# Math Help - Proof of path dependence for savings with interest?

1. ## Proof of path dependence for savings with interest?

Hello!

Consider saving an amount X each year on an account with annual compunding. the interest rate is some interest rate process, or you can even consider a discrete set of values the interest can take every year. How do you prove that the final amount is path dependent? i.e. there's no amount X, for which the final amount is the same regardless of which path the interests take. Is this also true if the amount saved each year varies?

Thank for any help.

2. Originally Posted by ksliljes
Hello!

Consider saving an amount X each year on an account with annual compunding. the interest rate is some interest rate process, or you can even consider a discrete set of values the interest can take every year. How do you prove that the final amount is path dependent? i.e. there's no amount X, for which the final amount is the same regardless of which path the interests take. Is this also true if the amount saved each year varies?

Thank for any help.
Is this not obvious? Consider the degenerate cases where the interest rate is constant at 10% and and another where it is constant at 20%, as long as X is non zero the final amounts in the two cases differ.

CB

3. Originally Posted by CaptainBlack
Is this not obvious? Consider the degenerate cases where the interest rate is constant at 10% and and another where it is constant at 20%, as long as X is non zero the final amounts in the two cases differ.

CB
I would consider it intuitive, but not obvious. If anyone can give a mathematical proof for non-trivial cases, I'd be grateful.

Consider the interest rate path (3,435% 1,230% 2,518% 3,000%)
This yields the same amount if received backwards (3,000% 2,518 % 1,230% 3,435%)
namely 4,26 * X ( the amount saved each year)... so it's not path dependent

4. Originally Posted by ksliljes
I would consider it intuitive, but not obvious. If anyone can give a mathematical proof for non-trivial cases, I'd be grateful.

Consider the interest rate path (3,435% 1,230% 2,518% 3,000%)
This yields the same amount if received backwards (3,000% 2,518 % 1,230% 3,435%)
namely 4,26 * X ( the amount saved each year)... so it's not path dependent
The fact that two paths give the same result does not prove that the process is not path dependent.

Path independence requires that the result is the same for all paths which is why a trivial case disproves it if it is permitted by the conditions of the problem.

CB

5. Originally Posted by ksliljes
Hello!

Consider saving an amount X each year on an account with annual compunding. the interest rate is some interest rate process, or you can even consider a discrete set of values the interest can take every year. How do you prove that the final amount is path dependent? i.e. there's no amount X, for which the final amount is the same regardless of which path the interests take. Is this also true if the amount saved each year varies?

Thank for any help.
There is such an amount and it is 0 you should have a condition X>0 somewhere.

What you mean by a path here could do with some clariffication as well would 1% 2% 2% cound as an interest rate path?

What about 0% 0% 1% ?

CB

6. Originally Posted by CaptainBlack
There is such an amount and it is 0 you should have a condition X>0 somewhere.

What you mean by a path here could do with some clariffication as well would 1% 2% 2% cound as an interest rate path?

What about 0% 0% 1% ?

CB
ok, let's add the condition X>0. Let's think of the interest rates like this:
There's t time intervals and for each time interval the interest rate can have any discrete value from r1....ri, so there's i values that r is allowed to have. The amount of possible paths for r to take from t=0....t is i^t
agree?
the above example rates are, yes, permutations, but

Originally Posted by CaptainBlack
The final sum is invariant under permutations of a finite set of interest rates.
CB
This does not hold if the rates were, say
(3,000 %1,230 %2,518 %3,435 %)
and yes, your paths mentioned above would count for t=3 with discrete space(example) R=[-1 0 1 2 3]

Maybe the question should be refraced.
How do you prove, for the Space R=[r1.....ri] (discrete or continuous) and time t, that all paths ri(t)= (ri1....rit) produces a different amount in the end for any X>0?

7. Originally Posted by ksliljes
ok, let's add the condition X>0. Let's think of the interest rates like this:
There's t time intervals and for each time interval the interest rate can have any discrete value from r1....ri, so there's i values that r is allowed to have. The amount of possible paths for r to take from t=0....t is i^t
agree?
the above example rates are, yes, permutations, but

This does not hold if the rates were, say
(3,000 %1,230 %2,518 %3,435 %)
and yes, your paths mentioned above would count for t=3 with discrete space(example) R=[-1 0 1 2 3]

Maybe the question should be refraced.
How do you prove, for the Space R=[r1.....ri] (discrete or continuous) and time t, that all paths ri(t)= (ri1....rit) produces a different amount in the end for any X>0?
That is false, as any permutation of a given path produces the same final total. But not all paths give the same result, since if r1 \ne r2 the path (r1, r1, ... , r1) gives a different result to (r2, r2, ..., r2)

CB

8. Originally Posted by CaptainBlack
That is false, as any permutation of a given path produces the same final total. But not all paths give the same result, since if r1 \ne r2 the path (r1, r1, ... , r1) gives a different result to (r2, r2, ..., r2)

CB
What is false?
that it's i^t? the first year there's i possible rate, the second there's i possible rates => i*i*.....*i=i^t
Remeber, that the fact that some path yields the same final result as another, does not make the paths identical.
And I'm not shure what you mean with "any permutation of a given path". I just gave you an example where one permutation of a given path gives a different final result than another.
For 4 years annual compound, 1$deposit each year, with interests 3,435 %1,230 %2,518 %3,000 % The result is 4,260$
and for the otherwise same savings but with interests 3,000 %1,230 %2,518 %3,435 %
The result is 4,274$Not all paths give the same result, exactly. But do some? 9. Originally Posted by ksliljes What is false? that it's i^t? the first year there's i possible rate, the second there's i possible rates => i*i*.....*i=i^t Remeber, that the fact that some path yields the same final result as another, does not make the paths identical. And I'm not shure what you mean with "any permutation of a given path". I just gave you an example where one permutation of a given path gives a different final result than another. For 4 years annual compound, 1$ deposit each year, with interests 3,435 %1,230 %2,518 %3,000 %
The result is 4,260$and for the otherwise same savings but with interests 3,000 %1,230 %2,518 %3,435 % The result is 4,274$

Not all paths give the same result, exactly. But do some?
Go back to the beginning.

What do you mean by path dependent?

What would path independent mean?

CB

10. Originally Posted by ksliljes
ok, let's add the condition X>0. Let's think of the interest rates like this:
There's t time intervals and for each time interval the interest rate can have any discrete value from r1....ri, so there's i values that r is allowed to have. The amount of possible paths for r to take from t=0....t is i^t
agree?
the above example rates are, yes, permutations, but

This does not hold if the rates were, say
(3,000 %1,230 %2,518 %3,435 %)
and yes, your paths mentioned above would count for t=3 with discrete space(example) R=[-1 0 1 2 3]

Maybe the question should be refraced.
How do you prove, for the Space R=[r1.....ri] (discrete or continuous) and time t, that all paths ri(t)= (ri1....rit) produces a different amount in the end for any X>0?
Ignore the permutation comment, its a brain fade.

CB

11. Originally Posted by CaptainBlack
Go back to the beginning.

What do you mean by path dependent?

What would path independent mean?

CB
Well, I've been thinking about that too. Depends on how you define it.
In this case I think it means that everything in the past makes a difference, starting point and intermediate points. There's never a point where you can be indifferent about steps and it wouldn't effect the result.
What I really want is a mathematical expression for what defines the subset of paths that lead to the same end result, or what they have in common.
Metaphor:
"I'm lost and want to find home. There are a myriad paths to choose from and I can start from whatever place I want. You know that there are only two paths that will lead me home and they're green. All the other paths are blue. How do you explain to me how to get home if I'm colorblind?"

Maybe I should move this thread to some forum that's more on the philosophical side... heh... I'm a researcher so sorry for that.

12. Originally Posted by ksliljes
Well, I've been thinking about that too. Depends on how you define it.
In this case I think it means that everything in the past makes a difference, starting point and intermediate points. There's never a point where you can be indifferent about steps and it wouldn't effect the result.
What I really want is a mathematical expression for what defines the subset of paths that lead to the same end result, or what they have in common.
Metaphor:
"I'm lost and want to find home. There are a myriad paths to choose from and I can start from whatever place I want. You know that there are only two paths that will lead me home and they're green. All the other paths are blue. How do you explain to me how to get home if I'm colorblind?"

Maybe I should move this thread to some forum that's more on the philosophical side... heh... I'm a researcher so sorry for that.
Path dependent - at least two different paths give different results. Note does not require that all paths give different results.

Path indpendent - all paths give the same result.

CB