Annuities Revision Notes for HSC SSCE Mathematics Standard

Using a Present Value Table

What is present value?

Present value is a fundamental concept in financial mathematics that helps us understand the time value of money. When we talk about the present value of an annuity, we're asking: "How much money would I need to invest right now to achieve the same result as making regular payments over time?"

An annuity involves making equal payments at regular intervals, such as monthly deposits into a savings account. The present value represents the lump sum amount you would need today to match what that series of payments will be worth in the future.

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In mathematical terms, present value is the principal ( $P$ ) in the compound interest formula, while the future value is represented as the amount ( $A$ or $FV$ ). These quantities are related through the compound interest formula, which forms the foundation of all present value calculations.

Understanding the formula

The relationship between future value and present value is expressed through the compound interest formula:

$FV = PV(1 + r)^n$

where:

$FV$ = future value (the total amount accumulated)
$PV$ = present value (the initial investment needed)
$r$ = interest rate per period (as a decimal)
$n$ = number of time periods

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If we rearrange this formula to make present value the subject, we divide both sides by $(1 + r)^n$ :

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

This formula shows that present value is the future value discounted by the compound interest factor. This is a critical concept: present value is always less than future value because we're accounting for the interest that will be earned over time.

However, calculating this repeatedly can be time-consuming, which is where present value tables become incredibly useful tools for quick and accurate calculations.

How to use a present value table

Present value tables simplify calculations by providing pre-calculated values for different combinations of time periods and interest rates. The process involves three straightforward steps:

Step 1: Determine the time period and rate of interest

Identify how many payment periods are involved and what interest rate applies per period. Remember to adjust the interest rate if compounding occurs more frequently than annually (for example, monthly or quarterly).

Step 2: Find the intersection of the time period and rate of interest in the table

Locate the row corresponding to your time period and the column corresponding to your interest rate. The value where they meet is your intersection value.

Step 3: Multiply the intersection value by the payment amount

Take the intersection value from the table and multiply it by the regular payment amount to find the present value.

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Quick Formula: $PV = \text{intersection value} \times \text{payment amount}$

This simple relationship makes present value tables extremely efficient for financial calculations. The intersection value represents all the complex discounting calculations already done for you!

Worked example: Finding present value

Let's examine a straightforward example of using a present value table to find the present value when we know the regular contribution amount.

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Worked Example: Calculate the present value of annual contributions

Question: Calculate the present value if $9600 is contributed per year for 7 years at 4% p.a. compounded annually.

End of year	3%	4%	5%	6%
6	5.4172	5.2421	5.0757	4.9173
7	6.2303	6.0021	5.7864	5.5824
8	7.0197	6.7327	6.4632	6.2098
9	7.7861	7.4353	7.1078	6.8017

Solution:

Step 1: Identify the time period and interest rate

Time period: 7 years
Interest rate: 4% p.a.

Step 2: Find the intersection value from the table

Looking at the row for period 7 and column for 4%
Intersection value is 6.0021

Step 3: Multiply the intersection value by the annual contribution

$PV = 6.0021 \times 9600$

$PV = 57620.16$

Answer: The present value is $57,620.16.

Interpretation: This means that if you invested $57,620.16 today at 4% p.a. compound interest, it would be equivalent to contributing $9,600 annually for 7 years.

Advanced applications

Present value tables can be used for more complex scenarios involving different compounding periods and even for finding unknown payment amounts when the present value is known. Understanding these applications will help you tackle a wide variety of financial mathematics problems.

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Critical Concept: Adjusting for Compounding Periods

When interest is compounded more frequently than annually, you must:

Divide the annual interest rate by the number of compounding periods per year
Multiply the number of years by the compounding periods per year

This adjustment is essential for accurate calculations and is a common source of errors in exams.

Multiple scenarios

Consider the following present value table with various interest rates and time periods:

Period	1%	2%	3%	4%	6%	8%	10%	12%
1	0.99	0.98	0.97	0.96	0.94	0.93	0.91	0.89
2	1.97	1.94	1.91	1.89	1.83	1.78	1.74	1.69
3	2.94	2.88	2.83	2.78	2.67	2.58	2.49	2.40
4	3.90	3.81	3.71	3.63	3.47	3.31	3.17	3.04

Let's work through several different types of problems using this table to demonstrate the versatility of present value calculations.

Example A: Monthly compounding

When payments and compounding occur monthly, we need to adjust both the interest rate and the number of periods accordingly.

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Worked Example: Present value with monthly compounding

Question: What is the present value of an investment of $500 per month for 2 months at 24% p.a. compounded monthly?

Solution:

Step 1: Determine the period interest rate

Since interest is 24% per annum but compounded monthly:

$r = \frac{24\%}{12} = 2\%$

Number of periods: $n = 2$ months

Step 2: Find the intersection value

Period: 2
Interest rate: 2%
Intersection value is 1.94

Step 3: Calculate present value

$PV = 1.94 \times 500$

$PV = 970$

Answer: The present value is $970.

Example B: Quarterly compounding

Quarterly compounding requires dividing the annual rate by 4 and multiplying the years by 4 to get the number of periods.

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Worked Example: Present value with quarterly compounding

Question: What is the present value of an investment of $7,000 per quarter for 1 year at 12% p.a. compounded quarterly?

Solution:

Step 1: Determine the period interest rate

Since interest is 12% per annum but compounded quarterly:

$r = \frac{12\%}{4} = 3\%$

Number of periods: $n = 4 \times 1 = 4$ quarters

Step 2: Find the intersection value

Period: 4
Interest rate: 3%
Intersection value is 3.71

Step 3: Calculate present value

$PV = 3.71 \times 7000$

$PV = 25970$

Answer: The present value is $25,970.

Example C: Finding the payment amount

Sometimes we know the present value and need to find what payment amount is required. The table can still help us with this reverse calculation by rearranging our formula.

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Worked Example: Finding the unknown payment amount

Question: Find the payment per period of an annuity with a present value of $33,240 at 6% p.a. compounded annually for 3 years.

Solution:

Step 1: Find the intersection value

Period: 3 years
Interest rate: 6%
Intersection value is 2.67

Step 2: Set up the equation

We know that:

$PV = \text{intersection value} \times PMT$

$33240 = 2.67 \times PMT$

Step 3: Solve for the payment amount

$PMT = \frac{33240}{2.67}$

$PMT = 12449$

Answer: The payment is $12,449.

Example D: Biannual compounding with unknown payment

This example combines both concepts: adjusting for compounding frequency and solving for an unknown payment.

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Worked Example: Unknown payment with biannual compounding

Question: Find the payment per period of an annuity with a present value of $10,000 at 8% p.a. compounded biannually for 1 year.

Solution:

Step 1: Determine the period interest rate

Since interest is 8% per annum but compounded biannually (twice per year):

$r = \frac{8\%}{2} = 4\%$

Number of periods: $n = 2 \times 1 = 2$ periods

Step 2: Find the intersection value

Period: 2
Interest rate: 4%
Intersection value is 1.89

Step 3: Set up and solve the equation

$PV = \text{intersection value} \times PMT$

$10000 = 1.89 \times PMT$

$PMT = \frac{10000}{1.89}$

$PMT = 5291$

Answer: The payment is $5,291.

Exam tips

When working with present value tables in your exam, following these strategies will help you avoid common mistakes and maximize your marks.

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Essential Exam Strategies:

Always check whether the interest rate needs adjusting for the compounding period
Calculate the number of periods carefully (multiply frequency by years)
Round your final answer appropriately (usually to the nearest dollar or cent)
Show all working steps clearly - this earns you method marks even if your final answer is incorrect
Double-check that you're reading from the correct row and column intersection
Remember the formula can work both ways: finding PV from payments or finding payments from PV

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Common Mistakes to Avoid:

Forgetting to adjust the interest rate when compounding is not annual
Using years instead of periods (e.g., using 2 years instead of 8 quarters)
Reading from the wrong intersection in the table
Forgetting to show the formula before substituting values

Summary

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Key Points to Remember:

Present value is the amount you need to invest now to match the future value of an annuity - it answers the question "how much do I need today?"
The fundamental relationship is expressed by: $PV = \frac{FV}{(1 + r)^n}$ , which shows present value as discounted future value
Three steps to use the table:
1. Identify time period and rate
2. Find intersection value
3. Multiply by contribution amount
Always adjust interest rates and time periods to match the compounding frequency - this is the most common source of errors
The table works in reverse too: divide present value by the intersection value to find the required payment amount using $PMT = \frac{PV}{\text{intersection value}}$
Present value calculations help you understand the true cost or value of financial commitments made over time, making them essential for real-world financial decision-making

Annuities (HSC SSCE Mathematics Standard): Revision Notes

Using a Present Value Table

What is present value?

Understanding the formula

How to use a present value table

Worked example: Finding present value

Advanced applications

Multiple scenarios

Example A: Monthly compounding

Example B: Quarterly compounding

Example C: Finding the payment amount

Example D: Biannual compounding with unknown payment

Exam tips

Summary

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