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Module 7 · L6 of 12 ~30 min MST-12-S2-03 ⚡ +50 XP available

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.

Today's hook, 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? Predict before calculating.

0/5QUESTS

Orient to present value

Connect the lump-sum choice to the lesson goals, formula and essential language.

Worksheets

Practise this lesson

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

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Think First, your gut answer first

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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.

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Key ideas for this lesson

reference

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 $M$.

The PV formula for an ordinary annuity and its variables.

Finding loan repayment: $M = PV \times \dfrac{r}{1 - (1+r)^{-n}}$. Total interest $= (M \times n) - PV$.

The $(1+r)^{-n}$ step

Calculate $(1+r)^{-n}$ first, then subtract from 1, then divide by $r$, then multiply by $M$. Don't skip steps.

PV vs FV relationship

FV = what you end up with (saving). PV = what it's worth today (borrowing). Use FV for savings, PV for loans.

Shorter term = higher payments

Reducing a 25-year mortgage to 20 years increases repayments but saves a large amount in total interest.

What you will master

Know

Key facts

PV formula for ordinary annuity
Finding loan repayment from PV
Total interest formula
Amortisation, how each payment splits

Understand

Concepts

Why money today is worth more than money tomorrow
How banks calculate loan repayments
Why early payments are mostly interest

Can do

Skills

Calculate PV of any annuity
Find loan repayments
Calculate total interest on loans
Build an amortisation table

Key terms

Present Value (PV)The current worth of a future stream of payments, discounted at a given interest rate.

Discount rateThe interest rate used to convert future payments back to today's value. Same as $r$ in the formula.

Loan repayment (M)The regular payment required to repay a loan of amount PV at rate $r$ over $n$ periods.

AmortisationThe process of gradually repaying a loan through regular payments, each split into interest and principal.

Principal componentThe portion of a repayment that reduces the outstanding loan balance.

Interest componentThe portion of a repayment that pays the interest on the current balance, interest on the remaining debt.

Calculate present value

Discount future payments and rearrange the model to find loan repayments.

The present value formula, discounting future payments

core concept

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.

Worked example: What is $500/month for 3 years at 6% p.a. compounded monthly worth today?
$r = 0.005$, $n = 36$
$PV = 500 \times \dfrac{1 - (1.005)^{-36}}{0.005} = 500 \times \dfrac{1 - 0.8356}{0.005} = 500 \times 32.87 = \$16{,}435$
A lump sum of $\$16{,}435$ today is equivalent to receiving $500 every month for 3 years at this interest rate.

Finding loan repayments: Rearrange to solve for $M$: $$M = PV \times \frac{r}{1 - (1+r)^{-n}}$$ Example: $300{,}000 mortgage at 5.4% p.a. compounded monthly over 25 years.
$r = 0.0045$, $n = 300$
$M = 300000 \times \dfrac{0.0045}{1 - (1.0045)^{-300}} = \dfrac{1350}{0.739973} = \$1{,}824.39$/month
Total repaid $\approx 1824.39 \times 300 = \$547{,}317$. Total interest $\approx \$247{,}317$.

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?

Build amortisation models

Split repayments into interest and principal, then track the changing loan balance.

Amortisation, how each payment is split

core concept

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.

Amortisation table example: $\$300{,}000$ loan at 5.4% p.a., $M = \$1{,}824.39$/month.

Month 1: Interest $= 300000 \times 0.0045 = \$1{,}350.00$. Principal $= 1824.39 - 1350.00 = \$474.39$. Balance $= \$299{,}525.61$.
Month 2: Interest $= 299525.61 \times 0.0045 = \$1{,}347.87$. Principal $= \$476.52$. Balance $= \$299{,}049.09$.
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!

Outstanding balance shortcut: After $k$ payments, the remaining balance equals the PV of the remaining $(n-k)$ payments: $$\text{Balance after } k \text{ payments} = M \times \frac{1 - (1+r)^{-(n-k)}}{r}$$

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.

Work loan examples

Calculate repayments, total interest and outstanding balances in worked applications.

Worked examples · reveal each step

PROBLEM 1 · LOAN REPAYMENTS

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.

$r = 0.072 \div 12 = 0.006$, $\quad n = 5 \times 12 = 60$

Convert rate and count periods

PROBLEM 2 · MORTGAGE REPAYMENTS

Find the monthly repayment on a $400{,}000$ mortgage at 4.8% p.a. compounded monthly over 30 years. Find total interest paid.

$r = 0.004$, $\quad n = 360$

Monthly rate and total periods

Compare loan options

Evaluate the repayment and total-interest trade-off across different loan terms.

Comparing loan options, term vs total interest

core concept

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.

Example, 20 vs 25 years: $\$180{,}000$ mortgage at 4.8% p.a.

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.

Trap 01

Using PV formula for savings

PV is for loans and "what is a stream of payments worth today?" questions. For savings accumulation, use the FV formula from Lesson 5.

Trap 02

Forgetting the negative exponent

$(1+r)^{-n}$ is a fraction (less than 1). If you get a number greater than 1, you used a positive exponent, check your calculator input.

Trap 03

Interest formula for amortisation

Interest for each period is calculated on the current balance, NOT the original loan. Always update the balance before computing the next period's interest.

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:

PV
M (loan)
Interest component
Principal component

Balance × r for that period
Repayment minus interest, reduces the balance
What a stream of future payments is worth today
Regular repayment, interest plus principal

Evaluate present value decisions

Compare lump sums with payment streams and revisit the opening decision.

Extended calculation, lottery choice

apply

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?

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Show answer

(1) $PV = 40000 \times \dfrac{1-(1.05)^{-20}}{0.05} = 40000 \times 12.462 = \$498{,}480$.
(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.

Revisit your thinking

$PV = 50000 \times \dfrac{1-(1.04)^{-25}}{0.04} = 50000 \times 15.622 = \$781{,}100$. 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?

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Apply present value

Calculate repayments, construct an amortisation table and justify financial choices.

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.

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:

Short answer

ApplyBand 42 marks

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)

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ApplyBand 42 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)

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AnalyseBand 53 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)

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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 $\approx \$263{,}860$.

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