Chapter 12

Compound Interest, Exponents, and Present Value

Move a single sum forward or backward on a timeline.

A cash amount means more when its date is clear. This chapter builds timeline models for future expenses and savings, using hypothetical people, rates, and amounts rather than promised returns. Money is rounded to cents at the end unless a minimum-goal deposit must be rounded upward.

12.1

Exponents and Compound Growth

Learning objectives

After this section you will be able to:

Place the relevant cash flows on a timeline.
Calculate the modeled value using a matching periodic rate.
Interpret the result and identify the model's assumptions.

Real-life situation

A student savings timeline

A Philadelphia student uses a hypothetical fixed-rate model to compare cash flows for future school or equipment costs.

A student savings timeline

Initial deposit: $1,000.00
Annual rate: 5%; annual compounding; 2 years
Year 1: $1,050.00; year 2: $1,102.50

Figure 12.1 A student savings timeline (hypothetical)

The concept

An exponent tells how many times a factor is multiplied by itself. In (1 + i)ⁿ, i is the rate per compounding period and n counts periods, not necessarily years. Square means two factors; it does not mean multiply the factor by two.

Compound interest adds each period's interest to the balance, so later interest is earned on both original principal and earlier interest. At a positive fixed rate this produces more growth than simple interest over multiple periods.

For a single deposit with no withdrawals or fees, multiply principal by the accumulation factor. Keep all calculator digits until the final result. These models round money only at the end; a provider that rounds each period can produce a slightly different result.

All rates and cash flows are hypothetical classroom assumptions, not investment advice or promised returns. Retain full precision through intermediate calculations and round final money to cents; for a minimum savings target, round the deposit upward instead. Draw the timeline before calculating and explain what the result means.

Compound interest timeline

Year 0
$1,000.00
Principal
Year 1
$1,050.00
Interest: $50.00
Year 2
$1,102.50
Interest: $52.50

Year 0: deposit $1,000.
Year 1: multiply by 1.05; $50 interest gives $1,050.
Year 2: multiply the new balance by 1.05; $52.50 interest gives $1,102.50. Growth is not guaranteed in an investment.

Original WealthWiseGuys diagram · Hypothetical teaching quantities.

Money Words

Plain-language words you will use in this lesson — learn them before we reach the formulas.

compound interest
Interest earned on both the original amount and the interest already earned.
present value
What a future amount of money is worth today at a given interest rate.
future value
What an amount of money today will be worth at a later date with interest.

Key terms

Exponent: The number of repeated factors.
Periodic rate: The rate applying in one compounding period.
Compound interest: Interest on principal and previously credited interest.
Future value: The modeled balance at a later date.

Formulas & rules

Exponents and Compound Growth
FV = P × (1 + i)ⁿ
Related relationship
Interest = FV − P

Worked examples

Example 1The timeline in the figure

Find future value of $1,000.00 at 5% annually compounded for 2 years.

Solution

1Use P = $1,000.00, i = 0.05, and n = 2.
2Growth factor = 1.05² = 1.1025.
3FV = 1000 × 1.1025 = $1,102.50.
4Interest = $1,102.50 − $1,000.00 = $102.50.

Answer: $1,102.50

Example 2A different amount

Find future value of $2,000.00 at 5% annually compounded for 2 years.

Solution

1Use P = $2,000.00, i = 0.05, and n = 2.
2Growth factor = 1.05² = 1.1025.
3FV = 2000 × 1.1025 = $2,205.00.
4Interest = $2,205.00 − $2,000.00 = $205.00.

Answer: $2,205.00

Practice exercises 12.1

Find future value of $1,000.00 at 5% annually compounded for 2 years.
Find future value of $2,000.00 at 5% annually compounded for 2 years.
Find future value of $500.00 at 5% annually compounded for 2 years.
Find future value of $3,000.00 at 5% annually compounded for 2 years.
Find future value of $4,000.00 at 5% annually compounded for 2 years.

Section 12.1 recap

Draw the timeline and match the rate to each interval.
Keep intermediate precision; distinguish modeled values from guaranteed outcomes.

Answer key with explanations: Section 12.1

Practice exercises

Use P = $1,000.00, i = 0.05, and n = 2. Growth factor = 1.05² = 1.1025. FV = 1000 × 1.1025 = $1,102.50. Interest = $1,102.50 − $1,000.00 = $102.50.
Use P = $2,000.00, i = 0.05, and n = 2. Growth factor = 1.05² = 1.1025. FV = 2000 × 1.1025 = $2,205.00. Interest = $2,205.00 − $2,000.00 = $205.00.
Use P = $500.00, i = 0.05, and n = 2. Growth factor = 1.05² = 1.1025. FV = 500.0 × 1.1025 = $551.25. Interest = $551.25 − $500.00 = $51.25.
Use P = $3,000.00, i = 0.05, and n = 2. Growth factor = 1.05² = 1.1025. FV = 3000 × 1.1025 = $3,307.50. Interest = $3,307.50 − $3,000.00 = $307.50.
Use P = $4,000.00, i = 0.05, and n = 2. Growth factor = 1.05² = 1.1025. FV = 4000 × 1.1025 = $4,410.00. Interest = $4,410.00 − $4,000.00 = $410.00.

Recap quiz

$1,102.50: Use P = $1,000.00, i = 0.05, and n = 2. Growth factor = 1.05² = 1.1025. FV = 1000 × 1.1025 = $1,102.50. Interest = $1,102.50 − $1,000.00 = $102.50.
Match the rate period to the timeline: The periodic rate and period count must describe the same interval; assumptions are not guarantees.
Retain full precision: Retaining calculator precision avoids accumulating rounding errors; round the final monetary result.

12.2

Compounding Frequency and Effective Rates

Learning objectives

After this section you will be able to:

Place the relevant cash flows on a timeline.
Calculate the modeled value using a matching periodic rate.
Interpret the result and identify the model's assumptions.

Real-life situation

A student savings timeline

A Philadelphia student uses a hypothetical fixed-rate model to compare cash flows for future school or equipment costs.

A student savings timeline

Initial deposit: $1,200.00
Nominal annual rate: 6%; compounded monthly
Monthly rate: 0.5%; periods: 12
One-year value: $1,274.01

Figure 12.2 A student savings timeline (hypothetical)

The concept

A nominal annual rate does not directly state the growth over a year when compounding occurs more often than annually. Divide the nominal rate by the number of equal compounding periods per year to find the periodic rate.

Monthly compounding uses m = 12 and n = 12 × years. Do not put the annual rate into every monthly period; doing so greatly overstates growth. A fractional year must correspond to the compounding schedule specified in the model.

The effective annual rate measures one full year of growth after compounding. At a positive nominal rate it exceeds the nominal rate if interest compounds more than once a year. Compare effective rates only after also checking fees, access restrictions, and risk.

Monthly compounding

1
Initial balance
$1,200
2
12 monthly periods
0.5% each → multiply by 1.005¹²
3
Future balance
$1,274.01

Figure 12.2.V Monthly compounding. Hypothetical teaching data.

Money Words

Plain-language words you will use in this lesson — learn them before we reach the formulas.

principal
The original amount of money borrowed or invested, before interest.
interest rate
The percent used to calculate how much interest is earned or charged.
investment
Money put into something with the hope it will grow or earn income later.

Key terms

Nominal annual rate: An annual quoted rate before within-year compounding.
Frequency: The number of compounding periods each year.
Effective annual rate: One-year proportional growth after compounding.
Period count: Total number of compounding intervals.

Formulas & rules

Compounding Frequency and Effective Rates
FV = P × (1 + r ÷ m)ⁿ
Related relationship
Effective annual rate = (1 + r ÷ m)ᵐ − 1

Worked examples

Example 1The timeline in the figure

Find the one-year value of $1,200.00 at 6% nominal annual interest compounded monthly.

Solution

1Use principal $1,200.00 and nominal r = 0.06.
2Monthly i = 0.06 ÷ 12 = 0.005; n = 12.
3FV = 1200 × 1.005¹² = $1,274.01.
4Effective annual rate = 1.005¹² − 1 ≈ 6.17%.

Answer: $1,274.01

Example 2A different amount

Find the one-year value of $2,400.00 at 6% nominal annual interest compounded monthly.

Solution

1Use principal $2,400.00 and nominal r = 0.06.
2Monthly i = 0.06 ÷ 12 = 0.005; n = 12.
3FV = 2400 × 1.005¹² = $2,548.03.
4Effective annual rate = 1.005¹² − 1 ≈ 6.17%.

Answer: $2,548.03

Practice exercises 12.2

Find the one-year value of $1,200.00 at 6% nominal annual interest compounded monthly.
Find the one-year value of $2,400.00 at 6% nominal annual interest compounded monthly.
Find the one-year value of $600.00 at 6% nominal annual interest compounded monthly.
Find the one-year value of $3,600.00 at 6% nominal annual interest compounded monthly.
Find the one-year value of $4,800.00 at 6% nominal annual interest compounded monthly.

Section 12.2 recap

Draw the timeline and match the rate to each interval.
Keep intermediate precision; distinguish modeled values from guaranteed outcomes.

Answer key with explanations: Section 12.2

Practice exercises

Use principal $1,200.00 and nominal r = 0.06. Monthly i = 0.06 ÷ 12 = 0.005; n = 12. FV = 1200 × 1.005¹² = $1,274.01. Effective annual rate = 1.005¹² − 1 ≈ 6.17%.
Use principal $2,400.00 and nominal r = 0.06. Monthly i = 0.06 ÷ 12 = 0.005; n = 12. FV = 2400 × 1.005¹² = $2,548.03. Effective annual rate = 1.005¹² − 1 ≈ 6.17%.
Use principal $600.00 and nominal r = 0.06. Monthly i = 0.06 ÷ 12 = 0.005; n = 12. FV = 600.0 × 1.005¹² = $637.01. Effective annual rate = 1.005¹² − 1 ≈ 6.17%.
Use principal $3,600.00 and nominal r = 0.06. Monthly i = 0.06 ÷ 12 = 0.005; n = 12. FV = 3600 × 1.005¹² = $3,822.04. Effective annual rate = 1.005¹² − 1 ≈ 6.17%.
Use principal $4,800.00 and nominal r = 0.06. Monthly i = 0.06 ÷ 12 = 0.005; n = 12. FV = 4800 × 1.005¹² = $5,096.05. Effective annual rate = 1.005¹² − 1 ≈ 6.17%.

Recap quiz

$1,274.01: Use principal $1,200.00 and nominal r = 0.06. Monthly i = 0.06 ÷ 12 = 0.005; n = 12. FV = 1200 × 1.005¹² = $1,274.01. Effective annual rate = 1.005¹² − 1 ≈ 6.17%.
Match the rate period to the timeline: The periodic rate and period count must describe the same interval; assumptions are not guarantees.
Retain full precision: Retaining calculator precision avoids accumulating rounding errors; round the final monetary result.

12.3

Present Value and Financial Equivalence

Learning objectives

After this section you will be able to:

Place the relevant cash flows on a timeline.
Calculate the modeled value using a matching periodic rate.
Interpret the result and identify the model's assumptions.

Real-life situation

A student savings timeline

A Philadelphia student uses a hypothetical fixed-rate model to compare cash flows for future school or equipment costs.

A student savings timeline

Future obligation: $1,210.00 in 2 years
Assumed annual discount rate: 10%
Present value: $1,000.00

Figure 12.3 A student savings timeline (hypothetical)

The concept

Present value asks how much today is mathematically equivalent to a specified future sum under a stated rate model. It reverses accumulation: divide by the growth factor instead of multiplying.

Place the future payment on a timeline and count the compounding intervals back to today. Use the rate per interval. Discounting at a positive rate makes present value smaller than future value; this is a useful reasonableness check.

A discount rate is an assumption, not a promised return. Financial equivalence in this model does not establish that an account can safely produce the assumed rate. Fees, inflation, taxes, and uncertainty are excluded unless specifically modeled.

Equivalent amounts at different dates

1
Present: year 0
$1,000
2
Assumed growth
10% annually for 2 years
3
Future: year 2
$1,210

Figure 12.3.V Equivalent amounts at different dates. Hypothetical teaching data.

Money Words

Plain-language words you will use in this lesson — learn them before we reach the formulas.

return
The money or gain an investment produces, often shown as a percent.
annuity
A series of equal payments made at regular intervals, such as monthly.
inflation
A general rise in prices over time, which makes money buy less.

Key terms

Present value: The equivalent amount at the start of a timeline.
Discounting: Moving a future amount backward using a rate model.
Discount rate: The assumed rate used to value later cash flows today.
Financial equivalence: Equality of values at a common date under specified assumptions.

Formulas & rules

Present Value and Financial Equivalence
PV = FV ÷ (1 + i)ⁿ
Related relationship
FV = PV × (1 + i)ⁿ

Worked examples

Example 1The timeline in the figure

Find present value of $1,210.00 due in 2 years using 10% annually compounded.

Solution

1Future value is $1,210.00; i = 0.10; n = 2.
2Accumulation factor = 1.10² = 1.21.
3PV = 1210 ÷ 1.21 = $1,000.00.
4Check by multiplying the unrounded PV by 1.21 to recover the future amount.

Answer: $1,000.00

Example 2A different amount

Find present value of $2,420.00 due in 2 years using 10% annually compounded.

Solution

1Future value is $2,420.00; i = 0.10; n = 2.
2Accumulation factor = 1.10² = 1.21.
3PV = 2420 ÷ 1.21 = $2,000.00.
4Check by multiplying the unrounded PV by 1.21 to recover the future amount.

Answer: $2,000.00

Practice exercises 12.3

Find present value of $1,210.00 due in 2 years using 10% annually compounded.
Find present value of $2,420.00 due in 2 years using 10% annually compounded.
Find present value of $605.00 due in 2 years using 10% annually compounded.
Find present value of $3,630.00 due in 2 years using 10% annually compounded.
Find present value of $4,840.00 due in 2 years using 10% annually compounded.

Section 12.3 recap

Draw the timeline and match the rate to each interval.
Keep intermediate precision; distinguish modeled values from guaranteed outcomes.

Answer key with explanations: Section 12.3

Practice exercises

Future value is $1,210.00; i = 0.10; n = 2. Accumulation factor = 1.10² = 1.21. PV = 1210 ÷ 1.21 = $1,000.00. Check by multiplying the unrounded PV by 1.21 to recover the future amount.
Future value is $2,420.00; i = 0.10; n = 2. Accumulation factor = 1.10² = 1.21. PV = 2420 ÷ 1.21 = $2,000.00. Check by multiplying the unrounded PV by 1.21 to recover the future amount.
Future value is $605.00; i = 0.10; n = 2. Accumulation factor = 1.10² = 1.21. PV = 605.0 ÷ 1.21 = $500.00. Check by multiplying the unrounded PV by 1.21 to recover the future amount.
Future value is $3,630.00; i = 0.10; n = 2. Accumulation factor = 1.10² = 1.21. PV = 3630 ÷ 1.21 = $3,000.00. Check by multiplying the unrounded PV by 1.21 to recover the future amount.
Future value is $4,840.00; i = 0.10; n = 2. Accumulation factor = 1.10² = 1.21. PV = 4840 ÷ 1.21 = $4,000.00. Check by multiplying the unrounded PV by 1.21 to recover the future amount.

Recap quiz

$1,000.00: Future value is $1,210.00; i = 0.10; n = 2. Accumulation factor = 1.10² = 1.21. PV = 1210 ÷ 1.21 = $1,000.00. Check by multiplying the unrounded PV by 1.21 to recover the future amount.
Match the rate period to the timeline: The periodic rate and period count must describe the same interval; assumptions are not guarantees.
Retain full precision: Retaining calculator precision avoids accumulating rounding errors; round the final monetary result.

Chapter 12 Organizer and Review

Chapter Organizer

Every section at a glance: key formulas, key terms, and the main idea.

12.1 Exponents and Compound Growth
Key formulas
- Exponents and Compound Growth: FV = P × (1 + i)ⁿ
- Related relationship: Interest = FV − P
Key terms
Exponent, Periodic rate, Compound interest, Future value
Remember
Draw the timeline and match the rate to each interval.
12.2 Compounding Frequency and Effective Rates
Key formulas
- Compounding Frequency and Effective Rates: FV = P × (1 + r ÷ m)ⁿ
- Related relationship: Effective annual rate = (1 + r ÷ m)ᵐ − 1
Key terms
Nominal annual rate, Frequency, Effective annual rate, Period count
Remember
Draw the timeline and match the rate to each interval.
12.3 Present Value and Financial Equivalence
Key formulas
- Present Value and Financial Equivalence: PV = FV ÷ (1 + i)ⁿ
- Related relationship: FV = PV × (1 + i)ⁿ
Key terms
Present value, Discounting, Discount rate, Financial equivalence
Remember
Draw the timeline and match the rate to each interval.

Cumulative Review

Mixed practice from this chapter and the two chapters before it.

Section 12.1
Find future value of $4,000.00 at 5% annually compounded for 2 years.
Show answer
Use P = $4,000.00, i = 0.05, and n = 2. Growth factor = 1.05² = 1.1025. FV = 4000 × 1.1025 = $4,410.00. Interest = $4,410.00 − $4,000.00 = $410.00.
Section 12.2
Find the one-year value of $4,800.00 at 6% nominal annual interest compounded monthly.
Show answer
Use principal $4,800.00 and nominal r = 0.06. Monthly i = 0.06 ÷ 12 = 0.005; n = 12. FV = 4800 × 1.005¹² = $5,096.05. Effective annual rate = 1.005¹² − 1 ≈ 6.17%.
Section 12.3
Find present value of $4,840.00 due in 2 years using 10% annually compounded.
Show answer
Future value is $4,840.00; i = 0.10; n = 2. Accumulation factor = 1.10² = 1.21. PV = 4840 ÷ 1.21 = $4,000.00. Check by multiplying the unrounded PV by 1.21 to recover the future amount.
Review of Section 10.3
A $700 face loan withholds $25 and requires $735 at maturity. Find proceeds and cost.
Show answer
Proceeds = 700 − 25 = $675.00; cost = 735 − 675 = $60.00.
Review of Section 11.3
On $1,200 at 9%, actual/360, pay $318 after 60 days. Find the new principal.
Show answer
I = 1,200 × 0.09 × 60 ÷ 360 = $18; principal reduction = 318 − 18 = $300; new principal = $900.00.

All prices, stores, people, and accounts in this chapter are hypothetical and for teaching only. Product references are generic categories; no brand endorses or is endorsed by this textbook.

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