Discount Curve Monotonicity and Implied Forward Rates

# Discount factors curve shapes


# Discount factors curve shapes












I have 2 discount factor curves;

DF 1

I expected every DF curve to have the shape of the 2nd one (almost a straight line), what does it mean economically when a DF curve has the shape of the 1st one? or maybe the curve is wrong?

How could a shorter CF 2036 offer a higher yield than the latter CF 2041

DF 2

EDIT: In the end this is the curve I got, I believe this shape is normal. THe issue I had was with the date formats, quanlib, matplotlib.

## Answer by oronimbus (score 2, accepted)

https://quant.stackexchange.com/a/74293

If the input data is correct and there aren't any calculation errors, then the discount curve should be decreasing (just like your second chart).

Using a no-arbitrage argument, Hagan & West (2007) state:

> As already mentioned, the discount factor curve must be monotonically decreasing whether the yield curve is normal, mixed or inverted. Nevertheless, many bootstrapping and interpolation algorithms for constructing yield curves miss this absolutely fundamental point.

This should be straight forward to see for a zero coupon bond $Z$ (assuming continuous compounding):

$$Z(0,t)=exp(-r(t)t) \ \Longleftrightarrow \ r(t)=-\frac{1}{t} ln Z(0,t)$$

Naturally, if you just randomly bump a zero coupon rate and recalculate the discount factors you will get a spike like in your first chart. That's why you should bump the traded instruments, then re-strip the curve and re-calculate your discount factors.

So to answer your question: my guess is that chart 1 doesn't show a consistent discount factor curve and there's either a calculation error or the rates $r(t)$ have been bumped like in the above example.

## Answer by Jamie Ballingall (score 1)

https://quant.stackexchange.com/a/74299

It might be helpful to think about the forward rates that your curve implies.

While instantaneous rates are not very intuitive they are mathematically simple, so if we have a discount curve of $Z(t)$ (with the index indicating the current time suppressed) then we can define the instantaneous forward rate as $$ f(t) = -\frac{d}{dt}\ln(Z(t)) $$ See, for example, Brigo and Mercurio equation 1.23.

Given this definition, a "bump" or locally increasing section of the discount curve represents negative forward rates.

I agree with most of what @oronimbus says but negative forward rates are, strictly speaking, allowed under no-arbitrage. It is certainly very odd for them to appear in the middle of the curve and you very well may wish to construct models that only produce positive forward rates but, unless you have access to costless storage of cash, they do not violate the no-arbitrage assumption.

Our definition of the instantaneous forward rate also helps use understand what a discount curve "should" look like. If we set $f(t)$ to equal some constant rate $r$ and then integrate we get that $Z(t)=\exp(C - rt)$ where $C$ is the constant of integration. We know that $Z(t)=1$ (that is arbitrage enforced) so $C=0$ and $$ Z(t)=\exp(-rt) $$ This has the shape of your third curve.