Unveiling the Math Behind the Contradictory Interest Rates
Have you ever wondered how a certificate of deposit (CD) with a lower interest rate can yield a higher return over a certain period compared to a loan with a higher interest rate? This puzzling financial scenario often sparks a lot of curiosity and a need to understand the underlying mathematical principles at play.
Comparing Fixed Investment and Loan Scenarios
Consider two scenarios: a 30,000 CD at 3.4% and a 30,000 loan at 3.9% over a 4-year period. The interest calculator makes it clear that the CD will net 3,221 in interest over 3 years, whereas the loan will cost 2,449 over 4 years. The question remains: how does this math work?
The Difference in Interest Calculation Methods
The discrepancy lies in the method by which interest is calculated. For the CD, the interest is already calculated on an increasing balance, meaning the interest earned each day, month, or year depends on how the bank compounds it. On the other hand, loan interest is calculated on a decreasing balance as the loan is paid down each month. This fundamental difference causes the yield to vary significantly between the two investments.
Breaking Down the CD Interest Calculation
A certificate of deposit (CD) involves lending a principal amount for a specific duration. In the 3-year CD, the 30,000 is lent out and earns interest over 36 months (3 years). Since the CD compounds daily, the interest is calculated based on the balance of the account at the end of each day. The formula for compound interest is:
Compounded Interest Principal × (1 Rate/Time)^Time - Principal
Using the given 30,000 CD and 3.4% annual rate:
First year: 30,000 × (1 0.034/365)^365 - 30,000 903.73 Second year: (30,000 903.73) × (1 0.034/365)^365 - (30,000 903.73) 913.75 Third year: (30,000 1,817.48) × (1 0.034/365)^365 - (30,000 1,817.48) 924.03The total interest over 3 years is approximately 903.73 913.75 924.03 2,741.51. The difference of the actual interest earned versus the advertised interest (3.4%) is due to the daily compounding and the non-compounding nature of simple interest.
Loan Interest Calculation with Principal Reduction
A loan with a higher interest rate, such as 3.9%, is different because it involves borrowing a principal amount and paying it back with interest over a specific period. Over the 4 years, the loan balance decreases each month as the principal is reduced. As a result, the interest paid on the loan decreases with each payment.
Effective Yield and APR
The CD will show an Annual Percentage Rate (APR) of 3.4%, which is the stated annual interest rate. However, the yield, which is the total return over the investment period, is higher due to the compounding effect. In the CD example, the yield can be calculated as follows:
After 1 year: 30,000 30,000 × 0.034 30,000.30 After 2 years: 30,000.30 30,000.30 × 0.034 30,000.61 After 3 years: 30,000.61 30,000.61 × 0.034 30,009.50The total earned interest over 3 years is approximately 9,50. The effective yield can be calculated by dividing the total interest earned by the principal and the investment period. In this case, the effective yield is around 3.17%, which is higher than the stated APR of 3.4%.
Why Preference Simple Interest on a Loan and Compound Interest on Investments
In contrast, a loan with simple interest offers a predictable and lower interest cost over time. With each monthly payment, the interest is based on the remaining principal, and as the principal is reduced, the interest paid decreases. This makes simple interest loans more straightforward and easier to budget for.
Therefore, it is recommended to choose simple interest on a loan and compound interest on an investment for higher returns over time. The intricacies of interest calculations and compounding effects make these financial instruments behave differently, as evident in the CD and loan scenarios.