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The Term Structure of Interest Rates

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Term Structure of Interest Rates

Time to maturity

rzero

0 3m 6m 1yr 3yr 5yr 10yr 30yr

Yield curve

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What is the Term Structure?

• Term Structure - the pattern of interest rates for different maturity securities.

• Yield Curve - a graphical representation of the term structure– series of period yields based on zero-coupon

bonds– Treasury strips are excellent securities to use for

finding the yield curve– securities need to have same default risk

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Term Structure of Interest Rates

If we knew the future interest rates:

0(Today) 8%

1 10%

2 11%

3 11%

)11.01)(11.01)(10.01)(08.01(

000,1$

P

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Spot Rates

• Spot rates are the current interest rates for a specified period of time– A two-year spot rate means that you can earn

this rate each year for the next two years– A five-year spot rate means that you can earn

this rate each year for the next five years

• Spot rates are the nominal yields to maturity that we observe in the market

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Short rates

• Short rates are short-term rates, usually one year– These rates can be current or future

– The current short rate is equal to the one-year spot rate

– You can also have future short rates, i.e., the two year short rate is the one-year rate in two years.

– We do not actually know what future short rates will be, but we can estimate what the market expects them to be

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Short versus Spot Rates

r1 = 8% r1 = 10% r3 = 11% r4 = 11%

Spot rate is the yield to maturity on zero-coupon bonds.

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Short versus Spot Rates

r1 = 8% r1 = 10% r3 = 11% r4 = 11%

y1= 8%

y2= 8.995%

y3= 9.66%

y4= 9.993%

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Forward Rates

• Forward rates are the markets’ expectations of future short and spot rates.

• Forward rates are derived from current spot rates.– From the law of one price, there must be an explicit

relationship between all spot rates and forward rates.

– If the relationship doesn’t hold, then the market is out of equilibrium and there are arbitrage profits to be made

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Forward RatesSuppose you will need a loan in two years

from now for one year.

How one can create such a loan today?

Go short a three-year zero coupon bond.

Go long a two-year zero coupon bond.

+1 0 0 -1.3187

-1 0 +1.188 0

0 1 2 3

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Forward Rates (1 + yn)n = (1 + yn-1)n-1(1 + fn)

(1 + yn)n

(1 + yn-1)n-1

+1 -1.3187

-1 +1.188

0 1 2 3fn

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Forward Rates

In other words we can lock now interest rate for a loan which will be taken in future.

To specify a forward interest rate one should provide information about

today’s date

beginning date of the loan

end date of the loan

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Term Structure Relationships

• (1+yn)n = (1+r1)(1+1f1)(1+2f1)…(1+n-1f1)

– where:• yn is the n-year spot rate

• r1 is the current short rate

• ifj is the j-year forward rate beginning in year i

– 1f1 = one-year forward rate beginning in year one

– 2f3 = three-year forward rate beginning in year two

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Example• Suppose we actually know the following

short rates– year short rate

– 1 10%

– 2 9%

– 3 8%

• We can compute spot rates– 1-year = 10%

– 2-year (1+y2)2 = (1.1)(1.09) y2 = 9.5%

– 3-year (1+y3)3 = (1.1)(1.09)(1.08) y3 = 9%

– or 3-year (1+y3)3 = (1.095)2(1.08) y3 = 9%

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Example

• Unfortunately, we don’t know future short rates, we can only infer them from current spot rates

• Suppose the current 2-year spot rate is 8% and the current short rate is 9%. What is the one-year rate expected in one year?– (1.08)2 = (1.09)(1+1f1)

1f1 = 7%

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Example continued

• Suppose the current 3-year spot rate is 7.5%. Given the information presented above, what is the one-year rate expected in two years?

– (1.075)3 = (1.09)(1.07)(1+2f1)

2f1 = 6.5%

• What is the two-year rate expected in one year?

– (1.075)3 = (1.09)(1+1f2)2

1f2 = 6.76%

• Note that the sum of the exponents on the right hand side have to equal the exponent on the left hand side.

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Example Continued

• Suppose the four-year spot rate is 7.75%. What is the one-year rate expected in three years?

– (1.0775)4 = (1.09)(1.07)(1.065)(1+3f1)

3f1 = 8.5%

– or (1.0775)4 = (1.09)(1.0676)2(1+3f1)

3f1 = 8.5%

• What is the two-year rate expected in year 2?

– (1.0775)4 = (1.09)(1.07)(1+2f2)2

2f2 = 7.51%

• What is the three-year rate expected in year 1?

– (1.0775)4 = (1.09)(1+1f3)3

1f3 = 7.34%

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More Examples

• What is the five period spot rate if the four period spot rate is 6% and the one-period rate in year 4 is 6.25%?

– (1+y5)5 = (1+y4)4(1+4f1)

– (1+y5)5 = (1.06)4(1.0625)

– y5 = 6.05%

• Suppose the six period spot rate is 8% and the ten period spot rate is 10%. What is the four period rate beginning in year 6?

– (1+y10)10 = (1+y6)6(1+6f4)4

– 6f4 = 13.07%

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Theories of the Term Structure

• Three major theories– Expectations– Liquidity Preference– Market Segmentation (Preferred Habitat)

• Each of these theories can be used to predict any shaped yield curve.

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Expectations Theory

• Long-term rates are determined by expectations of future short-term rates.

• All of the work we have just been doing is based on the expectations hypothesis.

• Explaining the yield curve– Upward sloping - future ST rates are expected to

increase, thereby increasing LT rates relative to ST

– Flat - future ST rates are expected to remain constant

– Downward sloping - future ST rates are expected to decrease, thereby decreasing LT rates relative to ST rates

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Liquidity Preference

• This is an extension of the expectations hypothesis.

• Long-term rates are based on expected future short-term rates plus a liquidity premium

• Investors prefer to invest in the more liquid short-term securities.

• To get investors to invest in long-term securities, a higher return, over and above what is expected from the expectations hypothesis, is required

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Liquidity Preference - Explaining the Yield Curve• upward sloping - Case 1

– Future ST rates are expected to increase– Liquidity premium causes LT rates to be even

higher

• upward sloping - Case 2– Future ST rates are expected to decrease– Liquidity premium is large enough to offset

decrease in LT rates from expected decline of future ST rates

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Liquidity Preference - Explaining the Yield Curve• downward sloping

– Future ST rates are expected to decrease– Liquidity premium is not large enough to offset

decrease in LT rates from expected decline in ST rates

• flat– Future ST rates are expected to decrease– Liquidity premium just offsets decrease in LT

rates from expected decline in ST rates

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Preferred Habitat (Market Segmentation)

• There are various groups that prefer to invest in a specific maturity range.

• The shape of the yield curve will depend on the supply and demand for each maturity range.

• Explaining the yield curve– upward sloping - greater demand for ST securities so

have to increase LT rates to induce investors to invest in LT securities

– downward sloping - greater demand for LT securities so have to increase ST rates to induce investors to invest in ST securities

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Modern Theories

Equilibrium Theories: CIR, BP

Non-equilibrium Theories: Dothan, Vasicek,

Ho-Lee, Hull-White, HJM

Most of them are based on a Brownian Motion as a source of market uncertainty.

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Brownian Motion

Time

B

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Brownian Motion

• Starts at the origin

• Is continuous

• Is normally distributed at each time

• Increments are independent

• Markovian property

• Technical conditions

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Diffusion Processes

General diffusion process:

drift noise

- volatility (diffusion parameter)

tt dBdtdS

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Diffusion Processes

- volatility

The major model for stock prices.

Why it can NOT be used for bonds?

tttt dBSdtSdS

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Dothan Model of IR

This model gives an analytic pricing formula for bonds, options, but it is not rich enough.

ttt dBrdr

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Vasicek Model (1977)

The Ornstein-Uhlenbeck process with mean reversion. is the long-run mean.

Merton first proposed this process for IR (1971).

Extended by Jamshidian for IR options.

ttt dBdtrdr )(

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Cox-Ingersoll-Ross Model

The Ornstein-Uhlenbeck process with mean reversion.

Merton first proposed this process for IR (1971).

tttt dBrdtrdr )(

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HJM=Heat-Jarrow-Morton

The most general approach based on a multifactor stochastic model.

Very difficult to implement, especially to calibrate.

However gives significant advantage in pricing and hedging IR sensitive instruments.

Standard implementation via Monte-Carlo.