Predicting Interest Rates
Statistical Models
Economic vs. Statistical Models
Economic models are designed to match
correlations between interest rates and
other economic aggregate variables
Pro: Economic (structural) models use all the
latest information available to predict interest
rate movements
Con: They require a lot of data, the equation
can be quite complex, and over longer time
periods are very inaccurate
Economic vs. Statistical Models
Statistical models are designed to match
the dynamics of interest rates and the
yield curve using past behavior.
Pro: Statistical Models require very little data
and are generally easy to calculate
Con: Statistical models rely entirely on the
past. They don’t incorporate new information.
The Yield Curve
6 5.07
4.3
4 2.94
2.12 2.61
2
0
1 yr 2 yr 5yr 10 yr 20yr
S(1) S(2) S(5) S(10) S(20)
Recall that the yield curve is a collection of current spot
rates
Forward Rates
Forward rates are interest rates for
contracts to be written in the future. (F)
F(1,1) = Interest rate on 1 year loans contracted 1
year from now
F(1,2) = Interest rate on 2 yr loans contracted 2
years from now
F(2,1) = interest rate on 1 year loans contracted 2
years from now
S(1) = F(0,1)
Spot/Forward Rates
S(3)
Spot
S(2) Rates
S(1)
Now 1yr 2yrs 3yrs 4yrs 5yrs
F(0,1) F(1,1) F(2,1)
F(0,2) F(2,2)
Forward
Rates
F(1,2)
F(1,3)
Calculating Forward Rates
Forward rates are not observed, but are implied in the
yield curve
Suppose the current annual yield on a 2 yr Treasury is
2.61% while a 1 yr Treasury pays an annual rate of
2.12%
S(2) 2.61%/yr
S(1) 2.12%/yr
Now 1yr 2yrs 3yrs 4yrs 5yrs
F(1,1)
Calculating Forward Rates
S(2) 2.61%/yr
S(1) 2.12%/yr
Now 1yr 2yrs 3yrs 4yrs 5yrs
F(1,1)
Strategy #1: Invest $1 in a Strategy #2: Invest $1 in a
two year Treasury 1 year Treasury and then
reinvest in 1 year
$1(1.0261)(1.0261) = 1.053 (5.3%) $1(1.0212)(1 + F(1,1))
For these strategies, to pay $1(1.0261)(1.0261) = $1(1.0261)(1+F(1,1)
the same return, the one year
$1(1.0261)(1.0261)
forward rate would need to 1+F(1,1) = $1(1.0212) =1.031
be 3.1%
Calculating Spot Rates
We can also do this in reverse. If we knew the path for
forward rates, we can calculate the spot rates:
S(3) ???
S(2) ???
S(1) 2%
Now 1yr 2yrs 3yrs 4yrs 5yrs
2% 3.3% 2.9%
Calculating Spot Rates
S(2) ???
Now 1yr 2yrs 3yrs 4yrs 5yrs
2% 3.3%
Strategy #1: Invest $1 in a Strategy #1: Invest $1 in a
two year Treasury 1 year Treasury and then
reinvest in 1 year
$1(1+(S(2))(1+S(2)) $1(1.02)(1.033) = 1.054 (5.4%)
2
For these strategies, to pay $1(1.02)(1.033) = $1(1+S(2))
the same return, the two year
1/2
spot rate would need to be 1+S(2) = ((1.02)(1.033)) =1.026
2.6%
Arithmetic vs. Geometric Averages
S(2) 2.6%
Now 1yr 2yrs 3yrs 4yrs 5yrs
2% 3.3%
In the previous example, we calculated the Geometric Average of
expected forward rates to get the current spot rate
1/2
1+S(2) = ((1.02)(1.033)) =1.026 (2.6%)
The Arithmetic Average is generally a good approximation
2% + 3.3%
S(2) = = 2.65%
2
Spot rates are equal to the averages of the
corresponding forward rates (expectations hypothesis)
S(3) 2.73%
S(2) 2.65%
S(1) 2%
Now 1yr 2yrs 3yrs 4yrs 5yrs
2% 3.3% 2.9%
2% + 3.3% 2% + 3.3% + 2.9% = 2.73%
S(2) = = 2.65% S(3) =
2 3
However, the expectations hypothesis assumes
that investing in long term bonds is an equivalent
strategy to investing in short term bonds
This rate is “locked
in” at time 0
S(2) 2.65%
Now 1yr 2yrs 3yrs 4yrs 5yrs
2% 3.3%
This rate is flexible at
time 0
Long term bondholders should be compensated for
inflexibility of their portfolios by adding a “liquidity
premium” to longer term rates (preferred habitat
hypothesis)
Statistical Models
Now 1yr 2yrs 3yrs 4yrs 5yrs
F(0,1) F(1,1) F(2,1) F(3,1) 3.3%
F(4,1)
First, write down a model to explain movements in
the forward rates
Then, calculate the yield curve implied by the forward
rates. Does it look like the actual yield curve?
S(3)
S(2)
S(1)
Now 1yr 2yrs 3yrs 4yrs 5yrs
Lattice Methods (Discrete)
Lattice models assume that the interest
rate makes discrete jumps between time
periods (usually calibrated monthly)
Binomial: Two Possibilities each Period
Trinomial: Three Possibilities each Period
An Example
At time zero, the interest rate 5%: F(0,1) = S(1)
An Example
In the first year, the interest rate has a 50%
chance of rising to 5.7% or falling to 4.8%: F(1,1)
An Example
In the second year, there is also a 50% chance of rising
or falling conditional on what happened the previous
year: F(2,1)
Calculating the Yield Curve
S(1)
5.7% Path 1: (1.05)(1.057) = 1.10985 (10.985%)
5%
4.8% Path 2: (1.05)(1.048) = 1.10040 (10.04%)
Expected two year
= (.5)(1.10985) + (.5)(1.10040) = 1.105125 (10.5125%)
cumulative return
1/2
Annualized Return = (1.105125) = 1.0512 (5.12%) = S(2)
6.4% Path 1: (1.05)(1.057)(1.064) = 1.181 (18.1%)
5.7%
Path 2: (1.05)(1.057)(1.052) = 1.168 (16.8%)
5% 5.2%
Path 3: (1.05)(1.048)(1.052) = 1.157 (15.7%)
4.8%
4.6% Path 4: (1.05)(1.048)(1.046) = 1.151 (15.1%)
Expected three year
= (.25)(1.181) + (.25)(1.168) + (.25)(1.157) +(.25)(1.151) = 1.164
cumulative return
1/3
Annualized Return = (1.164) = 1.0519 (5.19%) = S(3)
Future Yield Curves
6.4%
5.7%
5% 5.2% Path 1: (1.048)(1.052) = 1.1025 (10.25%)
4.8%
4.6% Path 2: (1.048)(1.046) = 1.096 (9.6%)
Suppose that
next months
interest rate (.5)(1.1025) + (.5)(1.096) = 1.0993(9.3%)
turns out to be 1/2
4.8% = S(1)’ S(2)’ = (1.099) = 1.049 (4.9%)
Volatility & Term Structure
A common form for a binomial tree is as follows:
1 it with probabilit y .5
it 1 it
(1 ) with probabilit y .5
Sigma is measuring volatility
Higher volatility raises the probability of very
large or very small future interest rates. This will
be reflected in a steeper yield curve
8.3
8.25
8.2
8.15
8.1
High Sigma
8.05
Low Sigma
8
7.95
7.9
7.85
1 2 3 4 5 6 7 8 9 10 11 12
Continuous Time Models
dit ait , t dt it , t dz
Random Error
Change in the term with N(0,1)
interest rate at distribution
time ‘t’
Deterministic (Non-Random)
component Random component
Vasicek
The Vasicek model is a particularly simple form:
dit it dt dz
Controls Persistence
Controls Variance
Controls Mean
Using the Vasicek Model
Choose parameter values
Choose a starting value
Generate a set of random numbers with mean 0 and
variance 1
dit .26 it dt 2dz
i0 6%
t=0 t=1 t=2 t=3 t=4
i 6% 6.8% 6.84% 4.202% 5.5616%
.2(6-i) 0 -.16 -.168 .3596
dz .4 .2 -1.1 .5
di .8 .04 -2.368 1.3596 -.9
Vasicek (sigma = 2, kappa = .17)
0.2
0.18
0.16
Path1
0.14
Path2
0.12
Path 3
0.1
Path 4
0.08
Path 5
0.06
Average
0.04
0.02
0
0 8 16 24 32 40 48 56 64 72 80 88 96
Vasicek (sigma = 4, kappa = .17 )
0.3
0.25
0.2
Path1
0.15
Path2
0.1 Path 3
0.05 Path 4
Path 5
0
Average
-0.05 0 8 16 24 32 40 48 56 64 72 80 88 96
-0.1
-0.15
Vasicek (sigma = 2, kappa = .4)
0.16
0.14
0.12 Path1
0.1 Path2
Path 3
0.08
Path 4
0.06 Path 5
0.04 Average
0.02
0
0 8 16 24 32 40 48 56 64 72 80 88 96
Cox, Ingersoll, Ross (CIR)
The CIR framework allows for volatility that
depends on the current level of the interest
rate (higher volatilities are associated with
higher rates)
drt r dt r dz
dit it dt i dz
Heath,Jarow,Morton (HJM)
Vasicek and CIR assume a process for a single forward
rate and then use that to construct the yield curve
In this framework, the correlation between different
interest rates of different maturities in automatically one
(as is the case with any one factor model)
HJM actually model the evolution of the entire array of
forward rates
df t, T at, T , f (t, T )dt t , T , f (t, T )dz
Change it the forward rate of maturity T
ant time t
Table 1
Summary Statistics for Historical Rates
Shape
Normal Inverted Humped Other
68.8% 11.6% 13.4% 6.3%
Yield Statistics
1 yr. 3 yr. 5 yr. 10 yr.
Mean 6.08 6.47 6.64 6.81
S.D. 3.01 2.88 2.84 2.81
Skewness 0.97 0.84 0.77 0.68
Exc. Kurtosis 1.10 0.69 0.48 0.16
Percent iles
1 yr. 3 yr. 5 yr. 10 yr.
1% 1.07 1.59 1.94 2.38
5% 2.05 2.52 2.72 2.90
50% 5.61 6.20 6.44 6.68
95% 12.08 12.48 12.59 12.56
99% 15.17 14.69 14.59 14.29
Corr (1 yr,10 yr) = 0.944
Tables 1-4 from Ahlgrim, D’Arcy, and Gorvett, CAS 1999 DFA Call Paper Program
Table 2
Summary Statistics for Vasicek Model
Shape
Normal Inverted Humped Other
41.6% 54.8% 3.6% 0.0%
Yield Statistics
1 yr. 3 yr. 5 yr. 10 yr.
Mean 8.81 8.75 8.68 8.52
S.D. 3.83 3.24 2.77 1.95
Skewness -0.16 -0.16 -0.16 -0.16
Exc. Kurtosis -0.19 -0.19 -0.19 -0.19
Percent iles
1 yr. 3 yr. 5 yr. 10 yr.
1% -0.38 0.97 2.04 3.84
5% 2.33 3.27 4.00 5.22
50% 8.94 8.86 8.77 8.59
95% 14.69 13.73 12.94 11.53
99% 17.22 15.87 14.76 12.82
Corr (1 yr,10 yr) = 1.000
Notes: Number of simulations = 10,000, = 0.1779, = 0.0866, = 0.0200
Table 3
Summary Statistics for CIR Model
Shape
Normal Inverted Humped Other
47.7% 47.6% 4.7% 0.0%
Yield Statistics
1 yr. 3 yr. 5 yr. 10 yr.
Mean 8.08 8.04 7.98 7.86
S.D. 2.89 2.31 1.88 1.20
Skewness 0.92 0.92 0.92 0.92
Exc. Kurtosis 1.49 1.49 1.49 1.49
Percentiles
1 yr. 3 yr. 5 yr. 10 yr.
1% 2.92 3.90 4.62 5.71
5% 3.95 4.73 5.29 6.14
50% 7.71 7.73 7.73 7.70
95% 13.42 12.31 11.45 10.09
99% 17.19 15.33 13.90 11.66
Corr (1 yr,10 yr) = 1.000
Notes: Number of simulations = 10,000, = 0.2339, = 0.0808, = 0.0854
Table 4
Summary Statistics for HJM Model
Yield Statistics
1 yr. 3 yr. 5 yr. 10 yr.
Mean 7.39 7.51 7.60 7.80
S.D. 2.26 2.27 2.31 2.44
Skewness 0.51 0.53 0.54 0.54
Exc. Kurtosis -0.88 -0.85 -0.85 -0.86
Percentiles
1 yr. 3 yr. 5 yr. 10 yr.
1% 4.45 4.48 4.52 4.59
5% 4.79 4.85 4.90 4.99
50% 7.48 7.58 7.65 7.83
95% 11.57 11.74 11.92 12.38
99% 12.09 12.26 12.44 12.89
Corr (1 yr,10 yr) = 0.999
Notes: Number of simulations = 100
