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.
Practise this lesson
Three printable worksheets that build from foundations to mastery — or build your own from any module’s questions.
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.
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:
This is a geometric series with first term $\dfrac{a}{1+r}$ and ratio $\dfrac{1}{1+r}$. Applying the GP sum formula gives:
Key facts
- The present value annuity formula
- How to discount future payments back to today
- The relationship between PV and FV
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
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
PV and FV answer different questions — using the wrong one reverses time and produces nonsense.
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 $r > 0$).
Worked examples · 3 in a row, reveal as you go
What is the present value of $500 per month for 3 years at 6% p.a. compounded monthly?
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.
You have $100,000 to fund 10 years of monthly withdrawals at 4.8% p.a. compounded monthly. What is the maximum monthly withdrawal?
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:
Common errors · the 3 traps that cost marks
Write the two-step argument for why banks use the PV formula when deciding how much to lend you, not the FV formula.
Quick-fire practice · 4 drills
Find $PV$ for $a = \$1{,}000$, $r = 6\%$ per period, $n = 20$ periods.
$\$800$ per quarter for 6 years at 5.2% p.a. compounded quarterly. Find $PV$.
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?
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.
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?
Earlier you predicted whether Option A ($2M lump sum) or Option B ($150K/year for 20 years) was worth more. Let's calculate:
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.
Pick your answer, then rate your confidence — that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
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)
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)
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)
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].
Five timed PV questions. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.
Enter the arenaClimb platforms by answering present value and annuity comparison questions. Pool: lessons 1–8.
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