How to Annualised 3months monthly returns?

Question

Hungary's monthly CPI growth rate has been as follows for the past 3 months:

Dec 18 - $2.8\%$

Jan 19 - $3.2\%$

Feb 19 - $3.5\%$

I am wondering how you'd calculate the annualized growth rate $r$ from the past 3 months, as inflation there is obviously accelerating and I would like to know the extent to which it has accelerated in the past 3 months on an annualized basis.

I was thinking that I could use the formula:

$$r = \left(1 + \frac{\text{End Value}}{\text{Start Value}}\right)^{12/3}-1$$

But I don't think that applies here?

Answer 1

I understand you posted monthly rate figures. If they are annualized, compute $r_m = (1+r_a)^{1/12} - 1$ where $r_m,r_a$ denote monthly and annual rates.

Assuming you start with $1$ in the beginning of December, at the end of December you have $1 \times 1.028$ and at the end of February $$1 \times 1.028 \times 1.032 \times 1.035 = 1.09802736.$$

This is what your growth is over 3 months, so if you would grow at that rate for a year, you get $12/3 = 4$ more such periods, so your final value would be $$ 1.09802736^{12/3} = 1.45362588178, $$ which amounts to approximately $45.36 \%$ annualized or a monthly rate of $$ r_m = \left(1.09802736^{12/3}\right)^{1/12} - 1 = 1.09802736^{1/3} - 1 \approx 3.166 \%. $$

Answer 2

The equation you quote assumes that the inflation over these three months is representative of what you will see for the year. The increase in the three months is a factor $1.028\cdot 1.032 \cdot 1.035\approx 1.098$. If we go four quarters we raise that to the fourth power and get a factor $1.454$, so the inflation would be projected to be $45.4\%$ for the year.

This ignores your perception that inflation is increasing. If you believe that, you need to make a model of how it is increasing to guess what it will be for the next three quarters. This is difficult, especially with only three months of data, because there is noise in the data. You could assume the fundamental growth in the rate is linear with time and do a least squares fit to get the growth rate. When I did that (taking the $0.028$ increase to be month $0$) I get the inflation is $0.0282+0.0035m$ where $m$ is the number of the month. You can plug $0$ to $11$ in for $m$, get the inflation for each month, add each to $1$, multiply, and subtract $1$ to get the inflation for the year. In nine months we will know which model is closer.

You could also fit a quadratic through the three points. It will be downward curving, so the inflation in that model will be less than the linear increase one.

The mathematics of all this is well defined. Once you decide on a model it just goes through the equations. There is noise in the data, so you cannot rely on the parameters you calculate. New events may change the trend even if your current version of it is correct.