A principal P becomes Q in 1 year when compounded half-yearly with R% annual rate of interest. If the same…
A principal P becomes Q in 1 year when compounded half-yearly with R% annual rate of interest. If the same principal P becomes Q in 1 year when compounded annually with S% annual rate of interest, then which one of the following is correct?
R = S
R > S
R < S
R \le S
Solution
When compounded half-yearly at annual rate R, the effective annual factor is $\left(1 + \dfrac{R}{200}\right)^2$. When compounded annually at rate S, the factor is $\left(1 + \dfrac{S}{100}\right)$. Since both yield the same amount Q from the same P, the effective annual rates are equal. Half-yearly compounding gives a higher effective rate than the nominal rate, so to match annual compounding the nominal half-yearly rate R must be lower than S. Hence $R < S$.