Raffles Bank pays 4% simple interest on its deposit account, whereas Bugis Bank pays interest on its deposit account compounded monthly.
Formulate the quoted and effective interest rates that Bugis Bank should set if it wants to match Raffles Bank, assuming a 5-years horizon period.
To determine the quoted and effective interest rates that Bugis Bank should set to match Raffles Bank's 4% simple interest rate over a 5-year horizon period, we need to understand the difference between simple interest and compound interest.
Simple interest is calculated on the initial principal amount only, while compound interest takes into account both the initial principal and the interest accumulated over time. Compound interest is typically more advantageous for depositors since it allows their money to grow faster.
To match Raffles Bank's 4% simple interest rate, we need to find the equivalent compound interest rate that Bugis Bank should offer. We can use the formula for compound interest:
A = P (1 + r/n)^(nt)
Where:
A = the total amount accumulated after n years, including interest
P = the principal amount (initial deposit)
r = the annual interest rate (in decimal form)
n = the number of times interest is compounded per year
t = the number of years
Let's assume Bugis Bank compounds interest monthly, so n = 12.
We want to find the interest rate that, when compounded monthly over 5 years, will give us the same final amount as a 4% simple interest rate.
Let's solve for r:
A = P (1 + r/n)^(nt)
After 5 years, the total amount accumulated with simple interest is:
A_simple = P + (P * r_simple * t_simple)
A_simple = P + (P * 0.04 * 5)
A_simple = P * (1 + (0.04 * 5))
A_simple = P * 1.2
Now let's find the equivalent compound interest rate for Bugis Bank:
A_compound = P (1 + r_compound/n)^(nt)
A_compound = P (1 + r_compound/12)^(12 * 5)
We want A_compound to equal A_simple:
P * (1 + r_compound/12)^(12 * 5) = P * 1.2
Canceling out the P's:
(1 + r_compound/12)^(60) = 1.2
Taking the 60th root of both sides:
1 + r_compound/12 = 1.2^(1/60)
Simplifying:
r_compound/12 = 1.00233517236 - 1
r_compound/12 = 0.00233517236
r_compound = 0.00233517236 * 12
r_compound ≈ 0.02802206832
To match Raffles Bank's 4% simple interest rate over a 5-year horizon period, Bugis Bank should set an effective compound interest rate of approximately 2.8% per year.