Relationship of par-curve and zero-curve/spot-curve

Question

I've been trying to bootstrap the zero-curve from a swap curve composed of ESTR OIS swaps. Theory says when the par-curve is upward sloping, the zero-curve will be above the par-curve and vice-versa. Moreover, I would imagine when the par-curve is perfectly flat, so would the zero-curve. However, applying different methodologies results in inconsistencies. I applied bootstrapping with discrete compounding and no day-count convention (i.e. ACT/ACT) and found theory to hold, e.g. that the zero-rates are above the par-rates when the par-rate curve is upward sloping etc. However, when applying bootstrapping with continuous compounding, there are zero-rates that are below the par-curve, even if it is upward sloping, which thus seems to go against theory.

For the discrete bootstrapping I use the following approach:

Assume the following swap-curve (i.e. fixed-leg par-rates) with annual payments:

Par-rates = $[0.035, 0.04, 0.045, 0.05, 0.055] $

Tenors = $[1y, 2y, 3y, 4y, 5y]$

Discrete bootstrap ($df_1$ being $\frac{1}{1+0.035}=0.96618)$:

$df_n=\frac{1-s_n\times\sum_{i=1}^{n-1}df_i}{1+s_n}$

then we find the spot-rates/zero-rates from the discount factors by applying:

$zeroRate_n=\sqrt[n]{\frac{1}{df_n}}-1$

For continuous bootstrapping, I apply the following method (with $r_1$ being $-ln(\frac{1}{1+0.035})/1=0.034401$:

$r_T=-\frac{ln\Big[\frac{1-\sum_{i=1}^{T-1}s_T*e^{-r_i*t_i}}{1+s_T}\Big]}{T}$

Once computed, I have the following:

As you can see, the continuous bootstrapped zero-rates are evidently below the par-rates. Is this how it is supposed to be?

Answer 1

I think I can answer this one.

You're effectively comparing different units for a lack of a better term.

Your discrete comparison works because you are comparing a par curve with a zero curve which are both expressed as annualized/discrete rates.

However for your continuously compounded comparison, it isn't valid because your par curve is a discrete rate, which you are comparing to a continuously compounded rate.

A continuously compounded rate is always a lower number than a discrete rate for the same amount of actual interest accrued.

Effectively you need to express your par rates as continuously compounded in order to compare them to continuously compounded zero rates.

as a second check:

If you convert your continuously compounded zero rates into their relevant discount factors by doing EXP(-r*t), they should be equal to your discrete discount factors.