The Term Structure of Interest Rates
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Transcript of The Term Structure of Interest Rates
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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.