FNCE 3020 Financial Markets and Institutions

1
FNCE 3020Financial Marketsand Institutions

• Lecture 5
• Term Structure of Interest Rates
• (AKA Yield Curves)

2
Relationship of Yields to Maturity

• In lecture 4 we noted various factors, such as
  risk of default, which can affect market interest
  rates.
• In this lecture we will look at how term to
  maturity may be factor explaining observed
  differences in market rates of interest.
• Term to maturity refers to the time before the
  asset matures.

3
Initial Observations Does Maturity Matter?

• Yes, generally long term rates are above short
  term rates

4
But, There Are Exceptions

• There are times when short term rates exceed long
  term rates.

5
Illustrating 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.
• OR
• (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
Yield Curves Defined

• Technique used to show 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 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 it we plot
• current interest rates on the Y axis, and
• corresponding term to maturity on the X axis,
  or
• i rate
• Term to maturity ?

8
First Possible Yield Curve Upward Sweeping
(Ascending)

• Assume following observed market interest rates
• Short term (st) rates are 4 and
• Long term (lt) rates are 8.
• Then the yield curve is
• i rate
• 8 o
• 4 o
• (st) Term to Maturity (lt)

9
Second Possible Yield Curve Downward Sweeping
(Descending)

• Assume following observed market interest rates
• Short term (st) rates are 7 and
• Long term (lt) rates are 3
• Them the yield curve is
• i rate
• 7 o
• 3 o
• (st) Term to Maturity (lt)

10
Third Possible Yield Curve Flat

• Assume following observed market interest rates
• Short term (st) rates are 7 and
• Long term (lt) rates are 7
• Then the yield curve is
• i rate
• 7 o o
• (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.
• Next slide illustrates these three basic shapes
  for historical U.S. data.

12
Historical U.S. Yield Curves
13
What Securities and What Interest Rates do we Use
for Data to Construct a Yield Curve?

• Security possibilities include
• Government debt and Corporate debt
• Recommendation
• Since we need to make sure that observed
  differences in interest rates are not being
  affected by credit risk (i.e., default risk) we
  would use data only from Government securities
  (no risk of default on U.S.).
• Interest rate possibilities include
• Coupon yield, Current yield and Yield to maturity
• Recommendation
• Use Yield to Maturity as it is the best
  representation of interest rate conditions at a
  point in time.

14
Yield Curve Reported in Wall Street Journal

• WSJ yield curve shown is for Friday, September
  28, 2007 (and a year ago)
• Source
• http//online.wsj.com/mdc/public/page/mdc_bonds.ht
  ml?mod2_0031

15
Yield Curves Reported by Bloomberg

• U.S. Yield Curve October 1, 2007
• http//www.bloomberg.com/markets/rates/index.html

16
Bloomberg Yield Curve a Year Ago

• U.S. Yield Curve September 6, 2006
• http//www.bloomberg.com/markets/rates/index.html

17
Theories to Explain the Shape of the Yield Curve

• There are three generally accepted theories or
  explanations of the yield curve, these are
• (Pure) Expectations Theory
• Liquidity Premium Theory
• Market Segmentations Theory
• However, we will focus on the Expectations Theory
  as it may also be the best theory for
• (1) Understanding the shape of the yield curve
  and
• (2) Forecasting future moves in interest rates.
• Note the other two theories are covered in
  Appendix 1 at the end of the slides.

18
The Expectations Theory

• Assumption The financial markets expectations
  regarding future interest rates shape a given
  yield curve.
• Financial markets are assumed to be very
  efficient.
• Widely disseminated information allows markets to
  form expectations about the future level of
  interest rates (referred to as the forward
  interest rate).
• Thus, at any point in time, there is a market
  consensus regarding future (forward) interest
  rates.
• Forward rates are based upon the markets
  analysis of all relevant events likely to affect
  interest rates in the future (central bank
  actions, inflationary expectations, business
  cycles).
• These expectations are incorporated into current
  market interest rates (referred to as spot
  interest rates).
• Important this consensus view is subject to
  constant revision and change!

19
The Expectations Theory

• The Expectations Theory assumes that the current
  long term spot interest rate is comprised of
• current short term (spot) rate and
• expected, future short-term (forward) rates.
• Theory assumes that an efficient market will
  refer to its expected future (forward) short term
  rate in setting current long term spot rates.
• Assume the current 1 year spot rate is 3 and the
  markets expect the 1 year rate, 1 year from now
  to be 5.
• What will the market lend now for 2 years?
• Answer 4.0
• What if the current 1 year spot rate is 7 and
  the markets expect the 1 year rate, 1 year from
  now to be 5

20
Expectations Formula for the Long-term
Interest Rate

• The current long term spot interest rate (ilsn
  where n years to maturity) is equal to the
  average of the current short term spot rate
  (isst) and all appropriate expected future short
  term (i.e., forward) rates (iet1, ien).
• Note 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, for
  time period t
• Forward rates (iet1, ien) are the markets
  expectations about where interest rates will be
  in the future.

21
The Markets Setting of Long Term Spot Rates
Example 1

• Assume
• Current (spot) one year rate is 5 and
• The markets forward one year rates over the next
  five years (years 2, 3, 4, and 5) are 6, 7,
  8, and 9, respectively.
• Given this data, what would be the markets long
  term spot rates, specifically
• Current (spot) two year bond rate (ils2)
• Current (spot) five year bond rate (ils5)

22
Markets Long Term Spot Rates

• Answer The markets current rate on a two-year
  (long-term) bond is as follows
• Current 1 year rate is 5 and expected 1 year
  rate, 1 year from now is 6, then
• 2 year bond rate (5 6)/2 5.5
• Answer The markets current rate on a five-year
  (long-term) bond is as follows
• Current 1 year rate is 5 and expected 1 year
  rates, 1 year from now through five years from
  now are 6, 7, 8, and 9, then
• 5 year bond rate (5 6 7 8 9)/5
  7

23
Expectations and Yield Curve

• 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
• Therefore, as the term to maturity increases,
  current spot rates increase, and thus the
  observed yield curve will be upward sloping!
• Yield Curve is upward sloping because market
  expects higher rates in the future.

24
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 ?

25
The Markets Setting of Long Term Spot Rates
Example 2

• Assume
• Current (spot) one year rate is 9 and
• The markets forward one year rates over the next
  five years (years 2, 3, 4, and 5) are 8, 7,
  6, and 5, respectively.
• Given this data, what would be the markets long
  term spot rates, specifically
• Current (spot) two year bond rate (ils2)
• Current (spot) five year bond rate (ils5)

26
Markets Long Term Spot Rates

• Answer The markets current rate on a two-year
  (long-term) bond is as follows
• Current 1 year rate is 9 and expected 1 year
  rate, 1 year from now is 8, then
• 2 year bond rate (9 8)/2 8.5
• Answer The markets current rate on a five-year
  (long-term) bond is as follows
• Current 1 year rate is 9 and expected 1 year
  rates, 1 year from now through five years from
  now are 8, 7, 6, and 5, then
• 5 year bond rate (9 8 7 6 5)/5
  7

27
Expectations and Yield Curve

• 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
• Therefore, as the term to maturity increases,
  current spot rates decrease, and thus the
  observed yield curve will be downward sloping!
• Yield Curve is downward sloping because market
  expects lower rates in the future.

28
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 ?

29
Expectations and Yield Curve

• When short rates expected to stay same in future,
  average of these expected short rates will be the
  same as today's spot rate.
• Thus the 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 ?
30
Summary of Expectations Regarding Future Interest
Rates

• The shape and slope of the yield curve reflects
  the markets expectations about future 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.

31
Forecasting Interest Rates Using the Expectations
Model

• The Expectations Model can be used to forecast
  expected future spot interest rates.
• If we assume the long term rate is an average of
  short term (spot and forward) rates, it is
  possible to derive the expected forward rate
  (ie), on a one-period bond for some future time
  period (n-t) through the following formula

32
Forecasting Example 1

• Assume current 1 year short term spot (iss1) and
  current 2 year long-term spot (ils2) rates are as
  follows
• iss1 5.0 and
• ils2 5.5
• Then the 1 year rate, 1 year from now (ien-t) is
  expected to be

33
Yield Curve Example 1

• i rate
• 6.0 oie
• 5.5 o
• 5.0 o
• 1y 2y
• Term to Maturity ?

34
Forecasting Example 2

• Assume current 1 year short term spot (iss1) and
  current 2 year long-term spot (ils2) rates are as
  follows
• iss1 7.0 and
• ils2 5.0
• Then the 1 year rate, 1 year from now (ien-t) is
  expected to be

35
Yield Curve Example 2

• i rate
• 7.0 o
• 5.0 o
• 3.0 oie
• 1y 2y
• Term to Maturity ?

36
Using the Current Yield Curve

• What is the current yield curve telling us about
  the markets expectation regarding future interest
  rates
• Going up or going down? Can you approximate some
  forward rates? (e.g., 6 month rate, 6 months from
  now)

37
Useful Yield Curve Web Sites

• http//www.bondsonline.com/Todays_Market/Treasury_
  Yield_Curve.php
• This site not only has a picture of the most
  recent yield curve, but data as well.

38
Appendix 1 Liquidity Premium and Market
Segmentations Theory of the Yield Curve
39
Liquidity Premium Theory

• The second explanation of the yield curve shape
  is referred to as the Liquidity Premium Theory.
• 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.
• Interest rate on a long term bond will equal an
  average of the expected short term rates PLUS a
  liquidity premium!
• What are these risks associated with illiquidity
• Price risk (a.k.a. interest rate risk).
• Risk of default (on corporate issues).

40
Price Risk (Interest Rate Risk) Revisited

• Observation Long term securities 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

41
Example of Price Risk

• 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 discount rate 6 (market rate or
  opportunity cost)
• What will happen to the prices of both issues?
• Both bonds should fall in price (sell below their
  par values). See new prices on next slide!

42
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

43
Price Change Comparisons

• Price Change over par (1,000)
• 1 year bond 9.43
• 2 year bond 17.33
• Note The long term (2 year) bond experienced
  greater price change!
• Thus, there is greater price risk with longer
  term bonds!
• Thus, investors want a higher return on long term
  bonds because of the potential for greater price
  changes.
• This is called a liquidity premium!!!

44
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 rates are higher than
  short term rates, the question is
• Are higher long term rates due to expectations of
  higher rates in the future (Expectations Theory),
  OR
• Are higher long term rates due to added on
  liquidity premiums (Liquidity Premium Theory)?
• There is no good answer to this question!!!

45
Liquidity Premium Theory Formula for Long Term
Interest Rates

• Need to modify the expectations theory formula to
  take into account liquidity premiums, or
• Where, Ln is the liquidity premium for holding a
  bond of n maturity.

46
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' preferences for holding
  short-term bonds so liquidity premium 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.

47
Calculations and Comparisons

• Market interest rate on the two-year bond (5
  6)/2 0.25 5.75
• Market interest rate on the five-year bond (5
  6 7 8 9)/5 1.0 8
• Compare Liquidity Premium rates to Pure
  Expectations Rates
• 2 year 5.75 (LP) 5.5 (PE)
• 5 year 8.00 (LP) 7.0 (PE)
• Thus
• liquidity premium theory produces yield curves
  more steeply upward sloped

48
Yield Curve Liquidity Premium

• 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

49
Forecasting Interest Rates Using the Liquidity
Premium Theory

• We can use the Liquidity Premium Theory to
  forecast future interest rates. But to do so
• We need to make some estimate as to the liquidity
  premium per maturity.
• We then subtract our estimated liquidity premium
  out of the forecast rate.
• Start with the Pure Expectations Forecast formula

50
Forecasting Example 3 Assuming a Liquidity
Premium

• Assume current 1 year short term spot (iss1) and
  current 2 year long-term spot (ils2) rates are as
  follows
• iss1 5.0 and
• ils2 5.75
• Also assume the liquidity premium on a two year
  bond is .25.
• Calculate the markets forecast for the 1 year
  rate, one year from now.
• Forecast both for the liquidity premium and
  assuming no liquidity premium (and compare the
  two).

51
Forecasting Example 3

• The 1 year rate, 1 year from now without a
  liquidity premium (ien-t) is expected to be
• The 1 year rate, 1 year from now with a 25 basis
  point liquidity premium (ien-t -lp) is expected
  to be

52
Forecasting Example 4

• Assume current 1 year short term spot (iss1) and
  current 2 year long-term spot (ils2) rates are as
  follows
• iss1 5.0 and
• ils2 5.75
• Also assume the liquidity premium on a two year
  bond is .75.
• Calculate the markets forecast for the 1 year
  rate, one year from now.
• Forecast both for the liquidity premium and
  assuming no liquidity premium.

53
Forecasting Example 4

• The 1 year rate, 1 year from now without a
  liquidity premium (ien-t) is expected to be
• The 1 year rate, 1 year from now with a 75 basis
  point liquidity premium (ien-t -lp) is expected
  to be

54
Forecasting Example 5

• Assume current 1 year short term spot (iss1) and
  current 2 year long-term spot (ils2) rates are as
  follows
• iss1 5.0 and
• ils2 5.75
• Also assume the liquidity premium on a two year
  bond is 1.00.
• Calculate the markets forecast for the 1 year
  rate, one year from now.
• Forecast both for the liquidity premium and
  assuming no liquidity premium.

55
Forecasting Example 5

• The 1 year rate, 1 year from now without a
  liquidity premium (ien-t) is expected to be
• The 1 year rate, 1 year from now with a 100 basis
  point liquidity premium (ien-t -lp) is expected
  to be

56
Differences in Forecasts

• Assuming Forecasted
  Forecasted Spot Rate Change in
  1 yr from Now Spot Rate
• No Liquidity Premium 6.5
  150bps
• LP of .25 6.0 100bps
• LP of .75 5.0 no change
• LP of 1.00 4.5 - 50 bps
• In basis points over current 1 year spot rate of
  5.0

57
Yield Curve Liquidity Premiums and Forecasts
(Oie)

• i rate
• 6.75
• 6.50 oie (No
  Liquidity Premium) 6.5
• 6.25
• 6.0 oie (.25
  LP) 6.0
• 5.75 o
• 5.5
• 5.25 Observed Yield
  Curve
• 5.0 o oie (.75 LP)
  5.0
• 4.75
• 4.5 oie (1.00
  LP) 4.5
• 1yr 2yr Years to Maturity

58
Liquidity Premium Conclusions

• If there are liquidity premiums on longer term
  rates, NOT subtracting them out will result in
  over forecasting errors.
• Question (Problem)
• Is there a liquidity premium, and if so
• HOW MUCH IS IT?

59
Market Segmentations Theory

• The third theory of the yield curve is the Market
  Segmentations Theory.
• Assumptions the yield curve is determined by the
  supply of and the demand of loanable funds (or
  securities) at a particular maturity.
• Begin with a neutral position
• What would be the natural tendencies of borrowers
  and lenders?
• Borrowers prefer longer term loans (or to supply
  longer term securities)
• Lenders prefer shorter term loans (or to demand
  shorter term securities)
• What type of yield curve would this neutral
  (natural) position result in?
• Upward sweeping!

60
Natural (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)

61
Market Segmentations and Business Cycles

• What do we know generally happens to interest
  rates 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 during a business
  expansion and why?
• What happens to interest rates during a business
  recession and why?
• Look at the charts on the next 2 slides for
  answers!

62
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

63
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

64
Observations From Two Previous Charts

• Short term rates are more volatile than long
  terms.
• During a business expansion interest rates
  gradually drift up (just before shaded area).
  Why?
• Increasing business activity is pushing up the
  demand for funds
• Corporates and individuals increasing borrowing
• Central bank likely to be raising interest rates
  (impact on short term rates)
• Inflationary expectations may be increasing
  (impact on long term rates)
• During a business recession interest rates come
  down. Why?
• Decreasing business activity is bring down the
  demand for funds.

65
Term to Maturity 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.

66
Near the End of a Business Expansion Explanation
of Yield Curve

• Short term rates exceeding long term.
• 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 natural tendencies.

67
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)

68
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.

69
Near the End of a Business Recession or Early
Expansion

• Short term rates below long term.
• (Severe) 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 natural tendencies.

70
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)

71
Yield Curves and Business Cycle
72
Forecasting with Market Segmentations Theory

• The Market Segmentations Theory CANNOT be used to
  forecast future spot rate (forward rates).
• The Market Segmentations Theory can be used to
  identify (signal) turning points in the movement
  of interest rates (and in the economy itself)
  based on the shape of the curve.
• Downward sweeping curve suggests a fall in
  interest rates, the end of an economic expansion,
  and a future economic (business) recession.
• Severe upward sweeping curve suggests a rise in
  interest rates, the end of an economic recession,
  and a future economic (business) expansion.

73
Lag Problem with Market Segmentations Theory

• Lags between what the yield curve is suggesting
  and what may eventually happen are variable and
  potentially very long.
• Upward sloping yield curve on Jan 2, 2002
  suggested the end of a recession.
• When did it end?
• A year later!!!

74
Upward Sweeping Yield Curve in Early 2002
Recession Ended in Early 2003