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FINS5513 Term Structure
Creation date：2024-05-04 16:06:22
Belonging category：金融Finance
Term Structure

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FINS5513 Lecture 8A

Term Structure and
Bond Risk
2❑ 8.1 Bond Risk: Default Spread and Term Spread
➢ Default spread
➢ Term Spread
❑ 8.2 Theories of Term Structure
➢ Expectations Hypothesis
➢ Liquidity Preference Theory
• Liquidity risk and reinvestment risk
➢ Interpretation of Yield Curve Theories
• Interpreting the term structure
• Inverted yield curves
❑ 8.3 Interest Rate Risk
➢ Bond Price Sensitivity to Changes in Interest Rates
• Maturity, coupon and YTM impacts on interest rate risk
Lecture Outline
8.1 Bond Risk: Default
Spread and Term
Spread
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4Default Spread
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5❑ The main reason that two bonds with the same term, coupon and face value may have
different yields to maturity is due to the relative likelihood of issuer default
❑ Default premiums and default spreads on risky bonds are often referred to as the risk
structure of interest rates
❑ There is a difference between the yield based on expected cash flows (which prices in the risk
of default) and the yield based on promised cash flows (with no default risk)
➢ The difference between the (default risk weighted) expected yield and the promised yield
(i.e. the yield of an otherwise identical riskless government bond) is the default risk
premium
• The yield to maturity is higher for bonds with higher default risk
• That is, bonds with greater default risk have a higher default risk premium
Default Risk
6❑ Determination of the default risk premium on a bond is mainly driven by its credit rating
❑ Rating companies
➢ Moody’s Investor Service, Standard & Poor’s, Fitch
❑ Rating Categories
➢ Highest rating is AAA or Aaa
➢ Investment grade bonds (BBB- or Baa3 and above)
➢ Speculative grade/junk bonds (below BBB- or Baa3)
❑ Determinants of bond safety and credit rating
➢ Coverage ratios; Leverage ratios; Liquidity ratios; Profitability ratios; Cash flow-to-debt ratios
Bond Credit Ratings
Further Reading 8AR1: “S&P Credit Rating Essentials”
7❑ The default spread is often derived by showing the difference between yields on AAA bonds
and yields on either low-level investment grade bonds (BBB-) or speculative grade high
yielding bonds
Default Premiums and Spreads
Further Reading
8AR2: “S&P Credit
Markets Update Asia
Pacific - Q3 2021”
8Term Spread
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9❑ How yields change based on maturity is the term structure of interest rates (or yield curve)
Term Premiums and Spreads
❑ The term spread is often a
proxy for the yield curve
➢ The term spread is the yield
on 10-year bonds less the
yield on 90-day bills over
time
➢ We have seen the yield
curve is upward sloping –
which means a positive term
spread
➢ Allows the yield curve to be
graphed over time (rather
than at one point in time) and
inversions easily identified
Further Reading 8AR3: “Predicting Recessions Using the Term Spread”
8.2 Theories of Term
Structure
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11
Expectations Hypothesis
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12
❑ We showed previously that under certainty, we can exactly replicate cash flows of long-term
bonds by reinvesting cash flows from short-term bonds:
1 + 2
2 = (1 + 1)(1 + 12)
➢ However, in reality 12 is not known with certainty at = 0
➢ The final realized cash flow of the 2-year zero (the LHS) is known at time 0. But the final
realized cash flow of the 1 year rollover strategy (the RHS) is not known with certainty at
time 0
❑ However it may be that the market has an expectation that the (certain) cash flow from the
2-year zero will equal the expected cash flow from the 1-year rollover. That is:
1 + 2
2 = (1 + 1)(1 + E(12))
❑ This is known as the Expectations Hypothesis
Expectations Hypothesis
13
❑ Since forward rates are inferred from the term structure, a common way to denote the
Expectations Hypothesis (EH) is:
= ()
❑ We know from the earlier generalised forward rate equation that:
1 +
= (1 + )
× (1 + )
(−)
Therefore under EH: 1 +
= (1 + )
× (1 + ())
(−)
❑ However, the yield curve is typically upward sloping
➢ If Expectations Hypothesis were to hold, this means the market expects actual interest rates
in the future to be higher than today almost all the time
➢ Perhaps there is some other explanation for the typically upward sloping yield curve?
Expectations Hypothesis
14
Liquidity Preference Theory
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15
❑ Example 8A1: You are an investor with a 1-year horizon to save $1000. You have a choice
of buying either:
A) a 1-year zero coupon bond and hold it to maturity; or
B) a 2-year zero coupon bond and selling after 1 year
a) Price a 1-year and a 2-year $1,000 zero (use the pure yield curve from 7B3)
b) How many 2-year bonds would you buy (include fractions) to outlay the same amount as
strategy A
c) Show that strategy A is riskless (excluding default risk) while strategy B contains risk.
What type of risk is this called?
➢ 1 year zero: 1 =
1000
(1.05)
= $952.38 2 year zero: 2 =
1000
(1.06)2
= $890.00
➢ Number of 2-year zeros bought to outlay the same $’s as the 1-year strategy:
952.38
890.00
= 1.0701 2-year zeros for every 1-year zero
Example: Liquidity Risk
16
➢ If the 1-year interest rate in 1 year 12 = 12 = 7.01% as predicted by the yield curve, the two
strategies are equivalent:
Strategy A = $1000 = Strategy B =
1,000
1.0701
× 1.0701
➢ But what if 1-year interest rate in 1 year’s time is more (or less) than 7%?
If actual rate is 6%:
1,000
1.06
× 1.0701 = $1009 > $1000
If actual rate is 8%:
1,000
1.08
x 1.0701 = $991 < $1000
➢ The actual realised return on Strategy A is certain but the actual return on the 2-year bond is
uncertain! Therefore Strategy B is uncertain - it is exposed to interest rate risk (namely
liquidity or price risk)
❑ When an investor’s time horizon is shorter than the term of the bond they are holding, they
are exposed to interest rate risk known as liquidity risk
❑ When an investor’s time horizon is longer than the term of the bond they are holding, they
are exposed to interest rate risk known as reinvestment risk
Example: Liquidity Risk
17
❑ Strategy B has uncertainty and is more risky. Strategy A is certain and therefore less risky
❑ The Liquidity preference theory refers to the case where most investors in the market have
shorter horizons and therefore prefer short-term investments to avoid the liquidity risk of
holding long-term bonds
➢ Therefore, investors with shorter term horizons will require a risk premium – called the
liquidity risk premium - to hold a longer-term bond (eg the 2 year bond). That is:
> ()
OR = +
❑ This LRP or liquidity risk premium compensates short-term investors for the uncertainty about
future interest rates (interest rate risk)
Liquidity Preference Theory
Further Reading 8AR4: “The Bond Market Term Premium”
18
Interpretation of Yield Curve
Theories
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19
Summary of Theories of Term Structure
Expectations Hypothesis Liquidity Preference Theory
❑ = () ❑ = + (LRP > 0)
❑ Forward rates are market expectations of
future spot rates
❑ Forward rates are market expectations of
future spot rates plus a liquidity premium
❑ The liquidity premium reflects the fact that
long term bonds are more risky
❑ Investors with long-term horizons offset
those with short-term horizons – therefore
no net liquidity premium in the market (the
liquidity risk premium is zero)
❑ Investors with short-term horizons dominate
the market requiring a liquidity premium to
avert liquidity risk
❑ Does not explain the typically upwards
sloping yield curve
❑ Explains the typically upward sloping yield
curve
❑ The yield curve has an upward bias built
into the long-term rates because of the
liquidity premium
20
❑ If there is a liquidity
premium the yield
curve is biased
upward, particularly
if the premium
increases with
maturity
Liquidity Premium Impact on the Yield Curve
21
❑ The yield curve reflects expectations of future interest rates
➢ But the forecasts of future rates are clouded by other factors, such as liquidity premiums
❑ Therefore, an upward sloping curve could indicate:
➢ Rates are expected to rise
and/or
➢ Investors require large liquidity premiums to hold long term bonds
❑ The yield curve is a good predictor of the business cycle
➢ Long term rates tend to rise in anticipation of economic expansion
➢ An inverted yield curve may indicate that interest rates are expected to fall and therefore
may signal a recession
• Read the lecture topic 7 case study carefully to understand exactly why
Interpreting the Term Structure
Mini Case Study Lecture Topic 7: “Dark Clouds of Yield Inversion”
Further Reading 8AR5: “Analysing the Effect of an Inverted Yield Curve on Investors”
8.3 Interest Rate Risk
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23
Bond Price Sensitivity to
Changes in Interest Rates
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24
❑ Interest rate sensitivity
① Bond prices and yields are inversely related YTM  P 
② Bond prices move up in response to an interest rate fall more than they move down in
response to the same size interest rate increase (convexity)
③ Long-term bond prices are more interest rate sensitive than short-term bond prices
T  ΔP 
④ As maturity increases, price sensitivity to interest rate changes increases at a decreasing
rate
⑤ High coupon bond prices are less interest rate sensitive than low coupon bond prices
C  ΔP 
⑥ Higher YTM bond prices are less interest rate sensitive than lower YTM bond prices
YTM  ΔP 
Bond Price Sensitivity to ∆ Interest Rates
25
Bond Price Sensitivity to ∆ Interest Rates
①Downward sloping left to
right
②Convex
③ B more sensitive than A
④ Slide 28
⑤C more sensitive than B
⑥D more sensitive than C
26
❑ Example 8A2: Rank these bonds in order of interest rate risk:
a) A 10 year bond with a 6% coupon and 8% yield to maturity
b) A 3 year bond with a 6% coupon and a 8% yield to maturity
c) A 10 year bond with a 4% coupon and a 5% yield to maturity
d) A 10 year bond with a 4% coupon and 8% yield to maturity
➢ c) 10 year 4% coupon YTM = 5% bond price is: more interest rate sensitive than
➢ d) 10 year 4% coupon YTM = 8% bond price (higher YTM) is: more interest rate sensitive
than
➢ a) 10 year 6% coupon YTM = 8% bond price (higher coupon) is: more interest rate sensitive
than
➢ b) 3 year 6% coupon YTM = 8% bond price (shorter maturity)
Example: Interest Rate Risk
27
❑ Example 8A3: Consider the following $1000 6% annual coupon bonds with maturities of 1
year, 10 years and 20 years, all trading at a yield to maturity of 6%.
a) Price each of the bonds at YTM = 6%. Price each bond assuming YTM changes to 7%.
b) What can you say about the relationship between maturity and interest rate risk (i.e. the
bond price sensitivity to changes in interest rates)? As the interest rate increases, does
the interest rate sensitivity for bonds of different tenor increase/decrease at an increasing
or decreasing rate?
➢ Long-term bonds prices are more interest rate sensitive than short-term bond prices
T  ΔP . That is, longer term bonds have more interest rate risk (principle 3, slide 25)
➢ As maturity increases, price sensitivity to interest rate changes increases at a decreasing
rate (principle 4, slide 25)
Example: Maturity Impact on Interest Rate Risk
YTM 1 year (CR = 6%) 10 year (CR = 6%) 20 year (CR = 6%)
6% $1,000.00 $1,000.00 $1,000.00
7% $990.65 $929.76 $894.06
Change in Price -0.9% -7.0% -10.6%
28
❑ Example 8A4: Now price $1000 1 year, 10-year and 20-year zero-coupon bonds at a 6%
and 7% yield to maturity. Are they more or less sensitive to a change in interest rates than
the coupon bonds? Can you explain why?
➢ Zero coupon bond prices are more interest rate sensitive than coupon bonds
➢ Why? More of the return on zeros is “back-ended” so higher interest rates erode present
values on zeros proportionately more
➢ If the return is back-ended, does this indicate zero-coupon bonds have longer effective
maturity than coupon bonds?
➢ We will discuss the concept of duration next lecture
Example: Coupon Impact on Interest Rate Risk
YTM 1 year (Zero) 10 year (Zero) 20 year (Zero)
6% $943.40 $558.39 $311.80
7% $934.58 $508.35 $258.42
Change in Price -0.9% -9.0% -17.1
29
❑ BKM Chapter 16
❑ 8.4 Measuring Interest Rate Risk: Duration
❑ 8.5 Managing Interest Rate Risk: Immunisation
Next Lecture
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270M CIS 3207 Programs CSCI 2132 CS270 CSE-381 COMP3230A CIS 212 CSI 333 FINM 326 FINM-36702 Introduction to Software Analysis EEE102 GNG1106 ECS 60 CS 161 CSCI851 CSE1222 EDPS0249 TCHR5003 ANTA02 BBUS1SBY SOSC 1140 MDIA2091 FINS3641 INSY 661 ITEC3040 CPSC 481 computer DEE3519 COMP304 CSE536 Microtheory OFFICE 280 CIVE50003 COSC2673 ECS784U CMPSC/MATH 455 AMA 505 CSE 158 BUS1040 FIT3179 FIT5202 Math 104A Math 111A MBUS107 COMPSCI 761 COMP3710 COMPSCI 316 COMP559 COMP557 CP1 1902 COMP 310 ECED 3403 COMPX523 AGCM 3103 EMET 7001 ENGR3821 CISC 124 IPC comp20005 CS 323 CS105 STAT 326 MATH4007 CS4551 COMP1000 ACIT 4640 ENVX3002 CS2034 FC712 algorithms CSCI 4155 CSI4142 POL 850 MATH20811 2B03 CSCI803 Economies COMP 2016 Architect TRADE INFO1110 FIT1043 networks Robotics ICS 31 ECEC-301 FIT2099 CS 3640 CSE1010 Computational COMP1001 CSE 231 CS10 FIT5166 CSE 400 CSCI561 Math 350 CO-496 ENGG6400 CSCI 1133 COMP3055 MATH 819 CSE 6363 CIS 555 CS4420 COMP0037 ECS 140A Spanning Tree Protocol network 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EBU5376 STAT6118 ECON 20110 COMU7000 average acceleration BUSI4412 BFF3121 BIT233 ECMM462 ETC3430 DESN2348 INTE2584 ENME502 MA3XJ INTE2627 MA2815 MA3AM L0101 Math 3MB3 CPCCBC4004 EEEM061 0502E220 APPLIED 6SSMN310 ENV 3310 PM2PCOL3 CONS6201 MATH502 FIT5003 EEEN313 MATH3001 MECH3460 ECON30009 COSC1252 Assembler COMP 3023 ENGG7601 COMP20005 COMP SCI 1103 ICSI 402 INF2C CS320 COMPSCI7064 ECE 3220 PROGRAMMAMING MIPS Computer Science 2413 AM3611 MSOA TE2101 FFT CMPSC-121 CSE 109 COMP SCI 7064 HCI CMP-5013A CCN2042 COMP3400 EEEN30052 CO4215 EE101 KIT107 CSE 421 CSE521 MCD4720 EECS 280 MIT 403 GRPC VG101 APiE EECE 1080C FIT3143 CSE 325 ISOM 3029 CS3307 State SIT255 Engineering MQF Modelling ACX3150 SIA322 ACCT90004 COMU7201 MGMT90018 WRIT1000 COMP3702 BISM1201 COMP7505 IDEA9106 CSE3SMT BFC5926 BFC5280 BFC5935 QBUS3330 ECON3700 IFN521 ECON2121 COSC2626 MA413 DDES9903 ENGG1300 COMP9337 CO2201 ACCT2002 ICM289 MMM054 MSBA-204 DIGM707 CSC2250 Econ312 EVAT python CAN404 LUBS2840 CSCI-GA.2250 COMSM0093 COMP281 AFM 333 IBUS1001 CP2501 Math 105A VE320 TEC101 SOCY2166 COM240 EFIM20033 ENGI4467 EFIM20034 EC 979 BE432 7EJ502 CSSE6400 COMP90025 Econ 312 ENG2031 ECO0006 EXERSCI 207 CAN209 ECO 137W FT312 ECO 2199 MATH32052 BDW4213 APS 360 SAS 6 CHME0017 PO92Q Economics 01AV MATH39512 MATH 5486 COMP8620 ISE102 ECON235 ECON 206 FI 345 PHAY0020 MATH0099 MATH0085 EFB106 PHY 134 MAE115 circuits Chemistry CSE12 MATH-UA 233 PHAS0038 COMP2140 CONMGNT 7049 COMP151101 ECON 485 ACX5903 ENGN6536 ARCH1162 Law PHAS0042 CCF1C03 MATH 351 Phys 139 AG942 MGT001371 APH106 ECON1051 MA4AM MA3Z7 NUMBER THEORY REST0006 SESS0026 STAT0023 TABL 2741 CMP4008Y EFIM10015 Acct 560 PSYC735 GEOG 5 ACCT3104 CSEC5616 PHIL1012 ITLS6111 ECON8022 5CCM226A compX123 MA4XJ R001 EWB LLP120 COMP643 ENG 103 ECON 23950 ACCT 207 FINC 616 Bio99 Math 137A Psychology BCPM0054 MATH20212 EC3450 CENV6141 PSY 205 CE335 CS365 Calculus ACSD INFT6800 MCIT INFO6030 PHYS 111LV H63EEJ ECON2151 ENGI 2181 MTRX1701 PLIT08009 CS240 Math 270 CMP7000A CMP-7038B Math 263 LAW3016 CS371 STAT 334 FIN0008 ACC3004 STAT 371 ​COMP1921 MTH201 ACCT2000 UN3412 CIV6000 MATHS 3021 MATE7014 ECON1 CIBC6035 LAW121G COMM 321 CSC8204R TTM2033 CSCI 2600 MAE 115 LNG302 CENV6175W1 ECON30019 CPS2015 BST969 DDES9015 AMATH242 CS 1151 N1592 ACCT600 EG503G SMM519 EC318 Ecstatistics EEE 471 MARK2051 Topology COM00037H ACCT3000 MTHM501 EG501V COM00055H BUSN9085 GGR273 MATH 2B ITD102 MA214 ECON0060 ECE3114 ECMM447 Mechanics CCT361H5S BSAN3210 WELF2001 Fin3001 RESP5001 STA 137 Linguistics 101 CSCI368 CMPSC497 LUBS3655 OMBA 5355 BUSM2562 ​BFF5902 CYBR7001 CULT2005 ECON 7720 MTRX3760 FSE598 ​LUE1001 MGSC 7100 CS3240 COSC7500 ACCT3091 INFOGR TRAN 2080 DESIGN CIVE215001 CIVE2360 CPSC 8430 FNCE2003 Stoichiometry INFS6023 Quantitative Method Math 21D ECE5179 Design ​ECON8022 Chemical DMS 213 ADM1370 BFF5902 BFF2701 AGRI10051 AMN400 ECON7322 ​FINC5090 ​MATH1052 ​SciComp Elec4622 ​FINS5516 ENG 101 MMM143 CS431CA2 BSAN2201 EBSC7070 ES9v9-10 PLIN0009 DATA 8 MATH 101 BCPM0097 ​MAT 237 BISM2201 EEE8099 Physics 11 ECE2238B Biostat 234 ​ECON30009 INF10025 CHEM252 CCMA4008 CIVE 3007 STAT0031 ELEC ENG 3108 MGMT90140 CHEM 252 STAT3016 MMM095 ​MMGT6016 Legal STAT 311 ELEC9764 COMP 370 TCS 3053 EECS 1021 FIT1051 C/C++ EE3C COMP2003J COM398 COMP51915 Simulator FIN42330 8PRO102 PGD ELEC5160 XJCO1921 COMP5123M INFS 2042 CHE1148H ECA5313 IN3329 Microsoft Modeling B15 2TT COMP3687 COMP2432 DMS2030 KAN4TSF CFKG CSE 5322 CSE 445 STATS 3DA3 CHC5223 COMP1048 EL3105 EECS 3221 CS231 ME 4434 ITP4510 ITP4501 CSCI 5708 Controller EECE 5699 CPEN 231L CPN programs CSC8016 MATU9D2 ENGN4213 2. SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum
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