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
Yield Curves Reported in WSJ

• WSJ yield curve is for Tuesday, September 5, 2006
• Source
• http//online.wsj.com/page/2_0031.html?mod2_0031

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
  (ilstn 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
  (iet1, 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 (iet1, ien) are the markets
  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 oei oie oie
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
  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/(1r)1 int/(1r)n
  principal/(1r)n

37
Example of Price Risk

• Given that
• Price int/(1r)1 int/(1r)n
  principal/(1r)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/(1r)1 principal/(1r)n
• Price 50/(1.06) 1,000/(1.06)
• Price 47.17 943.40
• Price 990.57
• 2 year bond
• Price int/(1r)1 int/(1r)2
  principal/(1r)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
  (ilstn) 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

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.