[RESOLVED] Salary raise

[Rassis]

You are willing to raise an employee current salary of 2.000 €/month by 10% (200 €/month) within a year, but want to give him this raise in two halves: one (x%) within 6 months and the other (x%) within 12 months. How much shall the x% be?

[[Pieter]]

Your x = 4,8809%

2000 * x^2 = 2200 (x^2 because of the 2 periods) ->
x^2 = 2200/2000 = 1,1 ->
x = sqrt(1,1) = 1,048809

So starting with 2000:
after first raise: 2000 * 1,048809 = 2097,617696
after second raise: 2097,617696 * 1,048809 = 2200


Pieter

[Rassis]

Right Pieter. Another short one:

Suppose now that you want to reward another employee for his increase in productivity, giving him a salary raise equal to the mean of the productivity rates that he managed to achieve during the past two years (3% and 17%). What should this amount be if he currently earns 2.000 €/month?

[[Pieter]]

Hi Rassis,

The mean of the productivity rates is (3+17)/2 = 10%, so I think it’s the same as the previous time.

Pieter

[Rassis]

Thanks Pieter for your repply.

If it were 10%, I wouldn´t make the question...don´t you think? Take into consideration that 3% refers to an increse between, say, December n-2 and December n-1 and 17% between December n and December n-1 (we are in December n and the raise is to be valid for the year ahead). There is some particular property about these percentages (3 and 17%). The right answer is not so far apart from 10%, but it is different nevertheless!

[[Pieter]]

I thought already that this is a too easy way, but I don't know in fact what the answer is.

Pieter

[zaza]

Is this a test? If the productivity increase is compound, i.e. an increase over the previous year, then surely the overall productivity increase is 1.03x1.17 = 1.2051 over two years. Hence this is a 20.51% increase, or 10.255% p.a.

zaza

[Rassis]

The problem is that the two percentages (3% and 17%) follow a geometric series and not an arithmetic series. Hence Zaza is right to a certain extent when he considers annual compounding but is wrong when it comes to calculate the average.

[zaza]

If I make 100 units in the first year and my productivity increases by 3%, then in the second year I will make 103 = 100 x 1.03.
In the third year my productivity increases by 17% over the second year. This means that I make 120.51 = 103 x 1.17 units. Thus, the number of units I have made is 223.51 over the two years, so I have actually made 23.51 more units than I would otherwise have done. This is 11.755% more units per year. Forgot to add on the extra 3.

zaza

[Rassis]

Zaza,

You are pretty close. If you take into account the fact that the actual tax has to be compounded twice, you easily see that 11,755% doesn’t work as it would return (1 + 0,1175)^2 – 1 = 0,2489 which is higher than (1 + 0,03).(1 + 0,17) – 1 = 0,2051. So you have to go the other way around and calculate what tax compounded twice returns 0,2051.

(sorry for the "he" in my previous post...)

[zaza]

Yes, of course. I wasn't intending this to be the entire answer. Clearly it has to be plumbed back into the equation given above by Pieter to work out how to raise the salary. This is just the productivity bit - the rest is a case of number crunching.

zaza

[Rassis]

Do you Pieter or Zaza want to give the definitive answer?

[[Pieter]]

the simple number crunching :

(1+x)^2 = 1,2051
-> x = 9,777%

So the salary is 2000 * 109,777% = 2195,54

@zaza: It's not difficult after you've done all the work

Pieter

[Rassis]

Correct. Let me add something more:

The geometric mean g can also be calculated this way:

g = [(1 + i1).(1 + i2)….(1 + in)]^(1/n) – 1 = [(1 + 0,03).(1 + 0,17)]^(1/2) – 1 = 0,09777

Or using Excel:

GEOMEAN((1 + 0,03);(1 + 0,17)) – 1 = 0,0977

Thanks

[deranged]

the simple number crunching :

(1+x)^2 = 1,2051
-> x = 9,777%

So the salary is 2000 * 109,777% = 2195,54

@zaza: It's not difficult after you've done all the work

Pieter

Why do you use commas instead of periods for decimals? It is very confusing.

[Rassis]

I could ask you the same question this time changing commas and periods! All Latin countries and, I think, most European countries, apart from England, use commas for decimal numbers and periods as thousands separators. Thanks god we are not all equal! It would be terribly boring...don’t you think?