CH -13 The Greek Rho

Can someone please explain the below line. what does it mean?

"a low value of p reduces risk relative to uncertainty in the risk-free rate of interest."

 

It simply means value of derivative is less sensitive to small changes in interest rate. So corresponding interest rate risk is reduced.

 

Not a brilliantly worded line. Let's break it down:
(1) "a low value of rho"
(2) "reduces risk"
(3) "relative to"
(4) "uncertainty in the risk-free rate of interest"

(1)
Rho is simply the (mathematical) derivative of the value of a (financial) derivative with respect to the variable r (the theoretical risk-free rate you can save or borrow at under the Black-Scholes assumptions). A (mathematical) derivative just describes how something changes when another quantity changes a very small amount. So a low value of rho here means that the value of the (financial) derivative changes by a low amount when r changes.

(2)
What exactly does a reduction in risk mean? Well, here 'risk' means 'how much the value might change'. If the value of the derivative will swing around wildly in response to other things happening in the market, such as changes in the underlying instrument or the risk-free rate, we can say the derivative is "risky". It makes sense, then, putting together (1) and (2), that a low value of rho reduces risk. Because if we know that rho is small then we know the value won't change much in response to small changes in r.

(3)
This is best understood in the context of the whole statement, I think. So let's go through it in full below.

(4)
Uncertainty in the risk-free rate here means fluctuations in r, the assumed rate we can borrow or lend at under the Black-Scholes framework.


Working backwards from (4) to (1), let's say we have some range of fluctuations in the risk-free rate: say r might go up or down 100bps (1 percentage point) - we'll say this makes the uncertainty on r a range of +/-100bps.
With a low value of rho, as per (1), we know that changes in r will lead to small changes in the value of the derivative. For example if rho=0.1, then we expect fluctuations in the value of the derivative of +/-10bps (+/-0.001) in respect of +/-100bps fluctuations in r.
Now remember, as per (2), the fluctuations in the value of the derivative is the 'risk'. So a range of +/-0.001 is the risk on the derivative relating to uncertainty of +/-100bps in the risk-free rate r.
So we have that a low value of rho (0.1) makes sure that the risk (+/-0.001) on the derivative is low *compared to* the uncertainty (+/-100bps) in the risk-free rates . The words "relative to" here can be substituted for *compared to*, which explains part (3) of the statement.


Addendum:
I don't love the statement because, taken by itself (sorry I don't have a copy of the core reading) it sounds as if low rho makes sure all risk is low when compared to potential changes in r. But of course there are plenty of other drivers of risk on derivatives, and in fact rho is pretty irrelevant these days for anything but really long dated options because of how low interest rates are. Apart from in emerging markets perhaps / places with higher interest rates like Mexico. The other greeks will build up a fuller picture of the risk on a derivative. Remember that changes in one greek change the other greeks too though, e.g. changes in Vega will impact delta and so on.