1. Derivatives: A Primer on Bonds
First Part: Fixed Income Securities
Bond Prices and Yields
Term Structure of Interest Rates
Second Part: TSOIR
Interest Rate Risk & Bond Portfolio Management

2. Bond Prices and Yields Time value of money and bond pricing
Time to maturity and risk
Yield to maturity
vs. yield to call
vs. realized compound yield
Determinants of YTM
risk, maturity, holding period, etc.

3. Bond Pricing Equation:
P = PV(annuity) + PV(final payment) =
Example: Ct = $40; Par = $1,000; disc. rate = 4%; T=60

4. Prices vs. Yields P   yield  convexity intuition
BKM6 Fig. 14.3; ; BKM4 Fig. 14.6
intuition: yield   P   price impact 

5. Measuring Rates of Return on Bonds
Standard measure: YTM
Problems
callable bonds: YTM vs. yield to call
default risk: YTM vs. yield to expected default
reinvestment rate of coupons
YTM vs. realized compound yield
Determinants of the YTM
risk, maturity, holding period, etc.

6. Measuring Rates of Return on Bonds 2
Yield To Maturity
definition
discount rate such that NPV=0
interpretation
(geometric) average return to maturity
Example: Ct = $40; Par = $1,000; T=60; sells at par

7. Measuring Rates of Return on Bonds 3
Yield To Call
definition
discount rate s.t. NPV=0, with TC = earliest call date
deep discount bonds vs. premium bonds
BKM6 Fig. 14.4; ; BKM4 Fig. 14.7
Example: Ct = $40, semi; Par = $900; T=60; P = $1,025; callable in 10 years (TC=20), call price = $1,000

8. Measuring Rates of Return on Bonds 4
Yield To Default
definition
discount rate s.t. NPV=0, with TD= expected default date
default premium and business cycle
economic difficulties and “flight to quality”
Example: Ct = $50, semi; Par = $1,000; T=10; P = $200; expected to default in 2 years (TC=4), recover $150

9. Measuring Rates of Return on Bonds 5
Coupon reinvestment rate
YTM assumption: average
problem: not often true
“solution”: realized compound yield
forecast future reinvestment rates
compute future value (BKM6 Fig.14.5; BKM4 Fig.14.9)
compute the yield (rcy) such that NPV = 0
practical?
need to forecast reinvestment rates

10. Bond Prices over Time Discount bonds vs. premium bonds
coupon rate < market interest rates
 built-in capital gain (discount bond)
coupon rate > market interest rates
 built-in capital loss (premium bond)
Behavior of prices over time
BKM6 Fig. 14.6; BKM4 Fig
Tax treatment
capital gains vs. interest income

11. Discount Bonds OID vs. par bonds Zeroes
original issue discount (OID) bonds
less common
coupon need not be 0
par bonds
most common
Zeroes
what? mostly Treasury strips
how? “certificates of accrual”, “growth receipts”, ...
annual price increase = 1-year disc. factor (BKM6 Fig. 14.7; BKM4 Fig )

12. OID tax treatment -- Discount Bonds 2
Idea for zeroes
built-in appreciation = implicit interest schedule
tax the schedule as interest, yearly
tax the remaining price change as capital gain or loss
Other OID bonds
same idea
taxable interest = coupon + computed schedule

13. OID tax treatment -- Discount Bonds 3
Example
30-year zero; issued at $57.31; Par = $1,000
compute YTM:
1st year taxable interest

14. OID tax treatment -- Discount Bonds 4
Example (continued)
interests on 30-year bonds fall to 9.9%
capital gain
tax treatment: taxable interest = $5.73; capital gain

15. Term Structure of Interest Rates
Basic question
link between YTM and maturity
Bootstrapping short rates from strips
forward rates and expected future short rates
Recovering short rates from coupon bonds
Interpreting the term structure
does the term structure contain information?
certainty vs. uncertainty

16. “Term”inology Term structure = yield curve (BKM6 Fig. 15.1)
= plot of the YTM as a function of bond maturity
= plot of the spot rate by time-to-maturity
Short rate vs. spot rate
1-period rate vs. multi-period yield
spot rate = current rate appropriate to discount a cash-flow of a given maturity
BKM6 Figure 15.3; BKM4 Figure 14.3

17. Extracting Info re:Short Interest Rates
From zeroes
non-linear regression analysis
bootstrapping
From coupon bonds
system of equations
regression analysis (no measurement errors)
Certainty vs. uncertainty
forward rate vs. expected future (spot) short rate

18. Bootstrapping Fwd Rates from Zeroes
Forward rate
“break-even rate” – BKM Fig. 15.4
equates the payoffs of roll-over and LT strategies
Uncertainty
no guarantee that forward = expected future spot
General formula
f1 = YTM1 and

19. Bootstrapping Fwd from Zeroes 2
Data
BKM Table 15.2 & Fig. 15.1
4 bonds, all zeroes (reimbursable at par of $1,000)
T Price YTM
1 $ %
2 $ %
3 $ %
4 $ %

20. Bootstrapping Fwd Rates from Zeroes 3
Forward interest rate for year 1
Forward interest rate for year 2

21. Bootstrapping Fwd Rates from Zeroes 4
Short rate for years 3 and 4
keep applying the method
you find f3 = 11% = f4
General Formula
f1 = YTM1

22. Yield, Maturity and Period Return
Data
2 bonds, both zeroes (reimbursable at par of $1,000)
T Price YTM
1 $ %
2 $ %
Question
investor has 1-period horizon; no uncertainty
does bond 2 (higher YTM) dominate bond 1?

23. Yield, Maturity and Period Return 2
Answer: Nope
Bond 1 HPR:
Bond 2 HPR:
f2 = 10%
price in 1 year = Par/(1+ f2) = $
capital gain at year-1 end =

24. Fwd Rate & Expected Future Short Rate
Interpreting the term structure
Short perspective
liquidity preference theory (investors)
liquidity premium theory (issuer)
Expectations hypothesis
Long perspective
Market Segmentation vs. Preferred Habitat
Examples

25. Fwd Rate & Exp. Future Short Rate 2
Short perspective
liquidity preference theory (“short” investors)
investors need to be induced to buy LT securities
example: 1-year zero at 8% vs. 2-year zero at 8.995%
liquidity premium theory (issuer)
issuers prefer to lock in interest rates
f2  E[r2]
f2 = E[r2] + risk premium

26. Fwd Rate & Exp. Future Short Rate 3
Long perspective
“long investors” wish to lock in rates
roll over a 1-year zero at 8%
or lock in via a 2-year zero at 8.995%
E[r2]  f2
f2 = E[r2] - risk “premium”

27. Fwd Rate & Exp. Future Short Rate 4
Expectation Hypothesis
risk premium = 0 and E[r2] = f2
idea: “arbitrage”
Market segmentation theory
idea: clienteles
ST and LT bonds are not substitutes
reasonable?
Preferred Habitat Theory
investors do prefer some maturities
temptations exist

28. Fwd Rate & Exp. Future Short Rate 5
In practice
liquidity preference + preferred habitat
hypotheses have the edge
Example
BKM Fig. 15.5

29. Fwd Rate & Exp. Future Short Rate 6
Example 2
short term rates: r1 = r2 = r3 = 10%
liquidity premium = constant 1% per year
YTM

30. Measurement: Zeroes vs. Coupon Bonds
ideal
lack of data may exist (need zeroes for all maturities)
Coupon Bonds
plentiful
coupons and their reinvestment
low coupon rate vs. high coupon rate
short term rates -> they may have different YTM

31. Short Rates, Coupons and YTM
Example
short rates are 8% & 10% for years 1 & 2; certainty
2-year bonds; Par = $1,000; coupon = 3% or 12%
Bond 1:
Bond 2:

32. Measurements with Coupon Bonds 2
Example
2-year bonds; Par = $1,000; coupon = 3% or 12%
Prices: $ (coupon = 3%); $1, (coupon = 12%)
Year-1 and Year-2 short rates
$ = d1 x d2 x 1,030
$ 1, = d1 x d2 x 1,120
Solve the system: d2 = , d1 =
Conclude ...

33. Measurements with Coupon Bonds 3
Example (continued)

34. Measurements with Coupon Bonds 4
Practical problems
pricing errors
taxes
are investors homogenous?
investors can sell bonds prior to maturity
bonds can be called, put or converted
prices quotes can be stale
market liquidity
Estimation
statistical approach

35. Rising yield curves Causes Interpretative assumptions
either short rates are expected to climb: E[rn]  E[rn-1]
or the liquidity premium is positive
Fig. 15.5a
Interpretative assumptions
estimate the liquidity premium
assume the liquidity premium is constant
empirical evidence
liquidity premium is not constant; past -> future?!

36. Inverted yield curve Easy interpretation Example
if there is a liquidity premium
then inversion  expectations of falling short rates
why would interest rates fall?
inflation vs. real rates
inverted curve  recession?
Example
current yield curve: The Economist

37. Arbitrage Strategies

38. Arbitrage Strategies

39. Fixed Income Portfolio Management
In general
bonds are securities just like other
-> use the CAPM
Bond Index Funds
Immunization
net worth immunization
contingent immunization

40. Bond Index Funds Idea US indices composition
Solomon Bros. Broad Investment Grade (BIG)
Lehman Bros. Aggregate
Merrill Lynch Domestic Master
composition
government, corporate, mortgage, Yankee
bond maturities: more than 1 year
Canada: ScotiaMcLeod (esp. Universe Index)

41. Bond Index Funds 2 Problems lots of securities in each index
portfolio rebalancing
market liquidity
bonds are dropped (maturities, calls, defaults, …)

42. Bond Index Funds 3 Solution: “cellular approach” idea effectiveness
classify by maturity/risk/category/…
compute percentages in each cell
match portfolio weights
effectiveness
average absolute tracking error = 2 to 16 b.p. / month

43. Special risks for bond portfolios
cash-flow risk
call, default, sinking funds, early repayments,…
solution: select high quality bonds
interest rate risk
bond prices are sensitive to YTM
solution
measure interest rate risk
immunize

44. Interest Rate Risk Equation: Yield sensitivity of bond Prices:
P = PV(annuity) + PV(final payment) =
Yield sensitivity of bond Prices:
P   yield 
Measure?

45. Interest Rate Risk 2 Determinants of a bond’s yield sensitivity
time to maturity
maturity   sensitivity  (concave function)
coupon rate
coupon   sensitivity 
discount bond vs. premium bond
zeroes have the highest sensitivity
intuition: coupon bonds = average of zeroes
YTM
initial YTM   sensitivity 

46. Duration Idea Measure maturity   sensitivity 
 to measure a bond’s yield sensitivity,
measure its “effective maturity”
Measure
Macaulay duration:

47. Duration 2 Duration = effective measure of elasticity Proof
Modified duration with

48. Duration 4 Interpretation 1 Interpretation 2
= average time until bond payment
Interpretation 2
% price change of coupon bond of a given duration
= % price change of zero with maturity = to duration

49. Duration 4 Example (BKM Table 15.3)
suppose YTM changes by 1 basis point (0.01%)
zero coupon bond with years to maturity
old price
new price

50. Duration 5 Example: BKM4 Table 15.3
suppose YTM changes by 1 basis point (0.01%)
coupon bond
either compare the bond’s price with YTM = 5.01% relative to the bond’s price with YTM = 5%
or simply compute the price change from the duration

51. Duration 6 Properties of duration (other things constant)
zero coupon bond: duration = maturity
time to maturity
maturity   duration 
exception: deep discount bonds
coupon rate
coupon   duration 
YTM
YTM   duration 
exception: zeroes (unchanged)

52. Duration 7 Properties of duration duration of perpetuity =
less than infinity!
coupon bonds (“annuities + zero”)
see book
simplifies if par bond

53. Duration 8 Importance simple measure
essential to implement portfolio immunization
measures interest rate sensitivity effectively

54. Possible Caveats to Duration
1. Assumptions on term structure
Macaulay duration uses YTM
only valid for level changes in flat term structure
Fisher-Weil duration measure

55. Possible Caveats to Duration 2
problems with the Fisher-Weil duration
assumes a parallel shift in term structure
need forecast of future interest rates
bottom line: same problem as realized compound yield
Cox-Ingersoll-Ross duration
bottom line: let’s keep Macaulay

56. Possible Caveats to Duration 3
2. Convexity
Macaulay duration
first-order approximation:
small changes vs. large changes
duration = point estimate
for larger changes, an “arc” estimate is needed
solution: add convexity

57. Possible Caveats to Duration 4
Convexity (continued)
second-order approximation:

58. Possible Caveats to Duration 5
Convexity: numerical example
P = Par = 1,000; T = 30 years; 8% annual coupon
computations give D*=11.26 years; convexity = years
suppose YTM = 8% -> YTM = 10%

59. Bottom Line on Duration
Very useful
But take it with a grain of salt for large changes

60. Immunization Why? How? obligation to meet promises (pension funds)
protect future value of portfolio
ratios, regulation, solvency (banks)
protect current net worth of institution
How?
measure interest rate risk: duration
match duration of elements to be immunized

61. Immunization What? Who? net worth immunization
match duration of assets and liabilities
target date immunization
match inflows and outflows
immunize the net flows
Who?
insurance companies, pension funds
banks

62. Net Worth Immunization
Gap management
assets vs. liabilities
long term (mortgages, loans, …) vs. short term (deposits, …)
match duration of assets and liabilities
decrease duration of assets (ex.: ARM)
increase duration of liabilities (ex.: term deposits)
condition for success
portfolio duration = 0 (assets = liabilities)

63. Target Date Immunization
Idea
Example: suppose interest rates fall
good for the pension fund
price risk
existing (fixed rate) assets increase in value
bad for the pension fund
reinvestment risk
PV of future liabilities increases
so more must be invested now

64. Target Date Immunization 2
Solution
match duration of portfolio and fund’s horizon
single bond
bond portfolio
duration of portfolio
= weighted average of components’ duration
condition: assets have equal yields

65. Target Date Immunization 3

66. Target Date Immunization 4

67. Target Date Immunization 5

68. Dangers with Immunization
1. Portfolio rebalancing is needed
Time passes  duration changes
bonds mature, sinking funds, …
YTM changes  duration changes
example: BKM4 Table 15.7
duration YTM % % %

69. Dangers with Immunization 2
2. Duration = nominal concept
immunization only for nominal liabilities
counter example
children’s tuition
why?
solution
do not immunize
buy assets

70. An Alternative? Cash-Flow Dedication
Buy zeroes
to match all liabilities
Problems
difficult to get underpriced zeroes
zeroes not available for all maturities
ex.: perpetuity

71. Contingent Immunization
Idea
try to beat the market
while limiting the downside risk
Procedure (BKM6 Fig ; BKM4 Fig. 15.6)
compute the PV of the obligation at current rates
assess available funds
“play” the difference
immunize if trigger point is hit