Chapter 13

Annuities, Sinking Funds, and Long-Term Savings

Model repeated deposits with clear timing and realistic assumptions.

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.

13.1

Ordinary Annuities and Deposit Timing

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

End-of-month deposit: $100.00 for 3 months
Assumed monthly rate: 1%
At month 3: $102.01 + $101.00 + $100.00 = $303.01

Figure 13.1 A student savings timeline (hypothetical)

The concept

An annuity is a sequence of equal payments at equally spaced times. An ordinary annuity places each payment at the end of its period. Deposits from a part-time job can be modeled this way when both the amounts and timing are regular.

Earlier deposits earn interest longer. At the third deposit date, the first of three end-of-month deposits has grown for two periods, the second for one, and the last for zero. The last deposit is included but earns no interest yet.

The ordinary-annuity factor sums these growth factors. Match the deposit interval to the interest period; these examples assume they match. At zero interest the factor is simply n, because the formula with division by i cannot be used directly.

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.

Value at month 3 of each $100 deposit

Month 1 deposit$ 102.01

Month 2 deposit$ 101

Month 3 deposit$ 100

Bars share a zero baseline.

Figure 13.1.V Value at month 3 of each $100 deposit. Hypothetical teaching data.

Money Words

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

annuity
A series of equal payments made at regular intervals, such as monthly.
sinking fund
Money set aside regularly so a future obligation can be paid without borrowing.
future value
What an amount of money today will be worth at a later date with interest.

Key terms

Annuity: Equal payments at equally spaced intervals.
Ordinary annuity: Payments occur at each period-end.
Annuity due: Payments occur at each period-beginning.
Sinking fund: Regular savings designated for a future obligation.
Valuation date: The date at which cash-flow values are compared.

Formulas & rules

Ordinary Annuities and Deposit Timing
FV = R × ((1 + i)ⁿ − 1) ÷ i
Related relationship
Total contributions = R × n

Worked examples

Example 1The timeline in the figure

Deposit $100.00 at each month-end for 3 months at an assumed 1% monthly rate. Find value just after deposit 3.

Solution

1R = $100.00; i = 0.01; n = 3.
2Factor = 1.01² + 1.01 + 1 = 3.0301.
3FV = 100 × 3.0301 = $303.01.
4Contributions = 3 × 100 = $300.00; interest = $3.01.

Answer: $303.01

Example 2A different amount

Deposit $200.00 at each month-end for 3 months at an assumed 1% monthly rate. Find value just after deposit 3.

Solution

1R = $200.00; i = 0.01; n = 3.
2Factor = 1.01² + 1.01 + 1 = 3.0301.
3FV = 200 × 3.0301 = $606.02.
4Contributions = 3 × 200 = $600.00; interest = $6.02.

Answer: $606.02

Practice exercises 13.1

Deposit $100.00 at each month-end for 3 months at an assumed 1% monthly rate. Find value just after deposit 3.
Deposit $200.00 at each month-end for 3 months at an assumed 1% monthly rate. Find value just after deposit 3.
Deposit $50.00 at each month-end for 3 months at an assumed 1% monthly rate. Find value just after deposit 3.
Deposit $300.00 at each month-end for 3 months at an assumed 1% monthly rate. Find value just after deposit 3.
Deposit $400.00 at each month-end for 3 months at an assumed 1% monthly rate. Find value just after deposit 3.

Section 13.1 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 13.1

Practice exercises

R = $100.00; i = 0.01; n = 3. Factor = 1.01² + 1.01 + 1 = 3.0301. FV = 100 × 3.0301 = $303.01. Contributions = 3 × 100 = $300.00; interest = $3.01.
R = $200.00; i = 0.01; n = 3. Factor = 1.01² + 1.01 + 1 = 3.0301. FV = 200 × 3.0301 = $606.02. Contributions = 3 × 200 = $600.00; interest = $6.02.
R = $50.00; i = 0.01; n = 3. Factor = 1.01² + 1.01 + 1 = 3.0301. FV = 50.0 × 3.0301 = $151.51. Contributions = 3 × 50.0 = $150.00; interest = $1.51.
R = $300.00; i = 0.01; n = 3. Factor = 1.01² + 1.01 + 1 = 3.0301. FV = 300 × 3.0301 = $909.03. Contributions = 3 × 300 = $900.00; interest = $9.03.
R = $400.00; i = 0.01; n = 3. Factor = 1.01² + 1.01 + 1 = 3.0301. FV = 400 × 3.0301 = $1,212.04. Contributions = 3 × 400 = $1,200.00; interest = $12.04.

Recap quiz

$303.01: R = $100.00; i = 0.01; n = 3. Factor = 1.01² + 1.01 + 1 = 3.0301. FV = 100 × 3.0301 = $303.01. Contributions = 3 × 100 = $300.00; interest = $3.01.
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.

13.2

Annuities Due and Present Value

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

Beginning-of-month deposit: $100.00 for 3 months
Assumed monthly rate: 1%; valuation at month 3
Future value: $306.04
PV of three $100 month-end payments: $294.10

Figure 13.2 A student savings timeline (hypothetical)

The concept

An annuity due makes payments at the beginning of each period. Relative to an ordinary annuity with the same number of payments and the same final valuation date, every deposit earns one extra period of interest. Multiply ordinary-annuity future value by 1 + i.

Timing must be explicit: three beginning-of-month deposits occur at times 0, 1, and 2, with valuation at time 3. Three month-end deposits occur at times 1, 2, and 3. Confusing these schedules changes the answer even when payment amounts match.

Present value of an ordinary annuity discounts each future payment to today and adds the results. For three $100 month-end payments at 1% per month, PV = 100 ÷ 1.01 + 100 ÷ 1.01² + 100 ÷ 1.01³ = $294.10. For a due annuity, multiply this ordinary present value by 1.01, giving $297.04.

Timing changes deposit value

1
End-of-month deposits
$303.01 future value
2
Beginning-of-month deposits
One extra period → $306.04
3
Ordinary annuity present value
$294.10

Figure 13.2.V Timing changes deposit value. Hypothetical teaching data.

Money Words

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

present value
What a future amount of money is worth today at a given interest rate.
compound interest
Interest earned on both the original amount and the interest already earned.
deposit
Money placed into a bank account.

Key terms

Annuity: Equal payments at equally spaced intervals.
Ordinary annuity: Payments occur at each period-end.
Annuity due: Payments occur at each period-beginning.
Sinking fund: Regular savings designated for a future obligation.
Valuation date: The date at which cash-flow values are compared.

Formulas & rules

Annuities Due and Present Value
FV due = R × ((1 + i)ⁿ − 1) ÷ i × (1 + i)
Related relationship
PV ordinary = R × (1 − (1 + i)⁻ⁿ) ÷ i

Worked examples

Example 1The timeline in the figure

Find value at month 3 of three beginning-of-month deposits of $100.00 at an assumed 1% monthly rate.

Solution

1R = $100.00, i = 0.01, n = 3; deposits occur at times 0, 1, and 2.
2Ordinary future-value factor is 3.0301.
3Due factor = 3.0301 × 1.01 = 3.060401.
4FV at time 3 = 100 × 3.060401 = $306.04.

Answer: $306.04

Example 2A different amount

Find value at month 3 of three beginning-of-month deposits of $200.00 at an assumed 1% monthly rate.

Solution

1R = $200.00, i = 0.01, n = 3; deposits occur at times 0, 1, and 2.
2Ordinary future-value factor is 3.0301.
3Due factor = 3.0301 × 1.01 = 3.060401.
4FV at time 3 = 200 × 3.060401 = $612.08.

Answer: $612.08

Practice exercises 13.2

Find value at month 3 of three beginning-of-month deposits of $100.00 at an assumed 1% monthly rate.
Find value at month 3 of three beginning-of-month deposits of $200.00 at an assumed 1% monthly rate.
Find value at month 3 of three beginning-of-month deposits of $50.00 at an assumed 1% monthly rate.
Find the present value of three $100 month-end payments at 1% per month.
Find present value of three $100 beginning-of-month payments at 1% per month.

Section 13.2 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 13.2

Practice exercises

R = $100.00, i = 0.01, n = 3; deposits occur at times 0, 1, and 2. Ordinary future-value factor is 3.0301. Due factor = 3.0301 × 1.01 = 3.060401. FV at time 3 = 100 × 3.060401 = $306.04.
R = $200.00, i = 0.01, n = 3; deposits occur at times 0, 1, and 2. Ordinary future-value factor is 3.0301. Due factor = 3.0301 × 1.01 = 3.060401. FV at time 3 = 200 × 3.060401 = $612.08.
R = $50.00, i = 0.01, n = 3; deposits occur at times 0, 1, and 2. Ordinary future-value factor is 3.0301. Due factor = 3.0301 × 1.01 = 3.060401. FV at time 3 = 50.0 × 3.060401 = $153.02.
PV = 100 × (1 − 1.01⁻³) ÷ 0.01 = $294.10, rounded at the end.
PV due = unrounded ordinary PV × 1.01 = $297.04. The first payment is at time 0, so it is not discounted.

Recap quiz

$306.04: R = $100.00, i = 0.01, n = 3; deposits occur at times 0, 1, and 2. Ordinary future-value factor is 3.0301. Due factor = 3.0301 × 1.01 = 3.060401. FV at time 3 = 100 × 3.060401 = $306.04.
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.

13.3

Sinking Funds and Savings Goals

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

Goal at month 3: $1,000.00
Three end-of-month deposits; assumed monthly rate: 1%
Exact modeled deposit: $330.022111…
Round up to $330.03; projected balance: $1,000.02

Figure 13.3 A student savings timeline (hypothetical)

The concept

A sinking fund sets aside regular deposits for a known future expense. A student might model equipment replacement or a course fee rather than wait until the entire bill arrives. Solving the ordinary-annuity formula for R finds the deposit needed for a target.

Divide the target by the accumulation factor. The result may require fractions of a cent, which cannot be deposited as ordinary cash. For a minimum-goal problem, round the required deposit up to the next cent and check the resulting future value.

A long-term plan is a model, not a guarantee. A lower realized rate, missed deposit, fee, or higher future price can create a shortfall. At a zero rate the required deposit equals target divided by the number of deposits; regular reviews allow the plan to adapt.

Build a sinking fund

1
Goal
$1,000 at month 3
2
Rounded deposit
$330.03 at each month-end
3
Projected balance
$1,000.02

Figure 13.3.V Build a sinking fund. Hypothetical teaching data.

Money Words

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

interest rate
The percent used to calculate how much interest is earned or charged.
inflation
A general rise in prices over time, which makes money buy less.
return
The money or gain an investment produces, often shown as a percent.

Key terms

Annuity: Equal payments at equally spaced intervals.
Ordinary annuity: Payments occur at each period-end.
Annuity due: Payments occur at each period-beginning.
Sinking fund: Regular savings designated for a future obligation.
Valuation date: The date at which cash-flow values are compared.

Formulas & rules

Sinking Funds and Savings Goals
R = target FV × i ÷ ((1 + i)ⁿ − 1)
Related relationship
Shortfall = target − available balance

Worked examples

Example 1The timeline in the figure

Find equal month-end deposits to reach at least $1,000.00 after 3 deposits at an assumed 1% monthly rate.

Solution

1Target = $1,000.00; i = 0.01; n = 3.
2Accumulation factor = (1.01³ − 1) ÷ 0.01 = 3.0301.
3Exact R = 1000 ÷ 3.0301 = 330.022111481… .
4Round up to cents: $330.03.
5Check: rounded deposit × 3.0301 = $1,000.02.

Answer: $330.03

Example 2A different amount

Find equal month-end deposits to reach at least $2,000.00 after 3 deposits at an assumed 1% monthly rate.

Solution

1Target = $2,000.00; i = 0.01; n = 3.
2Accumulation factor = (1.01³ − 1) ÷ 0.01 = 3.0301.
3Exact R = 2000 ÷ 3.0301 = 660.044222962… .
4Round up to cents: $660.05.
5Check: rounded deposit × 3.0301 = $2,000.02.

Answer: Required deposit rounded upward: $660.05

Practice exercises 13.3

Find equal month-end deposits to reach at least $1,000.00 after 3 deposits at an assumed 1% monthly rate.
Find equal month-end deposits to reach at least $2,000.00 after 3 deposits at an assumed 1% monthly rate.
Find equal month-end deposits to reach at least $500.00 after 3 deposits at an assumed 1% monthly rate.
With no interest, what month-end deposit reaches $900 in 6 deposits?
A $1,200 target has only $1,050 available. Find the shortfall and a six-month zero-interest catch-up amount.

Section 13.3 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 13.3

Practice exercises

Target = $1,000.00; i = 0.01; n = 3. Accumulation factor = (1.01³ − 1) ÷ 0.01 = 3.0301. Exact R = 1000 ÷ 3.0301 = 330.022111481… . Round up to cents: $330.03. Check: rounded deposit × 3.0301 = $1,000.02.
Target = $2,000.00; i = 0.01; n = 3. Accumulation factor = (1.01³ − 1) ÷ 0.01 = 3.0301. Exact R = 2000 ÷ 3.0301 = 660.044222962… . Round up to cents: $660.05. Check: rounded deposit × 3.0301 = $2,000.02.
Target = $500.00; i = 0.01; n = 3. Accumulation factor = (1.01³ − 1) ÷ 0.01 = 3.0301. Exact R = 500.0 ÷ 3.0301 = 165.011055740… . Round up to cents: $165.02. Check: rounded deposit × 3.0301 = $500.03.
At i = 0 use target ÷ n: 900 ÷ 6 = $150.00.
Shortfall = 1,200 − 1,050 = $150.00; 150 ÷ 6 = $25.00 per month, assuming no other changes.

Recap quiz

$330.03: Target = $1,000.00; i = 0.01; n = 3. Accumulation factor = (1.01³ − 1) ÷ 0.01 = 3.0301. Exact R = 1000 ÷ 3.0301 = 330.022111481… . Round up to cents: $330.03. Check: rounded deposit × 3.0301 = $1,000.02.
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 13 Organizer and Review

Chapter Organizer

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

13.1 Ordinary Annuities and Deposit Timing
Key formulas
- Ordinary Annuities and Deposit Timing: FV = R × ((1 + i)ⁿ − 1) ÷ i
- Related relationship: Total contributions = R × n
Key terms
Annuity, Ordinary annuity, Annuity due, Sinking fund, Valuation date
Remember
Draw the timeline and match the rate to each interval.
13.2 Annuities Due and Present Value
Key formulas
- Annuities Due and Present Value: FV due = R × ((1 + i)ⁿ − 1) ÷ i × (1 + i)
- Related relationship: PV ordinary = R × (1 − (1 + i)⁻ⁿ) ÷ i
Key terms
Annuity, Ordinary annuity, Annuity due, Sinking fund, Valuation date
Remember
Draw the timeline and match the rate to each interval.
13.3 Sinking Funds and Savings Goals
Key formulas
- Sinking Funds and Savings Goals: R = target FV × i ÷ ((1 + i)ⁿ − 1)
- Related relationship: Shortfall = target − available balance
Key terms
Annuity, Ordinary annuity, Annuity due, Sinking fund, Valuation date
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 13.1
Deposit $400.00 at each month-end for 3 months at an assumed 1% monthly rate. Find value just after deposit 3.
Show answer
R = $400.00; i = 0.01; n = 3. Factor = 1.01² + 1.01 + 1 = 3.0301. FV = 400 × 3.0301 = $1,212.04. Contributions = 3 × 400 = $1,200.00; interest = $12.04.
Section 13.2
Find present value of three $100 beginning-of-month payments at 1% per month.
Show answer
PV due = unrounded ordinary PV × 1.01 = $297.04. The first payment is at time 0, so it is not discounted.
Section 13.3
A $1,200 target has only $1,050 available. Find the shortfall and a six-month zero-interest catch-up amount.
Show answer
Shortfall = 1,200 − 1,050 = $150.00; 150 ÷ 6 = $25.00 per month, assuming no other changes.
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.
Review of Section 12.3
Find present value of $1,210.00 due in 2 years using 10% annually compounded.
Show answer
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.

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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