000
1
$30,000
0.9091
$27,273
2
$30,000
0.8264
$24,793
3
$30,000
0.7513
$22,539
4
$30,000
0.6830
$20,490
Total PV
$200,000
—
$175,095
Without discounting, total cost is $200K. With it, the same stream is worth $175K — a 12% reduction. The cost arrives later, so it costs less.
§ VExample 217 / 24
Solving for unknown rate (IRR preview)
What rate makes them equal?
$50,000 today, or $80,000 in 5 years. At what discount rate are these two equivalent in PV?
This break-even discount rate is the precursor of the Internal Rate of Return (IRR) we will see in Lecture 5. It is the rate that makes alternatives equivalent.
If your firm's discount rate is below 9.86%, take the $80K in 5 years. If above, take $50K now. Knowing your discount rate is half the decision.
§ VExample 318 / 24
Compounded short periods
Quarterly billing vs annual billing.
A SaaS vendor offers $1,200 / year billed annually upfront, or $300 / quarter billed in advance. If your discount rate is 8% annual:
Plan
Cash flow (yr 1)
Effective PV
Annual
$1,200 at t=0
$1,200.00
Quarterly
$300 at t=0, ¼, ½, ¾
$1,165.93
Quarterly billing saves the customer ≈ $34 per year in PV terms. Modest, but the principle scales: any time you can defer a payment, you reduce its PV.
This is exactly why every SaaS vendor offers a 10–20% discount for annual upfront payment — they are pricing the customer's option to defer.
BridgeTo Lecture 419 / 24
Where this is going
Tomorrow we string cash flows together.
Single sums are the unit. But software projects produce streams — uniform, growing, irregular. Tomorrow we learn the closed-form formulas that make these streams tractable.
Uniform series (equal payments) — the P / A factor.
Bring tomorrow: today's PV / FV intuition and your TCO worksheet. We will combine them.
HomeworkDue Lecture 520 / 24
Homework 03 — due Sunday 7 June, before class
Make the formulas yours.
Six PV / FV exercises (see course site). Show the formula and the substitution; not just the final answer.
Convert a quoted 24% APR (monthly compounding) into the effective annual rate. Convert it again into continuous-compounding equivalent.
Take your TCO worksheet from Lecture 2 and apply a 10% discount rate. Compute the PV of total cost. Compare with the undiscounted total. One paragraph: what changed about your reading of the project?
ReadingFurther21 / 24
If you want to go deeper
To read this week.
Park, Contemporary Engineering Economics, chapter on time value of money.