Unit 4 · Investing & Planning
📈 Compound Interest Investments
You already know compound interest from Unit 3. Unit 4 pushes further: how does compounding frequency change things? How do you compare two investment accounts with different rates and different compounding? The effective annual rate is the key tool.
Builds on Unit 3 compound interest
Effective annual rate is new here
CAS Finance solver used
Before starting: Make sure you're comfortable with
compound interest from Unit 3. This page uses the same formula structure but introduces compounding more than once per year and the effective annual rate for comparisons.
1
Compounding frequency, why it matters
The compound interest formula from Unit 3 assumed interest was calculated annually. In real life, banks compound monthly, quarterly, or even daily. More frequent compounding = slightly more interest earned.
Annual
Once per year
m = 1
Semi-annual
Twice per year
m = 2
Quarterly
4× per year
m = 4
Monthly
12× per year
m = 12
To use the formula with any frequency: convert the annual rate to a per-period rate and count total periods instead of years.
Compound Interest, any compounding frequency
FV = PV × (1 + r/m)^(m × t)
r = annual rate (decimal) · m = periods per year · t = years
Or think of it as: rate per period = r/m · total periods N = m × t
Then: FV = PV × (1 + rate per period)^N
2
Calculating future value
Worked Example A · annual compounding
$1,000 invested at 10% p.a. compounded annually for 3 years. Find the future value.
Given:PV = $1,000 · r = 0.10 · m = 1 · t = 3
Rate/period:0.10 ÷ 1 = 0.10
Periods N:1 × 3 = 3
FV:1,000 × (1.10)³ = 1,000 × 1.331 = $1,331
Worked Example B · quarterly compounding
$2,000 invested at 6% p.a. compounded quarterly for 2 years.
Rate/period:6% ÷ 4 = 1.5% = 0.015
Periods N:4 × 2 = 8
FV:2,000 × (1.015)⁸ ≈ 2,000 × 1.1265 ≈ $2,253
| Compounding | m | Rate/period | Periods | FV |
|---|
| Annually | 1 | 6% | 2 | $2,247.60 |
| Quarterly | 4 | 1.5% | 8 | $2,253.00 |
| Monthly | 12 | 0.5% | 24 | $2,254.32 |
More frequent compounding = slightly higher FV. The difference grows with higher rates and longer time.
3
Effective annual rate (EAR)
The effective annual rate (EAR) converts any compounding frequency into an equivalent annual rate. This lets you directly compare investments that compound at different frequencies.
Effective Annual Rate
EAR = (1 + r/m)^m − 1
r = nominal annual rate · m = compounding periods per year · answer is a decimal (× 100 for %)
The EAR answers: "if this account compounded annually instead, what rate would give the same outcome?" A higher EAR = a better investment.
Worked Example C · comparing investments
Two accounts: Account A offers 6% p.a. compounded quarterly. Account B offers 6.1% p.a. compounded annually. Which has the higher effective rate?
EAR (A):(1 + 0.06/4)⁴ − 1 = (1.015)⁴ − 1 = 1.06136 − 1 = 6.14%
EAR (B):(1 + 0.061/1)¹ − 1 = 6.10%
Winner:Account A (6.14% > 6.10%), despite the lower nominal rate!
Key insight: A lower nominal rate with more frequent compounding can beat a higher nominal rate with less frequent compounding. Always compare EAR, not nominal rates.
4
Finding n, how long to reach a target?
If you know the target FV and want to find when it's reached, you can solve algebraically using logarithms, or use trial and error / the CAS Finance Solver.
Finding n using logarithms
FV = PV × (1 + i)^n
→ n = log(FV/PV) ÷ log(1 + i)
where i is the rate per period
Worked Example D · finding the time
How many years for $1,000 to grow to $2,197 at 30% p.a. compounded annually?
FV/PV:2,197 ÷ 1,000 = 2.197
Log method:n = log(2.197) ÷ log(1.30)
Check:1.30¹ = 1.3 · 1.30² = 1.69 · 1.30³ = 2.197 ✓ → n = 3 years
When the numbers work out cleanly like this, checking by trial-and-error is often faster in an exam.
Practice Questions
1. Find the future value of $1,000 invested at 10% p.a. compounded annually for 3 years.
▶
FV = 1,000 × (1.10)³ = 1,000 × 1.331 = $1,331
2. $2,000 is invested at 20% p.a. compounded annually for 2 years. Find the FV and the interest earned.
▶
FV = 2,000 × (1.20)² = 2,000 × 1.44 = $2,880
Interest = 2,880 − 2,000 = $880
3. An account pays 6% p.a. compounded monthly. How many compounding periods are there in 4 years?
▶
N = m × t = 12 × 4 = 48 periods
Rate per period = 6% ÷ 12 = 0.5% = 0.005
4. $4,000 is invested at 10% p.a. compounded annually for 3 years. What is the total interest earned?
▶
FV = 4,000 × (1.10)³ = 4,000 × 1.331 = $5,324
Interest = 5,324 − 4,000 = $1,324
5. Two investments both start with $10,000: Plan X at 8% p.a. compounded annually, Plan Y at 7.8% p.a. compounded monthly. Which has the higher effective annual rate? (EAR of Y ≈ 8.08%)
▶
EAR of X: (1 + 0.08)¹ − 1 = 8.00%
EAR of Y: (1 + 0.078/12)¹² − 1 = (1.0065)¹² − 1 ≈ 8.08%
Plan Y wins, despite the lower nominal rate, monthly compounding pushes its effective rate above Plan X.
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