02Lesson 2 · Module 2

Some cashflows repeat. Finance has shortcuts for them.

Perpetuities, annuities, and compounding are not just formulas. They are patterns of cash over time.

M2Module map

Present Value Relations

Four connected lessons turn future cashflows into value today. Complete the first three to unlock the integrated CFO Decision Room.

Node 01

Cashflow Canyon

Assets, timelines, and NPV

Ready

Node 02

Perpetuity Port

Special cashflows and compounding

Ready

Node 03

Inflation Ridge

Real vs nominal value

Ready

Node 04

CFO Decision Room

Integrated final challenge

Ready

01Part II

Perpetuities, Annuities, and Compounding

Lesson objectives

What you should understand in this lesson

Present value is now applied to practical investments and special cashflow patterns. You will value projects, perpetuities, growing perpetuities, annuities, and loans with frequent compounding.

How $1 grows

A dollar invested today compounds forward. At 5%, one dollar becomes $1.05 after one year, $1.103 after two, and $1.158 after three. The same force runs in reverse: a future dollar is worth less the farther out it sits.

Year 1

$1 × 1.05 = $1.05

Year 2

$1 × 1.05² = $1.103

Year 3

$1 × 1.05³ = $1.158

Future value

FV = 1 × (1+r)^t

Present value of $1 received in the future

$0.926

Year 1

$0.681

Year 5

$0.463

Year 10

$0.215

Year 20

At 8%, a dollar arriving in Year 20 is worth only $0.215 today. PV declines as the receipt year moves farther into the future.

Try itDiscount beam visualizer

PV = CF ÷ (1 + r)^t

A future cashflow sits out on the timeline. Run it back through the Year 0 conversion gate and watch it shrink: the higher the rate, the faster distant dollars shrink into present value.

Cashflow amount (CF)Year (t)10Discount rate (r)8.0%

Year 0 · PV gate

Year 0 · PV $0.463Year 10 · CF $1.00

Replays the beam from Year 10 back through the gate.

Present value

PV = CF(1+r)¹⁰ = $1.00(1+0.080)¹⁰ = $0.463

Try itWorked example 03

Step 1 of 4

Worked Example: Lighting system investment

A building spends $800,000 a year on electricity. A new lighting system costs $230,000 today and saves $90,000 in each of Years 1–3. The interest rate is 4%.

1Draw the timeline
Year 0
−$230,000
Year 1
+$90,000
Year 2
+$90,000
Year 3
+$90,000

Try itWorked example 04

Step 1 of 4

Worked Example: CNOOC cheap-loan subsidy

CNOOC receives a $2.5B zero-interest 2-year loan and a $4.5B 3.5% 30-year loan. CNOOC's normal borrowing rate is 8%. How much are these cheap loans worth today?

1Identify the subsidized loans
Loan 1 · 2-year, zero interest
$2.5B at 0% vs. a normal 8% rate. Annual subsidy = 2.5 × (0.08 − 0.000) = $0.2B / yr for 2 years.
Loan 2 · 30-year, 3.5%
$4.5B at 3.5% vs. a normal 8% rate. Annual subsidy = 4.5 × (0.08 − 0.035) = $0.2025B / yr for 30 years.

Perpetuity

Definition · Perpetuity

A perpetuity pays a constant cashflow C every period forever, starting one period from now.

Perpetuity

PV = Cr

The value is finite because distant cashflows are discounted more heavily. Even an infinite stream collapses to a single number today.

Try itPerpetuity lab

PV = C ÷ r

A perpetuity pays the same cashflow every period, forever. Even though the stream never ends, its value is finite — and it rises fast as the discount rate falls.

Cashflow (C)$100Discount rate (r)10.0%

Infinite cashflow stream

Each bar is $100. They fade as distance grows — exactly what discounting does.

Present value$1,000

Lower r → larger PV. At r = 10.0% the meter reads 25% of the reference ceiling.

Perpetuity

PV = Cr = $1000.100 = $1,000

Growing perpetuity

Definition · Growing perpetuity

A growing perpetuity pays cashflows that grow at rate g forever.

Growing perpetuity

PV = Cr − g, where r > g

Growth raises value, but the formula only works when r is greater than g. If growth meets or beats the discount rate, the series never converges.

Try itGrowing perpetuity meter

PV = C ÷ (r − g)

Growth pushes value up, but the formula only holds while the discount rate stays above the growth rate. Drag g toward r and watch the stability gauge turn cautionary — past r, the value stops converging.

Cashflow (C)$100Discount rate (r)10.0%Growth rate (g)3.0%

Stability gauger − g = 7.0 pp

g = 0%limit: g = r (10.0%)

Growing perpetuity

PV = Cr − g = $1000.070 = $1,429

Annuity

Definition · Annuity

An annuity pays a constant cashflow C for T periods and then stops.

Annuity

PV = Cr × [ 1 − 1(1+r)^T ]

An annuity is a perpetuity that stops

The annuity formula is not magic — it is a perpetuity with the infinite tail removed. Subtract a second perpetuity that begins at date T and the tails cancel.

Perpetuity

…∞

Minus date-T perpetuity−

…∞

= T-period annuity

stops

An annuity equals a perpetuity that starts today, minus a second perpetuity that starts at date T. The infinite tails cancel, leaving a finite stream.

Try itAnnuity builder

PV = C × (1/r) × [1 − 1/(1+r)^T]

An annuity pays a fixed cashflow for a finite number of years, then stops. The longer it runs, the closer it gets to a perpetuity — but only the forever version reaches the full perpetuity value.

Cashflow (C)$100,000Discount rate (r)10.0%

Term

20 years$851,356

50 years$991,481

Foreverperpetuity limit$1,000,000

Selected value: $851,356. The 50-year bar nearly reaches the forever bar; the 20-year bar sits well below it.

With C = $100,000 and r = 10.0% — are you a millionaire today?

Compounding

Interest may be credited or charged more often than annually. Bank accounts may compound daily. Mortgages and leases often compound monthly. Bonds often use semiannual conventions.

Effective annual rate

EAR = (1 + rn)ⁿ − 1

Definitions

r = APR (quoted annual rate)
n = compounding periods per year
r/n = per-period rate
EAR = effective annual rate

More frequent compoundingEAR > APRQuoted vs. true rate

Try itCompounding simulator

EAR = (1 + r/n)^n − 1

When interest compounds more than once a year, you earn interest on interest. The quoted APR understates the true annual growth — the effective annual rate (EAR) is what you actually receive.

PrincipalAPR (r)6.75%

Compounding frequency

n = 365 periods / year

Per-period rate

0.0185%

Final balance

$10,698.24

EAR

6.982%

APR (quoted)

6.750%

EAR 6.982% > APR 6.750% — compounding 365× per year adds 0.232% of extra annual growth.

Balance curve over one year

Start · $10,000.00End · $10,698.24

Reflection — why does compounding make EAR higher than APR?

Try itPart II mastery check

Pass with 5 of 6 correct

Answer all questions, then check your work. You can retry any time — mastery is based on correctness, not speed.

01
Calculate the future value of $1 at 5% for 3 years.
02
In the lighting example, the NPV at 4% is closest to:
03
Which cashflow pays a constant amount every period forever?
04
A perpetuity pays C = 100 at r = 10%. Its value is:
05
An annuity pays C = 100,000 for 20 years at r = 10%. PV is:
06
Which statement is true?

Lesson summary

1Assets are sequences of cashflows.
2Cashflows at different dates are different economic units.
3Present value converts future cashflows into today's dollars.
4NPV is the present value of benefits minus costs.
5Positive-NPV projects create value.
6Perpetuities and annuities are special cashflow patterns.
7Compounding affects the true annual rate.
8Inflation changes purchasing power.
9Real and nominal cashflows must be discounted consistently.

Back to Cashflows and NPV Replay this lesson Continue to Real vs Nominal Value