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hscscience Maths Adv · Y12
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Module 7 · L8 of 20 ~40 min ⚡ +95 XP available

Present Value of an Annuity

Would you take $\$1$ million today or $\$50,000$ every year for 25 years? The answer depends on present value, the lump sum that is mathematically equivalent to a stream of future payments. You'll learn to discount future cash flows back to today's dollars: the skill behind every pension valuation, loan approval, and lottery payout decision in the world.

Today's hook, A lottery winner can choose $\$2$ million now or $\$150,000$ per year for 20 years. That's $3 million total, but is it really worth more? Present value is how you find the actual answer, not the misleading one.
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Get oriented

Recall what you already know, meet the key ideas and settle the terms.

Worksheets

Practise this lesson

Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.

Build Foundations & guided practice Apply Application practice Master Mastery challenge Build custom Build your own from any module question
01
Recall, your gut answer first
+5 XP warm-up

A lottery winner can choose:

Option A: $2 million lump sum today.

Option B: $150,000 per year for 20 years.

Assume the money could be invested at 5% p.a. Which option is worth more in today's dollars? Make a prediction before reading on.

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02
The present value concept
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Present value answers one question: What lump sum today is equivalent to a series of future payments?

Each future payment is worth less in today's money because money now can be invested. We discount each payment:

Each payment $a$ received at the end of period $k$ is worth $\dfrac{a}{(1+r)^k}$ today. Sum all $n$ discounted payments:

$$PV = \dfrac{a}{(1+r)} + \dfrac{a}{(1+r)^2} + \cdots + \dfrac{a}{(1+r)^n}$$

This is a geometric series with first term $\dfrac{a}{1+r}$ and ratio $\dfrac{1}{1+r}$. Applying the GP sum formula gives:

$$PV = a \times \dfrac{1 - (1 + r)^{-n}}{r}$$
Money has a time value. A dollar today is worth more than a dollar tomorrow because you can invest it now. Discounting reverses compounding, it asks: "If I could invest this sum today at rate $r$, what payment stream would it produce?"
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What you'll master
Know

Key facts

  • The present value annuity formula
  • How to discount future payments back to today
  • The relationship between PV and FV
Understand

Concepts

  • Why a dollar today is worth more than a dollar tomorrow
  • How discounting reverses compounding
  • When to use PV vs FV in decision-making
Can do

Skills

  • Calculate PV for any ordinary annuity
  • Compare lump sums vs payment streams
  • Transpose to find payment $a$ given PV
  • Evaluate loan and pension products using PV
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Key terms
Present Value (PV)The lump sum today that is mathematically equivalent to a future stream of payments.
DiscountingThe process of finding today's value of a future amount, the reverse of compounding.
Discount rate ($r$)The interest rate per period used to discount future payments, must match payment frequency.
Time value of moneyThe principle that money available now is worth more than the same amount in the future.
PV–FV relationship$PV = FV / (1+r)^n$, moving money backwards through time divides by the growth factor.
Loan principalThe amount borrowed, equal to the PV of all future repayments discounted at the loan rate.

Choose between PV and FV

Read a question and decide which annuity formula it is asking for.

05
When to use PV vs FV
core concept

PV and FV answer different questions using the wrong one reverses time and produces nonsense.

Use FV when…
Your regular savings are growing into a future balance. Example: superannuation projections, savings goals.
Use PV when…
You need today's lump sum. Example: loan amounts, pension valuations, lottery lump-sum equivalents.
The link
$PV = FV / (1+r)^n$. FV accumulates forward; PV discounts backward. Same maths, opposite direction.
Cash Flow Diagram: PV vs FV PV (lump sum) Regular payments a each period FV (accumulated) Today Future (period n) Time →

PV and FV are two ends of the same timeline. PV discounts backwards; FV accumulates forwards.

PV annuity formula: $PV = a \times \dfrac{1 - (1+r)^{-n}}{r}$; Derived from GP sum with first term $\dfrac{a}{1+r}$, ratio $\dfrac{1}{1+r}$, $n$ terms

Pause, copy the PV annuity formula $PV = a \times \dfrac{1-(1+r)^{-n}}{r}$, derived from the GP sum with first term $\dfrac{a}{1+r}$ and ratio $\dfrac{1}{1+r}$, into your book.

Did you get this? True or false: the present value of an annuity is always less than the total of all payments (when $r > 0$).

Work the examples

Follow three worked present value problems end to end.

Worked examples · 3 in a row, reveal as you go

PROBLEM 1 · BASIC PV CALCULATION

What is the present value of $500 per month for 3 years at 6% p.a. compounded monthly?

1
$r = \dfrac{0.06}{12} = 0.005$, $\quad n = 3 \times 12 = 36$
Convert annual rate to monthly; convert years to months. Frequency must match.
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PROBLEM 2 · PENSION VALUATION

A pension pays $2,000 per month for 15 years. The discount rate is 4.8% p.a. compounded monthly. Find the present value of this pension stream.

1
$r = \dfrac{0.048}{12} = 0.004$, $\quad n = 15 \times 12 = 180$
Monthly payments → monthly rate and total months.
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PROBLEM 3 · FIND THE PAYMENT $a$

You have $100,000 to fund 10 years of monthly withdrawals at 4.8% p.a. compounded monthly. What is the maximum monthly withdrawal?

1
$r = 0.004$, $n = 120$, $PV = 100{,}000$
PV is the lump sum available today; we need to find $a$ (the monthly withdrawal).
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Quick check: A bank approves a home loan by calculating the PV of your future repayments. If your monthly repayment is $2,000 for 25 years at 6% p.a. compounded monthly, the loan amount equals:

Avoid the traps

Fix the slips that come from reaching for the wrong formula.

Common errors · the 3 traps that cost marks

Trap 01
Using FV when PV is needed (or vice versa)
PV and FV answer different questions. PV finds today's lump sum; FV finds tomorrow's accumulated total. Using the wrong formula produces an answer that is off by a factor of $(1+r)^n$, a massive error for large $n$.
Trap 02
Forgetting the negative exponent
The PV formula uses $(1+r)^{-n}$, not $(1+r)^n$. Entering a positive exponent on your calculator gives a number much larger than 1, making the bracket negative, a sure sign something went wrong.
Trap 03
Comparing nominal totals instead of PV
$150,000 \times 20 = \$3$ million looks better than $\$2$ million lump sum, but that ignores time value. The PV of the annuity at 5% is only $\$1.87$ million. Always compare in the same time units (today's dollars).

Write the two-step argument for why banks use the PV formula when deciding how much to lend you, not the FV formula.

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Drill it and revisit

Run the drill, then return to your opening answer and name what has changed.

Quick-fire practice · 4 drills

1

Find $PV$ for $a = \$1{,}000$, $r = 6\%$ per period, $n = 20$ periods.

2

$\$800$ per quarter for 6 years at 5.2% p.a. compounded quarterly. Find $PV$.

3

A lottery offers $\$50{,}000$/year for 20 years or a lump sum. The discount rate is 5%. Find the fair lump sum using PV. If the lottery offers only $\$550{,}000$, is it a good deal?

4

Zoe borrows to buy a car. She can repay $\$600$/month for 5 years at 6% p.a. compounded monthly. What is the maximum loan she can afford?

Fill in the blanks. The present value of an ordinary annuity is $PV = a \times \dfrac{1 - (1+r)^{\underline{\phantom{-n}}}}{r}$. This is less than $n \times a$ because each future payment is __________. The relationship $PV = FV / (1+r)^{\underline{\phantom{n}}}$ shows that PV moves money __________ through time.

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Match each financial question to PV or FV.

A. How much will my super grow to in 30 years? → PV or FV?

B. What lump sum do I need to fund a pension of $3,000/month for 20 years? → PV or FV?

C. How much can I borrow if I can repay $1,500/month for 25 years? → PV or FV?

D. I save $500/month, what will I have in 15 years? → PV or FV?

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11
Revisit your thinking

Earlier you predicted whether Option A ($\$2$M lump sum) or Option B ($\$150$K/year for 20 years) was worth more. Let's calculate:

$$PV = 150{,}000 \times \dfrac{1 - (1.05)^{-20}}{0.05} = 150{,}000 \times 12.462 = \$1{,}869{,}300$$

Option B (the annuity) is worth approximately $1.87 million in today's dollars less than the $2 million lump sum. Option A is the better deal. Despite paying $3 million total, the distant payments are so heavily discounted that they are worth less right now. This is why lottery winners who take the lump sum are often making the rational choice.

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Practise present value

Write full short-answer responses, then check them against the model answers.

01
Multiple choice
+5 XP per correct · +25 XP all-correct

Pick your answer, then rate your confidence. That tells the system what to drill next. Each retry pulls a fresh mix from the bank.

02
Short answer
ApplyBand 43 marks

Q1. Sofia wants to buy a car with monthly repayments of $600 for 5 years at 6% p.a. compounded monthly. What is the maximum car price she can afford? (3 marks)

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ApplyBand 43 marks

Q2. A retiree is offered a pension of $\$80,000$ per year for 25 years or a lump sum. The appropriate discount rate is 6% p.a. (annual compounding). Find the fair lump-sum value. If the insurer offers only $\$900,000$, is this a good deal for the retiree? (3 marks)

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AnalyseBand 54 marks

Q3. A pension stream pays $\$2,500$ per month for 15 years at 4.8% p.a. compounded monthly. (a) Find the PV of this stream. (b) A lump sum of $\$350,000$ is offered instead. Which is better and by how much? (c) Explain why changing the discount rate to 3% p.a. would change your decision. (4 marks)

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Comprehensive answers (click to reveal)

Drill 1: $PV = 1{,}000 \times [1-(1.06)^{-20}]/0.06 = \$11{,}469.92$   2: $r=0.013$, $n=24$, $PV = \$16{,}146.40$   3: Fair lump sum $= 50{,}000 \times [1-(1.05)^{-20}]/0.05 = \$623{,}111$. The $\$550{,}000$ offer is $\$73{,}111$ below fair value, not a good deal.   4: $r=0.005$, $n=60$, $PV = 600 \times [1-(1.005)^{-60}]/0.005 = \$31{,}015.59$

Match: A: FV | B: PV | C: PV | D: FV

Q1 (3 marks): $r=0.005$, $n=60$ [1]. $PV = 600 \times [1-(1.005)^{-60}]/0.005 = \$31{,}015.59$ [2].

Q2 (3 marks): $PV = 80{,}000 \times [1-(1.06)^{-25}]/0.06 = \$1{,}022{,}962$ [2]. The $\$900{,}000$ offer is $\$122{,}962$ below fair value, a bad deal for the retiree [1].

Q3 (4 marks): (a) $r=0.004$, $n=180$. $PV = 2{,}500 \times [1-(1.004)^{-180}]/0.004 = \$320{,}250$ [2]. (b) $\$350{,}000 > \$320{,}250$; lump sum is better by $\$29{,}750$ [1]. (c) At a lower discount rate, future payments are discounted less heavily, increasing the PV of the annuity above $\$350{,}000$, the annuity stream would then be the better choice [1].