Present Value of Annuities
A bank offers you a $500,000 mortgage at 5% over 30 years. Your monthly repayment is $2,684. Where does that number come from? It comes from the present value of an annuity — the mathematical process of converting a stream of future payments into a single lump sum today. When you borrow money, the loan amount IS the present value of all your future repayments. This concept underpins every loan, mortgage, and pension calculation in the financial 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 $1 million today or $50,000 per year for 25 years. At 4% p.a., which is worth more in today's dollars?
Before reading on — write your gut feeling. We will revisit this at the end of the lesson.
The present value of an annuity tells you what a stream of future payments is worth right now. It is the foundation of all loan calculations.
Present value (ordinary): the lump sum today that is equivalent to a series of future equal payments, discounted at rate $r$ per period.
Loan repayment: when you borrow money, the loan amount IS the PV of all your future repayments. Rearrange to find the required payment $M$.
Key facts
- PV formula for ordinary annuity
- Finding loan repayment from PV
- Total interest formula
- Amortisation — how each payment splits
Concepts
- Why money today is worth more than money tomorrow
- How banks calculate loan repayments
- Why early payments are mostly interest
Skills
- Calculate PV of any annuity
- Find loan repayments
- Calculate total interest on loans
- Build an amortisation table
The present value of an ordinary annuity tells you what a future stream of equal payments is worth today:
$$PV = M \times \frac{1 - (1+r)^{-n}}{r}$$Each future payment is "discounted" back because money received later is worth less than money received now. The formula sums all those discounted values.
$r = 0.005$, $n = 36$
$PV = 500 \times \dfrac{1 - (1.005)^{-36}}{0.005} = 500 \times \dfrac{1 - 0.8356}{0.005} = 500 \times 32.87 = \$16{,}435$
A lump sum of $\$16{,}435$ today is equivalent to receiving $500 every month for 3 years at this interest rate.
$r = 0.0045$, $n = 300$
$M = 300000 \times \dfrac{0.0045}{1 - (1.0045)^{-300}} = \dfrac{1350}{0.7387} = \$1{,}827.54$/month
Total repaid $= 1827.54 \times 300 = \$548{,}262$. Total interest $= \$248{,}262$.
What to write in your book
- $PV = M \times \dfrac{1 - (1+r)^{-n}}{r}$ — present value (ordinary annuity).
- Loan repayment: $M = PV \times \dfrac{r}{1 - (1+r)^{-n}}$.
- Total interest $= (M \times n) - PV$.
- The loan amount IS the PV of all future repayments.
Quick check: A $20{,}000$ car loan at 6% p.a. compounded monthly over 4 years. What is $r$ in the repayment formula?
Every loan repayment has two components:
- Interest component: interest charged on the current outstanding balance: $\text{Interest} = \text{Balance} \times r$
- Principal component: the portion that reduces the loan balance: $\text{Principal} = M - \text{Interest}$
Early in the loan, most of each payment is interest because the balance is large. As the balance falls, the interest component shrinks and the principal component grows.
Month 1: Interest $= 300000 \times 0.0045 = \$1{,}350.00$. Principal $= 1827.54 - 1350.00 = \$477.54$. Balance $= \$299{,}522.46$.
Month 2: Interest $= 299522.46 \times 0.0045 = \$1{,}347.85$. Principal $= \$479.69$. Balance $= \$299{,}042.77$.
Month 3: Interest $= 299042.77 \times 0.0045 = \$1{,}345.69$. Principal $= \$481.85$. Balance $= \$298{,}560.92$.
After 12 months, the balance is only about $\$294{,}500$ — only $\$5{,}500$ paid off despite $\$21{,}930$ in payments!
What to write in your book
- Interest for period $= \text{Balance} \times r$. Principal $= M - \text{Interest}$. New balance $= \text{Old balance} - \text{Principal}$.
- Early payments are mostly interest; later payments are mostly principal.
- Remaining balance $= M \times \dfrac{1-(1+r)^{-(n-k)}}{r}$ after $k$ payments made.
True or false: For a standard loan, the interest component of each repayment decreases over time while the principal component increases.
Worked examples · reveal each step
Find the monthly repayment on a $25{,}000$ car loan at 7.2% p.a. compounded monthly over 5 years. Find total interest and the outstanding balance after 2 years.
Find the monthly repayment on a $400{,}000$ mortgage at 4.8% p.a. compounded monthly over 30 years. Find total interest paid.
Reducing the loan term increases repayments but reduces total interest. Paying extra each period also cuts the effective loan term.
25-year term: $r=0.004$, $n=300$. $M = 180000 \times 0.004 / [1-(1.004)^{-300}] = \$1{,}017.27$/month. Total $= \$305{,}181$. Interest $= \$125{,}181$.
20-year term: $n=240$. $M = 180000 \times 0.004 / [1-(1.004)^{-240}] = \$1{,}168.11$/month. Total $= \$280{,}346$. Interest $= \$100{,}346$.
Saving: $\$24{,}835$ in interest by paying $\$151$ extra per month for 5 fewer years.
What to write in your book
- Shorter loan term = higher repayments but much less total interest.
- The interest "saved" by reducing term is often tens of thousands of dollars.
- Outstanding balance after $k$ payments $= M \times \dfrac{1-(1+r)^{-(n-k)}}{r}$.
Fill the gap: For a $\$20{,}000$ loan at 6% p.a. compounded monthly over 4 years, $n =$ .
Quick-fire practice · 2 activities
(a) Find the PV of $300/month for 4 years at 5.4% compounded monthly. (b) Find the monthly repayment on a $20{,}000 loan at 6% over 4 years, compounded monthly. (c) Find total interest on a $350{,}000 mortgage at 5% over 25 years, compounded monthly.
For a $15{,}000 loan at 8% p.a. compounded monthly over 3 years: (a) find the monthly repayment, (b) calculate the first 3 months' interest and principal components in a table, (c) explain why the interest component decreases each month.
Match each term to its description:
A lottery winner can receive $\$40{,}000$ per year for 20 years, or a lump sum of $\$450{,}000$ today. The relevant interest rate is 5% p.a.
- Calculate the present value of the annuity.
- Compare to the lump sum. Which is the better financial choice?
- At what annual rate would the two options be equivalent?
Show answer
(2) PV of annuity ($\$498{,}480$) > lump sum ($\$450{,}000$). Taking the annuity is better value — assuming guaranteed payments and a 5% discount rate.
(3) When $r$ is higher (e.g. 7%+), the lump sum becomes preferable because future payments are more heavily discounted.
Top 3 list: List THREE decisions where the present value of an annuity helps you choose. For each, state what PV tells the decision-maker.
$PV = 50000 \times \dfrac{1-(1.04)^{-25}}{0.04} = 50000 \times 15.622 = \$781{,}100$. The annuity is worth $\$781{,}100$ in today's dollars — but the lump sum is $\$1{,}000{,}000$. The $1 million lump sum is worth more. At low discount rates, long annuities are very valuable, but the lump sum still wins here. The key insight: a higher discount rate makes future payments less valuable, which is why the annuity's present value can vary dramatically with the assumed rate.
What has changed in your understanding? What did you get right? What surprised you?
Pick your answer, then rate your confidence — that tells the system what to drill next.
Q1. A $20{,}000$ loan at 6% p.a. compounded monthly over 4 years. Which expression gives the monthly repayment?
Q2. The first month's interest on a $\$300{,}000$ mortgage at 5.4% p.a. compounded monthly is:
Q3. For a loan with $n = 60$ periods and $M = \$500$/month, the total interest paid is $\$4{,}000$. What was the original loan amount?
Q4. For a standard amortising loan, as time passes:
Q5. A $\$180{,}000$ mortgage at 4.8% p.a. compounded monthly has monthly repayments of $\$1{,}168$. After 15 years, the outstanding balance is found using:
SA 1. (a) Find the monthly repayment on a $\$180{,}000$ mortgage at 4.8% p.a. compounded monthly over 20 years. (b) Find total interest paid. (c) If the term is reduced to 15 years, find the new repayment and total interest saved. (2 marks)
SA 2. A lottery winner chooses $\$40{,}000$ per year for 20 years instead of a lump sum. At 5% p.a., what is the present value of this annuity? If the lump sum offered was $\$450{,}000$, did they make the right mathematical choice? (2 marks)
SA 3. For a $\$500{,}000$ mortgage at 4.5% p.a. compounded monthly over 30 years: (a) find the monthly repayment. (b) Create an amortisation table for the first 3 months. (c) After 15 years (half the term), what percentage of the original loan has been paid off? Explain why this is less than 50%. (3 marks)
Comprehensive answers (click to reveal)
MC 1 — C: Repayment formula is $M = PV \times r / [1-(1+r)^{-n}]$.
MC 2 — B: Interest $= 300000 \times (0.054/12) = 300000 \times 0.0045 = \$1{,}350$.
MC 3 — D: PV $= M \times n - \text{Interest} = 500 \times 60 - 4000 = 30000 - 4000 = \$26{,}000$.
MC 4 — A: As the balance falls, less interest is charged each period, so more of the fixed repayment goes to principal.
MC 5 — C: After 180 payments, 60 periods remain. Balance = PV of remaining 60 payments.
SA 1 (2 marks): (a) $M = 180000 \times 0.004/[1-(1.004)^{-240}] \approx \$1{,}167.89$/month [0.5]. (b) Total $= \$280{,}294$. Interest $= \$100{,}294$ [0.5]. (c) New $M \approx \$1{,}405.15$. Saved $\approx \$27{,}367$ [1].
SA 2 (2 marks): $PV = 40000 \times 12.462 = \$498{,}480$ [1]. PV ($\$498{,}480$) > lump sum ($\$450{,}000$) — annuity is the better choice [1].
SA 3 (3 marks): (a) $M = 500000 \times 0.00375/[1-(1.00375)^{-360}] = \$2{,}533.43$ [0.5]. (b) Month 1: I=$1{,}875$, P=$658.43$, CB=$499{,}341.57$; Month 2: I=$1{,}872.53$, P=$660.90$, CB=$498{,}680.67$; Month 3: I=$1{,}870.05$, P=$663.38$, CB=$498{,}017.29$ [1]. (c) Balance after 180 payments $\approx \$300{,}200$. Paid off $\approx \$199{,}800 = 40\%$. Less than 50% because early payments are mostly interest — the principal reduction accelerates only in later years [1].
Drill 1: (a) $PV \approx \$12{,}891$. (b) $M \approx \$469.70$/month. (c) Total $\approx \$613{,}860$. Interest $\approx \$263{,}860$.
Five timed questions on present value, loan repayments, amortisation, and total interest. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.
⚔ Enter the arenaClimb platforms using present value and loan calculations. Pool: lesson 6.
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