Why do increasing spot rates have to be equal to or larger than the corresponding par rates?

Question

Definitions

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.

Example data

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.

My intuition

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.

Problem

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?

Answer 1

To answer this question, we must fix a bit of the vocabulary, first. I will try to stick as close as possible to your conventions:

Spot rate: (also called zero rate) is the annualised rate of return on a non-coupon-bearing bond (hence zero coupon bond). For a given maturity $t$, let us call $r_t$ the corresponding spot rate, i.e. we have

$$ PV_0(t)=\left(\frac{1}{1+r_t}\right)^t $$

For the par rate, we need not only some rates but also corresponding cash flows -- e.g. coupon rates if we are talking about bonds. The par rate, then, is the rate $y_t$ that corresponds to a maturity $t$ that prices a coupon-bearing bond with coupon $c$ (which is itself usually a function of the time to maturity among others) at par. For the sake of argument, let us assume that our bonds are priced at par if their modeled present value equals 100%. Then:

$$ 1=c\sum_{k=1}^{N}\left(\frac{1}{1+y}\right)^{t_k}+\left(\frac{1}{1+y}\right)^{t_N} $$

Trivially, if, at emission, the bond is valued at par ($PV=100\%$), then the par yields $y$ must equal the corresponding coupon rates $c$, i.e. the coupon of a 1yr-bond implies the 1yr par rates, the coupon of a 2yr-bond implies the 2yr-pars and so on.

How do spot (zero) and par relate? In practice, you want to back out meaningful spot (or zero) rates from observed prices in order to apply them for discounting other cash flows, e.g. from other products, internal projects etc. The process is called bootstrapping and iteratively fixes spot rates (1yr, 2yr, 3yr and so on) so that we can use these - instead of the par rates - to value the coupon bearing bonds.

Thus, the first spot and par rates must coincide, as - from a mathematical perspective - it does not matter whether we analyse a discount bond or a coupon bearing 1yr bond:

$$ \frac{1}{1+r_1}\left(1+c_1\right)=\frac{1}{1+y_1}\left(1+c_1\right)=1 \Leftrightarrow c_1=r_1=y_1 $$

For the 2yr bond, par and spot rates must diverge if the yield curve is not flat:

$$ \label{qq1} \tag{1} 1=\frac{1}{1+r_1}c_2+\left(\frac{1}{1+r_2}\right)^2(1+c_2) $$

Where $r_1$ has already been fixed to $c_1$. If now the par rate structure, as embedded in the $c_i$ in our example, is upward sloping, we have that $c_2>c_1=r_1$, hence $r_2$ mus increase a bit more to allow for the fact that the first coupon cash flow is already above $r_1$ in our example:

$$ \label{qq2} \tag{1*} (1+r_2)^2=\frac{(1+c_1)(1+c_2)}{1+c_1-c_2} $$

Thus, if $c_1=c_2=c=y$, then $r_1=r_2=y$ as well. If $c_2>c_1$, then $r_2>r_1$ as well, and a bit more so!

The same line of reasoning is valid for all other par / spot rates.