# What is the definition of Curve

I recently joined an Investment Bank and I see lot of mentions of a term called Curve. Typically, a text book does not put such an emphasis on this term.

So my question, what is the definition of Curve in IB world? Is it basically, a group of risk factors like say discount factor at time 1/2/3/... year

Appreciate if someone share any detailed explanation

• There are many curves, the most important is this one ustreasuryyieldcurve.com (and it is in every textbook) Commented Sep 5, 2023 at 5:30
• You can read the methodology for Euro area yield curves here. The idea of Interest rate swap curves is outlined in this answer. As @nbbo2 mentioned, any textbook should discuss curves though. Commented Sep 5, 2023 at 6:16
• A function of time. Simple. Can be discount, projection, survival, volatility, mean reversion, break even, funding etc... Just a function of time. Commented May 6 at 21:27

Actually I think this is quite a good question. When I was in your shoes almost 20years ago I had exactly the same concern "what, exactly, is a curve?".

In my books, mainly about IRS trading, I actually go into quite of lot of detail about curves. But it suffices to say that a Curve in finance represents datetime indexed values. for example a set of (date, discount factor) pairs: $$\{(d_1, v_1), .., (d_n, v_n)\}$$. Ideally one requires a pair of (date, values) for every possible date. Practically curves tend to employ interpolation to fill in gaps when the curve is parametrized by a smaller set of pairs.

As with many things in finance there tends to be some overlap with different people's use of the word Curve. As an analogy I am European, if someone says the temperature is going to be 90 degrees tomorrow that is meaningless to me. I will always convert that value (which is Fahrenheit) to Celsius, before making any decisions, such as whether to wear a coat etc..

Similarly some Curves are isomorphisms of one another. For example a set of (date, discount factors) is isomorphic to a set of (date, continuously compounded zero coupon rates) which is isomorphic to a set of (date, overnight rates).

All that is required to return the isomorphic form is some kind of transformation between them, like the Fahrenheit-Celsius transformation. I have never used a (date, continuously compounded zero coupon rates) curve for example but regularly use a (date, discount factor) curve and visualize it with (date, overnight rates).

Other curves that I tend to avoid because they do not provide any relevant value for pricing instruments are the 'par tenors curves' where one might plot (date, yield-to-maturity) for bonds or (date, swap rate) for swaps. There are simply better and more refined curves that can be used for both visualization and pricing and which are more worthy of being monitored.