1. FNCE 3020 Financial 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. Yield Curves Reported in WSJ
WSJ yield curve is for Tuesday, September 5, 2006
Source:

16. Yield Curves Reported by Bloomberg
U.S. Yield Curve: September 6, 2006

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
oei
8.5
oei
7.5
oei o
6.5
oei o
5.0 o
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 o
oei
7.5
oei o
6.5
oei
5.5
oei
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 oei oie oie o
6.5
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 greater premiums (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 stream discounted at some interest rate, or:
Price = int/(1+r)1 + int/(1+r)n + … principal/(1+r)n

37. Example of Price Risk Assume two fixed income securities: 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 = $ $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 = $ $ $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%)/ % = 5.5% % = 5.75%
Five-year bond: (5% + 6% + 7% + 8% + 9%)/ % = 7.0% %
= 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
o LP Yield Curve
7.75
Difference is the liquidity premium
7.25
o PE Yield Curve
6.75
6.50
6.25
6.0 o o
5.25
5.0
2yr yr 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

51. Three Observations From Two Previous Charts
(1) Short term rates are more volatile than long term.
(2) During a business expansion interest rates gradually drift up (just before shaded area). Why?
Increasing business activity is pushing up the demand for loanable funds:
Corporates and individuals increasing their borrowing.
Central bank likely to be raising interest rates (especially important for short term interest rates)
Inflationary expectations may be increasing (especially important for long term interest rates)
(3) During a business recession interest rates come down. Why?
Decreasing business activity is reducing the demand for loanable funds.

52. Yield Curve Observations From Business Cycle Charts
Near the end of a business expansion (period before shaded areas) short term rates exceed long term rates.
Thus, during this period we would observe a downward sloping yield curve.

53. Explanation of Yield Curve Near the End of a Business Expansion:
Observation: Short term rates exceed long term near the end of a business expansion.
Producing a downward sweeping yield curve.
Why this shape?
Interest rates have risen during the expansionary period and are now “relatively” high.
Borrowers realizing that rates are relatively high, finance in the short term (not wanting to lock in long term liabilities at high interest rates).
Lenders realizing that rates are relatively high, lend in the long term (wanting to lock in long term assets at high interest rates)
Note: Both borrowers and lenders move away from their neutral (or, natural) tendencies.
This causes the yield curve to invert.

54. Market Segmentations Yield Curve Near the End of an Expansion
i rate
o Lenders supplying longer
term funds (pushes down rates)
Borrowers demanding shorter o
term funds (pushes up rates)
(st) Term to Maturity (lt)

55. Term to Maturity Observations From Business Cycle Charts
Into a recession (shaded area), short term rates come down faster than long term and eventually, near the end of the recession or beginning of the expansion, short term rates fall below long rates.
Thus, during this period we would observe an upward sweeping yield curve.

56. Explanation of Yield Curve Near the End of a Business Recession or Early Expansion
Observation: Short term rates below long term.
Producing an upward sweeping yield curve.
Why this shape?
Interest rates have fallen during the recessionary period and are now “relatively” low.
Borrowers realizing that rates are relatively low, finance in the long term (wanting to lock in long term liabilities at low interest rates).
Lenders realizing that rates are relatively low, lend in the short term (not wanting to lock in long term assets at low interest rates)
Note: Both borrowers and lenders accentuate their neutral (i.e., natural) tendencies.
This produces an accentuated upward sweeping yield curve.

57. Market Segmentations Yield Curve Near the End of Recession
i rate
Lenders supplying shorter
term funds (pushes down rates) o
Borrowers demanding longer
o term funds (pushes up rates)
(st) Term to Maturity (lt)

58. Yield Curves Over the Business Cycle

59. Summary of Yield Curve Explanations
The liquidity premium theory is probably the most widely accepted theory of the term structure of interest rates.
Why?
The simple spread between long and short term interest rates does not always predict the future spot rate.
Reason: The presence of and fluctuations in the liquidity premium.
Intuitively, it is also the most appealing.