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Chapter 22: Modeling the Term Structure
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Objectives of this chapter
Produce a smooth curve of the term structure
x-axis: maturity of bond cash flow
y-axis: pure-discount interest rate
Make sure that this curve fits existing bond data in the best possible fashion
Ultimately: explain the use of the Nelson-Siegel (1985) model
Basic concepts
Term structure: relationship between bond yield or interest rates and the term to maturity
Yield to maturity (YTM): internal rate of return of the bond price and its future anticipated payments
Pure-discount interest rate [see next]
Pure-discount discount factor [see next]
In this example:
15 bonds, each with different maturity
We compute the YTM for each bond
First example
Yield to Maturity (YTM)
YTM: same discount rate applied to all cash flows of a bond such that their sum equals the bond price i.e., IRR
A bond with price P and payments C1, C2, …Cn:
YTM = IRR of the bond payments
Pure discount rate
Pure discount rate assumes that each bond cash flow is discounted at its own rate
This is the concept usually favored by economists
A bond with price P and payments C1, C2, …Cn:
Each payment is discounted at its own rate, the pure discount rate
Pure discount factors
Instead of writing y1, y2…, write
Example: finding the discount factors
Bond 1: 96.6 = 102d1 (2% coupon, 1-year maturity)
Bond 2: 93.71 = 2.5d1 + 102.5d2 (2.5% coupon, 2-year maturity)
Bond 3: 91.56 = 3d1 + 3d2 + 103d3 (3% coupon, 3-year maturity)
In general: price =
Example: finding the discount factors
Once the discount factors, d1, d2, …, have been found, you can translate them into interest rates
Discount factors determine a zero-coupon term structure:
where yt = continuously-compounded pure-discount rate at time t
Solving discount factors for zero-coupon interest rate gives:
Example: finding the discount factors
Changing the example slightly produces
different term structure
In reality…
Reality is much more complex!!
Bonds with same maturities have different coupons with different prices
We could call this “noisy prices”
Maturities are not usually whole numbers!
We will deal with these problems one at a time
Multiple bonds, same maturity
We have 15 bonds but only 12 maturities
We’ll find the least-squares approximation to the best discount factors d1, d2, … , d12
Some more computations on next page
This is a pretty good method – the one objection might be that the derived term structure is not a smooth curve. If this is what you want, use the Nelson-Siegel method
Multiple bonds, same maturity
Nelson-Siegel: Fitting a functional form to the term structure
BondPriceMaturity
196.6012.0%5.59%
293.7122.5%5.93%
391.5633.0%6.17%
490.2443.5%6.34%
589.7454.0%6.47%
690.0464.5%6.56%
791.0975.0%6.63%
892.8285.5%6.69%
995.1996.0%6.73%
1098.14106.5%6.76%
11101.60117.0%6.79%
12105.54127.5%6.81%
13109.90138.0%6.83%
14114.64148.5%6.84%
15119.73159.0%6.85%
INITIAL EXAMPLE
5.50%5.70%5.90%6.10%6.30%6.50%6.70%6.90%0246810121416YTM OF BONDS AS FUNCTION OF MATURITY
ABCDEFGHIJKLMNO
BondPriceMaturity
196.6012.0%0.94715.44%<-- =-LN(F3)/C3 1-96.60102.000.000.000.00
293.7122.5%0.89115.76%<-- =-LN(F4)/C4 2-93.712.50102.500.000.00
391.5633.0%0.83546.00%<-- =-LN(F5)/C5 3-91.563.003.00103.000.00
490.2443.5%0.78156.16%<-- =-LN(F6)/C6 4-90.243.503.503.50103.50
589.7454.0%0.73006.29% 5-89.744.004.004.004.00
690.0464.5%0.68146.39% 6-90.044.504.504.504.50
791.0975.0%0.63586.47% 7-91.095.005.005.005.00
892.8285.5%0.59306.53% 8-92.825.505.505.505.50
995.1996.0%0.55306.58% 9-95.196.006.006.006.00
1098.14106.5%0.51576.62% 10-98.146.506.506.506.50
11101.60117.0%0.48096.66% 11-101.607.007.007.007.00
12105.54127.5%0.44846.68% 12-105.547.507.507.507.50
13109.90138.0%0.41816.71% 13-109.908.008.008.008.00
14114.64148.5%0.38986.73% 14-114.648.508.508.508.50
15119.73159.0%0.36356.75% 15-119.739.009.009.009.00
INITIAL EXAMPLE
=IF(L$2<$C3,$D3*100,IF(L$2=$C3,(1+$D3)*100,0))
{=MMULT(MINVERSE(L3:Z17),-K3:K17)}
5.0%5.2%5.4%5.6%5.8%6.0%6.2%6.4%6.6%6.8%7.0%0123456789101112131415Pure Discount Term Structure
ABCDEFGHIJKLMN
BondPriceMaturity
196.6012.0%0.94715.44%<-- =-LN(F3)/C3 1-96.60102.000.000.00
293.7122.5%0.89115.76%<-- =-LN(F4)/C4 2-93.712.50102.500.00
391.5633.0%0.83546.00%<-- =-LN(F5)/C5 3-91.563.003.00103.00
490.2443.5%0.78156.16%<-- =-LN(F6)/C6 4-90.243.503.503.50
589.7454.0%0.73006.29% 5-89.744.004.004.00
690.0464.5%0.68146.39% 6-90.044.504.504.50
791.0974.8%0.64636.24% 7-91.094.804.804.80
892.8284.9%0.62735.83% 8-92.824.904.904.90
995.1995.0%0.61425.42% 9-95.195.005.005.00
1098.14105.1%0.60605.01% 10-98.145.105.105.10
11101.60115.3%0.59444.73% 11-101.605.305.305.30
12105.54125.7%0.56954.69% 12-105.545.705.705.70
13109.90136.5%0.51175.15% 13-109.906.506.506.50
14114.64147.0%0.48035.24% 14-114.647.007.007.00
15119.73157.3%0.46845.06% 15-119.737.307.307.30
{=MMULT(MINVERSE(L3:Z17),-K3:K17)}
A DIFFERENT EXAMPLE OF THE TERM STRUCTURE
4.0%4.5%5.0%5.5%6.0%6.5%7.0%0123456789101112131415Pure Discount Term Structure
Bond #Price
191.896712.0%
283.256422.5%
376.000033.0%
476.234733.2%
571.211043.5%
667.967254.0%
766.000064.5%
866.162564.2%
965.488175.0%
1065.700385.5%
1164.000095.8%
1266.615896.0%
1368.0989106.5%
1470.0480117.0%
1572.3857127.5%
MULTIPLE BONDS,
SAME MATURITY
Formula for the least squares approximat
Minverse(MMultTransposeCashflows,Cashflo
MMultTransposeCashflows,Prices
dCashflowsCashflowsCashflowsPrices
Bond #Price
Maturityd(t)y(t)
191.896712.0%10.900910.43%<-- =-LN(G4)/F4
283.256422.5%20.790311.77%
376.000033.0%30.687412.49%
476.234733.2%40.607612.46%
571.211043.5%50.538712.37%
667.967254.0%60.486312.01%
766.000064.5%70.432711.97%
866.162564.2%80.391111.74%
965.488175.0%90.347311.75%
1065.700385.5%100.323111.30%
1164.000095.8%110.294511.11%
1266.615896.0%120.268710.95%
1368.0989106.5%
1470.0480117.0%
1572.3857127.5%
Equations/formulas
LEAST SQUARES SOLUTION TO TERM STRUCTURE
Least squares and term structure
Column G: {=MMULT(MINVERSE(MMULT(TRANSPOSE(cashflows),cashflows)),
MMULT(TRANSPOSE(cashflows),prices))}
10.0%10.5%11.0%11.5%12.0%12.5%13.0%123456789101112MaturityLeast Squares Term Structure
JKLMNOPQRSTUV
Time -->123456789101112
10200000000000
2.5102.50000000000
33103000000000
3.23.2103.2000000000
3.53.53.5103.500000000
44441040000000
4.54.54.54.54.5104.5000000
4.24.24.24.24.2104.2000000
55555510500000
5.55.55.55.55.55.55.5105.50000
5.85.85.85.85.85.85.85.8105.8000
66666666106000
6.56.56.56.56.56.56.56.56.5106.500
77777777771070
7.57.57.57.57.57.57.57.57.57.57.5107.5
Bond cash flows
10.5%11.0%11.5%12.0%12.5%13.0%13.5%123456789101112Least squares rateNelson-Siegel rate
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