Time Series Analysis of the term structure of interest rates
Xi Wang 5002-3922
Abstract
This paper selected the data of the yield curve from Jan 1980 to Dec 2011 to analysis
the term structure of the United State. For the dramatic change in the economic
condition in the past decades, the time series data is divided into two periods: the
first period is from Jan, 1980 to Dec, 1999, while the second periods is from Jan,
2000 to Dec, 2011. Then the paper analysis the term structure from Xxx
perspectives: first to check the yield curve trend for the past decades to obtain the
economic trends and the average yield curve for each periods. Then illustrate the
change in the yield and the relationship between yield with different maturities.
Finally test two classic term structure theories.
1. Yield curve trends
To analysis the term structure of the interest rate, we can check the trend of the
yield with different maturity first.
1
yield curve trend for 1980~2000
20
15
10
5
1/1/80
12/1/80
11/1/81
10/1/82
9/1/83
8/1/84
7/1/85
6/1/86
5/1/87
4/1/88
3/1/89
2/1/90
1/1/91
12/1/91
11/1/92
10/1/93
9/1/94
8/1/95
7/1/96
6/1/97
5/1/98
4/1/99
0
3m
6m
1y
2y
3y
4y
5y
7y
10y
30y
yield curve for 2000~2011
3m
6m
1y
2y
3y
4y
5y
7y
10y
30y
9/1/11
2/1/11
7/1/10
12/1/09
5/1/09
10/1/08
3/1/08
8/1/07
1/1/07
6/1/06
11/1/05
4/1/05
9/1/04
2/1/04
7/1/03
12/1/02
5/1/02
10/1/01
3/1/01
8/1/00
1/1/00
8
7
6
5
4
3
2
1
0
For the economic of USA had changed significantly in the past decades, I divide the
data of yield curve from 1980-2011 into two parts: 1980~1999 and 2000~2011.
The entire trend for the YTM with different maturities is decreasing with fluctuates
since 1980. The peak value is approximately 17 in Aug 1981 and the trough is 0 for
short term in recently years. If the economic condition is good, the interest rate is
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high for the central bank need to prohibit the inflation caused by over invest, while
in the bad economic like recently, the interest will be very low to incentive the
invertor to invest at a low cost.
2. Average yield curve
Then graph the average yield curve of different periods:
yield curve
10
yields(%)
8
6
4
2
0
0.25
0.5
1
2
1980~1999
3
4
years
5
7
10
30
2000~2011
From the chart above we can see that the average yield curve of two periods are
both increasing with the maturity and the average yield curve of the period of
1980~1999(which concentrate in 7%) is higher than that of the recently period
(which concentrate in 3%). When looking back to the data of the yield curve, there is
a significant trend of decreasing in the yield curve. The current yield curve is
extremely low even compare to the yield of other years in the second period. This
phenomenon can reflect the economic situation changing in the USA.
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standard deviation of changes in yields for
different maturities
0.8000
0.7000
0.6000
0.5000
0.4000
1980-1999
0.3000
2000-2011
0.2000
0.1000
0.0000
0.25 0.5
1
2
3
4
5
7
10
30
Standard deviation
3m
6m
1y
2y
3y
4y
5y
7y
10y
30y
1980-1999
0.6677
0.6795
0.6696
0.5935
0.5287
0.5127
0.4963
0.4556
0.4256
0.3722
2000-2011
0.2465
0.2367
0.2444
0.2750
0.2877
0.2927
0.3027
0.2974
0.2852
0.2553
The stander deviation for different maturities as shown in the above chart is
decreasing with maturities. The short-term interest rates are more volatility than
the long-term interest rates. In addition, for the same maturity, the first period yield
is more volatility than the second periods. For the recently periods, we can see that
the standard deviation of the short-term is not significant different from the long
term. The reason for this is that recently yield curve is almost close to zero and
there is really less room for the short-term yields to volatility.
3. Correlation analysis
4
Correlation Matrix
3m
6m
1y
2y
3y
4y
5y
7y
10y
30y
3m
1
6m
0.9290
1
1y
0.8571
0.9659
1
2y
0.7513
0.8874
0.9558
1
3y
0.6741
0.8192
0.9067
0.9803
1
4y
0.6516
0.7961
0.8857
0.9690
0.9905
1
5y
0.6166
0.7637
0.8564
0.9502
0.9795
0.9925
1
7y
0.5481
0.6967
0.7964
0.9039
0.9476
0.9715
0.9804
1
10y
0.5301
0.6805
0.7737
0.8766
0.9225
0.9475
0.9615
0.9818
1
30y
0.4778
0.6241
0.7110
0.7996
0.8438
0.8696
0.8852
0.9213
0.9613
1
Above the correlation matrix of changes in yields with different maturities, there is
an obvious trends that more close two maturities are, higher correlation of change
between the two maturities. The Federal Reserve always use the Federal funds
which has great correlation with the short term interest rate to infect the short term
interest rate. However, this method has less influence on the long term interest
rates.
4. Term structure theory tests
There are three classic term structure theories:
(1)Expectations theory is that market’s expectations about future short-term rates
are given by current forward rates, no other systematic factor affect forward rates
and they are unbiased predictors of the future spot rates.
(1+𝑆0,2)2
E(S1,2)=F(S1,2)= (1+𝑆0,1) – 1
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(2)For liquidity theory, interest rates are biased because of liquidity premium and
the liquidity premium is rises with maturity. This phenomenon is results from the
loss of liquidity in the long-term bond. The investor cannot sell the bond at it fair
value so they ask for some compensation for this loss. In the liquidity theory
(1 + 𝑆0,2)2 =(1+S0,1+P)(1+E(S1,2))
P is the premium
(3) Market segmentation theory. The yields for bonds of different maturities are
determined by relative supply and demand of bonds; there is no compensation for
the risk, as investors do not change their preference on market for the market is
segment
Here I test the first two term structure theories
If the expectation theory holds, the E(S1,2)=S1,2. So I run the regression of actual
change on the one-year interest rate (S1,2-S0,1) on the forecast change on the oneyear interest rate(S1,2-F1,2).
The regression results is show as below
Linear Regression
Regression Statistics
R
R Square
Adjusted R Square
Standard Error
Total Number Of Cases
0.91322
0.83396
0.83351
0.75645
372
S1,2-S0,1 = 0.6517 + 0.8982 * S1,2-F1,2
ANOVA
d.f.
Regression
Residual
Total
1.
370.
371.
SS
1,063.40992
211.7177
1,275.12762
Coefficients
Standard Error
6
MS
1,063.40992
0.57221
F
1,858.42601
LCL
UCL
p-level
0.E+0
t Stat
p-level
H0 (2%) rejected?
Intercept
S1,2-F1,2
0.65172
0.89818
0.04608
0.02083
0.54406
0.8495
0.75939
0.94686
14.14336
43.10947
0.E+0
0.E+0
Yes
Yes
For the pure expectation theory, the spot interest rate should be equal to the
forward interest rate. That is the actual change of interest rate is same as the
forecast change of the interest rate. If the pure expectation theory holds, the
regression equation: (S1,2-S0,2)=𝛼+𝛽*(S1,2-F1,2)+𝜀 should have results that 𝛼 = 0
𝛽 = 1 statistically. From the regression results above that we can seen that both of
the intercept and the coefficient of the equation’s t statistic is greater than 2, which
means that they are statistically significantly. The 𝛼 value makes the forwards
interest rate is no longer an unbiased forecast for the spot interest rate in next
periods. The R-square is very high which means the actual change is highly
correlated with the forecast change and this is a good model. Here we can conclude
that the pure expectation theory is not holds in the real worlds.
For the liquid premium theory
F1,2 is the sum of two unobservable variables
F1,2=E1,2+P
Different from the pure expectation theory, the F1,2 not equal to E1,2. So a positive
premium (p) value can show the investor’s request of the compensation for liquidity,
in other words, the liquidity theory holds.
(1+𝑆0,2)2
The average of S1,2 is 5.45. Use the equation E(S1,2)=F(S1,2)= (1+𝑆0,1) – 1 to get F1,2,
then get the average of F1,2 is 6.61. Consequently,
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F1,2-S1,2= 1.16. The positive result 1.16 is the premium that request by the investors.
Then we can conclude that liquidity theory is hold in the real worlds for invertor in the
real world is risk adverse and they need the compensation for the risk they undertake.
5. Conclusion
The interest rate for United States has a decreasing trend since 1980 for the
dramatically economic condition change in the past decades. The average yield curve is
increasing with maturities for the longer the maturities are, the higher liquidity risk it
has. The investor in the real world ask for premium for the risk they burden and this
perspective is proofed in the test of the liquidity term structure theory. The changes in
the yield are more volatile in the first periods from 1980 to 1999 than the recently
periods. Affected by the current economic, the short-term interest rate is near to zero
and this limit the volatility of the changes in the short-term interest rate. In addition, the
correlation between two maturities that close to each other is higher than that between
two further maturities. The Federal Reserve uses this character to control the shortterm interest rate on the market.
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