Interest Rate Curves

Overview

When money is borrowed or lent for a given period of time, the money earns a pre-agreed to interest rate. However, the rate of interest earned is usually dependent on the period over which the money is lent. This means that interest rates are best represented as being a curve, that is, a function of the time to maturity.

Types of Curves

Even though it is typical to just refer "the curve", there are actually multiple curves, the difference being how the curve is being quoted or measured. For instance, the yield curve typically refers to a plot of current yield to maturies versus period.

There are problems with yield to maturity however, the primary issue being that even in efficient markets and accounting for all mitigating factors like tax treatment and credit risk, two bonds structured differently (for example amortized versus bullet) will trade at different yield to maturities.

• Zero Coupon Curve - the zero coupon curve is the interest rate that is charged when receiving a single cash flow at various maturities. (that is, this bond pays no coupons, only principal at maturity)
  
  Using continuous time compounding, the value {% B %} of a zero coupon bond with principal {% P %} today is
  {% B \times e^{r(t) \times t} = P %}
  where here, {% r(t) %} is the zero coupon rate for maturity {% t %}
• Discount Curve - the present value of a cash flow occurring at time {% t %}, given the zero coupon rate curve, is simply
  {% B = e^{-r(t) \times t} P %}
  the factor {% e^{-r(t) \times t} %} is referred to as the discount. As an alternative to quoting the zero coupon curve, some will quote instead the discount curve.
  {% D(t) = e^{-r(t) \times t} %}
• Forward Curve the forward curve is a curve that is derived from the zero coupon curve which gives the rates at which one could enter into a arbitrage free forward rate agreement.

Topics

• Theories of the Term Structure : various theories have been put forward to explain the shape of the curves discussed above.
• Estimating the Yield Curve refers to a process whereby a smooth curve, (in this case, a function of time) is calculated from a discrete set of bond prices. The smooth yield curve is then used in fixed income models to price other instruments.
• Derivations of the various rate relationships on the curve, including forward rate relationships.
• Spread Analysis
• Instrument Curves
  • Treasury Curve
  • Swap Rate Curve