Q.Mr. Y has two investment options - either at 10% per annum compounded semi-annually or 9.5% per annum compounded continuously. Which option is preferable and why?

The key idea is to compare the Effective Annual Rate (EAR) for each option — the rate that tells you the actual annual growth after accounting for compounding frequency.

Step 1: EAR for semi-annual compounding (10% p.a., compounded semi-annually)

Formula: $\text{EAR} = \left(1 + \frac{r}{n}\right)^n - 1$

Here $r = 0.10$, $n = 2$:

$\text{EAR} = \left(1 + \frac{0.10}{2}\right)^2 - 1 = (1.05)^2 - 1 = 1.1025 - 1 = 0.1025$ or $10.25\%$.

Step 2: EAR for continuous compounding (9.5% p.a., compounded continuously)

Formula: $\text{EAR} = e^{r} - 1$

Here $r = 0.095$: …

Why Effective Annual Rate?

When interest compounds more than once a year, the nominal rate (the stated percentage) doesn't tell the full story. Compounding means you earn interest on interest, so the actual yearly growth is higher than the nominal rate. The Effective Annual Rate (EAR) converts any compounding frequency into a single, comparable annual percentage — it’s the true annual return.

For semi-annual compounding, interest is added twice a year, each time at half the nominal rate. For continuous compounding, interest is added at every instant, using the mathematical constant $e$. To compare fairly, we compute the EAR for both.

Step-by-step comparison

1. Option A: 10% per annum compounded semi-annually

Here, $r = 0.10$ and $n = 2$ (twice a year). Each half-year, the interest rate applied is $r/n = 0.05 = 5\%$.

The growth factor after one year is:

So the EAR is:

2. Option B: 9.5% per annum compounded continuously

Here, $r = 0.095$. Continuous compounding uses the exponential function $e^{r}$ as the growth factor after one year.

The growth factor is:

We can compute this. A useful approximation: $e^{0.095} \approx 1.0996$ (more precisely, using a calculator: $e^{0.095} = 1.099618...$).

So the EAR is:

3. Compare the two EARs