1. October 12, 2010 MATH 2510: Financial Mathematics 2 1 Interest Rates: The Big Picture So far, we have assumed a single, constant interest rate. We may quote it discretely (i in CT1) or continuously (”as a force” in CT1). That is not realistic.

2. October 12, 2010 MATH 2510: Financial Mathematics 2 2 Danish Interest Rates

3. October 12, 2010 MATH 2510: Financial Mathematics 2 3 Interest rates are ”non-constant in two dimensions”: Term (or time to maturity) and calendar time. In CT1, Units 13 and 14 we relax the assumption

4. October 12, 2010 MATH 2510: Financial Mathematics 2 4 The Term Structure of Interest Rates CT1, Unit 13 Sec. 1-3: Definitions and concepts (n-year) zero coupon bonds and (discrete) spot rates Discrete forward rates; intepretation and realtion to spot rates Continuous versions of the rates Instantaneous forward rates

5. October 12, 2010 MATH 2510: Financial Mathematics 2 5 Zero coupon bonds An n-term zero coupon bond (ZCB) pays £1 after n years – and nothing else. Its price is denoted by P n. ZCBs differ by term. If we know ZCB prices, we can easily price anything else w/ deterministic payments:

6. October 12, 2010 MATH 2510: Financial Mathematics 2 6 Spot Rates/Zero Coupon Rates The discrete spot rate, y n, is the yield on an n-term zero coupon bond: The function that maps term n to spot rate y n is called the (zero coupon) yield curve or the term structure of interest rates

7. October 12, 2010 MATH 2510: Financial Mathematics 2 7 Forward Rates The discrete forward rate f t,r is the interest rate that can be agreed upon today for a future loan that runs between time t and time t+r. At time t, we recieve £1, at time t+r we pay back £1*(1+ f t,r ) r. (And no money changes hands today.)

8. October 12, 2010 MATH 2510: Financial Mathematics 2 8 An (absence of) arbitrage argument shows that forward rates and zero coupon bond prices are related We write f t for the one-period forward rates f t,1, by repeatedly using the formula above we see that Spot rates are (geometric’ish) averages of forward rates.

9. October 12, 2010 MATH 2510: Financial Mathematics 2 9 The Yield Curve CT1, Unit 13, Sec. 4.1 Real-world yield curves come in many shapes Increasing w/ term (”normal”; Fig. 2, Unit 13) Decreasing w/ term (”inverted”; Fig. 1) Humped (Fig. 3)

10. October 12, 2010 MATH 2510: Financial Mathematics 2 10 Five Danish Yield Curves

11. October 12, 2010 MATH 2510: Financial Mathematics 2 11 Why do Yield Curves Look the Way the Do? Popular explanations (not ”models”) Expectations Theory. Rates will fall. I’d better lock in a good rate on my long term investments. Higher demand for long term bonds -> higher prices -> lower yields. Good for inverted yield curves.

12. October 12, 2010 MATH 2510: Financial Mathematics 2 12 Liquidity Preference: Higher long term rates is a compensation for risk. (Or: Lower short term rates is a reward for short term funding.) Does actually have ”theoretical merit”. Market Segmetation. Supply and demand are important. But also ”Humped curve? I give up!”