Math · College algebra · Worked example
Compare compound and continuous interest
$1000 is invested at 5% a year for 10 years. Find the balance with yearly compounding, monthly compounding and continuous compounding.
Yearly compounding
With n = 1, the balance is multiplied by 1.05 ten times.
Monthly compounding
With n = 12, each month multiplies the balance by 1 + 0.05/12, 120 times.
Continuous compounding
Letting n grow without bound gives Pe^(rt) = 1000e^0.5.
Compare the results
More frequent compounding earns a little more, but the gains shrink: monthly beats yearly by $18.12, while continuous beats monthly by only $1.71. Simple interest, which never earns interest on interest, gives just $1500.
Result
$1628.89 yearly, $1647.01 monthly and $1648.72 continuously.
Your turn
How much must be invested now at 4% compounded continuously to have $5000 in 8 years?
Show the answer and explanation
About $3630.75.
Solve 5000 = Pe^(0.04 × 8): P = 5000e^(−0.32) ≈ 5000 × 0.726149 ≈ $3630.75.
Keep exploring
Open the graph: the compound and continuous curves nearly coincide, while the simple-interest line falls further behind every year.
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