QF5213 Chap.3 Annuities and Effective Annual Rates
Annuities and Effective Annual Rates
From one cash flow to a repeating stream
Once a single sum can be moved backwards and forwards in time, the next question is what happens when the same amount repeats. Rent, salaries, loan instalments, insurance premiums and regular savings all share that shape.
An annuity is the name for a fixed payment made once per period for a set number of periods, and its value can be found in a single step rather than by discounting every payment separately.
Timing matters more than students expect. An ordinary annuity pays at the end of each period, so the first flow lands one period after today. An annuity due pays at the start, so the first flow lands today.
Because every payment of an annuity due arrives one period sooner, its present and future values are the ordinary values multiplied by one plus the period rate. The count of payments stays the same; only their position on the timeline shifts.
The Week 2 class extends the idea in two directions. A perpetuity lets the payments continue forever, and its value reduces to the payment divided by the rate.
A growing annuity lets each payment rise by a constant percentage, and it can be built as the gap between two growing perpetuities that start at different dates. Both forms reappear later when stocks are valued from dividend streams.
Quoted rates and the rate you actually pay
Most real contracts run monthly, so the formulas are rewritten with a rate per month and a count of months.
A lender quotes an annual percentage rate, which is simply the period rate scaled up by the number of periods in a year. That headline figure ignores compounding inside the year. The effective annual rate captures it, and it is the only fair basis for comparing a card that compounds monthly against a bank loan that compounds quarterly.
A second complication appears when payments and compounding run on different clocks.
If deposits are yearly but interest compounds monthly, or instalments are monthly but the quote is semi-annual, the rate has to be converted before any annuity formula is used.
The cleanest route goes through the effective annual rate: find it from the quoted terms, then convert it to the period rate that matches the payment schedule.
Why the basis of interest changes the answer
Two loans with the same quoted percentage can cost very different amounts. Under a reducing balance, each month’s interest is charged only on what is still owed, so the quote and the true rate agree.
A flat rate charges interest on the original principal for the whole term, even after much of it has been repaid, and the true rate ends up close to double the quote. Annual rest sits between the two, recomputing the balance only once a year.
Processing fees push the true cost higher again, since the borrower receives less cash than the face amount.
Working through these cases teaches one habit that carries across the whole course: identify when each payment occurs, choose a period that matches the payments, convert the rate to that period, and only then apply a formula.
That sequence underpins later topics on capital budgeting, stock valuation and bond pricing.
What this chapter covers
- 01
Ordinary annuities and annuities due: present and future value
- 02
Perpetuities, deferred perpetuities and growing annuities
- 03
Monthly annuities for mortgages, car loans and savings plans
- 04
Annual percentage rate versus effective annual rate
- 05
Converting rates when payment and compounding frequencies differ
- 06
Reducing balance, flat rate and annual rest loan quotes
- 07
Amortisation: splitting an instalment into interest and principal
Worked example · free
Monthly savings under a quarterly compounding quote
- 1Convert the quarterly quote to an effective annual rate.
- 1Convert the effective annual rate to a rate per month that matches the deposits.
- 2Apply the future value of an ordinary annuity over 96 months.
- 1Compare with the unadjusted monthly rate and state the error.
Key terms
- Ordinary Annuity
- A stream of equal payments made at the end of each period for a fixed number of periods, so the first payment falls one period from today.
- Annuity Due
- A stream of equal payments made at the start of each period, worth the ordinary annuity value multiplied by one plus the period rate.
- Perpetuity
- A level payment that continues forever; when the first payment is one period away its present value equals the payment divided by the rate.
- Growing Annuity
- A finite stream whose payments rise by a constant rate each period, valued as the difference between two growing perpetuities starting at different dates.
- Annual Percentage Rate
- The quoted yearly rate, equal to the rate per period multiplied by the number of periods in a year, ignoring compounding within the year.
- Effective Annual Rate
- The rate actually earned or paid over a year once compounding within the year is included, used to compare quotes with different frequencies.
- Flat Rate Basis
- A method that charges interest on the original principal for the full loan term, making the true borrowing rate far higher than the quote.
- Reducing Balance Basis
- A method that charges each period's interest only on the balance still outstanding, so the quoted and true period rates coincide.
Annuities and Effective Annual Rates FAQ
How do I tell whether a question describes an ordinary annuity or an annuity due?
Look for when the first payment happens. Phrases such as starting immediately, today, or at the beginning of each month signal an annuity due, while end of the month, starting next year, or salary style payments signal an ordinary annuity. Draw the timeline first and place the first arrow before choosing a formula.
Why is the effective annual rate higher than the APR?
The APR multiplies the period rate by the number of periods and so ignores interest earned on interest within the year. The effective annual rate compounds the period rate across the year, so whenever compounding happens more than once a year it exceeds the APR, and the gap widens with higher rates and more frequent compounding.
What should I do when payments are monthly but interest compounds semi-annually?
Convert the quote to an effective annual rate, then take the twelfth root to find the equivalent monthly rate, and use that rate in the annuity formula with the number of months. Dividing the semi-annual quote by twelve gives a slightly wrong rate and a payment that is off by a few dollars.
Why does a flat rate loan cost so much more than its advertised rate?
Flat interest is computed on the full original principal for every year of the loan, even though instalments steadily repay that principal. Solving the annuity equation for the rate that matches the instalments reveals the true cost, which is typically close to twice the advertised figure on a loan of a few years.
How do I value a perpetuity whose first payment comes several years from now?
Apply the perpetuity formula to get a value one period before the first payment, then discount that single amount back to today. For example, a stream starting at year six is valued at year five, then discounted five years. Discounting six years is a frequent slip.
Exam move
Build a one page conversion sheet with three columns: the quote as written, its effective annual rate, and the period rate matching the payments. Fill it for six quotes, mixing monthly, quarterly and semi-annual compounding with yearly and monthly payments.
Then take one loan and price it three ways, on reducing balance, flat and annual rest bases, solving for the true rate with your financial calculator each time, and rank the results before checking them.
Working through Annuities and Effective Annual Rates in QF5213? Sia is AskSia’s AI Finance tutor — ask any QF5213 Annuities and Effective Annual Rates question and get a clear, step-by-step explanation grounded in how QF5213 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
