Comparing Quarterly and Continuous Compounding
A student has €2,500 from a summer job and is comparing two savings products before starting university. Account A pays a nominal annual rate of 4.8% compounded quarterly. Account B pays a nominal annual rate of 4.6% compounded continuously. The student wants the money to grow to €4,000 for a deposit, and a bank clerk claims that because 4.8>4.6, Account A must always be better. The student writes the balance after t years as 2500(1+0.048/4)^{4t} for A and 2500e^{0.046t} for B, but is unsure whether t should be in years or quarters in the first formula. A calculator is permitted; the student must give times to the nearest 0.1 year and explain any modelling assumptions. The final answer must identify which account reaches €4,000 first and by how long. The student also needs to compare effective annual rates, because the nominal rates alone do not settle the comparison.
1. Find the time, in years to one decimal place, for Account A to reach €4,000. State the equation you solve and show the logarithm step.
Read the worked answer
For Account A the balance after t years is 2500(1+0.048/4)^{4t}=2500(1.012)^{4t}. Set this equal to 4000:
2500(1.012)^{4t}=4000
(1.012)^{4t}=1.6
Take natural logarithms:
4t ln(1.012)=ln(1.6)
t=ln(1.6)/(4 ln(1.012))
t=0.4700/(4×0.0119286)=0.4700/0.047714≈9.85.
So Account A reaches €4,000 after about 9.9 years. The exponent 4t counts quarters; if the unknown is the number of quarters n, then (1.012)^n=1.6 gives n≈39.4 quarters, and dividing by 4 returns t≈9.85 years.
2. Find each account's effective annual rate and use it to assess the clerk's claim that Account A must be better simply because 4.8>4.6.
Read the worked answer
Account A: effective annual rate =(1+0.048/4)^4−1=(1.012)^4−1≈0.04887, or 4.887%.
Account B: effective annual rate =e^{0.046}−1≈0.04707, or 4.707%.
The clerk's conclusion is correct for these particular products because 4.887%>4.707%, so Account A genuinely has the higher effective rate. The reasoning, however, is not generally valid: comparing nominal rates alone ignores compounding frequency. A nominal rate of 4.8% compounded quarterly can be beaten by a lower nominal rate compounded continuously if the continuous rate is high enough. The sound comparison is between effective annual rates, not nominal rates.
3. Find the time for Account B to reach €4,000 and hence state which account reaches the target first, and by how much. Discuss one assumption.
Read the worked answer
For Account B:
2500e^{0.046t}=4000
e^{0.046t}=1.6
0.046t=ln(1.6)
t=ln(1.6)/0.046≈0.4700/0.046≈10.22 years.
Account A takes about 9.85 years and Account B about 10.22 years, so Account A reaches €4,000 first, by about 10.22−9.85=0.37 years, roughly 4.4 months.
One modelling assumption is that both interest rates remain fixed for the whole period and that there are no fees, tax or withdrawals. The calculation also assumes the advertised compounding conventions are applied exactly: quarterly credits for A and continuous compounding for B.