Chapter 1: Goldilocks is ousted by the bears
Once upon a time, the banks used a fixing called LIBOR as a measure of the risk-free interest rate. Then the big hairy crisis came along and ate all our assumptions, leaving just the bones of the fixing (upon which everything else still fixes) and the mantle of risk-free rate proxy was passed on to a family of Overnight fixings, called Sonia, Eonia and -ahem- FedFundEffective.
Since everything (e.g. FRAs, 3m Interest Rate Futures, OTC IRS, Deliverable Swap Futures, cross-currency basis swaps, 3m-OIS basis, etc) still fixes on LIBOR (or other xIBOR depending on the currency), the questions now are:
- What do you expect the xIBOR fixing to be for any given term?
- What do you expect the Overnight fixing to be for any given term?
- How much is your CVA desk going to charge you to cover the credit risk?
So to value a forward-starting IRS, I need both an Overnight curve for discounting, but also a curve of forecasted fixings to estimate the cash flows themselves. When there was 1 curve, it was far simpler.
Summary: Once there was 1 fixing, and it was a proxy for the risk free rate, so there was 1 curve of discount factors. Now it is no longer such a proxy, but because liquid instruments fix on these other fixings, you have to build curves to work out what those fixings are expected to be as well.
-- Edit for great good and new RFR --
Chapter 2: Goldilocks returns
Regulatory changes (viz: the ARRC, EMIR, the BoE) have deemed that even the mantle of Libor (and cousins Euribor, etc) must now be offered up to the be burnt in the witch's oven. So instead of having a 1m Libor, 3m Libor and 6m Libor curves, the instruments which use those fixings must be replaced by new RFR (Risk Free Rates, or, Daughter of Goldilocks).
This is a return to the single curve world - all your multicurves are belong to RFR. FedFunds is being replaced (in usage) by a comfy SOFR, Eonia by the inpronouncable €STR (in October 2019) and TOIS has already become SARON. Sonia is dead, long live Sonia.
Chapter 2.5: Goldilocks and the mirror
Goldilocks II gives us backward looking 'in arrears' fixings - they are fixed in light of actual trades. This is in contrast to the forward-looking originals in Libor and Euribor.
Once Goldilocks II returns and kills all the bears (prophesied to be 2022), there is still a shadow in the wings (I mean, how else can we make the 3rd and 4th films of the trilogy): the buy side and the FI markets both want a forward looking fixing like the fixing they once knew, so they may well make up a sister fixing to Goldilocks II, a mirror fixing which looks forward. This would then be either a second curve or a lag adjustment.
So in summary, most of the textbooks were written during Goldilocks I's reign, some have since been written during the Bear Junta, and the next chapter is as yet unwritten.