Spot rate: the interest rate applied to a given spot investment to be repaid at maturity, as a single cash flow.
Par rate: the interest rate such that the PV of the cash flows (lets say semi-annually) between spot and maturity is equal to par.
For 6-monthly rates:
Spot rates: 0.705; 0.875; 1.045; 1.238; 1.450
Par rates: 0.705; 0.875; 1.043; 1.235; 1.445
As you can see, both are increasing with the par rates sitting slightly less than, or equal to, the spot rates.
Consider the spot rate at $T=2.5$, which is 1.450% (with a corresponding par rate of 1.445%), and the question is why the par rate needs to be lower than the spot rate.
There is a discount factor $d(T=2.5)$ applied to the spot rate of 1.450%, which gives the present value of the return on a 2.5-year loan at 1.450%.
If the 2.5-year par rate is equal to 1.450%, then in order for it to be a valid par rate we need the present value of the 5 cash flows to be equal to par.
We need the par rates to be lower in order for the discounting effects to be lower, in order to account for the fact that there is more discounting involved in the calculation of par rates vs spot rates. In other words, in order to get the 2.5-year spot rate, we are discounting one cash flow, whereas to get the 2.5-year par rate we are discounting 5 cash flows, meaning that the discounting effect needs to be less across the cash flows (and subsequently rates lower) in order to make the two curves consistent.
I am able to come up with a reasonable intuition here that comes to the correct conclusion, but I'm wondering if my intuition actually makes sense? Or have I forced the right conclusion with the wrong reasoning?
If the former, could someone please put my reasoning more succinctly? If the latter, could someone please explain?