Lecture Four · 6 June 2026

04Lecture Four

Cash Flow
& Equivalence

Drawing money in time — and proving when two very different streams are, in fact, worth the same.

Instructor

Dr. Zhijiang Chen

Session

No. 04 of 16

Date

6 June 2026

Room

SY109

Time

16:20 – 18:10

Duration

110 minutes

Builds on Lecture 3 (PV / FV) · sets up NPV / IRR in Lecture 5.

Lecture IVAgenda 02 / 31

Today's Plan

Six movements — four shapes — one principle.

By the end of today you should be able to draw any project's cash flow, name its pattern, price it, and prove its equivalence to any other.

§	Topic	Minutes
I.	Cash flow diagrams — a visual language for money in time	15
II.	The uniform series — annuities, mortgages, SaaS subscriptions	20
III.	Arithmetic gradient — linear growth and decline	15
IV.	Geometric gradient — exponential growth, inflation, SaaS ramps	15
V.	Equivalence — when two different streams are the same thing	15
VI.	Composition — decomposing real software-project cash flows	20
	Discussion, homework, questions	10

Part OneCash flow diagrams 03

Part — One

Before you compute a cash flow,
you should be able to draw it.

"The cash flow diagram is to the financial engineer what the free-body diagram is to the mechanical one."
— after Park, Contemporary Engineering Economics

§ I · DiagramsLecture Four

§ IWhat & why 04

Working definition

A cash flow diagram is a picture of money in time.

A cash flow diagram (CFD) plots every receipt and every payment of a project on a single time axis, distinguishing each by sign, magnitude, and timing.

It is the bridge between a narrative ("we'll pay $100K up front, then receive $40K a year for four years") and a computation (PV, FV, NPV, IRR).

A CFD does not solve the problem. It makes the problem solvable by forcing every ambiguity onto the page.

It records, for each cash flow:

When it occurs — end of which period.
How much — in absolute terms.
Direction — into or out of the project.
Whose perspective — buyer, seller, lender, borrower.

Change perspective, and every arrow flips.

§ I · Diagrams04 / 31

§ IConventions 05

Four conventions to memorise

The grammar of the diagram.

Convention	Meaning	Why it matters
Horizontal axis	Time, in discrete periods 0, 1, 2, … , n. Period 0 is "now."	Forces you to choose a period (month vs. year); all rates must match it.
Arrows up	Cash into the decision-maker (revenue, savings, salvage).	Visual sign convention — green / positive.
Arrows down	Cash out (cost, payment, investment).	Visual sign convention — red / negative.
End-of-period rule	Unless stated otherwise, every cash flow occurs at the end of its period.	An "annual" cash flow at year 1 means month 12, not month 0.

Continuous and beginning-of-period conventions exist (annuity-due, continuous compounding) — we will name them when we need them.

§ I · Diagrams05 / 31

§ IReading a diagram 06

A first picture

A small software project, drawn in full.

$100K invested up-front; $40K of net cash flow returned each year for four years. From the investor's perspective.

Cash flow diagram · investor's perspective · annual periods

Naive total: −100 + 4·40 = +$60K — looks like a win.

At i = 10%: PV of inflows = $126.79K; PV of outflow = $100K; NPV ≈ +$26.79K.

§ I · Diagrams06 / 31

§ ICommon pitfalls 07

What goes wrong

Five mistakes that quietly invalidate a CFD.

Mixing perspectives mid-problem — half the arrows are the customer's, half the vendor's.
Mixing periods — monthly cash flow on a yearly rate. The factor will silently lie.
Including sunk costs — past payments belong in history, not in the diagram.

Ignoring opportunity cost — using your own time "for free" because no cash changes hands.
Mis-timing the first cash flow — an annuity-due at t = 0 looks the same on paper as an ordinary annuity at t = 1, but they differ by a factor of (1+i).

When in doubt: write the perspective and the period at the top of the diagram. Two extra words save a wrong answer.

§ I · Diagrams07 / 31

Part TwoUniform series 08

Part — Two

The first shape:
the same number,
every period.

Mortgages, leases, salaries, SaaS subscriptions — the world is full of uniform series.

§ II · Uniform seriesLecture Four

§ IIFour factors 09

A is the constant cash flow at each period 1 … n

Four directions, four factors.

Factor	Reads as	Formula	Used to
P / A	Present value of a uniform series.	$ P = A \cdot \dfrac{(1+i)^n - 1}{i\,(1+i)^n} $	Price a stream of equal future payments today.
F / A	Future value of a uniform series.	$ F = A \cdot \dfrac{(1+i)^n - 1}{i} $	Project a savings plan or sinking fund.
A / P	Annual equivalent of a present sum.	$ A = P \cdot \dfrac{i\,(1+i)^n}{(1+i)^n - 1} $	Convert a loan into level payments.
A / F	Annual equivalent of a future sum.	$ A = F \cdot \dfrac{i}{(1+i)^n - 1} $	Set aside equal amounts to hit a target.

P/A and A/P are reciprocals; F/A and A/F are reciprocals. Memorise one of each pair.

§ II · Uniform series09 / 31

§ IIWhere it comes from 10

A 60-second derivation

P/A is just a geometric series
in a smart disguise.

Discount each of the n equal payments of A back to today:

Step 1 — sum of discounted payments

$ P = \displaystyle\sum_{t=1}^{n} \dfrac{A}{(1+i)^t} $

Step 2 — factor and sum the geometric series

$ P = A \cdot \dfrac{1 - (1+i)^{-n}}{i} = A \cdot \dfrac{(1+i)^n - 1}{i\,(1+i)^n} $

Why this matters

The formula is not arbitrary — it's the closed form of n single-sum PVs.
As n → ∞, the factor collapses to 1/i — the price of a perpetuity.
As i → 0, the factor collapses to n — undiscounted total.

Both limits are useful sanity checks on any spreadsheet answer.

§ II · Uniform series10 / 31

§ IIWorked example · P/A 11

Pricing a stream — P/A

What is a $1,500/month rental promise worth today?

A landlord receives $1,500 at the end of each month for the next five years (60 months). The opportunity cost of capital is 9% per year, compounded monthly — so i = 0.75% per month.

Apply P/A

$ P = 1500 \cdot \dfrac{(1.0075)^{60} - 1}{0.0075 \cdot (1.0075)^{60}} $

$ P \approx 1500 \cdot 48.173 \approx \$72{,}260 $

What the answer tells you

The undiscounted total is $90,000 (60 × $1,500) — the time-value gap is $17.7K, about 20%.
If a buyer offers $70K cash today to take over the lease, the landlord should refuse — the stream is worth more.
At i = 0 (no time-value), the same stream is worth $90K; the gap is entirely the cost of waiting.

Always check that i and the period of A are in the same unit. Mixing kills the answer.

§ II · Uniform series11 / 31

§ IIWorked example · A/P 12

Sizing a payment — A/P

Lump-sum server vs. monthly cloud bill.

A team can buy a server outright for $60,000, or pay a hyperscaler monthly for the same compute. At 9%/yr over a 3-year life, what monthly bill would make the two equivalent?

A/P at i = 0.75% per month, n = 36

$ A = 60{,}000 \cdot \dfrac{0.0075 \cdot (1.0075)^{36}}{(1.0075)^{36} - 1} $

$ A \approx 60{,}000 \cdot 0.03180 \approx \$1{,}908 \,/\, \text{month} $

How to read the result

Any cloud bill below $1,908/month makes leasing cheaper than owning, ignoring everything else (depreciation, maintenance, salvage).

A real comparison would add: power, ops, scaling head-room, optionality — we will layer those in Lecture 11 (Build / Buy / Reuse).

§ II · Uniform series12 / 31

Part ThreeArithmetic gradient 13

Part — Three

III

When the same number
becomes the same increment.

Maintenance costs that creep up by $5K every year are the canonical arithmetic gradient.

§ III · Arithmetic gradientLecture Four

§ IIIThe shape 14

A constant additive step

An arithmetic gradient adds G every period.

A pure arithmetic gradient · base 0 · step G

A pure arithmetic gradient starts at zero at period 1, grows by a constant G each period, and is rarely a stream you actually receive in isolation — almost always you see it added to a uniform series base A₁.

§ III · Arithmetic gradient14 / 31

§ IIIP/G and A/G 15

Two factors, two uses

Price the tilt — or convert it to an equivalent uniform.

P / G — present value of a pure arithmetic gradient

$ P_G = G \cdot \dfrac{(1+i)^n - i\,n - 1}{i^{2}\,(1+i)^n} $

A / G — equivalent uniform-series amount of a gradient

$ A_G = G \cdot \left[\,\dfrac{1}{i} - \dfrac{n}{(1+i)^n - 1}\,\right] $

"A flat series of A_G per period is worth the same as a gradient stepping by G."

For a stream that starts at A₁ at period 1 and grows by G per period, the PV is the sum of two pieces: a uniform series of A₁ (use P/A) plus a pure gradient of G (use P/G).

§ III · Arithmetic gradient15 / 31

§ IIIWorked example 16

Rising maintenance cost

Year 1: $20K. Then $5K more every year, for six years.

A legacy system costs $20K to maintain in year 1, and that cost grows by $5K per year through year 6. At i = 8%, what is the PV of the maintenance burden?

Decompose: uniform $20K base + pure gradient G = $5K

$ P = 20K \cdot (P/A, 8\%, 6) + 5K \cdot (P/G, 8\%, 6) $

$ P \approx 20K \cdot 4.6229 + 5K \cdot 10.523 $

$ P \approx \$92{,}460 + \$52{,}615 \approx \$145{,}080 $

Why this matters for software economics

Almost every long-lived system has a creeping maintenance gradient — the codebase ages, dependencies decay, the team rotates. Quoting only year-1 cost ($20K) understates the lifetime burden by more than seven times.

A 60% increase in year-1 cost over six years is roughly an 8%/yr arithmetic gradient on $20K base — not unusual in practice.

§ III · Arithmetic gradient16 / 31

Part FourGeometric gradient 17

Part — Four

When the same number
becomes the same percentage.

Inflation, SaaS price escalators, salary bands, AI token spend — the modern economy runs on geometric gradients.

§ IV · Geometric gradientLecture Four

§ IVThe shape 18

A constant proportional step

A geometric gradient grows by a constant factor (1+g).

Geometric gradient · base A₁ · rate g per period

In a geometric series, each period's cash flow is (1+g) times the previous. g > 0 is a ramp; g < 0 is a decay (declining licence renewal, for example).

§ IV · Geometric gradient18 / 31

§ IVClosed form 19

One factor — two cases

One formula, two regimes: i ≠ g and i = g.

Case 1 — i ≠ g

$ P = A_1 \cdot \dfrac{1 - \left(\dfrac{1+g}{1+i}\right)^{n}}{i - g} $

Case 2 — i = g (degenerate)

$ P = A_1 \cdot \dfrac{n}{1+i} $

The growth exactly cancels the discount: each year's PV is the same constant.

Mnemonic: define i* = (i − g)/(1 + g), then a geometric stream with growth g and rate i is just a flat stream with rate i*. P/A with i* gives the answer.

§ IV · Geometric gradient19 / 31

§ IVWorked example · SaaS 20

The syllabus problem

A SaaS subscription growing 10% per year, for five years.

Year-1 payment $24,000; annual price escalator g = 10%; discount rate i = 9%; term n = 5. What is the PV of the five-year payment stream?

Apply Case 1 — i ≠ g

$ P = 24{,}000 \cdot \dfrac{1 - \left(\dfrac{1.10}{1.09}\right)^{5}}{0.09 - 0.10} $

$ P = 24{,}000 \cdot \dfrac{1 - 1.0462}{-0.01} = 24{,}000 \cdot 4.6243 $

$ P \approx \$110{,}983 $

Sanity check — the i* trick

Effective rate i* = (0.09 − 0.10) / 1.10 ≈ −0.91%. P/A factor for n = 5 at i* ≈ −0.91% ≈ 5.139; times $24K × (1/1.10) ≈ $112K. Close to the closed form, modulo rounding.

Vendor's framing vs. customer's truth

The vendor quotes "$24K with a modest 10% annual increase" — the customer commits to a stream whose undiscounted total is $146.5K, and whose PV at 9% is $111K. Both numbers are real, and only the diagram reveals the gap.

§ IV · Geometric gradient20 / 31

Part FiveEquivalence 21

Part — Five

Two streams are equivalent
when they share the same PV
at the same rate.

Every financial decision in this course is, at root, an equivalence statement.

§ V · EquivalenceLecture Four

§ VDefinition 22

The central principle

Equivalence is what gives "worth the same" a precise meaning.

Two cash-flow streams X and Y are equivalent at interest rate i if

$ PV_i(X) = PV_i(Y) $

Equivalence is — in this strong sense — the only meaningful definition of "same value" between two streams that occur at different times.

Three properties

Rate-dependent. Equivalence at 5% is not equivalence at 10%.
Symmetric & transitive. If X ≡ Y and Y ≡ Z (at the same i), then X ≡ Z.
Additive. If X₁ ≡ Y₁ and X₂ ≡ Y₂, then X₁+X₂ ≡ Y₁+Y₂.

Additivity is what makes decomposition (Part VI) work.

§ V · Equivalence22 / 31

§ VWorked example 23

A lump sum vs. an annuity

$10,000 today — or $2,400 a year for five years?

Discount rate i	PV of $10K today	PV of $2,400 × 5 yrs	Which wins?
0%	$10,000	$12,000	Annuity
4%	$10,000	$10,683	Annuity
6.40%	$10,000	$10,000	Equivalent
8%	$10,000	$9,582	Lump sum
12%	$10,000	$8,651	Lump sum

The two streams are equivalent at exactly one rate: i ≈ 6.40%. Below it, the annuity is worth more; above it, the lump sum is. The rate is the deciding fact.

§ V · Equivalence23 / 31

Part SixComposition 24

Part — Six

Real cash flows aren't
clean — but they're always
a sum of clean pieces.

§ VI · CompositionLecture Four

§ VIThe toolkit 25

Every cash flow is a sum of these four

Four shapes, all the financial world.

Single sum

One payment, one period. Up-front investment, salvage value, one-time bonus.

Factor: P/F, F/P

Uniform series

The same amount, every period. Subscriptions, loan payments, salaries.

Factor: P/A, A/P, F/A, A/F

Arithmetic gradient

Constant additive step. Maintenance creep, ramping headcount, depreciating savings.

Factor: P/G, A/G

Geometric gradient

Constant proportional step. SaaS escalators, inflation, exponential growth or decay.

Factor: closed-form (1+g)/(1+i)

Decomposition rule — by additivity of equivalence: split the stream into clean pieces · price each at the same i · sum the PVs.

§ VI · Composition25 / 31

§ VIWorked example · SaaS contract 26

An enterprise software contract, decomposed

Three shapes, one PV.

Year 0: implementation fee of $30K. Years 1–5: base subscription of $48K, with annual escalator g = 6%. Year 3: one-time professional-services credit of +$10K. i = 9%.

Piece	Shape	Factor	PV at i = 9%
Implementation at t=0	Single sum (outflow)	P/F × 1	−$30,000
Subscription, A₁=$48K, g=6%, n=5	Geometric gradient (outflow)	closed form	−$201,690
PS credit at t=3	Single sum (inflow)	P/F, 9%, 3	+$7,722
Total contract PV (customer's perspective)			−$223,968

Same numbers, different framing: the vendor's "$30K + $48K/yr (with a small 6% increase)" pitch is a present-day commitment of roughly $224K. That is the number that belongs in the comparison with the in-house alternative.

§ VI · Composition26 / 31

§ VICheat sheet 27

One slide to carry forward

The factor map.

Convert	From	To	Factor	Formula
Single → PV	F at t=n	P at t=0	P/F	1 / (1+i)ⁿ
Single → FV	P at t=0	F at t=n	F/P	(1+i)ⁿ
Uniform → PV	A × n	P	P/A	[(1+i)ⁿ−1] / [i(1+i)ⁿ]
PV → Uniform	P	A × n	A/P	i(1+i)ⁿ / [(1+i)ⁿ−1]
Uniform → FV	A × n	F	F/A	[(1+i)ⁿ−1] / i
FV → Uniform	F	A × n	A/F	i / [(1+i)ⁿ−1]
Arith. gradient → PV	0, G, 2G, … ,(n−1)G	P	P/G	[(1+i)ⁿ−in−1] / [i²(1+i)ⁿ]
Geo. gradient → PV	A₁, A₁(1+g), …	P	closed form	A₁·[1−((1+g)/(1+i))ⁿ] / (i−g)

Allowed on the final exam: one A4 cheat sheet, written by you. This table is a reasonable first draft.

§ VI · Composition27 / 31

DiscussionThree prompts 28

Open floor — 10 minutes

Three questions for the room.

Recall one financial commitment from your own life — a loan, lease, subscription, salary band. Which of the four shapes does it fit? Was that shape transparent at the time you signed?
In a software project you have worked on, where did the cash flows depart from a clean pattern — a support spike at month six, a license renegotiation at year two, a sudden cloud bill? How would you have decomposed it in advance?
Why might a vendor prefer to quote a geometric ramp rather than a lump sum, even when the equivalent PVs are identical? What does the answer say about how budgets — not finance — actually get decided?

Discussion28 / 31

Open the floor. Aim for 4–5 voices. If the room is shy, start with Q1 — everyone has signed a loan or a subscription. On Q2 — the goal is to surface the gap between textbook patterns and reality. The answer is almost always: "we didn't model it in advance; we ate the surprise later." That gap is exactly what this course is meant to close. Tie back to the TCO worksheet from Lecture 2: identify any line items that are secretly gradients. On Q3 — the deep answer is psychological: a quoted $48K/yr feels smaller than an equivalent $224K lump sum, even when they are the same wealth. Budget-holders sign per-year; CFOs sign per-PV. That asymmetry is exploited routinely in enterprise sales. Do NOT moderate too tightly. Let students disagree. ~10 minutes.

HomeworkDue Lecture 5 29

Homework 04 — due Sunday 7 June, before class

Decompose, then price.

Decomposition. For each of three real-world cash-flow patterns — (a) a front-loaded data-migration project, (b) a ramping SaaS subscription with a $5K loyalty rebate in year 3, (c) a declining license-maintenance contract — draw the cash-flow diagram, identify the shapes, and write the PV expression as a sum of standard factors. Numerical PV not required.
Equivalence. Solve the four equivalence problems on the handout. For each, draw both streams, state i, and show the equivalence at that rate (or compute the rate at which they become equivalent).
Reflection (½ page). Choose one decomposition above and explain — in plain English — how the diagram would change the way the decision-maker should talk about the contract internally.

Submit as a single PDF to /submissions/HW4/<your-name>.pdf. Late: −10%/day, three-day max.

Homework 0429 / 31

RecapOne sentence 30

Every project cash flow is some combination of single sums, uniform series, arithmetic gradients, and geometric gradients; once you can decompose, you can price — and once you can price, you can prove equivalence.

— the whole lecture, in one line

Lecture Four · Recap30 / 31

EndLecture Four 31

Questions & conversation.

Dr. Zhijiang Chen
Software Engineering Economics · Summer 2026
frostburg-state-university.github.io/bju

Software Engineering EconomicsChen · FSU 2026