Can I go from a stated annual rate compounded annually to a semestral effective interest rate?

Question

If the compounding period is one year, so in 6 months you haven't earned any interests yet, therefore you wouldn't be able to use the formula for the equivalent rate, one should use the formula for stated interest rate, right?

Answer 1

You can certainly use the formula for the effective rate. The effective six-month rate is the rate of interest, compounded every six months, you would need to earn in order to earn the same amount of interest as an investment of the same principal, compounded annually.

For example, if the interest rate r_annual, compounded annually, is 10%, you calculate the six-month effective rate r_sixmonth as shown:

Solving this equation for r_sixmonth yields a six-month effective rate of 4.881%. You can verify that this gives the same investment as the annual rate with this calculation:

where $100 is the principal, i.e. the initial investment. You can cancel it in this equation, but I left it in for the sake of clarify. This means that an investment earning 4.881% interest, compounded every six months, would be worth the same as an investment earning a 10% interest rate, compounded every year.

In other words, the fact that the six-month compounding period is smaller than the annual compounding period doesn't matter. The annual rate and the six-month rate both represent an answer to this question: Given some compounding period, what interest rate do I need to earn 10% on my initial investment over the course of a year?

In my example, say you start with $100 and want to have $110 by the end of the year. A single interest payment of $10 at the end of a year ("an annual rate of 10%") would work, but two interest payments of 4.881%, the first paid after six months based on your initial investment, and the second paid after a year based on your initial investment and the previous interest payment, would also work. Both interest rate structures give you the same amount after a year. The amounts and times at which they pay you are different, but the end result is the same.

I'm not sure which formula for nominal interest rates you're referring to, but usually this refers to interest rates that don't take inflation into account. That formula (real rate + inflation) isn't directly related to compounding, and usually if you see the term "interest rate" listed without specifying nominal or real, it's referring to the nominal rate. In the calculations above, I'm exclusively referring to nominal rates.