Showing posts with label Inflation. Show all posts
Showing posts with label Inflation. Show all posts
Approximating the impact of inflation
The other day someone mentioned to me a rule of thumb that he was using to estimate the number of years \(n\) it would take for inflation to destroy half of the purchasing power of today's money:
\[ n = \frac{70}{p}\]
Here \(p\) is the inflation in percent, e.g. if the inflation rate is \(2\%\) then today's money would buy only half of today's goods and services in 35 years. You can also think of a saving account with an interest rate of \(2\%\) that would double your money in 35 years.
It is not difficult to derive this formula. The starting point is:
\[
2K = K (1 + \frac{p}{100})^n
\]
This is equivalent to:
\[
2 = (1 + \frac{p}{100})^n
\]
Taking the log gives:
\[
\log(2) = n \log(1 + \frac{p}{100})
\]
The first term of the Taylor series approximation of \(\log(1+x)\) for small \(x\) is \(x\). Hence for small \(p\) I can set:
\[
\log(2) \doteq n \, \frac{p}{100}
\]
Next I have to estimate the value for \(\log(2)\). Writing it as an integral leads to:
\[
\log(2) = \int_1^2 \frac{1}{x} \,dx
\]
Using Simpson's rule I can approximate the integral with:
\[
\int_1^2 \frac{1}{x} \,dx \doteq \frac{2-1}{6} (1+4\frac{2}{1+2}+\frac{1}{2} )
= \frac{25}{36} \doteq 0.7
\]
Thus,
\[
n \doteq \frac{70}{p}
\]
Plotting the two formulas against each other reveals that the approximation works pretty well, even for inflation rates up to 10%.
\[ n = \frac{70}{p}\]
Here \(p\) is the inflation in percent, e.g. if the inflation rate is \(2\%\) then today's money would buy only half of today's goods and services in 35 years. You can also think of a saving account with an interest rate of \(2\%\) that would double your money in 35 years.
It is not difficult to derive this formula. The starting point is:
\[
2K = K (1 + \frac{p}{100})^n
\]
This is equivalent to:
\[
2 = (1 + \frac{p}{100})^n
\]
Taking the log gives:
\[
\log(2) = n \log(1 + \frac{p}{100})
\]
The first term of the Taylor series approximation of \(\log(1+x)\) for small \(x\) is \(x\). Hence for small \(p\) I can set:
\[
\log(2) \doteq n \, \frac{p}{100}
\]
Next I have to estimate the value for \(\log(2)\). Writing it as an integral leads to:
\[
\log(2) = \int_1^2 \frac{1}{x} \,dx
\]
Using Simpson's rule I can approximate the integral with:
\[
\int_1^2 \frac{1}{x} \,dx \doteq \frac{2-1}{6} (1+4\frac{2}{1+2}+\frac{1}{2} )
= \frac{25}{36} \doteq 0.7
\]
Thus,
\[
n \doteq \frac{70}{p}
\]
Plotting the two formulas against each other reveals that the approximation works pretty well, even for inflation rates up to 10%.
R Code
Here is the R code to reproduce the plot.curve(70/x, from=1, to=10,
xlab="Inflation rate p%",
ylab="Number of years for purchaing power to half",
main="Impact of inflation on purchasing power",
col="blue",
type="p", pch=16, cex=0.5)
curve(log(2)/(log(1+x/100)),
from=1, to=10, add=TRUE,
col="red")
legend("topright",
legend=c("70/p","log(2)/log(1+p/100)"),
bty="n",
col=c("blue", "red"),
pch=c(16,16), pt.cex=c(1,1))Claims Inflation - a known unknown
Over the last year I worked with two colleagues of mine on the subject of inflation and claims inflation in particular. I didn't expect it to be such a challenging topic, but we ended up with more questions than answers. The key question and biggest challenge is to define what inflation, or indeed claims inflation actually is and how to measure it. We published a summary of our thoughts and findings in this month's issue of The Actuary.
Last year's discussion about the differences between the retail price index (RPI) and consumer price index (CPI) in the UK only exemplified the challenge. The economist Tim Harford illustrated the differences between the RPI and CPI with a simple example of price changes for a shirt and blouse in his Radio 4 programme More or Less. The radio podcast is still available from the BBC. Start listening after about 18 minutes into the show.
Last year's discussion about the differences between the retail price index (RPI) and consumer price index (CPI) in the UK only exemplified the challenge. The economist Tim Harford illustrated the differences between the RPI and CPI with a simple example of price changes for a shirt and blouse in his Radio 4 programme More or Less. The radio podcast is still available from the BBC. Start listening after about 18 minutes into the show.
