NSWMaths Standard 2Syllabus dot point

What is the present value of an annuity, and how do future value and present value annuity tables let you find or back-solve a contribution?

Use future value and present value annuity tables, and calculate the present value of an annuity

The HSC Maths Standard 2 dot point on annuity tables and present value. What the present value of an annuity is, the link PV = FV / (1+r)^n, reading future-value and present-value annuity tables at the period by rate intersection, back-solving a contribution or rate from a table, converting an annual rate to the per-period row, and a loan as the present value of its repayments.

Generated by Claude Opus 4.814 min answerUpdated 2026-06-21

Reviewed by: AI editorial process; not yet individually human-reviewed

Have a quick question? Jump to the Q&A page

What this dot point is asking

NESA wants you to do two related things. First, understand and calculate the present value of an annuity: the single lump sum today that is worth exactly the same as a stream of equal future payments. Second, read annuity factor tables. There are two, the future-value-of- $1$ table and the present-value-of- $1$ table. You find the value where the right period row and rate column meet, then multiply it by the contribution. You also need to back-solve from a table for a required contribution or interest rate. This is the table partner to the future-value formula. The formula gives one answer; the table gives a whole grid of ready-made answers you simply read off.

The answer

Money has a time value. A dollar you will get in three years is worth less than a dollar in your hand today, because today's dollar can be banked to earn interest. The present value of an annuity uses that idea on a whole stream of payments. (An annuity is a series of equal payments made at regular times.) It asks a simple question: if I am promised an equal payment at the end of every period for $n$ periods, what single amount today is worth the same? You find the answer by discounting each future payment back to today, that is, working out what it is worth now, and then adding the results. The diagram below does this for four yearly payments of $9000 at $6\%$ . Each future $9000 shrinks as it is pulled back to the present, and the four shrunken amounts add up to a single present value of $31,186.

Present value of a single payment, then of a whole stream

Everything here is built on one rearrangement of the compound interest formula. A lump sum $PV$ invested now grows to $FV = PV(1 + r)^n$ . Make $PV$ the subject:

PV = \frac{FV}{(1 + r)^n}.

This is discounting: dividing a future amount by $(1 + r)^n$ to find what it is worth today. For example, $10,000 due in $3$ years at $5\%$ per annum is worth $\dfrac{10000}{1.05^3} = \dfrac{10000}{1.157625} \approx 8638.38$ , i.e. $8638.38 today, because that amount banked now at $5\%$ would grow back to $10,000.

The present value of an annuity just applies this to every payment in the stream and adds them up. If a payment $M$ falls at the end of each of $n$ periods, the payment $k$ periods away is worth $\dfrac{M}{(1 + r)^k}$ today, so

PV = \frac{M}{(1 + r)} + \frac{M}{(1 + r)^2} + \cdots + \frac{M}{(1 + r)^n}.

That is a geometric series, a sum where each term is a fixed multiple of the one before. Just as the future-value series folds up into a neat formula, so does this one. You are not expected to add it up by hand in the exam, because the table does it for you. But seeing it as "the sum of each payment discounted back to today" is what makes a table answer make sense, rather than feel like magic.

A loan is the present value of its repayments

The clearest place this idea shows up is a loan. When a bank lends you money today, the amount it hands over is exactly the present value of all the repayments you promise to make. The lender is happy with this, because those future repayments, discounted at the loan rate, are worth exactly what they gave you now. So for a reducing-balance loan (one paid off by equal instalments),

\text{amount borrowed} = \text{(present-value factor)} \times \text{repayment per period}.

That one sentence lets you back-solve a repayment straight from a present-value table: just divide the amount borrowed by the table factor. It is the same method as a savings goal, only read in the other direction. That is why the present-value table and reducing-balance loans are two views of one idea.

Two tables, one method

There are two annuity tables and they are easy to confuse, so fix the difference firmly.

The future value of $1$ table answers: if I pay in $1 at the end of each period, how much have I accumulated by the end? The factor is $\dfrac{(1 + r)^n - 1}{r}$ . These factors are bigger than $n$ (interest has been added on).
The present value of $1$ table answers: a stream of $1 payments at the end of each period is worth how much as a single amount today? The factor is $\dfrac{1 - (1 + r)^{-n}}{r}$ . These factors are smaller than $n$ (future money is discounted down).

The procedure is identical for both: find the row for the number of periods, the column for the rate per period, read the factor at the intersection, and multiply by the contribution. The only decision is which table, and that is decided by the word "today" or "now" (present value) versus "after $n$ periods", "accumulated" or "final balance" (future value).

Here are small tables for $1, with the factors computed to $4$ decimal places. First the future value of $1 at the end of each period:

Period	$2\%$	$4\%$	$6\%$	$8\%$	$10\%$	$12\%$
$1$	$1.0000$	$1.0000$	$1.0000$	$1.0000$	$1.0000$	$1.0000$
$2$	$2.0200$	$2.0400$	$2.0600$	$2.0800$	$2.1000$	$2.1200$
$3$	$3.0604$	$3.1216$	$3.1836$	$3.2464$	$3.3100$	$3.3744$
$4$	$4.1216$	$4.2465$	$4.3746$	$4.5061$	$4.6410$	$4.7793$
$5$	$5.2040$	$5.4163$	$5.6371$	$5.8666$	$6.1051$	$6.3528$
$6$	$6.3081$	$6.6330$	$6.9753$	$7.3359$	$7.7156$	$8.1152$

And the present value of $1 paid at the end of each period:

Period	$2\%$	$4\%$	$6\%$	$8\%$	$10\%$	$12\%$
$1$	$0.9804$	$0.9615$	$0.9434$	$0.9259$	$0.9091$	$0.8929$
$2$	$1.9416$	$1.8861$	$1.8334$	$1.7833$	$1.7355$	$1.6901$
$3$	$2.8839$	$2.7751$	$2.6730$	$2.5771$	$2.4869$	$2.4018$
$4$	$3.8077$	$3.6299$	$3.4651$	$3.3121$	$3.1699$	$3.0373$
$5$	$4.7135$	$4.4518$	$4.2124$	$3.9927$	$3.7908$	$3.6048$
$6$	$5.6014$	$5.2421$	$4.9173$	$4.6229$	$4.3553$	$4.1114$

Notice the first row of the future-value table is all $1.0000$ : a single payment at the end of period $1$ in a $1$ -period annuity earns no interest, so $1 paid is $1 accumulated. In the present-value table the first row is the plain one-period discount factor $\dfrac{1}{1 + r}$ , for example $\dfrac{1}{1.06} = 0.9434$ at $6\%$ .

Converting an annual rate to the per-period row

Tables are labelled by the rate and the count per period, not per year. So when interest is added more than once a year (called sub-annual compounding), you must convert before you read the table, exactly as you do with the formula. Divide the yearly rate by the number of compounding periods in a year to get the column. Multiply the number of years by that same number to get the row.

Yearly contributions, yearly compounding: column $= R$ , row $=$ years.
Quarterly: column $= R/4$ , row $= 4 \times$ years.
Monthly: column $= R/12$ , row $= 12 \times$ years.

So $2000 per quarter for $1$ year at $8\%$ per annum compounded quarterly is read at column $8\% \div 4 = 2\%$ and row $4 \times 1 = 4$ , not at $8\%$ and $1$ . Getting this conversion wrong is the most common table error, because the headings tempt you to read the yearly figures straight off.

How exam questions ask about present value and tables

The wording is the whole game. Each phrasing points at one table and one operation. Learn to translate:

"What single amount today is equivalent to..." or "What lump sum must be invested now to provide..." A present-value question: read the present-value table and multiply, $PV = \text{factor} \times M$ .
"What will the account be worth after $n$ years / at the end of the term?" A future-value question: read the future-value table and multiply.
"Use the table to find the future value of an annuity of $X..." Straight table read: convert to the per-period row and column, read the factor, multiply by $X$ .
"Find the payment / contribution per period of an annuity whose present value (or future value) is $X." Back-solve: divide the target by the table factor, because $\text{target} = \text{factor} \times M$ means $M = \dfrac{\text{target}}{\text{factor}}$ .
"$A invested at the end of each year for $n$ years grows to $B. Find the interest rate." Read across a row: compute the factor you need as $\dfrac{B}{A}$ , then look along the period- $n$ row of the future-value table for the column whose factor matches.
"How much can be borrowed if the repayment is $R per period..." A loan is the present value of its repayments: amount borrowed $= \text{present-value factor} \times R$ . Reverse it to find the repayment from a known loan.

Annuity tables and present value: reading, back-solving, and a loan

Five problems read a future-value table and a present-value table forwards, back-solve a contribution and a rate, and treat a loan as the present value of its repayments. Every factor is taken from the tables above, computed to $4$ decimal places.

Find the present value of a retirement income stream

A retiree will draw a pension of $9000 at the end of each year for $4$ years from a fund earning $6\%$ per annum compounded annually. What single amount must be in the fund today to provide this?

Choose the table: The phrase "single amount today" means present value, so use the present-value-of- $1$ table.
Read the factor at the intersection: Compounding is annual, so the column is $6\%$ and the row is period $4$ . The factor is $3.4651$ .
Multiply by the payment: The present value is the factor times the annual draw.

PV = 3.4651 \times 9000 = 31185.90

i.e. about $31,185.90. (Using the unrounded factor gives $31,185.95; the $4$ decimal place table introduces a few cents of rounding, which is expected.)

Cross-check against the diagram. Discounting each payment by hand gives $\dfrac{9000}{1.06} + \dfrac{9000}{1.06^2} + \dfrac{9000}{1.06^3} + \dfrac{9000}{1.06^4} = 8490.57 + 8009.97 + 7556.57 + 7128.84 = 31185.95$ , which matches the table to the cent. This is the calculation drawn in the timeline at the top of the page.

Final answer: about $31,186 must be in the fund today.

Read a future value table for a quarterly investment

Priya invests $2500 at the end of each quarter into an account paying $8\%$ per annum compounded quarterly, for $1$ year. Use the future-value table to find what she has at the end of the year.

Convert to the per-period row and column: Compounding is quarterly, so the column is $8\% \div 4 = 2\%$ and the row is $4 \times 1 = 4$ quarters. This conversion is the step the headings tempt you to skip.
Read the factor: At period $4$ , rate $2\%$ , the future-value factor is $4.1216$ .
Multiply by the contribution: The future value is the factor times the quarterly payment.

FV = 4.1216 \times 2500 = 10304.02

i.e. $10,304.02. Total contributed is $4 \times 2500 = 10000$ , i.e. $10,000, so the interest earned is $304.02.

Final answer: Priya has $10,304.02 at the end of the year, including $304.02 of interest.

Back-solve the contribution needed for a savings goal

The Nguyen family wants $50,000 in $5$ years for a house deposit, saving at the end of each year into an account paying $10\%$ per annum compounded annually. Use the future-value table to find the annual contribution.

Set up the table relationship: The future value equals the future-value factor times the unknown payment: $FV = \text{factor} \times M$ .
Read the factor: At period $5$ , rate $10\%$ , the future-value factor is $6.1051$ .
Divide the target by the factor: Rearranging $50000 = 6.1051 \times M$ for $M$ means dividing.

M = \frac{50000}{6.1051} \approx 8189.87

i.e. about $8189.87 per year.

Final answer: the family must save about $8190 at the end of each year to reach $50,000 in $5$ years.

A car loan as the present value of its repayments

Jordan borrows $24,000 for a car at $10\%$ per annum compounded annually and repays it in equal instalments at the end of each year for $6$ years. Use the present-value table to find the annual repayment.

See the loan as a present value: The amount borrowed is the present value of the repayment stream: $\text{amount borrowed} = \text{present-value factor} \times \text{repayment}$ .
Read the factor: At period $6$ , rate $10\%$ , the present-value-of- $1$ factor is $4.3553$ .
Divide the loan by the factor: Rearranging $24000 = 4.3553 \times R$ for the repayment $R$ means dividing.

R = \frac{24000}{4.3553} \approx 5510.53

i.e. about $5510.53 per year. (The unrounded factor gives $5510.58; the difference is table rounding.)

Final answer: Jordan repays about $5510.53 at the end of each year, so over $6$ years pays back about $33,063, of which roughly $9063 is interest.

Read across a row to find the interest rate

An annuity of $20,420 invested at the end of each year for $3$ years grows to a future value of $63,743.07. Use the future-value table to find the annual interest rate.

Compute the factor you need. Since $FV = \text{factor} \times M$ , the factor must be $\dfrac{FV}{M}$ .

\text{factor} = \frac{63743.07}{20420} = 3.1216

Match the factor along the period- $3$ row: Look across the row for period $3$ in the future-value table for the column whose factor is $3.1216$ . The $2\%$ column is $3.0604$ , the $4\%$ column is $3.1216$ , the $6\%$ column is $3.1836$ .
Read off the rate: The exact match is the $4\%$ column.
Final answer: the annuity earns $4\%$ per annum.

Edge case: which table when the question gives you both numbers

Some questions hand you a future value and ask for a present value, or vice versa, to see whether you discount or accumulate. Anchor on the timeline. If the money you are given sits at the end of the term and you are asked for an equivalent amount now, you are discounting, so use the present-value table (or divide a future value by $(1 + r)^n$ for a single sum). If you are given regular contributions and asked for the final balance, you are accumulating, so use the future-value table. The two tables are linked: the present value of an annuity grown forward for $n$ periods equals its future value, since $PV \times (1 + r)^n = FV$ . In the pension example, $31,185.95 grown at $6\%$ for $4$ years gives $31185.95 \times 1.06^4 = 39371.54$ , which is exactly the future value of the same $9000 annuity, $9000 \times 4.3746 = 39371.54$ .

Edge case: tables only go so far

A printed table has a fixed set of rows and columns. Sometimes a question needs a rate or a period the table does not list, like a $7\%$ column when the table jumps from $6\%$ to $8\%$ , or period $30$ when it stops at $6$ . Then the table cannot answer it and you must fall back on the formula. NESA picks table questions so the period and rate you need land on the grid. So if your converted row or column is not in the table, recheck the conversion before assuming you need the formula. The table and the formula always agree where both apply, because the table is just the formula worked out ahead of time for common values.

Exam technique

State which table you are using and why in one short line: "present value, single amount needed today" or "future value, balance after the term". Markers award method marks for choosing the right table and the right intersection even if the final arithmetic slips. Always convert to the per-period row and column before reading: write the rate per period and the number of periods as your first step when compounding is quarterly or monthly. To read forwards, multiply the factor by the contribution; to back-solve a contribution or a loan repayment, divide the target by the factor; to find a rate, compute the needed factor as target over contribution and scan along the correct row. Keep the table factor to its full $4$ decimal places until the last step and round only the final money answer, to cents, noting that table rounding can leave a few cents difference from the formula.

Common traps

Reading the wrong table: Future value accumulates (factors bigger than $n$ ); present value discounts (factors smaller than $n$ ). The words "today" and "now" signal present value; "final balance" and "after $n$ years" signal future value.
Forgetting to convert the rate and period: A table is labelled per period. For $8\%$ per annum compounded quarterly over $1$ year, read column $2\%$ and row $4$ , not column $8\%$ and row $1$ .
Multiplying when you should divide: To find the value of an annuity, multiply the factor by the contribution. To find the required contribution or repayment from a known target, divide the target by the factor.
Discounting a single sum with the annuity factor: The present value of one future lump sum uses $PV = \dfrac{FV}{(1 + r)^n}$ , not an annuity table factor. The annuity factor is only for a stream of equal payments.
Expecting table and formula to agree to the last cent: Table factors are rounded to $4$ decimal places, so a table answer can differ from the formula by a few cents. That is normal; quote the table answer for a table question.

Exam-style practice questions

Practice questions written in the style of NESA exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

2021 HSC-style (table-style)3 marksA retiree will draw $9600 at the end of each year for

7

years from a fund earning

4\%

per annum compounded annually. Using the present-value-of-

1

annuity table, find the single amount that must be in the fund today.

Show worked answer →

The present value of the annuity is the lump sum today that is equivalent to the stream of future withdrawals.

Read the present-value-of- $1$ table at period $7$ , rate $4\%$ : the factor is $6.0021$ .

$PV = 6.0021 \times 9600 = 57619.72$ , i.e. $57619.72.

Markers reward selecting the present-value (not future-value) table, reading the correct intersection, multiplying by the contribution, and rounding to cents.

2022 HSC-style (table-style)3 marksAn annuity has a present value of $33240 at

6\%

per annum compounded annually for

3

years. Using the present-value-of-

1

annuity table, find the payment per period.

Show worked answer →

Read the present-value-of- $1$ table at period $3$ , rate $6\%$ : the factor is $2.6730$ .

The present value equals the factor times the payment, so $33240 = 2.6730 \times \text{PMT}$ .

Divide: $\text{PMT} = \dfrac{33240}{2.6730} \approx 12435.47$ , i.e. about $12435.

Markers reward writing $PV = \text{factor} \times \text{PMT}$ , dividing the target by the table factor, and rounding sensibly.

Practice questions

Original practice questions graded from foundation to exam level, each with a full worked solution. Try them before revealing the solution.

foundation1 marksAn annuity will pay $5000 at the end of each year for

3

years, with money worth

4\%

per annum compounded annually. Using the present-value-of-

1

annuity table, the factor at period

3

, rate

4\%

2.7751

. Find the present value of the annuity.

Show worked solution →

Choose the operation: A present value is wanted from a stream of payments, so multiply the table factor by the payment.
Use the given factor: Compounding is annual, so the factor is the one read at period $3$ , rate $4\%$ , namely $2.7751$ .
Multiply by the payment: The present value is the factor times the annual payment.

PV = 2.7751 \times 5000 = 13875.50.

State the answer. Answer: the present value of the annuity is $13,875.50.

foundation2 marksA retiree will draw a pension of $7500 at the end of each year for

4

years from a fund earning

6\%

per annum compounded annually. Using the present-value-of-

1

annuity table, find the single amount that must be in the fund today.

Show worked solution →

Choose the table: The phrase "single amount today" signals present value, so use the present-value-of- $1$ table.
Read the factor at the intersection: Compounding is annual, so the column is $6\%$ and the row is period $4$ . The factor is $3.4651$ .
Multiply by the payment: The present value is the factor times the annual draw.

PV = 3.4651 \times 7500 = 25988.25.

State the answer. About $25,988.25 must be in the fund today. Check. Discounting each payment by hand gives $\dfrac{7500}{1.06} + \dfrac{7500}{1.06^2} + \dfrac{7500}{1.06^3} + \dfrac{7500}{1.06^4} = 7075.47 + 6674.97 + 6297.14 + 5940.70 = 25988.29$ , matching the table to a few cents of rounding.

foundation2 marksMia invests $4000 at the end of each year into an account paying

10\%

per annum compounded annually, for

3

years. Using the future-value-of-

1

annuity table, find the balance at the end of the term.

Show worked solution →

Choose the table: "Balance at the end of the term" signals future value, so use the future-value-of- $1$ table.
Read the factor: Compounding is annual, so the column is $10\%$ and the row is period $3$ . The factor is $3.3100$ .
Multiply by the contribution: The future value is the factor times the annual payment.

FV = 3.3100 \times 4000 = 13240.00.

State the answer. The balance is $13,240.00. Check. The total paid in is $3 \times 4000 = 12000$ , i.e. $12,000.00, so the interest earned is $13240 - 12000 = 1240$ , i.e. $1240.00, which is positive as it must be for an account that earns interest.

foundation2 marksSam pays $6000 into an account at the end of each year for

2

years at

12\%

per annum compounded annually. Using the future-value-of-

1

annuity table, the factor at period

2

, rate

12\%

2.1200

. Find the balance at the end of the term.

Show worked solution →

Choose the operation: A final balance is wanted from regular payments, so multiply the future-value factor by the contribution.
Use the given factor: Compounding is annual, so the factor read at period $2$ , rate $12\%$ is $2.1200$ .
Multiply by the contribution: The future value is the factor times the annual payment.

FV = 2.1200 \times 6000 = 12720.00.

State the answer. Answer: the balance at the end of the term is $12,720.00.

core3 marksA fund must provide a payment of $11\,000 at the end of each year for

5

years to a beneficiary, with the fund earning

8\%

per annum compounded annually. Using the present-value-of-

1

annuity table, find the single amount needed in the fund today.

Show worked solution →

Choose the table: The payments fall in the future and a single amount is wanted now, so this is a present value: use the present-value-of- $1$ table.
Read the factor: Compounding is annual, so the column is $8\%$ and the row is period $5$ . The factor is $3.9927$ .
Multiply by the payment: The present value is the factor times the annual payment.

PV = 3.9927 \times 11000 = 43919.70.

State the answer. About $43,919.70 is needed in the fund today. Check. The total drawn over the five years is $5 \times 11000 = 55000$ , i.e. $55,000.00, and the present value $43,919.70 is sensibly less than this, because future money is discounted back to today.

core3 marksPriya invests $1500 at the end of each quarter into an account paying

8\%

per annum compounded quarterly, for

1

year. Using the future-value-of-

1

annuity table, find the balance at the end of the year and the interest earned.

Show worked solution →

Convert to the per-period row and column: Compounding is quarterly, so the column is $8\% \div 4 = 2\%$ and the row is $4 \times 1 = 4$ quarters, not column $8\%$ and row $1$ .
Read the factor: At period $4$ , rate $2\%$ , the future-value factor is $4.1216$ .
Multiply by the contribution: The future value is the factor times the quarterly payment.

FV = 4.1216 \times 1500 = 6182.40.

Find the interest. The total paid in is $4 \times 1500 = 6000$ , i.e. $6000.00, so the interest is $6182.40 - 6000 = 182.40$ .

State the answer. The balance is $6182.40, including $182.40 of interest. Check. Reading the unconverted row $1$ at $8\%$ would give a factor of $1.0000$ and a balance of only $1500, which is clearly too small for four deposits, confirming the conversion to row $4$ , column $2\%$ was needed.

core3 marksThe Lee family wants $40\,000 in

5

years for a house deposit, saving an equal amount at the end of each year into an account paying

6\%

per annum compounded annually. Using the future-value-of-

1

annuity table, find the annual contribution required.

Show worked solution →

Set up the table relationship: The future value equals the future-value factor times the unknown payment, $FV = \text{factor} \times M$ , so to find the payment you divide.
Read the factor: Compounding is annual, so the column is $6\%$ and the row is period $5$ . The future-value factor is $5.6371$ .
Divide the target by the factor: Rearranging $40000 = 5.6371 \times M$ for $M$ gives

M = \frac{40000}{5.6371} \approx 7095.85.

State the answer. The family must save about $7095.85 at the end of each year. Check. Multiplying back, $5.6371 \times 7095.85 \approx 40000$ , recovering the target, so the division was set up correctly.

core3 marksA retirement fund must pay an equal amount at the end of each year for

6

years to a member, and the fund earns

10\%

per annum compounded annually. The single amount in the fund today is $25\,000. Using the present-value-of-

1

annuity table, the factor at period

6

, rate

10\%

4.3553

. Find the annual payment the fund can provide.

Show worked solution →

See the relationship: The single amount today is the present value of the payment stream, so $PV = \text{factor} \times \text{PMT}$ , and the payment is found by dividing.
Use the given factor: Compounding is annual, so the present-value factor is the one at period $6$ , rate $10\%$ , namely $4.3553$ .
Divide the present value by the factor: Rearranging $25000 = 4.3553 \times \text{PMT}$ for the payment gives

\text{PMT} = \frac{25000}{4.3553} \approx 5740.13.

State the answer. Answer: the fund can pay about $5740.13 at the end of each year. Check. Multiplying back, $4.3553 \times 5740.13 \approx 25000$ , recovering the present value, so the division was set up correctly.

exam4 marksJordan borrows $28\,000 for a car at

6\%

per annum compounded annually and repays it in equal instalments at the end of each year for

6

years. Using the present-value-of-

1

annuity table, find the annual repayment, and hence the total interest paid over the life of the loan.

Show worked solution →

See the loan as a present value: The amount borrowed is the present value of the repayment stream, so $\text{amount borrowed} = \text{present-value factor} \times \text{repayment}$ , and the repayment is found by dividing.
Read the factor: Compounding is annual, so the column is $6\%$ and the row is period $6$ . The present-value-of- $1$ factor is $4.9173$ .
Divide the loan by the factor: Rearranging $28000 = 4.9173 \times R$ for the repayment $R$ gives

R = \frac{28000}{4.9173} \approx 5694.18.

Find the total interest. Over $6$ years the total repaid is $6 \times 5694.18 = 34165.08$ , i.e. about $34,165.08, so the interest is $34165.08 - 28000 = 6165.08$ , about $6165.08.

State the answer. Jordan repays about $5694.18 at the end of each year, paying about $6165.08 in interest in total. (Using the unrounded factor gives a repayment of about $5694.15; the few cents difference is table rounding.) Check. The total repaid $34,165.08 exceeds the $28,000 borrowed, as it must for a loan that charges interest.

exam3 marksAn annuity of $12\,500 invested at the end of each year for

3

years grows to a future value of $40\,580. Using the future-value-of-

1

annuity table, find the annual interest rate.

Show worked solution →

Compute the factor you need. Since $FV = \text{factor} \times M$ , the required factor is the future value divided by the contribution.

\text{factor} = \frac{40580}{12500} = 3.2464.

Match the factor along the period- $3$ row: Scan the row for period $3$ in the future-value table for the column whose factor is $3.2464$ . The $6\%$ column is $3.1836$ , the $8\%$ column is $3.2464$ , and the $10\%$ column is $3.3100$ .
Read off the rate: The exact match is the $8\%$ column.
State the answer: The annuity earns $8\%$ per annum. Check. Reading forwards at $8\%$ , $3.2464 \times 12500 = 40580$ , which is the given future value, confirming the rate.

exam4 marksAn annuity has a present value of $30\,000 at

12\%

per annum compounded annually for

4

years. Using the present-value-of-

1

annuity table, find the payment per period.

Show worked solution →

Identify the table and operation: A present value is given and the payment is wanted, so use the present-value-of- $1$ table and back-solve, since $PV = \text{factor} \times \text{PMT}$ means you divide.
Read the factor: Compounding is annual, so the column is $12\%$ and the row is period $4$ . The present-value-of- $1$ factor is $3.0373$ .
Divide the present value by the factor: Rearranging $30000 = 3.0373 \times \text{PMT}$ for the payment gives

\text{PMT} = \frac{30000}{3.0373} \approx 9877.19.

State the answer. The payment per period is about $9877.19. Check. Multiplying back, $3.0373 \times 9877.19 \approx 30000$ , recovering the present value, so the back-solve was set up correctly.

exam5 marksAisha saves $3000 at the end of each year for

5

years into an account paying

8\%

per annum compounded annually. From the annuity tables, the future-value-of-

1

factor at period

5

, rate

8\%

5.8666

, and the present-value-of-

1

factor at the same intersection is

3.9927

. (a) Find the balance at the end of the

5

years and the total interest earned. (b) Find the single amount Aisha could instead invest today to provide the same future balance, and confirm it is the present value of the annuity.

Show worked solution →

Part (a): choose the future-value table. A final balance from regular savings means future value, so multiply the future-value factor by the contribution.

FV = 5.8666 \times 3000 = 17599.80.

The total paid in is $5 \times 3000 = 15000$ , i.e. $15,000.00, so the interest earned is $17599.80 - 15000 = 2599.80$ .

Part (a) answer. The balance is $17,599.80, including $2599.80 of interest.

Part (b): use the present-value table. The lump sum today equivalent to the same stream is the present value of the annuity, found by multiplying the present-value factor by the payment.

PV = 3.9927 \times 3000 = 11978.10.

Confirm the link. Growing this present value forward for $5$ years gives $11978.10 \times 1.08^5 \approx 17599.76$ , matching the balance in part (a) to a few cents of table rounding, so it is indeed the present value of the annuity.

Part (b) answer. Answer: (a) the balance is $17,599.80 with $2599.80 interest; (b) the equivalent single amount today is $11,978.10, the present value of the annuity.

What this dot point is asking

The answer

Present value of a single payment, then of a whole stream

A loan is the present value of its repayments

Two tables, one method

Converting an annual rate to the per-period row

How exam questions ask about present value and tables

Edge case: which table when the question gives you both numbers

Edge case: tables only go so far

Exam-style practice questions

Practice questions

Related dot points