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FNCE 4070: FINANCIAL MARKETS AND INSTITUTIONS Lecture 5: The Term Structure of Interest Rates

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## FNCE 4070: FINANCIAL MARKETS AND INSTITUTIONS Lecture 5: The Term Structure of Interest Rates

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### FNCE 4070: FINANCIAL MARKETS AND INSTITUTIONS Lecture 5: The Term Structure of Interest Rates

### Appendix 1

Yield Curves: How do we Construct a Yield Curve and What are the Theories which Explain Yield Curves

Grand Cayman Islands as an Offshore Financial Center

- Offshore Financial Center: Countries or jurisdictions with financial centers that contain financial institutions that deal primarily with nonresidents and foreign currency transactions on a scale out of proportion to the size of its economy (OECD definition).
- Characterized by a low (or zero) tax environment and specializing in providing corporate and commercial services (including investment services) to non-resident entities on a confidential basis. .

- Grand Cayman Islands (population 52,000):
- 533 banks, with approximately US$415 billion in deposits (making it fifth largest financial center in the world). Other financial companies include insurance, cash management, asset management (including hedge funds).
- Foreign companies generally register as an “exempted company” because they are guaranteed against any future taxes for at least 30 years. In addition, its list of shareholders is not open to public inspection, they do not have to file annual returns and no annual audits are required.
- Total start-up costs to form an exempted company range between $2,300 and $3,000. Annual maintenance costs and government fees thereafter run about $2,000
- There are no personal income taxes, no corporate income taxes, no capital gains taxes, no withholding taxes, no estate, gift or inheritance taxes, no sales taxes in the Cayman Islands. The Caymans have no tax treaties with any nation.

Relationship of Yields to Maturity

- In previous lectures we have discussed various factors which can account for differences in market interest rates:
- Inflationary expectations, risk of default, business cycles, flight to safety, and term to maturity.
- In this lecture we will expand on term to maturity as a factor and do so in the context of yield curves (i.e., the “term structure of interest rates”).

Initial Observation Regarding Market Rates and Maturity

- “Normally,” longer term interest rates are above shorter term interest rates.
- Even when we adjust for default risk (e.g., just looking at Treasuries – see right hand panel, this is the case.
- Why do you think this is a normal relationship?

Why Might we Assume that the Long Term Rate will be Above the Short Term?

- Because of the risks associated with committing one’s capital for longer periods of time:
- Uncertainty about inflation.
- Uncertainty about economic activity.
- Knowledge of the price (i.e., interest rate) risk relationship regarding longer term issues.
- And if dealing with debt which is not free from default risk, uncertainty about future cash flows, risk of default and credit ratings.

A Second Look at the Market Rate and Maturity Relationship

- On occasion, the long term interest rate falls below the short term interest rate.
- See right hand panel.
- Not a typical relationship – see next slide.

How can we Illustrate the Relationship Between Interest Rates and Maturity?

- (1) We can look at interest rates over time.
- Compare movements (or differentials) of long term rates to short term rates over some historical period.
- OR
- (2) We can look at interest rates at a point in time, i.e., on a particular date.
- What is the short term interest rate, the intermediate term rate, and the long term interest rate on a specific date?
- This last approach is referred to as a yield curve.

Graphing a Yield Curve

- A yield curve is simply a graphic presentation of the relationship of term to maturity and yields to maturity (interest rate) on a given date. To construct it we plot:
- Term to maturity on the X axis, with its
- Corresponding interest rate on the Y axis, or:

interest rate

Term to maturity →

First Possible Yield Curve: Upward Sweeping (Ascending, Positive)

Assume following observed market interest rates:

Short term (st) interest rates are 4% and

Long term (lt) interest rates are 8%.

Then the yield curve is upward (positive) sweeping:

i rate

8% o

4% o

(st) Term to Maturity (lt)

Second Possible Yield Curve: Downward Sweeping(Descending, Inverted, Negative)

Assume following observed market interest rates:

Short term (st) interest rates are 7% and

Long term (lt) interest rates are 3%

The the yield curve is downward sweeping (inverted):

i rate

7% o

3% o

(st) Term to Maturity (lt)

Third Possible Yield Curve: Flat

Assume following observed market interest rates:

Short term (st) interest rates are 7% and

Long term (lt) interest rates are 7%

Then the yield curve is (relatively) flat:

i rate

7% o o

(st) Term to Maturity (lt)

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 Yield Curves (“Upward Sloping; Positive”) : Long term interest rates higher than short interest term
- Descending Yield Curves (“Downward Sloping; Inverted; Negative”) : Short term interest rates higher than long term interest rates.
- (Relatively) FlatYield Curves: Long term and short term rates essentially the same.
- Next slide illustrates these three basic shapes for historical U.S. data.

Observations About Yield Curves Over Long Periods of Time

- More variation (i.e., basis point change) associated with the shorter term segment of the yield curve.
- Also look at

the dynamic

yield curve

over time:

http://stockcharts.com/freecharts/yieldcurve.html

How Do We Construct a Yield Curve? What Debt Instrument Should We Use?

- Debt instrument possibilities include:
- Federal government debt,
- Municipal debt,
- Corporate debt
- Recommendation:
- Since we need to make sure that observed differences in market interest rates are not being affected by credit risk (i.e., default risk), we generally use data only from Government securities.
- We assume that these securities carry no risk of default.
- We can, however, construct non-government yield curves.
- Link to: http://www.bondsonline.com/Todays_Market/Composite_Bond_Yields.php

How Do We Actually Construct a Yield Curve? What Interest Rate Should We Use?

- Interest rate possibilities include:
- Coupon yield = coupon payment/par value
- Current yield = coupon payment/market price of bond.
- Yield to maturity = internal rate of return on a bond’s cash flow.
- Recommendation:
- Use Yield to Maturity as it is the best representation of interest rate conditions at a point in time.
- Because it takes into account the time value of money
- If you don’t have yield to maturity data, use the current yield as this is a better approximation of the yield to maturity than is the coupon yield
- Never use the coupon yield.

Constructing a Yield Curve

Bloomberg Data: U.S. Treasuries, Feb 24, 2011

Construct a Yield Curve for Feb 24, 2011

Yield Curves for Foreign Countries

- United Kingdom (and U.S.)
- http://www.yieldcurve.com/marketyieldcurve.asp
- This site lets you compare UK and US yield curves.
- Euro-Zone
- http://www.ecb.int/stats/money/yc/html/index.en.html
- Note: Select “spot rate” curve for current Euro-zone yield curve
- Other countries:
- http://www.bloomberg.com/
- Link to Markets, then to Government Bonds (current yield curves for U.K., UK, Germany, Japan, Australia and Brazil)

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
- These theories are potentially important because they provide a framework for:
- (1) Understanding the shape of the yield curve and
- (2) Forecasting future interest rates.
- (3) Forecasting future changes in economic activity (i.e., business cycle turning points)
- Note: Forecasting is covered in next lecture.

The (Pure) Expectations Theory

- Assumption: Financial market’s expectations regarding future interest rates will shape the current yield curve.
- Model assumes that financial markets are efficient.
- What does this mean?
- The existence of widely disseminated information allows market participants to form expectations about future interest rates (referred to as forward interest rates).
- Forward rates are based upon the markets’ analysis of all relevant events likely to affect interest rates in the future (e.g., central bank actions, inflationary expectations, business cycles).
- These forward rates are incorporated into current market interest rates (also referred to as spot interest rates).
- These spot interest rates are the interest rates represented in observed yield curves.

The Expectations Theory: Long Term Spot Interest Rates

- The Expectations Theory assumes that the current long term spot interest rate is comprised of 2 components
- (1) current short term (i.e., spot) interest rate and
- (2) expected, or forward, interest rate.
- Assume:
- (1) the current 1 year spot interest rate is 3% and
- (2) the forward 1 year interest rate, 1 year from now is 5%.
- Also, assume there is no risk of default and no required premium for longer term financial assets.
- Given the above, what will the market set as the current 2 year spot interest rate?

The Expectations Theory: Solving for the Long Term Spot Interest Rate

- Assumptions:
- (1) the current 1 year spot interest rate is 3% and
- (2) the forward 1 year interest rate, 1 year from now is 5%.
- And there is no risk of default and no required premium for longer term financial assets.
- What is the 2 year spot interest rate?
- Answer: 4.0%
- Why?
- At 4%, the financial markets are indifferent to investing (or lending) for 1 year or 2 years.
- A series of 2, one year instruments will offer an average annual return or 4% and the 2 year instrument will offer an annual return of 4%
- Another example: 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%.
- Solve for the 2 year rate.

Illustrating the Expectations Theory

- Assume:
- (1) it, the current 1 year spot interest rate is 3% and
- (2) iet+1, the forward 1 year interest rate, 1 year from now is 5%.
- Then:
- (3) i2t, the current 2 year spot will equal 4%

Expectations Formula for the Long-term Interest Rate

Where: 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 (iet+1, … ien).

Note: The long term spot rate (ilsn) is the observable long term market interest rate.

The current short term spot rate (isst) is the observable short term market interest rate, for time period t.

Forward rates (iet+1, … ien) are the market’s expectations about where interest rates will be in the future.

The Market’s Setting of Long Term Spot Rates: Example 1

- Assume the following:
- Current (spot) one year rate is 5% and
- The market’s forward one year rates over the next five years (years 2, 3, 4, and 5) are:
- Beginning of year 2: 6%,
- Beginning of year 3: 7%,
- Beginning of year 4: 8%, and
- Beginning of year 5: 9%.
- Given this data, calculate the following long term spot rates:
- Current (spot) two year bond rate (ils2)
- Current (spot) five year bond rate (ils5)

Market’s Long Term Spot Rates

- 2 Year Interest Rate: The market’s current rate on a two-year bond is calculated as follows:
- Current (spot)1 year rate is 5% and forward 1 year rate, 1 year from now is 6%, then:
- Current 2 year bond rate = (5% + 6%)/2 = 5.5%
- 5 year Interest Rate: The market’s current rate on a five-year bond is calculated as follows:
- Current (spot)1 year rate is 5% and forward 1 year rates, 1 year from now through five years from now are: 6%, 7%, 8%, and 9%, then:
- Current 5 year bond rate = (5% + 6% + 7% + 8% + 9%)/5 = 7%

Expectations For Rising Interest Rates and the Observed Yield Curve

- If interest rates are expected to rise in future, forward rates will be above today's spot rates.
- The higher forward rates will result in higher observed spot rates.
- Recall previous example:
- The current (spot)1 year rate was 5% and forward 1 year rate, 1 year from now was 6%.
- The forward rate is higher because the market expects higher rates in the future
- Question: Given this situation where forward rates are higher than current spot rate (because the market expects higher rates in the future), what will the current yield curve look like?

Yield Curve When Market Expects Higher Interest Rates in the Future

- If the current 1 year spot rate (S1) is 5.0% and
- Forward 1 year rates are:
- if2 = 6%,
- if3 = 7%,
- If4 = 8%
- If5 = 9%
- Then observed long term spot rates are:
- Spot 2 year (S2) = 5.5%
- Spot 3 year (S3) = 6.0%
- Spot 4 year (S4) = 6.5%
- Spot 5 year (S5) = 7.0%
- Conclusion: The observed yield curve is upward sloping because the market expects higher interest rates in the future.

Spot interest rate

9.0% oif5

8.5

8.0 oif4

7.5

7.0 oif3 os5

6.5 Os4

6.0 oif2Os3

5.5 os2

5.0 os1

1 2 3 4 5 year

Term to Maturity →

Expectations For Falling Interest Rates and the Observed Yield Curve

- If interest rates are expected to fall in future, forward rates will be below today's spot rates.
- Assume:
- The current (spot)1 year rate is 9% and forward 1 year rate, 1 year from now is 8%
- Calculate the 2 year spot rate:
- Answer: Current 2 year bond rate = (9% + 8%)/2 = 8.5%
- Thus, lower forward rates result in lower observed spot rate.
- Question: Given this situation where forward rates are lower than current spot rate (because the market expects lower rates in the future), what will the current yield curve look like?

Yield Curve When Market Expects Lower Interest Rates in the Future

- If the current 1 year spot rate (S1) is 9%, and :
- Forward 1 year rates are:
- if2 = 8%,
- if3 = 7%,
- If4 = 6%
- If5 = 5%
- Then observed long term spot rates are:
- Spot 2 year (S2) = 8.5%
- Spot 3 year (S3) = 8.0%
- Spot 4 year (S4) = 7.5%
- Spot 5 year (S5) = 7.0%
- Conclusion: The observed yield curve is downward sloping because the market expects lower interest rates in the future.

Spot interest rate

9.0% oS1

8.5 oS2

8.0 oif2 Os3

7.5 os4

7.0 oif3 os5

6.5

6.0 oif4

5.5

5.0 oif5

1 2 3 4 5 year

Term to Maturity →

Yield Curve When Market Expects No Change in Interest Rates in the Future

- If the current 1 year spot rate (S1) is 7%, and :
- Forward 1 year rates are:
- if2 = 7%,
- if3 = 7%,
- If4 = 7%
- If5 = 7%
- Then observed long term spot rates are:
- Spot 2 year (S2) = 7.0%
- Spot 3 year (S3) = 7.0%
- Spot 4 year (S4) = 7.0%
- Spot 5 year (S5) = 7.0%
- Conclusion: The observed yield curve is flat because the market expects no change in interest rates in the future.

Spot interest rate

7.5%

7.0 os1oei2oie3oie4 os5

6.5

1 2 3 4 5 year

Term to Maturity →

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, Positive) Yield Curves:
- Future (forward) interest rates are expected to increase above existing spot rates.
- Downward Sloping (Descending, Inverted, Negative) 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.

Liquidity Premium Theory

- The second explanation of the yield curve is referred to as the Liquidity Premium Theory.
- Assumptions of Liquidity Premium Theory: Long term securities carry a greater risk and therefore investors require greater returns to invest for longer periods of time.
- What are these risks associated with longer term securities:
- Price risk (i.e., interest rate risk).
- Risk of default (on corporate issues).
- Inflation risk

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 interest rates due to expectations of higher interest rates in the future (i.e., the Expectations Theory), OR
- Are higher long term interest rates due to liquidity premiums (i.e., the Liquidity Premium Theory), OR
- Some combination of both of the above?

Liquidity Premium Theory Formula for Long Term Interest Rates

- The Liquidity Premium needs to modify the pure expectations theory formula to take into account a liquidity premium, or
- Where, Ln is the liquidity premium for holding a bond of n maturity.

Liquidity Premium Example

- Assume: One-year (spot and forward) interest rates over the next five years as follows:
- 1 year spot interest rate = 5%
- 1 year forward rates:
- Beginning year 2 = 6%,
- Beginning year 3 = 7%,
- Beginning year 4 = 8%, and
- Beginning year 5 = 9%
- Now assume investors' liquidity premiums for holding one- to five-year bonds as follows:
- 1 year bond = 0.00%
- 2 year bonds = 0.25%
- 3 year bonds = 0.50%,
- 4 year bonds = 0.75%, and
- 5 year bonds = 1.00%
- With this information, calculate the market interest rate on:
- 1) a two year bond (with a Ln = .25%)
- 2) a five year bond (with a Ln = 1.0%)
- And, compare these calculated long term rates with those for the pure expectations theory formula.

Calculation of Spot Interest Rates With Liquidity Premium

- Market interest rate on the two-year bond with a liquidity premium on .25% on a 2 year bond: (5% + 6%)/2 + 0.25% = 5.5 + .25 = 5.75%
- Market interest rate on the five-year bond with a liquidity premium of 1.0% on a five year bond: (5% + 6% + 7% + 8% + 9%)/5 + 1.0% = 7.0 + 1.0 = 8%
- Compare Liquidity Premium spot rates to Expectations spot rates:
- 2 year spot rate: 5.75% (Liquidity Premium); 5.5% (Expectations)
- 5 year spot rate: 8.00% (Liquidity Premium); 7.0% (Expectations)
- As you can see:
- The liquidity premium theory will produce yield curves more steeply upward sloping (see next slide).

Yield Curves: Liquidity Premium and Pure Expectations Compared

interest rate

8.0% oLiquidity Premium Yield Curve

7.75

7.50 Difference is the liquidity premium

7.25

7.0 oPure Expectations 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

Liquidity Premium Example When Interest Rates are Expected to Fall

- Assume: One-year (spot and forward) interest rates over the next five years as follows:
- 1 year spot interest rate = 5%
- 1 year forward rates:
- Beginning year 2 = 4%,
- Beginning year 3 = 3%,
- Beginning year 4 = 2%, and
- Beginning year 5 = 1%
- Now assume investors' liquidity premiums for holding one- to five-year bonds as follows:
- 1 year bond = 0.00%
- 2 year bonds = 0.25%
- 3 year bonds = 0.50%,
- 4 year bonds = 0.75%, and
- 5 year bonds = 1.00%
- With this information, calculate the market interest rate on:
- 1) a two year bond (with a Ln = .25%)
- 2) a five year bond (with a Ln = 1.0%)
- And, compare these calculated long term rates with those for the pure expectations theory formula.

Calculation of Spot Interest Rates With Liquidity Premium

- Market interest rate on the two-year bond with a liquidity premium on .25% on a 2 year bond: (5% + 4%)/2 + 0.25% = 4.50 + 0.25 = 4.75%
- Market interest rate on the five-year bond with a liquidity premium of 1.0% on a five year bond: (5% + 4% + 3% + 2% + 1%)/5 + 1.0% = 3.0 + 1.0 = 4%
- Compare Liquidity Premium spot rates to Expectations spot rates:
- 2 year spot rate: 4.75% (Liquidity Premium); 4.5% (Expectations)
- 5 year spot rate: 4.00% (Liquidity Premium); 3.0% (Expectations)
- As you can see:
- The liquidity premium theory will produce yield curves less inverted (see next slide).

Yield Curves: Liquidity Premium and Pure Expectations Compared

interest rate

5.0%

4.75

4.50

4.25

4.00 Liquidity Premium Yield Curve

3.75

3.50Difference is the liquidity premium

3.25

3.0Pure Expectations Yield Curve

2yr 5yr Years to Maturity

Market Segmentations Theory

- The third theory of the yield curve is the Market Segmentations Theory.
- Assumptions of the Market Segmentations Theory: Each maturity segment of the yield curve (short term through long term) has its interest rate determined by the supply of and the demand for debt instruments within that maturity segment.
- Thus, within the short segment, there will be a demand and supply schedule for debt instruments, and in the intermediate term and long term segments there will also be demand and supply schedules.
- These schedules are independent of one another.

Market Segmentations Theory

- The Market Segmentations Theory begins with a determination of a “typical” yield curve, based on:
- What would be the typical tendencies, or preferences, of borrowers and lenders?
- Question: In which maturity segment would borrowers prefer to borrow?
- Answer: Borrowers would prefer to issue (i.e., supply) longer term debt instruments to lock in interest rate obligations.
- Question: In which maturity segment would lenders prefer to lend (or invest)?
- Answer: Lenders prefer to buy (i.e., demand) shorter term debt instruments to avoid longer term risk issues.
- Question: What type (i.e., shape) of yield curve would these typical tendencies produce?

Typical Upward Sweeping Market Segmentations Yield Curve

Spot interest rate

Lenders demanding short term debt instruments

(result: pushes up price and pushes down short term interest rates)

o

o Borrowers supplying long term debt instruments

(result: pushes down price and pushes up long term interest rates)

(short term) Term to Maturity (long term)

Market Segmentations Theory and the Shape of the Yield Curve

- According to the market segmentations theory, the shape of the yield curve is determined by shifts in the demand for and supply of debt instruments along various segments of the maturity range.
- Model starts with a typical yield curve, which is assumed to be upward sloping.
- Changes in this curve reflect changes in demand and supply as borrowers and lenders move away from their typical preferences.

Shift in the Yield Curve Under the Market Segmentations Theory

Shift from typical to inverted yield curve

Explanation

Borrowers shifted to the short term

Now a preference for issuing short term securities.

Pushes down prices and pushes up interest rates

Lenders shifted to the long term

Now a preference for demanding long term securities.

Pushes up prices and pushes down interest rates

Why might this shift in preference occur?

Typical (Upward) Yield Curve

Yield Curve web sites.

Yield Curves on Line

- Visit the following sites:
- (1) Bloomberg: U.S. Treasury and Selected Foreign Country Yield Curves
- http://www.bloomberg.com/markets/rates/index.html
- (2) Dynamic U.S. Treasury Yield Curve
- http://stockcharts.com/charts/yieldcurve.html
- You can view yield curve changes over time.
- (3)Yield Curves for Treasuries and Corporates
- http://www.bondsonline.com/Todays_Market/Composite_Bond_Yields.php
- (4) Nominal and Real U.S. Treasury Yield Curves
- http://www.treasury.gov/resource-center/data-chart-center/interest-rates/Pages/Historic-Yield-Data-Visualization.aspx

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