Study Guide: Annuities – Future Value (Chapter 5.3)

Study Guide: Annuities – Future Value (Chapter 5.3)

What Is an Annuity?

• Definition: An annuity is defined as a series of equal payments (referred to as rents) made or received at equal time intervals, with interest compounded once per interval.

• Requirements of an annuity:

  1. Equal periodic payments or receipts (rents).

  2. Equal time intervals between payments.

  3. Interest is compounded once per interval.

Types of Annuities

1. Ordinary Annuity

   • Description: Payments are made at the end of each period.

   • Implication: No interest is earned in the same period as the deposit.

   • Example: Rent paid at the end of each month.

2. Annuity Due

   • Description: Payments are made at the beginning of each period.

   • Implication: Each payment earns one extra period of interest compared to an ordinary annuity.

   • Example: Rent paid at the beginning of each month.

Future Value of an Ordinary Annuity (FV-OA)

• Formula: $FV_{oa} = R \times \frac{(1 + i)^{n} - 1}{i}$

  • Where:

  • $FV_{oa}$ = future value of an ordinary annuity

  • $R$ = periodic rent (payment)

  • $i$ = interest rate per period

  • $n$ = number of periods

• Key Idea: Since payments are made at the end of each period, there is one fewer compounding period than the number of payments.

Examples of FV-OA

• Example 1:

  • Facts: 5 deposits of $5,000 each year at 6% interest.

  • Find: Future value.

  • Calculation:

    • $FV_{oa} = 5,000 \times \frac{(1 + 0.06)^{5} - 1}{0.06}$

    • $FV_{oa} = 5,000 \times 5.63709$

  • Result: $FV = 28,185.45$

• Example 2:

  • Facts: 6 semiannual deposits of $75,000 at 10% annual (therefore, 5% per half-year).

  • Find: Future value.

  • Calculation:

    • $FV_{oa} = 75,000 \times \frac{(1 + 0.05)^{6} - 1}{0.05}$

    • $FV_{oa} = 75,000 \times 6.80191$

  • Result: $FV = 510,143.25$

Future Value of an Annuity Due (FV-AD)

• Because payments occur at the beginning of each period, each payment earns one additional period of interest.

• Relationship:

  • FV_ad = FV_oa * (1 + i)

• Formula:
  $FV_{ad} = R \times \frac{(1 + i)^{n} - 1}{i} \times (1 + i)$

Examples of FV-AD

• Example 3:

  • Facts: 8 deposits of $800 at 6% interest (with the first deposit made immediately).

  • Find: Future value.

  • Steps:

  1. Calculate FV-OA factor:

     • From Table 5.3, for 8 periods and 6%: $9.89747$

  2. Calculate FV-AD factor:

     • $9.89747 \times (1 + 0.06) = 10.49132$

  3. Calculate PV:

     • $FV_{ad} = 800 \times 10.49132$

  • Result: $FV = 8,393.06$

• Example 4:

  • Facts: $2,500 deposited annually for 30 years at 9% interest (the first deposit today).

  • Find: Future value.

  • Steps:

  1. Calculate FV-OA factor for 30 years and 9%: $136.30754$

  2. Calculate FV-AD factor:

     • $136.30754 \times 1.09 = 148.57522$

  3. Calculate FV:

     • $FV_{ad} = 2,500 \times 148.57522$

  • Result: $FV = 371,438$

Solving for Unknowns in FV of Annuity Problems

• It is possible to find any single variable (R, i, n, FV) if the other three are known.

Example 5 — Solving for Payment (R)

• Goal: Accumulate $14,000 in 5 years, earning 8% compounded semiannually.

  • Given:

  • $n = 10$ (periods),

  • $i = 4\%$ (per period).

  • Equation:

  • $FV_{oa} = R \times \frac{(1 + i)^{n} - 1}{i}$

  • Solve:

  • $14,000 = R \times 12.00611$

  • $R = \frac{14,000}{12.00611} = 1,166.07$

Example 6 — Solving for Number of Periods (n)

• Goal: Accumulate $117,332 by depositing $20,000 yearly at 8%.

  • Equation:

  • $117,332 = 20,000 \times \frac{(1 + 0.08)^{n} - 1}{0.08}$

  • Rearranging:

    • $FV_{oa} = \frac{117,332}{20,000} = 5.8666$

  • From the FV table at 8%:

  • Factor 5.8666 corresponds to $n = 5$ years.

Quick Comparison Table

Summary Formulas

1. Future Value of an Ordinary Annuity:

   • $FV_{oa} = R \times \frac{(1 + i)^{n} - 1}{i}$

2. Future Value of an Annuity Due:

   • $FV_{ad} = R \times \frac{(1 + i)^{n} - 1}{i} \times (1 + i)$

3. To find Rent (R):

   • $R = \frac{FV \times i}{[(1 + i)^{n} - 1]}$

4. To find Number of Periods (n):

   • Use tables or logarithms to solve:

   • $\frac{FV}{R} = \frac{[(1 + i)^{n} - 1]}{i}$