Help Needed for Formula Looking at Impact of Inflation

[Guest]

Hello (ps sorry I couldn’t format formulas correctly):

I’ll start saying I am no maths expert at all (that’s why I’m here), just in my spare time I am interested in learning about and designing financial algorithms and formulas and recently came across one I could reconfigure to calculate how much compound inflation one’s wages could take in the space of ‘n’ years before being eclipsed.

i = Income e = Expenditure n = Years

(((i/e)^1/n )-1)x100

Eg: (((25,000/20,000)^1/5 )-1)x100 = can sustain an inflation rate, year on year of 4.564% for 5 years

Then I changed it slightly to include the savings of the individual.

s = savings

((((i+(s/n))/e)^1/n )-1)x100

But I realised that this wouldn’t work, as each year one’s savings would be a little more as you would add the result of the leftover of i - e, but even this figure would change each year as inflation ate into what would be left over.

As I said, I’m no expert, just someone who enjoys basic financial calculations, so I cannot work out a neat (or a messy one!) formula that could be used to encapsulate all of this - all the changes in savings year on year and therefore produce a correct answer.

Any help would be most welcome!

[Math Tutor]

Your first formula answers "when does my annual spending overtake my annual income". Once savings are in the picture the right question is different: "when does my cumulative surplus run out".

With income [tex]i[/tex] each year, first-year expenditure [tex]e[/tex] growing at rate [tex]r[/tex], and savings [tex]s[/tex], total spending over [tex]n[/tex] years is a geometric sum, so you stay solvent as long as

[tex]s+ni\ \ge\ e\cdot\frac{(1+r)^n-1}{r}[/tex]

The break-even inflation rate is the [tex]r[/tex] that makes the two sides equal. There is no closed form for it (it is a degree-[tex]n[/tex] polynomial), so just goal-seek it in a spreadsheet - but that one line is the whole model, savings and all.

On your example ([tex]i=25000[/tex], [tex]e=20000[/tex], [tex]s=0[/tex], [tex]n=5[/tex]) it gives about 11.2%, not 4.56% - the surplus from the early years pays for the deficit in the later ones.

[nycmath]

Hello (ps sorry I couldn’t format formulas correctly):

I’ll start saying I am no maths expert at all (that’s why I’m here), just in my spare time I am interested in learning about and designing financial algorithms and formulas and recently came across one I could reconfigure to calculate how much compound inflation one’s wages could take in the space of ‘n’ years before being eclipsed.

i = Income e = Expenditure n = Years

(((i/e)^1/n )-1)x100

Eg: (((25,000/20,000)^1/5 )-1)x100 = can sustain an inflation rate, year on year of 4.564% for 5 years

Then I changed it slightly to include the savings of the individual.

s = savings

((((i+(s/n))/e)^1/n )-1)x100

But I realised that this wouldn’t work, as each year one’s savings would be a little more as you would add the result of the leftover of i - e, but even this figure would change each year as inflation ate into what would be left over.

As I said, I’m no expert, just someone who enjoys basic financial calculations, so I cannot work out a neat (or a messy one!) formula that could be used to encapsulate all of this - all the changes in savings year on year and therefore produce a correct answer.

Any help would be most welcome!

You are talented.

[nycmath]

Your first formula answers "when does my annual spending overtake my annual income". Once savings are in the picture the right question is different: "when does my cumulative surplus run out".

With income [tex]i[/tex] each year, first-year expenditure [tex]e[/tex] growing at rate [tex]r[/tex], and savings [tex]s[/tex], total spending over [tex]n[/tex] years is a geometric sum, so you stay solvent as long as

[tex]s+ni\ \ge\ e\cdot\frac{(1+r)^n-1}{r}[/tex]

The break-even inflation rate is the [tex]r[/tex] that makes the two sides equal. There is no closed form for it (it is a degree-[tex]n[/tex] polynomial), so just goal-seek it in a spreadsheet - but that one line is the whole model, savings and all.

On your example ([tex]i=25000[/tex], [tex]e=20000[/tex], [tex]s=0[/tex], [tex]n=5[/tex]) it gives about 11.2%, not 4.56% - the surplus from the early years pays for the deficit in the later ones.

You are gifted.