FNCE 3020 Financial Markets and Institutions Fall Semester 2006

1. FNCE 3020Financial Markets and Institutions Fall Semester 2006 Lecture 5: Part 1 Explaining the Term Structure of Interest Rates Yield Curves

2. Relationship of Yields to Maturity • In lecture 4 we noted various factors, other than maturity of a financial asset, which can affect interest rates. These factors included: • Risk of default • Liquidity (i.e., secondary market impacts) • Tax status (municipal securities versus fully taxable securities) • However, we did not include term to maturity as a possible factor explaining observed differences in market rates of interest. • Term to maturity refers to the time before the bond matures.

3. Initial Observations: Does Maturity Matter? • Yes, generally long term rates are above short term rates

4. But, Are There Exceptions to the Long Rate Above Short Rates? • Yes, there are times when short term rates exceed long term rates.

5. So How can we Illustrate the Relationship Between Interest Rates and Maturity? • (1) We can look at interest rates over time. • Compare short term to long term rates. • See last two slides. • (2) We can look at interest rates at a point in time, i.e., on a particular date. • Where is the short term and the long term rate on a selected date? • This last approach is referred to as a yield curve.

6. What is a Yield Curve? • Technique used to view the relationship between maturity and yields (interest rates) on a particular date. • Specifically, a yield curve shows the relationship between market interest rates and term to maturity on outstanding debt issues. • Done for a given date (i.e., for a point in time). • And using bonds of the same credit quality. • This way you avoid differences in risk of default. • Yield curves help us to observe what we call the term structure of interest rates

7. Graphing (Plotting) a Yield Curve • A yield curve is simply a graphic presentation of the relationship of term to maturity and yields on a given date. To construct a yield curve, we plot: • The current interest rates on the Y axis, and • Corresponding term to maturity on the X axis, or: i rate Term to maturity →

8. Possible Yield Curves: Upward Sweeping (Ascending) Assume following observed market interest rates: short term (st) rates are 4% and long term (lt) rates are 8%. i rate 8% o 4% o (st) Term to Maturity (lt)

9. Possible Yield Curves: Downward Sweeping(Descending) Assume following observed market interest rates: short term (st) rates are 7% and long term (lt) rates are 3% i rate 7% o 3% o (st) Term to Maturity (lt)

10. Possible Yield Curves: Flat Assume following observed market interest rates: short term (st) rates are 7% and long term (lt) rates are 7% i rate 7% o o 3% (st) Term to Maturity (lt)

11. Summary: Three Yield Curve Shapes • As illustrated in the last three slides, there are three basic shapes that yield curves can take. These are: • Ascending (Upward Sloping; positive) Yield Curves: Long term rates higher than short term • Descending (Downward Sloping; negative) Yield Curves: Shorter term rates higher than longer term. • Flat Yield Curves: Long term and short term rates essentially the same. • Examine the next slide to identify these three basic shapes for U.S. data and specific periods of time.

12. Historical U.S. Yield Curves

13. Constructing a Yield Curve: What Securities do we Need? • Question: What financial assets should we use to construct a yield curve? • Possibilities include: • Government debt • Corporate debt • Issue: • We want to make sure that differences in credit risk (i.e., default risk) is not affecting the plotted yield curve. • Thus, in practice we use: • Only Government securities (no risk of default on U.S.), or • Only corporate debt, and if you do, use similar risk classes: • Aaa Yield Curve • Baa Yield Curve • Important: Do NOT mix Governments with corporate debt, or mix within the corporate market.

14. Constructing a Yield Curve: What Interest Rates do we Need? • Question: What interest rates should be used in constructing a yield curve? • Possible interest rates include: • Coupon yield • Current yield • Yield to maturity • In practice, we use Yield to Maturity • Takes into account the time value of money and the complete “cash” flow associated with the asset! • Important: Do NOT mix the above possible rates in a yield curve.

15. WSJ yield curve is for Tuesday, September 5, 2006 Source: http://online.wsj.com/page/2_0031.html?mod=2_0031 Yield Curves Reported in WSJ

16. Yield Curves Reported by Bloomberg U.S. Yield Curve: September 6, 2006 http://www.bloomberg.com/markets/rates/index.html

17. Change in Yield Curve: U.S. a Year Ago U.S. Yield Curve: October 3, 2005

18. Three Main Theories to Explain the Shape of the Yield Curve • There are three main theories or explanations of the yield curve. • These theories attempt to explain why a yield curve has the shape that it does. These theories are: • (Pure) Expectations Theory • Liquidity Premium Theory • Market Segmentations Theory • Additionally, as we will see in the next lecture, these theories may be used to forecast (predict)future moves and levels of interest rates!

19. The Expectations Theory • Assumption: Expectations regarding future interest rate shape a given yield curve. • Financial markets are assumed to be efficient. • There is widely disseminated knowledge and information. • Thus, the market forms expectations about the future level of interest rates. • These future expected rates are called forward rates. • At any point in time, there is a market consensus regarding future interest rates. • Again, based upon the markets’ analysis of all relevant events affecting interest rates in the future. • These expectations are incorporated into current interest rates. • These current interest rates are called spot rates.

20. The Expectations Theory (Continued) • Current observable long term rates (i.e., spot rates) represent averages of • current short term (spot) rate and • expected, future short-term (forward) rates. • Efficient market incorporates expected future rates in setting current long term rates. • This way, the market will be satisfied to provide longer term funds to borrowers.

21. Quick Example • Assume you know the 1 year rate is 5% and someone comes to you asking for a 2 year rate. • How would you set this 2 year rate? • You have the rate for one year • This is the spot rate • Now you need the 1 year rate, 1 year from now. • This is the forward rate. • If you think the 1 year rate 1 year from now will be 7%, what would you set as the 2 year rate now? • What if you thought the 1 year rate 1 year from now will be 3, what would you set as the 2 year rate now?

22. Expectations Formula for the Long-term Interest Rate The current (t) long term spot interest rate (ilst+n where n = years to maturity)is assumed to be equal to the average of the current (t) short term spot rate (isst) and all appropriate expected future short term (i.e., forward) rates (iet+1, … ien). Long term spot rate (ilsn) is the observable long term market interest rate. Current short term spot rate (isst) is the observable short term market interest rate. Forward rates (iet+1, … ien) are the market’s expectations about where interest rates will be in the future.

23. Expectations Model Example #1 • Assume the following: • Current (spot) one year interest rate is 5% and • The (forward) one year interest rates over the next five years (years 2, 3, 4, and 5) are expected to be: 6%, 7%, 8%, and 9%, respectively. • Given this data, calculate the: • (1) Current (spot) two year bond rate (il2) • (2) Current (spot) five year bond rate (il5)

24. Calculated Two Year Spot Rate • Answer: • The calculated current (spot) rate on a two-year bond is as follows: Given the current 1 year spot rate of 5% and expected 1 year forward rate, 1 year from now of 6%, then: The 2 year spot bond rate = (5% + 6%)/2 = 5.5% • Note: Investor would be indifferent to holding a 2 year bond, versus a series of 1 year bonds. • Why? Both will yield 5.5% over the two year period.

25. Calculated Five Year Spot Rate • Answer: • The calculated current (spot) rate on a five-year bond is as follows: Given the current 1 year spot rate of 5% and expected 1 year forward rates, 1 year from now through five years from now of: 6%, 7%, 8%, and 9%, then: The 5 year spot bond rate = (5% + 6% + 7% + 8% + 9%)/5 = 7% • Note: Investor would be indifferent to holding a 5 year bond, versus a series of 1 year bonds. • Why: Both will yield 7.0% over the five year period.

26. Expectations and Yield Curve • So, when short term rates are expected to rise in future, the average of these expected (forward) short rates will be above today's short term spot rate. • Recall from previous example: • One year spot rate = 5% • Two year spot rate = 5.5% • Why 5.5%: forward rate (6%) higher than 5% • Five year spot rate = 7.0% • Why 7%: forward rates higher (6, 7, 8, 9%) than 5% • In these cases, as the term to maturity increases, current spot rates increase. • Therefore, the observed yield curve will be upward sloping!

27. Upward Sweeping Yield Curve i rate 9.0 oei 8.5 8.0 oei 7.5 7.0 oei o 6.5 6.0 oei 5.5 o 5.0 o 1 2 3 4 5 year Term to Maturity →

28. Expectations Model Example #2 • Assume: • Current (spot) one year rate is 9% and • The (forward) one year rates over the next five years (years 2, 3, 4, and 5) are expected to be: 8%, 7%, 6%, and 5%, respectively. • Given this data, calculate the • (1) Current (spot) two year bond rate (il2) • (2) Current (spot) five year bond rate (il5)

29. Calculated Two Year Spot Rate • Answer: • The calculated current spot rate on a two-year bond is as follows: Given the current 1 year spot rate of 9% and expected 1 year rate, 1 year from now of 8%, then: Then the 2 year spot bond rate = (9% + 8%)/2 = 8.5% • Note: Investor would be indifferent to holding a 2 year bond, versus a series of 1 year bonds. • Why? Both will yield 8.5% over the two year period.

30. Calculated Five Year Rate • Answer: • The calculated current rate on a five-year bond is as follows: Given the current 1 year rate of 9% and expected 1 year rates, 1 year from now through five years from now are: 8%, 7%, 6%, and 5%, then: The 5 year spot bond rate = (9% + 8% + 7% + 6% + 5%)/5 = 7% • Note: Investor would be indifferent to holding a 5 year bond, versus a series of 1 year bonds. • Why? Both will yield 7.0% over the five year period.

31. Expectations and Yield Curve • So, when short term rates are expected to fall in future, the average of these expected (forward) short rates will be below today's short term spot rate. • Recall from previous example: • One year spot rate = 9% • Two year spot rate = 8.5% • Why 8.5%: forward rate (8%) lower than 9% • Five year spot rate = 7.0% • Why 7%: forward rates lower (8, 7, 6, 5%) than 9% • In these cases, as the term to maturity increases, current spot rates decrease. • Therefore, the observed yield curve will be downward sloping.

32. Downward Sweeping Yield Curve i rate 9.0 o 8.5 o 8.0 oei 7.5 7.0 oei o 6.5 6.0 oei 5.5 5.0 oei 1 2 3 4 5 year Term to Maturity →

33. Expectations Model: Example #3 • When short rates are expected to stay same in future (i.e., all forward rates are equal to the current spot rate) , average of these expected short rates will be the same as today's spot rate. • Thus the plotted yield curve will be flat. i rate 7.5 7.0 o oeioieoie o 6.5 1 2 3 4 5 year Term to Maturity →

34. Summary of Expectations Regarding Future Interest Rates • The shape and slope of the yield curve reflects the markets’ expectations about future (forward) interest rates. • Upward Sloping (Ascending) Yield Curves: • Future (forward) interest rates are expected to increase above existing spot rates. • Downward Sloping (Descending) Yield Curves: • Future (forward) interest rates are expected to decrease below existing spot rates. • Flat Yield Curves • Future (forward) interest rates are expected to remain the same as existing spot rates.

35. Liquidity Premium Theory • This is the second explanation of the yield curve shape. • Assumptions: Long term securities carry a greater risk and therefore investors require greaterpremiums (i.e., returns) to commit funds for longer periods of time. • That is for being less liquid. • The interest rate on a long term bond will equal the spot rate plus an average of the expected short term rates, with: • The expected rates including a premium for illiquidity. • What are the risks associated with long term bonds? • Price risk (a.k.a. interest rate risk). • Risk of default (on corporate issues). • Inflation risk (offsetting the nominal return).

36. Price Risk (Interest Rate Risk) Revisited • Recall from a previous lecture (lecture 2): • Long term debt instruments vary more in price than shorter term. • Why? • Recall: The price of a fixed income security is the present value of the future income streamdiscounted at some interest rate, or: Price = int/(1+r)1 + int/(1+r)n + … principal/(1+r)n

37. Example of Price Risk • Given that: • Price = int/(1+r)1 + int/(1+r)n + … principal/(1+r)n • Assume two fixed income securities: • A 1 year, 5% coupon, par $1,000 • A 2 year, 5% coupon, par $1,000 • Assume market rate (or opportunity cost) rises to 6% • What will happen to the prices of both issues? • Both bonds will fall in price (sell below their par values). See new prices on next slide! • But long term bond will fall more in price.

38. Price Changes and Maturity • 1 year bond: • Price = int/(1+r)1 + … principal/(1+r)n • Price = $50/(1+.06) + $1,000/(1+.06) • Price = $47.17 + $943.40 • Price = $990.57 • 2 year bond • Price = int/(1+r)1 + int/(1+r)2 + … principal/(1+r)n • Price = $50/(1+.06) + $50/(1+.06)2 + 1,000/(1+.06)2 • Price = $47.17 + $44.50 + $890.00 • Price = $982.67 • Price Change over par ($1,000) • 1 year bond = $ 9.43 • 2 year bond = $17.33 • Note: The long term bond experienced greater price change!

39. Impact of Price Risk and other Risks on Market Requirements • The financial markets know that there is the potential for greater price changes on longer term bonds. • Investors also know there are additional risks they face in with long term securities: • Risk of default • Inflation risk • So, investors will want a higher return on long term bonds than on short term because of these potential risk factors. • This required higher return is called a liquidity premium.

40. Adding in a Liquidity Premium • Liquidity Premium is added by market participants to longer term bonds. • It is actually a premium for giving up the liquidity associated with shorter term issues. • Thus, if observed long term spot rates are higher than short term spot rates, the question we need to address is: • Are observed higher long term spot rates due to expectations of higher rates in the future (i.e., the Expectations Theory), OR • Are observed higher long term spot rates due to added on liquidity premiums (Liquidity Premium Theory)? • Unfortunately, there is no good answer to this question

41. Liquidity Premium Theory Formula for Long Term Interest Rates • This model contents that we need to modify the expectations theory formula to take into account the presence of liquidity premiums, or • In the above formula, the long term spot rate (ilst+n) includes a liquidity premium (Ln), for holding a bond of “n” maturity. • “n” maturity assumed to be longer term.

42. Liquidity Premium Examples • Assume: One-year spot and forward interest rates over the next five years as follows: • one year spot = 5% • (one year) forwards = 6%, 7%, 8%, and 9% • Assume: Investors' liquidity premium requirements for one- to five-year bonds as follows: 0%, 0.25%, 0.5%, 0.75%, and 1.0% • Calculate the market interest rate on: • 1) a two year bond (Ln = .25%) • 2) a five year bond (Ln = 1.0%) • Compare calculated long term rates with those for the pure expectations theory formula.

43. Calculations and Comparisons • Calculation of spot rates with liquidity premium added: • Two-year bond: (5% + 6%)/2 + 0.25% = 5.5% + 0.25% = 5.75% • Five-year bond: (5% + 6% + 7% + 8% + 9%)/5 + 1.0% = 7.0% + 1.0% = 8% • Comparison of Liquidity Premium spot interest rates to Pure Expectations spot interest rates: • 2 year Liquidity Premium: 5.75%, 2 year Expectations: 5.5% • 5 year Liquidity Premium: 8.00%, 5 year Expectations: 7.0% • Thus: • liquidity premium theory will produced yield curves more steeply upward sloping than the expectations model.

44. Yield Curve Comparisons: Liquidity Premium Versus Expectations Model i rate 8.0 o LP Yield Curve 7.75 7.50 Difference is the liquidity premium 7.25 7.0 o PE Yield Curve 6.75 6.50 6.25 6.0 5.75 o 5.5 o 5.25 5.0 2yr 5yr Years to Maturity

45. Liquidity Premium Theory Summary • Since the liquidity premium is always positive and grows as the term to maturity increases, this theory generally offers an explanation of an upward sweeping yield curve. • But how can the theory explain a downward sloping yield curve? • Easy: If forward rates are expected to be lower in the future, and if the difference between these forward rates and the current spot rate is greater than the liquidity premium, the yield curve will be downward sloping.

46. Market Segmentations Theory • The third model to explain the yield curve is the market segmentations theory. • Assumptions: the yield curve is determined by the supply of and the demand for loanable funds (or demand and supply of securities, i.e., bonds). • Begin with the economy in a “neutral” position. • What would be the natural tendencies of borrowers and lenders given the risks that each face? • Borrowers would “prefer” to borrow longer term (or to supply longer term securities) • I.e., they would demand long term loanable funds. • Lenders would “prefer” (demand) to lend shorter term (or to demand shorter term securities) • I.e., they would supply short term loanable funds. • What type of yield curve would this neutral position result in? • Look at the next slide.

47. “Neutral” Upward Sweeping Market Segmentations Yield Curve i rate Lenders supplying shorter term funds (pushes down rates) o o Borrowers demanding longer term funds (pushes up rates) (st) Term to Maturity (lt)

48. Relating the Market Segmentations Theory to Business Cycles • What did we observe about how interest rates generally respond over the course of a business cycle? • Specifically: • Which interest rates (short or long term) fluctuate more over a business cycle? • What happens to interest rates in general during a business expansion and why? • What happens to interest rates in general during a business recession and why? • Look at the charts on the next 2 slides for answers!

49. Cyclical Movement of Interest Rates, 1972 - 1984 • Note: Shaded areas represent business recessions • Blue line is short term bank lending rate • Red line is long term corporate (AAA) bond rate

50. Cyclical Movement of Interest Rates, 1990 - 2003 • Note: Shaded area represents business recession • Blue line is short term bank lending rate • Red line is long term corporate (AAA) bond rate