CUSCR sensitivity

[CLK]

I've come across the following general formula for the standard contribution rate:

In the same source document that I'm consulting, with reference to this formula, there is a comment which says:

"You will notice that if r = e, the SCR will change only due to second order effects if i and e both change.
This means that there will be cases where CUSCR is sensitive to changes in the discount rate, even if the gap between i and e remains constant."
​

Could someone possibly help me understand these comments?

1) I assume that by "second order effects" it is meant that i and e can both change with no impact on the SCR as long as the difference between i and e remains unchanged, in the case of r = e; and
2) With respect to the second sentence, I assume this is supposed to refer to when r and e differ?
In that case, the CUSCR is further sensitive to changes in r regardless of whether the relationship between i and e is maintained.​

Is that the full intended meaning?
I would appreciate any direction.

 

[Stuart Underwood]

The e=r restriction is presumably so that the second term is 0, to remove complication.
If e=r then if i and e change by the same amount, the first term and the SCR may be broadly the same.

(I'm not sure they'd be exactly the same - whether r=e or not...
... you could set up the CUSCR in a spreadsheet for different values of i, e, r and Y to test why that may be ...
... offhand, it may be because:
1) do you have the exact number of copies of (1+i) and (1+e) in the formula, applied in the exact same way in the numerator and denominator? does it depend on the control period Y?
2) second order here may mean in terms if you change i and e by the same (arithmetic) amount the (geometric) (1+e)/(1+i) will be unchanged in first order Taylor expansion but different at second order
But in my opinion life's too short and the exam too close to worry about this sort of stuff - I think this much too detailed for the purposes of passing the SP4 exam, which has never gone into this kind of detail.)

 

[CLK]

Thanks Stuart! Sorry for the delayed response.
I’m fortunately not writing this session already and your comments are very helpful.
You mention the Taylor expansion: I’ve frequently encountered references to “nth order effects” and often it’s unclear whether the author is referring to the formal mathematical structure or is just providing a loose description. In many cases, whether deliberate or not, they at least describe the same phenomenon.
In this case, unfortunately, I don’t have an intuitive sense of the implications so I probably need to think about it some more.