The Term Structure and
Volatility of Interest Rates
Fin 284
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Treasury Yield Curve
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The most commonly investigated and used term
structure is the treasury yield curve.
Treasuries are used since they are considered
free of default, and therefore differ mainly in
maturity. Also the treasury is the benchmark
used to set base rates.
The treasury market is also very liquid so there
are no problems with liquidity
Current Treasury Yield Curve
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The most straightforward way to represent the
yield curve is by graphing the combinations of
yield and maturity
The problem with this measure is that it does not
account for differences in coupon rates across
bonds of similar maturities. Therefore there are
some alternative methods we need to explore
such as using the zero spot rates.
Investigating the Yield Curve
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The best theoretical measure is looking at the
spot rte (zero coupon) yield curve.
However often the yields for the on the run
treasuries are used as a proxy for this.
The recent yield curve based upon the on the run
yields are given in the next few slides.
Yield Curves Previous 6 Months
0.06
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0.05
Yield
0.04
7/31/2003
0.03
8/29/2003
9/30/2003
0.02
10/31/2003
0.01
11/28/2003
12/31/2003
0
0.00
5.00
10.00
15.00
Maturity (Years)
20.00
Yield Curves Previous 5 quarters
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0.06
0.05
Yield
0.04
0.03
12/31/2002
3/31/2003
0.02
6/30/2003
9/30/2003
0.01
12/31/2003
0
0.00
5.00
10.00
Maturity (Years)
15.00
20.00
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0 .1
US Treas Interest Rates
Jan 1990- Dec 2003
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Fin 284
1-mo
0 .0 9
3-mo
0 .0 8
6-mo
0 .0 7
1-yr
0 .0 6
2-yr
0 .0 5
3-yr
0 .0 4
5-yr
0 .0 3
7-yr
0 .0 2
10-yr
0 .0 1
20-yr
0
12 /8 /19 8 9
30-yr
9 /3 /19 9 2
5/3 1/19 9 5
2 /2 4 /19 9 8
11/2 0 /2 0 0 0
8 /17/2 0 0 3
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US Treas Rates Jan 1990 Dec 2003
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0.1
0.09
Downward sloping yield curve
0.08
0.07
3-mo
Yield
0.06
0.05
1-yr
0.04
5-yr
0.03
0.02
20-yr
0.01
0
12/ 8/ 1989
9/ 3/ 1992
5/ 31/ 1995
2/ 24/ 1998
Date
11/ 20/ 2000
8/ 17/ 2003
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Yield Curve 2000
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Fin 284
0.07
0.065
0.06
Yield
0.055
0.05
12/31/1999
0.045
3/31/2000
6/30/2000
0.04
9/29/2000
12/29/2000
0.035
0.00
5.00
10.00
15.00
20.00
Maturity (years)
25.00
30.00
35.00
Shifts in the yield curve
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Using the Arbitrage free valuation approach of a
security requires an estimation of the zero
coupon treasury yield curve.
As the shape of the yield curve changes so will
the rate corresponding with each respective time
frame.
Shifts in the Yield Curve
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The shape of the yield curve and changes in the
shape can provide information to the market
concerning future interest rates.
Want to investigate two things, overall shifts in
the curve and changes in its slope.
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Parallel Shifts
Short
Short
Intermediate
Maturity
Intermediate
Maturity
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Long
Long
Approximate Parallel Shift
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0.06
0.05
Yield
0.04
0.03
Change in rates is approximately
the same for all maturities
0.02
0.01
0
0.00
6/28/2002
12/31/2002
5.00
10.00
Maturity
15.00
20.00
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Twists
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Flattening Twist
Short
Intermediate
Maturity
Long
Steepening Twist
Short
Intermediate
Maturity
Long
Flattening of the curve
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0.07
0.065
Yield
0.06
0.055
Change for short term is
greater than for long term
0.05
0.045
0.04
0.00
8/31/1999
2/29/2000
5.00
10.00
Maturity
15.00
20.00
Steepening of the Curve
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0.06
0.05
Yield
0.04
0.03
Change for long term is
greater than for short term
0.02
0.01
0
0.00
6/30/2003
12/31/2003
5.00
10.00
Maturity
15.00
20.00
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Butterfly Shifts
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Positive Butterfly
Short
Intermediate
Maturity
Long
Negative Butterfly
Short
Intermediate
Maturity
Long
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Positive Butterfly shift
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0.07
0.065
Yield
0.06
0.055
Increase for short term and long term is
greater than for intermediate term
0.05
8/31/1999
11/30/1999
0.045
0.00
5.00
10.00
15.00
Maturity
20.00
25.00
30.00
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Negative Butterfly Shift
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0.055
0.05
0.045
Yield
0.04
0.035
0.03
Decrease for short term and long term is
greater than for intermediate term
0.025
0.02
0.015
0.01
0.00
10/31/2002
4/30/2003
5.00
10.00
Maturity
15.00
20.00
Why does the Yield Curve
usually slope upwards?
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Three things are observed empirically concerning
the yield curve:
Rates across different maturities move together
More likely to slope upwards when short term
rates are historically low, sometimes slope
downward when short term rates are
historically high
The yield curve usually slopes upward
Three Explanations of the Yield
Curve
The Expectations Theories
Pure Expectations
Local Expectations
Return to Maturity Expectations
Segmented Markets Theory
Biased Expectations Theories
Liquidity Preference
Preferred Habitat
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Pure Expectations Theory
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Long term rates are a representation of the short
term interest rates investors expect to receive in
the future. In other words, the forward rates
reflect the future expected rate.
Assumes that bonds of different maturities are
perfect substitutes
In other words, the expected return from holding
a one year bond today and a one year bond next
year is the same as buying a two year bond
today. (the same process that was used to
calculate our forward rates)
Pure Expectations Theory:
A Simplified Illustration
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Let
rt = today’s time t interest rate on a one
period bond
ret+1 = expected interest rate on a one period bond
in the next period
r2t = today’s (time t) yearly interest rate on a two
period bond.
Investing in successive one
period bonds
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If the strategy of buying the one period bond in
two consecutive years is followed the return is:
(1+rt)(1+ret+1) – 1 which equals
rt+ret+1+ (rt)(ret+1)
Since (rt)(ret+1) will be very small we will ignore it
that leaves
rt+ret+1
The 2 Period Return
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If the strategy of investing in the two period bond
is followed the return is:
(1+r2t)(1+r2t) - 1 = 1+2r2t+(r2t)2 - 1
(r2t)2 is small enough it can be dropped
which leaves
2r2t
Set the two equal to each other
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2r2t = rt+ret+1
r2t = (rt+ret+1)/2
In other words, the two period interest rate is the
average of the two one period rates
Applying the model
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The 2 year rate is an average of the current 1 year
rate and the expected rate one year in the future.
This implies that the yield curve will slope upward
when the expected one year rate is expected to
increase compared to the current one year rate.
Similarly the yield curve will slope downward when
the expected rate is less than the current rate.
Expectations Hypothesis
r2t = (rt+ret+1)/2
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If you assume that the expected rate is
representative of the long run average and that
rates will move toward the average, empirical fact
two is explained.
When the yield curve is upward sloping (R2t>R1t)
The current rate would be less than the long run
average and it is expected that short term rates will
be increasing.
Likewise when the yield curve is downward sloping
the current rate would be above the long run
average (the expected rate).
Expectations Hypothesis
r2t = (rt+ret+1)/2
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As short term rates increase the long term rate will
also increase and a decrease in short term rates will
decrease long term rates. (Fact 1)
This however does not explain Fact 3 that the yield
curve usually slopes up. Given the explanation of
Fact 2 the yield curve should slope up about half of
the time and slope down about half of the time.
Problems with Pure Expectations
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The pure expectations theory also ignores the
fact that there is reinvestment rate risk and
different price risk for the two maturities.
Consider an investor considering a 5 year horizon
with three alternatives:
buying a bond with a 5 year maturity
buying a bond with a 10 year maturity and holding it 5
years
buying a bond with a 20 year maturity and holding it 5
years.
Price Risk
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The return on the bond with a 5 year maturity is
known with certainty the other two are not.
The longer the maturity the greater the price
risk.
If interest rates change the return and the 10
and 20 year bonds will be determined in part by
the capital gains resulting from the new price at
the end of five years.
Reinvestment rate risk
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Two new options:
Investing in a 5 year bond
Investing in 5 successive 1 year bonds
Investing in a two year bond today followed by a three
year bond in the future.
Again the 5 year return is known with certainty,
but the others are not.
Local expectations
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Local expectations theory says that returns of
different maturities will be the same over a very
short term horizon, for example 6 months.
This assumes that all the forward rates currently
implied by the spot yield curve are realized.
Local Expectations
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Previously we calculated the zero spot rates
using the bootstrapping method for the on the
run treasury securities given below.
Maturity
YTM
Maturity
YTM
0.5
4%
2.5
5.0%
1.0
4.2%
3.0
5.2%
1.5
4.45%
3.5
5.4%
2.0
4.75%
4.0
5.55%
Zero spot Review
(local expectations example)
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Given the assumption that all of the on the run
treasury securities were selling at par we found the
1.5 year zero coupon rate by discounting the coupons
by the respective zero coupon rates.
2.225
2.225
102.225
100 


2
3
(1.02) (1.021)
(1  z3 )
z3  .022293 semiannaul
z3  .022293(2)  .0444586 annual
Zero spot curve
(local expectations example)
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Continuing the same process for future rtes we
started to build a zero spot yield curve
Time
0.5
1.0
1.5
2.0
YTM
4%
4.2%
4.45%
4.75%
Zero Spot
4%
4.2%
4.4459%
4.7666%
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Forward Rates
(local expectations example)
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Fin 284
Given the zero spot rates it is possible to find the
forward rates.
Let 1fm be the 1 period (six month) forward rate from
time m to time m+1.
The forward rate can then be found as:
(1  zm ) (11 f m )  (1  zm1 )
m
m 1
(1  zm1 )

1
1 fm 
m
(1  zm )
m 1
Forward Rate
(local expectations example)
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Given the 6 mo. zero spot rate of 4% and the 1
year zero spot rate of 4.2%, the one period (6mo.)
forward rate from 6 months to 1 year would be:
(1+.021)2 = (1+.02)(1+1f1) 1f1 = .022
Similarly the 6 month forward rate from 1 year to
1.5 years can be found from the 1 year zero spot
rate of 4.2% and the 1.5 zero spot rate of 4.4459%
(1+.021)2(1+ 1f2) = (1.022293)3 1f2 = .024884
Likewise 1f3 = .02847
Local expectations example
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Local expectations theory says that returns of
different maturities will be the same over a very
short term horizon, for example 6 months.
The return from buying the 2 year 4.45% coupon
bond that makes semiannual payments selling at
par and selling it in six months should be equal
to the return on a 1 year coupon bond with a
YTM of 4.2% if you hold it 6 months.
6 mo return on 1 year bond
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The 1 year bond has a current YTM of 4.2%.
This means that a equivalent bond selling at par
would make a $2.10 coupon payment at eh end
of 6 months and at the end of 1 year.
If you bought this bond at t = 6 months and sold
it at t=1 year you should earn 1f1 (the 6 mo.
forward rate) over the time you own the bond.
The price of the bond at t=6 mos should reflect
this.
6 mo return on 1 year bond
At time t=1 year the bond makes payments of
102.10. The total return from owning the bond is
the capital gains yield and interest yield and it
should equal 1f1=.022
100  P0 2.10 102.10  P0
0.022 


P0
P0
P0
102.10
P0 
 99.90125
1.022
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6 mo return on 1 year bond
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Buying the bond at time 0 and selling it at the
end of the first 6 months would then produce a
return of
99.90215  100 2.10

 0.00098  .021  .02002
100
100
Which is equal to the spot (zero coupon) six month rate
Return on the 2 year bond
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The price of the 2 year bond at the end of 6
months should also equal the PV of its expected
cash flows discounted back at the forward rate
(otherwise there would be an arbitrage
opportunity).
By finding the price at the end of 6 months we
can again find the return from owning the bond
for 6 months form t=0 to t=6 mos.
6 mo return on 2 year bond
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There are three coupon payments left from time t = 6
mos to t= 2 years, using the forward rtes the price of
the bond at t=6mos should be:
P6mos
2.375
2.375
102.375



(11 f1 ) (11 f1 )(11 f 2 ) (11 f1 )(11 f 2 )(11 f 3 )
2.375
2.375
102.375



(1.022) (1.022)(1.024884) (1.022)(1.024884)(1.02847)
 99.62509
Total Return on holding
2 year bond for 6 months
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The 2 year bond originally sold for par and it made a
$2.375 coupon payment at t=6mos. The total return
from owning it is then:
99.62509  100 2.375
Total Return 

100
100
 .003749  .02375  .0200009
Which is the same as the 6 month return on the 1 year
bond (and the same as the 6 month spot rate)
Local Expectations
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Similarly owning the bond with each of the
longer maturities should also produce the same 6
month return of 2%.
The key to this is the assumption that the
forward rates hold. It has been shown that this
interpretation is the only one that can be
sustained in equilibrium.*
Cox, Ingersoll, and Ross 1981 Journal
of Finance
Return to maturity expectations
hypothesis
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This theory claims that the return achieved by
buying short term and rolling over to a longer
horizon will match the zero coupon return on the
longer horizon bond. This eliminates the
reinvestment risk.
Expectations Theory and
Forward Rates
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The forward rte represents a “break even” rate
since it the rte that would make you indifferent
between two different maturities
According to the pure expectations theory and its
variations are based on the idea that the forward
rte represents the market expectations of the
future level of interest rates.
However the forward rate does a poor job of
predicting the actual future level of interest rates.
Segmented Markets Theory
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Interest Rates for each maturity are determined
by the supply and demand for bonds at each
maturity.
Different maturity bonds are not perfect
substitutes for each other.
Implies that investors are not willing to accept a
premium to switch from their market to a
different maturity.
Therefore the shape of the yield curve depends
upon the asset liability constraints and goals of
the market participants.
Biased Expectations Theories
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Both Liquidity Preference Theory and Preferred
Habitat Theory include the belief that there is an
expectations component to the yield curve.
Both theories also state that there is a risk
premium which causes there to be a difference in
the short term and long term rates. (in other
words a bias that changes the expectations
result)
Liquidity Preference Theory
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This explanation claims that the since there is a price
risk and liquidity risk associated with the long term
bonds, investor must be offered a premium to invest
in long term bonds
Therefore the long term rate reflects both an
expectations component and a risk premium.
This tends to imply that the yield curve will be
upward sloping as long as the premium is large
enough to outweigh an possible expected decrease.
Preferred Habitat Theory
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Like the liquidity theory this idea assumes that there
is an expectations component and a risk premium.
In other words the bonds are substitutes, but savers
might have a preference for one maturity over
another (they are not perfect substitutes).
However the premium associated with long term
rates does not need to be positive.
If there are demand and supply imbalances then
investors might be willing to switch to a different
maturity if the premium produces enough benefit.
Preferred Habitat Theory
and The 3 Empirical Observations
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Thus according to Preferred Habitat theory a rise in
short term rates still causes a rise in the average of
the future short term rates. This occurs because of
the expectations component of the theory.
Therefore the long and short rates move together
and fact 1 is explained.
Preferred Habitat Theory
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The explanation of Fact 2 from the expectations
hypothesis still works. In the case of a downward
sloping yield curve, the term premium (interest rate
risk) must not be large enough to compensate for
the currently high short term rates (Current high
inflation with an expectation of a decrease in
inflation). Since the demand for the short term
bonds will increase, the yield on them should fall in
the future.
Preferred Habitat Theory
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Fact three is explained since it will be unusual for
the term premium to be so small or negative,
therefore the the yield curve usually slopes up.
Measuring Yield Curve Risk
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Key Rate Duration, Calculating the change in
value for a security or portfolio after changing
one key interest rate keeping other rates
constant.
Each point on the spot yield curve has a separate
duration associated with it.
If you allowed all rates to change by the same
amount, you could measure the response to the
security or portfolio to a parallel shift in the yield
curve.
Key Rates and Portfolios
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By focusing on a group of key rates it is possible
to investigate the impact of changes in the shape
of the yield curve on specific parts of a portfolio,
we will cover this in more detail in the portfolio
section of the course.
Yield Volatility
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One of the most important determinants of
interest rate risk is the volatility of the yield.
There fore it is important to measure the amount
of volatility present, this requires a review of
basic statistics.
Measuring Historical Volatility
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The easiest way to measure the historical volatility is to
calculate the variance and standard deviation of
historical data.
Variance of a random variable:
T
Variance 
2
(
X

X
)
 t
t 1
T 1
where; X t  observatio n t of variable X
X  the smaple mean for variab le X
T  the # of Observatio ns in the sample
Standard Deviation
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The standard deviation is then simply the square
root of the variance.
Generally, the variance of returns will be used to
measure volatility.
Change in Daily Yield
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We want to look at the daily change in yield,
calculated as the percentage change in yield.
This assumes simple compounding on the yield
Assume that the yield on a 20 year treasury was 4%
and it increased to 4.05% the next day.
The percentage change in yield would be:
0403  .04
 .0075
.04
.04(1.0075)  .0403
Continuous Compounding
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The change could also be calculated assuming
continuous compounding
In this case the change in the daily yield
would be equal to the natural log of the ratio
of the two rates:
 .0403 
ln 
  .007472
 .04 
Calculating Volatility
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Using historical data, the number of observations
used will have a large bearing on the calculation.
Often the choice depends upon the position that
you are concerned with. For example a trader
concerned with the change in value over a short
period of time might look at a very short period
such as 10 days. In contrast a portfolio manager
might look at 20 or 30 days.
Daily vs. Annual
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The daily standard deviation needs to be
adjusted if you want to consider the annual
volatility.
The daily standard deviation can be scaled by
multiplying it by the square root of the number
of days in a year. (250, 260, or 365). The choice
of the number of days to use will make a
difference if the yearly standard deviation is used
in valuation or measuring risk exposure.
Interpreting Standard Deviation
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Assume that the annual standard deviation of a
20 year treasury is 10% and its yield is 5%. This
implies that the annual standard deviation is 50
Bp ((.10)(.05) = .005)
Assuming that distribution of returns is normally
distributed there is a 68.3% probability that the
yield will be within a range one standard
deviation below and above the expected value.
Prob Ranges for Normal Dist.
68.26%
95.46%
99.74%
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Interpreting Standard Dev
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In our example the expected return was 5% with
a standard deviation of 50Bp. This implies that
there is a 68.3% probability that the yield will be
be between 4.5% and 5.5%. And a 95.5%
chance that he yield will be between 4% and 5%
Implied Volatility
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Instead of calculating the volatility based upon
past observations of the yield it is also possible to
look at interest rate options available in the
market and the volatility implied by their current
price.
Volatility is a key component in most option
pricing models . This idea reverses the process
by using the market price to calculate the
volatility that would have needed to be used to
arrive at the market price.