# Relationship between x and y

Studying a financial model from a third party we see:

36month - 1.2 multipler
48month - 1.3333 multipler
60month - 1.5 multipler


I am trying to determine if there is a relationship between x (month) and y (multiplier), which would allow us to determine the multipler at say 52 months etc.. I've tried my algerba but I cant see anything as simple, possibly something to the power or log? Whilst the answer would be great I'd also like to know the most pragmatic steps to approach this too.

Note: There might not be a correlation, they might be set manually

Thinking out loud, given the three points on a graph this will represent part of a curve.. now where

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It's not linear, that's for sure, and not logarithm either. What you're looking for could be a very flat exponential function. In fact, there probably exists several different "good" choices and all of them would probably be equally bad in predicting other values. It would require more datapoints to make an accurate predictor. –  Thomas E. Jan 30 '12 at 12:02

The function

$${\rm multiplier} = 1 + \frac{\textrm{number of years}}{60} + \frac{\textrm{(number of years)}^2}{60}$$

fits your data perfectly. Admittedly, so do many other functions, but the roundness of the denominators in this one suggests to me that this could well have been the formula your third party used to generate the multipliers.

I would be very careful using this formula to extrapolate multipliers (i.e. using it for number of months > 60) but you should be safe if you use it to interpolate.

I'll expand this answer to explain how I came up with this formula a bit later.

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+1 That was quick! –  user21436 Jan 30 '12 at 12:12

There is not a correct answer from the data given, though Chris Taylor may have spotted what actually happened, which would give a figure of about $1.385$ for 52 months.

If you believe this is an increasing function then for 52 months it will be between $1.3333$ and $1.5$. If you believe it is a convex function, then for 52 months it will be between the blue and green lines in the diagram below, for 52 months between about $1.3778$ and $1.3889$.

Or you could just draw a smooth freehand curve through the three points as with the red line in the diagram: for 52 months this looks just over $1.385$.

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