How to interpolate a curve between two points?

#1
I have two points, for e.g. let's take an equity Index like SPX 500. I have the values of the Index 2021 Q1 and a value for 2022 Q2. Now, I want to interpolate the values for the rest of the quarters which is 4 values but not in a straight line. I want put a curve is place. Is there any methodology for this?
 
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Dason

Ambassador to the humans
#2
You would need to be more specific about the kind of curve you want. With only two values how should we know how the curve should look?
 
#3
Lets assume the start value of the equity Index is 100 and end value is 50. I need to fit 4 points in between. And it should be decreasing concave curve similar to the bond price vs yield curves.
 

Dason

Ambassador to the humans
#4
Any other restrictions you can come up with? There are an infinite number of curves that meet your current criteria.
 
#5
Restrictions would be:

1) All 4 points should lie between 100 & 50.
2) They should be in a descending order i.e. 100> p1 > p2 > p3 > p4 > 50.
3) If possible the relative difference between 100 & p1 should be maximum as compared any other 2 adjacent points such as p2 & p3 or p3 & p4 etc.
 

noetsi

No cake for spunky
#7
When you only have two points you can as dason said fit an infinite number of curves through them (you can try this to see it). Nothing you added as a restraint really changes this. I think if you draw the two points on a graph and start fitting lines through them this will be obvious.
 

hlsmith

Less is more. Stay pure. Stay poor.
#8
Well unless you have more data to inform the process you actually aren't interpolating, just guessing. How do you truly know, say point 3 isn't 155? This really feels fishy. How do you know the relationship isn't approximately linear? If you had a series of values you could then fit a curve linear line if it was appropriate with say a slope coefficient that is multiplicative. But given two points with no other data you are just making things, up right?