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 onwrite 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 $= \$248{,}262$.
PV = M × [1 − (1 + r)^{−n}] / r. Present value discounts each future payment back to today's dollars. Use PV when you need to find the lump sum equivalent of a payment stream, or the maximum affordable loan.
Pause, copy PV = M × [1 − (1 + r)^(−n)] / r with all variable labels, and the two applications: finding the maximum affordable loan and comparing a lump sum to a payment stream into your book.
Quick check: A $20{,}000$ car loan at 6% p.a. compounded monthly over 4 years. What is $r$ in the repayment formula?
PV = M × [1 − (1 + r)^(−n)] / r gives the lump-sum value of a payment stream, and for a loan, PV is the amount you borrow. Inside each repayment, not all of the money reduces the debt: the interest component equals r × remaining balance, and only the rest chips away at the principal. An amortisation schedule tracks this split for every payment.
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!
In amortisation, each repayment covers interest on the remaining balance plus a principal reduction. Interest component = r × remaining balance. As the balance falls, less of each repayment is interest and more is principal.
Pause, copy the amortisation interest formula (interest = r × opening balance) and note how the split shifts: early payments are mostly interest; later payments are mostly principal, because the balance falls into your book.
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.
The amortisation table shows that early repayments are mostly interest and later repayments are mostly principal, because the balance falls over time. When comparing loans, what matters is not the monthly repayment but the total interest paid over the full term: total interest = (repayment × n) − principal, and a longer term always produces a higher total interest cost.
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.
To compare loans: a shorter term means higher repayments but less total interest; a longer term means lower repayments but far more total interest paid. Total interest = (repayment × n) − principal.
Pause, copy total interest = (repayment × n) − principal, and note the trade-off: shorter term = higher repayments but less total interest; longer term = lower repayments but far more total interest paid into your book.
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:
Total interest = total repaid − principal is the comparison tool for loans. The same logic extends to any lump-sum-versus-payments decision (like a lottery prize): calculate PV = M × [1 − (1 + r)^(−n)] / r for the payment stream at the given interest rate, then compare that PV directly to the lump sum on offer.
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.
Comparing a lump sum versus an annuity: calculate PV of the annuity stream at the given rate. If PV > lump sum offered, take the payments; if PV < lump sum, take the cash now.
Pause, copy the decision rule: calculate PV of the payment stream using PV = M × [1 − (1 + r)^(−n)] / r; if PV > lump sum offered, take the payments; if PV < lump sum, take the cash into your book.
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 confidencethat 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.
Climb platforms using present value and loan calculations. Pool: lesson 6.
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