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Annuities
Key Terms
- An annuity is a sequence of equal payments made at regular intervals
- Future value (FV) annuity
- regular deposits into an account earning interest (savings/investment plan)
- FV recurrence: Vn+1 = Vn(1 + r) + D (where D = regular deposit)
- Present value (PV) annuity
- regular withdrawals from a lump sum (e.g. retirement pension)
- PV recurrence: Vn+1 = Vn(1 + r) − W (where W = regular withdrawal)
- The annuity ends when Vn = 0 (funds exhausted) or Vn reaches a target value
Future Value Annuity formula:
FV = D × ((1 + r)n − 1) / r
Present Value Annuity formula:
PV = W × (1 − (1 + r)−n) / r
where r = interest rate per period, n = number of periods, D = deposit per period, W = withdrawal per period
FV = D × ((1 + r)n − 1) / r
Present Value Annuity formula:
PV = W × (1 − (1 + r)−n) / r
where r = interest rate per period, n = number of periods, D = deposit per period, W = withdrawal per period
Worked Example: Emma deposits $200 per month into an account earning 6% p.a. (0.5% per month). How much does she have after 12 months?
Recurrence: Vn+1 = Vn(1.005) + 200, V0 = 0
V1 = 0(1.005) + 200 = 200
V2 = 200(1.005) + 200 = 201 + 200 = 401
… iterate or use formula:
FV = 200 × (1.00512 − 1) / 0.005 = 200 × (1.0617 − 1) / 0.005 = 200 × 12.336 = $2 467.11
Recurrence: Vn+1 = Vn(1.005) + 200, V0 = 0
V1 = 0(1.005) + 200 = 200
V2 = 200(1.005) + 200 = 201 + 200 = 401
… iterate or use formula:
FV = 200 × (1.00512 − 1) / 0.005 = 200 × (1.0617 − 1) / 0.005 = 200 × 12.336 = $2 467.11
Hot Tip: The key difference between the two types: FV annuity builds up (you’re adding deposits), PV annuity runs down (you’re making withdrawals). The recurrence relation looks the same except for the sign — +D for FV, −W for PV.
What Is an Annuity?
An annuity is any financial arrangement involving a series of equal, regular payments. The two most common types in this course are:
- Future value (FV) annuity — you make regular deposits into an interest-earning account. The balance grows over time. Used for savings plans, superannuation contributions.
- Present value (PV) annuity — you start with a lump sum and make regular withdrawals. The balance shrinks over time. Used for retirement pensions, insurance payouts.
Future Value Annuity
Recurrence: Vn+1 = Vn(1 + r) + D, with V0 = 0 (or an opening balance)
Each period, the existing balance earns interest AND you add a new deposit D. Over time, both your contributions and the compound interest accumulate.
Formula: FV = D × ((1 + r)n − 1) / r
Present Value Annuity
Recurrence: Vn+1 = Vn(1 + r) − W, with V0 = starting lump sum
Each period, the remaining balance earns interest, but a withdrawal W is taken out. This is mathematically identical to a reducing balance loan with the loan amount as the starting “present value”.
Formula: PV = W × (1 − (1 + r)−n) / r
Finding Unknown Values
The formulas can be rearranged to find: the payment amount (D or W), the number of periods (n by iteration), or the interest rate (by trial). In practice, use the recurrence relation iteratively for most exam questions.
Strategy: Always identify the type (FV or PV) first. Ask: is the balance growing (+D) or shrinking (−W)? Then set up the correct recurrence and iterate, or use the formula if n is large.
Mastery Practice
- Fluency An account starts at $0. A deposit of $300 is made each month. The monthly interest rate is 0.4%. Write the recurrence relation and find the balance after 2 months.
- Fluency A retirement fund has $80 000. Monthly withdrawals of $1 200 are made. The monthly interest rate is 0.35%. Write the recurrence relation and find the balance after 1 withdrawal.
- Fluency Use the future value formula to find the accumulated value of $150 monthly deposits at 6% p.a. (0.5% monthly) after 24 months.
- Fluency A retiree has $200 000 and withdraws $1 500 per month. The monthly interest rate is 0.3%. What is the balance after 3 months?
- Understanding Mitchell deposits $500 per quarter into a savings account earning 4% p.a. compounded quarterly (1% per quarter). Use iteration to find the balance after 5 quarters.
- Understanding How much must be deposited monthly into an account earning 6% p.a. (0.5% per month) to accumulate $10 000 in 24 months? Use the FV formula and solve for D.
- Understanding A present value annuity of $150 000 pays out $2 000 per month at a monthly interest rate of 0.5%. Use iteration to determine how many months the fund lasts.
- Understanding A superannuation fund has $320 000 at retirement. The fund earns 4.8% p.a. (0.4% monthly). Monthly pension payments are $1 800. Will the fund last 20 years (240 months)? Use the PV formula to check.
- Problem Solving Anya wants to save for a $15 000 car. She can invest $400 per month in an account earning 5.4% p.a. (0.45% per month).
- (a) How many months will it take her to reach $15 000? Use iteration.
- (b) How much of the $15 000 comes from her deposits, and how much from interest?
- Problem Solving James retires with $500 000 in a fund earning 6% p.a. compounded monthly (0.5% monthly). He plans to withdraw $3 000 per month.
- (a) Use the PV formula to find how many months (to the nearest month) the fund will last if he withdraws $3 000/month.
- (b) If he reduces his withdrawal to $2 500/month, how much longer does the fund last? (Use the formula or iterate.)