The Term Structure and Volatility of Interest Rates

1. The Term Structure and Volatility of Interest Rates Fin 284

2. Treasury Yield Curve • The most commonly investigated and used term structure is the treasury yield curve. • Treasuries are used since they are considered free of default, and therefore differ mainly in maturity. Also the treasury is the benchmark used to set base rates. • The treasury market is also very liquid so there are no problems with liquidity

3. Current Treasury Yield Curve • The most straightforward way to represent the yield curve is by graphing the combinations of yield and maturity • The problem with this measure is that it does not account for differences in coupon rates across bonds of similar maturities. Therefore there are some alternative methods we need to explore such as using the zero spot rates.

4. Investigating the Yield Curve • The best theoretical measure is looking at the spot rte (zero coupon) yield curve. • However often the yields for the on the run treasuries are used as a proxy for this. • The recent yield curve based upon the on the run yields are given in the next few slides.

5. Yield Curves Previous 6 Months

6. Yield Curves Previous 5 quarters

8. US Treas Rates Jan 1990 Dec 2003 Downward sloping yield curve

9. Yield Curve 2000

10. Shifts in the yield curve • Using the Arbitrage free valuation approach of a security requires an estimation of the zero coupon treasury yield curve. • As the shape of the yield curve changes so will the rate corresponding with each respective time frame.

11. Shifts in the Yield Curve • The shape of the yield curve and changes in the shape can provide information to the market concerning future interest rates. • Want to investigate two things, overall shifts in the curve and changes in its slope.

12. Parallel Shifts Short Intermediate Long Maturity Short Intermediate Long Maturity

13. Approximate Parallel Shift Change in rates is approximately the same for all maturities

14. Twists Flattening Twist Short Intermediate Long Maturity Steepening Twist Short Intermediate Long Maturity

15. Flattening of the curve Change for short term is greater than for long term

16. Steepening of the Curve Change for long term is greater than for short term

17. Butterfly Shifts Positive Butterfly Short Intermediate Long Maturity NegativeButterfly Short Intermediate Long Maturity

18. Positive Butterfly shift Increase for short term and long term is greater than for intermediate term

19. Negative Butterfly Shift Decrease for short term and long term is greater than for intermediate term

20. Why does the Yield Curve usually slope upwards? Three things are observed empirically concerning the yield curve: • Rates across different maturities move together • More likely to slope upwards when short term rates are historically low, sometimes slope downward when short term rates are historically high • The yield curve usually slopes upward

21. Three Explanations of the Yield Curve • The Expectations Theories • Pure Expectations • Local Expectations • Return to Maturity Expectations • Segmented Markets Theory • Biased Expectations Theories • Liquidity Preference • Preferred Habitat

22. Pure Expectations Theory • Long term rates are a representation of the short term interest rates investors expect to receive in the future. In other words, the forward rates reflect the future expected rate. • Assumes that bonds of different maturities are perfect substitutes • In other words, the expected return from holding a one year bond today and a one year bond next year is the same as buying a two year bond today. (the same process that was used to calculate our forward rates)

23. Pure Expectations Theory: A Simplified Illustration Let rt = today’s time t interest rate on a one period bond ret+1 = expected interest rate on a one period bond in the next period r2t = today’s (time t) yearly interest rate on a two period bond.

24. Investing in successive one period bonds If the strategy of buying the one period bond in two consecutive years is followed the return is: (1+rt)(1+ret+1) – 1 which equals rt+ret+1+ (rt)(ret+1) Since (rt)(ret+1) will be very small we will ignore it that leaves rt+ret+1

25. The 2 Period Return If the strategy of investing in the two period bond is followed the return is: (1+r2t)(1+r2t) - 1 = 1+2r2t+(r2t)2 - 1 (r2t)2 is small enough it can be dropped which leaves 2r2t

26. Set the two equal to each other 2r2t = rt+ret+1 r2t = (rt+ret+1)/2 In other words, the two period interest rate is the average of the two one period rates

27. Applying the model • The 2 year rate is an average of the current 1 year rate and the expected rate one year in the future. • This implies that the yield curve will slope upward when the expected one year rate is expected to increase compared to the current one year rate. • Similarly the yield curve will slope downward when the expected rate is less than the current rate.

28. Expectations Hypothesis r2t = (rt+ret+1)/2 • If you assume that the expected rate is representative of the long run average and that rates will move toward the average, empirical fact two is explained. • When the yield curve is upward sloping (R2t>R1t) The current rate would be less than the long run average and it is expected that short term rates will be increasing. • Likewise when the yield curve is downward sloping the current rate would be above the long run average (the expected rate).

29. Expectations Hypothesis r2t = (rt+ret+1)/2 • As short term rates increase the long term rate will also increase and a decrease in short term rates will decrease long term rates. (Fact 1) • This however does not explain Fact 3 that the yield curve usually slopes up. Given the explanation of Fact 2 the yield curve should slope up about half of the time and slope down about half of the time.

30. Problems with Pure Expectations • The pure expectations theory also ignores the fact that there is reinvestment rate risk and different price risk for the two maturities. • Consider an investor considering a 5 year horizon with three alternatives: • buying a bond with a 5 year maturity • buying a bond with a 10 year maturity and holding it 5 years • buying a bond with a 20 year maturity and holding it 5 years.

31. Price Risk • The return on the bond with a 5 year maturity is known with certainty the other two are not. • The longer the maturity the greater the price risk. • If interest rates change the return and the 10 and 20 year bonds will be determined in part by the capital gains resulting from the new price at the end of five years.

32. Reinvestment rate risk • Two new options: • Investing in a 5 year bond • Investing in 5 successive 1 year bonds • Investing in a two year bond today followed by a three year bond in the future. • Again the 5 year return is known with certainty, but the others are not.

33. Local expectations • Local expectations theory says that returns of different maturities will be the same over a very short term horizon, for example 6 months. • This assumes that all the forward rates currently implied by the spot yield curve are realized.

34. Local Expectations • Previously we calculated the zero spot rates using the bootstrapping method for the on the run treasury securities given below. Maturity YTM Maturity YTM 0.5 4% 2.5 5.0% 1.0 4.2% 3.0 5.2% 1.5 4.45% 3.5 5.4% 2.0 4.75% 4.0 5.55%

35. Zero spot Review (local expectations example) • Given the assumption that all of the on the run treasury securities were selling at par we found the 1.5 year zero coupon rate by discounting the coupons by the respective zero coupon rates.

36. Zero spot curve (local expectations example) • Continuing the same process for future rtes we started to build a zero spot yield curve Time YTM Zero Spot 0.5 4% 4% 1.0 4.2% 4.2% 1.5 4.45% 4.4459% 2.0 4.75% 4.7666%

37. Forward Rates(local expectations example) • Given the zero spot rates it is possible to find the forward rates. • Let 1fm be the 1 period (six month) forward rate from time m to time m+1. • The forward rate can then be found as:

38. Forward Rate(local expectations example) • Given the 6 mo. zero spot rate of 4% and the 1 year zero spot rate of 4.2%, the one period (6mo.) forward rate from 6 months to 1 year would be: (1+.021)2 = (1+.02)(1+1f1) 1f1 = .022 • Similarly the 6 month forward rate from 1 year to 1.5 years can be found from the 1 year zero spot rate of 4.2% and the 1.5 zero spot rate of 4.4459% (1+.021)2(1+ 1f2) = (1.022293)3 1f2 = .024884 • Likewise 1f3 = .02847

39. Local expectations example • Local expectations theory says that returns of different maturities will be the same over a very short term horizon, for example 6 months. • The return from buying the 2 year 4.45% coupon bond that makes semiannual payments selling at par and selling it in six months should be equal to the return on a 1 year coupon bond with a YTM of 4.2% if you hold it 6 months.

40. 6 mo return on 1 year bond • The 1 year bond has a current YTM of 4.2%. This means that a equivalent bond selling at par would make a $2.10 coupon payment at eh end of 6 months and at the end of 1 year. • If you bought this bond at t = 6 months and sold it at t=1 year you should earn 1f1 (the 6 mo. forward rate) over the time you own the bond. • The price of the bond at t=6 mos should reflect this.

41. 6 mo return on 1 year bond • At time t=1 year the bond makes payments of 102.10. The total return from owning the bond is the capital gains yield and interest yield and it should equal 1f1=.022

42. 6 mo return on 1 year bond • Buying the bond at time 0 and selling it at the end of the first 6 months would then produce a return of • Which is equal to the spot (zero coupon) six month rate

43. Return on the 2 year bond • The price of the 2 year bond at the end of 6 months should also equal the PV of its expected cash flows discounted back at the forward rate (otherwise there would be an arbitrage opportunity). • By finding the price at the end of 6 months we can again find the return from owning the bond for 6 months form t=0 to t=6 mos.

44. 6 mo return on 2 year bond • There are three coupon payments left from time t = 6 mos to t= 2 years, using the forward rtes the price of the bond at t=6mos should be:

45. Total Return on holding 2 year bond for 6 months • The 2 year bond originally sold for par and it made a $2.375 coupon payment at t=6mos. The total return from owning it is then: • Which is the same as the 6 month return on the 1 year bond (and the same as the 6 month spot rate)

46. Local Expectations • Similarly owning the bond with each of the longer maturities should also produce the same 6 month return of 2%. • The key to this is the assumption that the forward rates hold. It has been shown that this interpretation is the only one that can be sustained in equilibrium.* Cox, Ingersoll, and Ross 1981 Journal of Finance

47. Return to maturity expectations hypothesis • This theory claims that the return achieved by buying short term and rolling over to a longer horizon will match the zero coupon return on the longer horizon bond. This eliminates the reinvestment risk.

48. Expectations Theory and Forward Rates • The forward rte represents a “break even” rate since it the rte that would make you indifferent between two different maturities • According to the pure expectations theory and its variations are based on the idea that the forward rte represents the market expectations of the future level of interest rates. • However the forward rate does a poor job of predicting the actual future level of interest rates.

49. Segmented Markets Theory • Interest Rates for each maturity are determined by the supply and demand for bonds at each maturity. • Different maturity bonds are not perfect substitutes for each other. • Implies that investors are not willing to accept a premium to switch from their market to a different maturity. • Therefore the shape of the yield curve depends upon the asset liability constraints and goals of the market participants.

50. Biased Expectations Theories • Both Liquidity Preference Theory and Preferred Habitat Theory include the belief that there is an expectations component to the yield curve. • Both theories also state that there is a risk premium which causes there to be a difference in the short term and long term rates. (in other words a bias that changes the expectations result)