Investments Analysis
• Last 2 Weeks: Fixed Income Securities
– Bond Prices and Yields
– Term Structure of Interest Rates
• Last & this Weeks:
– Performance benchmarking
– Bond-specific Risks
• Interest rate risk
• Credit Risk
Interpreting the Yield Curve
• TSOIR and interest rate uncertainty
• link between
» interest rates
» liquidity premia
• Various cases
• possible cases
• interpretation
“TSOIR” & Interest Rate Uncertainty
• Interpreting the term structure
– Short perspective
– liquidity preference theory (investors)
– liquidity premium theory (issuer)
– Long perspective
– Expectations hypothesis
– Market Segmentation vs . Preferred Habitat
– Examples
“TSOIR” & Interest Rate Uncertainty 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
• f
2
≥
E[ r
2
]
• f
2
=
E[ r
2
] + risk premium
“TSOIR” & Interest Rate Uncertainty 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[ r
2
]
≥ f
2
• f
2
=
E[ r
2
] - risk “ premium”
• Expectation Hypothesis
• E[ r
2
]
= f
2
(risk premium = 0)
• idea: “ arbitrage”
1

“TSOIR” & Interest Rate Uncertainty 4
• Market segmentation theory
• idea: clienteles
(ST and LT bonds are not substitutes)
• (un)reasonable?
• Preferred Habitat Theory
• investors prefer some maturities but can be tempted
• In practice
• liquidity preference + preferred habitat
» these hypotheses are thought more reasonable
“TSOIR” & Interest Rate Uncertainty 5
• Example 1
• short term rates: r
1
= r
2
= r
3
=
10%
• liquidity premium = constant 1% per year
• YTM y
1
= r
1
=
10 % y
2
=
( 1
+ r
1
)( 1
+ f
2
)
−
1
=
( 1
+
10 %)( 1
+
10 %
+
1 %)
−
1
=
10 .
5 % y
3
=
3 ( 1
+ r
1
)( 1
+ f
2
)( 1
+ f
3
)
−
1
=
3 ( 1
+
10 %)( 1
+
11 %)( 1
+
11 %)
−
1
=
10 .
67 %
“TSOIR” & Interest Rate Uncertainty 6
• Example 2: “Quick & dirty” forward rates
• Zero-Coupon Rates Bond Maturity (1yr) Fwd Rate
•
•
•
•
• y y y y y
1
2
3
4
5
= 12.00%
= 11.75%
= 11.25%
= 10.00%
= 9.25%
1
2
3
4
5 f
1 f
2 f
4 f
5 f
3
= y
1
= 12%
≈
11.5%
≈
10.25% *
≈
6.25%
*
≈
6.25% *
*:
If computed exactly, f
3
=10.26%; f
4
=6.33%; f
5
=6.30% (we ’ll show this below)
“TSOIR” & Interest Rate Uncertainty 7
• Example 2: “Quick” expected future short rates
• Period (1yr) Fwd Rate Expected short rate
*
•
•
•
•
•
1
2
3
4
5 f
3 f
4 f
1 f
2 f
5
= y
1
= 12%
≈
11.5%
≈
10.25%
≈
6.25%
≈
6.25%
*: Assumes a constant 0.5% per year liquidity premium
N-A
E(y
1
’) = r
2
≈
11%
E(y
1
’’) = r
3
≈
9.75%
E(y
1
’’’) = r
4
≈
5.75%
E(y
1
’’’’) = r
5
≈
5.75%
“TSOIR” & Interest Rate Uncertainty 8
• Example 2: “ Quick” expected future bond prices in 2 years
• Period Exp. Short Rate
• 1 N-A
•
•
•
•
2
3
4
5
Fwd Rate
N-A
N-A N-A
E(y
E(y
1
1
’’) = r
3
’’’) = r
≈ 9.75%
4
≈
5.75% f
1
’’ ≈ 9.75% * f
2
’’
≈
6.25% * *
E(y
1
’’’’) = r
5
≈
5.75% f
3
’’
≈
6.25% * *
Zero-Coupon Rates
N-A
N-A y
1
’’ ≈ 9.75 % y
2
’’
≈
8 % y
3
’’
≈
7.42 %
*:
In two years, this will be the current year
−> no LP need be added/subtracted
**: Assumes a constant 0.5% per year liquidity premium
“TSOIR” & Interest Rate Uncertainty 9
• Example 2: “Formal” forward rates
• Zero-Coupon Rates Bond Maturity
•
•
•
•
• y
1
= 12.00% y
2
= 11.75% y
3
= 11.25% y
4
= 10.00% y
5
= 9.25%
1
2
3
4
5
*:
If computed quickly, f
3
=10.25%; f
4
=6.25%; f
5
=6.25%
(1yr) Fwd Rate f
1
= y
1
= 12% f
2
= 11.5% f
3
= 10.26% * f
4
= 6.33% * f
5
= 6.30% *
2

“TSOIR” & Interest Rate Uncertainty 10
• Example 2: “Formal” forward rates
1yr Forward Rates
1yr from now : f
2
= [(1.1175) 2 / 1.12] - 1 = 0.115006
2yrs from now : f
3
= [(1.1125) 3 / (1.1175) 2 ] - 1 = 0.102567
3yrs from now : f
4
= [(1.1) 4 / (1.1125) 3 ] - 1 = 0.063336
4yrs from now : f
5
= [(1.0925) 5 / (1.1) 4 ] - 1 = 0.063008
“TSOIR” & Interest Rate Uncertainty 11
• Example 2: “Formal” expected future bond prices in 2 years
• Period Exp. Short Rate
• 1 N-A
• 2
• 3
•
•
4
5
E(y
N-A
1
’’) = r
3
= 9.76%
Fwd Rate
N-A
N-A f
1
’’ = 9.76% *
E(y
1
’’’) = r
4
= 5.83% f
2
’’ = 6.33% * *
E(y
1
’’’’) = r
5
= 5.80% f
3
’’ = 6.30% * *
Zero-Coupon Rates
N-A
N-A y y y
1
’’ = 9.76 %
2
’’ = 8.03 %
3
’’ = 7.36 %
*:
In two years, this will be the current year
−> no LP need be added/subtracted
**: Assumes a constant 0.5% per year liquidity premium
“TSOIR” & Interest Rate Uncertainty 8
• Example 2: “ Quick” expected future bond prices in 2 years
• Period Exp. Short Rate
• 1
• 2
• 3
•
•
4
5
N-A
Fwd Rate
N-A
E(y
E(y
1
1
N-A
’’) = r
3
’’’) = r
≈
9.75%
4
≈
5.75%
N-A f
1
’’
≈
9.75% * f
2
’’
≈
6.25% * *
E(y
1
’’’’) = r
5
≈ 5.75% f
3
’’ ≈ 6.25% * *
Zero-Coupon Rates
N-A
N-A y
1
’’
≈
9.75 % y
2
’’
≈
8.00 % y
3
’’ ≈ 7.42 %
*:
In two years, this will be the current year
−> no LP need be added/subtracted
**: Assumes a constant 0.5% per year liquidity premium
Interpreting the TSOIR
• Types of yield curve
• cases
» upward sloping (most common)
» downward sloping
» hump-shaped
• Fig 15.1
• Interpretative assumptions
• either short rates are the culprit
• or the liquidity premium is positive
Interpretation 2: Rising yield curves
• Causes
• either short rates are expected to climb: E[ r n
]
≥
E[ r n-1
]
• or the liquidity premium is positive
• Fig. 15.1B
• Interpretative assumptions
• estimate the liquidity premium
• assume the liquidity premium is constant
• empirical evidence
» liquidity premium is not constant; past
−> future?!
Interpretation 3: Inverted yield curve
• Easy interpretation
• 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
• 2000 yield curve
3

Interpretation 4: Hump-Shape curve
• Interpretation
• liquidity?
• Example
• Spring 00 yield curve
Arbitrage Strategies
Question:
The YTM on 1-year-maturity zero coupon bonds is 5%
The YTM on 2-year-maturity zero coupon bonds is 6%.
The YTM on 2-year-maturity coupon bonds with coupon rates of 12% (paid annually) is
5.8%.
What arbitrage opportunity exists for an investment banking firm? What is the arbitrage profit?
Arbitrage Strategies
Answer:
The price of the coupon bond, based on its YTM, is:
120 PA(5.8%, 2) + 1000 PF(5.8%, 2) = $1,113.99.
If the coupons were stripped and sold separately as zeros, then based on the YTM of zeros with maturities of one and two years, the coupon payments could be sold separately for
[120/1.05] + [1,120/1.06
2
] = $1,111.08.
The arbitrage strategy is to:
2 years buy zeros with face values of $120 and $1,120 and respective maturities of 1 and simultaneously sell the coupon bond.
The profit equals $2.91 on each bond.
Fixed Income Portfolio Management
• In general, bonds are just securities
−>
CAPM
• Specific issues:
– Benchmarking bond-fund managers’ performance
• Bond indices & Cellular approach ( NOT Exam Material )
– Risk
• Cash-flow risk (incl. default risk)
» Not Exam Material
• Interest rate risk: Measurement + Management
Bond Index Funds – NOT Exam Mat’l
• Idea
– Similar to that behind stock market indices
• Problem & Solution
– Cellular approach
Bond Index Funds 2 – NOT Exam Mat’l
• US indices for investment-grade, 1+ year bonds:
• BIG = Citigroup/Solomon Smith Barney Broad Investment Grade
– treasuries , agency debt, corporates , 144As , mortgage-backed securities (MBS), and asset-backed securities (ABS)
• “ The AGG ” = Lehman U.S. Aggregate Bond
– Same + municipals and commercial mortgage-backed securities (CMBS)
• Merrill Lynch
– Domestic Master (Ticker D0A0): U.S. gvt & corporate public bonds
– U.S. Broad Market : also includes MBS, global bonds, Yankee bonds , but excludes ABS, tax -exempt munis, etc.
– None of those indices include TIPS
4

Bond Index Funds 3 – NOT Exam Mat’l
• Problems
• Lots of securities in each index
» 4 to 5,000 bonds
» difficult to buy all the bonds
• Portfolio rebalancing
» market liquidity
» bonds are dropped (maturities, calls, defaults, …)
Bond Index Funds 4 – NOT Exam Mat’l
• Solution:
– “cellular approach” (Fig. 16.9)
– idea
• classify by maturity/risk/category/…
• compute percentages in each cell
• match portfolio weights
– effectiveness
• average absolute tracking error = 2 to 16 b.p. / month
Bond Index Funds 5 – NOT Exam Mat’l
– “Cellular approach” (Fig. 16.9)
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
Interest Rate Risk
• Equation:
• P = t
T ∑
=
1 coupon
+
( 1
+ r ) t
( 1
Par
+ r )
T
• Yield sensitivity of Prices:
• P
↑ ⇔ yield
↓
• Measure?
Interest Rate Risk 2
• Determinants of a bond’s yield sensitivity
5

Interest Rate Risk 3
• Determinants of a bond’s yield sensitivity
Interest Rate Risk 4
• 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 ↓
Duration
• Idea
• maturity
↑ ⇒ sensitivity
↑
•
⇒ to measure a bond’s yield sensitivity, measure its “ effective maturity”
• Measure
• Macaulay duration: D
=
T ∑ t
=
1 t .
w t w t
=
C t
P .( 1
+
YTM ) t
T ∑ t = 1 w t
=
1
P
T ∑ t = 1
( 1
+
C t
YTM ) t
=
P
P
=
1
Duration 2
• Duration = effective measure of elasticity
∆
P
P
= −
D .


∆
( 1
1
+
+
YTM
YTM
)


• Proof
• Modified duration
∆
P
=
P
−
D * .
[
∆
YTM
] with D * =
D
1
+ y
Duration 3
• Interpretation 1
• D
= t
T ∑
=
1 t .
w t
= average time until bond payment
» Price risk & reinvestment risk offset if bond sold at D
• Interpretation 2
• % price change of coupon bond of a given duration
• = % price change of zero with maturity = to duration
Duration 4
• Example
• bond : 2-year, 10% ytm, 8% coupon, annual payments
8%
Bond
Time years
Payment PV of CF
(10%)
Weight Col.1
times Col.4
1
2
80
72.73 = 80/1.10
0.0753
= 72.73 / 965.29
0.0753
1080 sum
892.56
965.29
0.9247
1.000
1.8494
1.9247
6

Duration 5
• Example
(Continued)
• suppose YTM changes by 1 basis point (0.01%)
– either compare relative to the bond ’s price with YTM = 10.01% the bond ’s price with YTM = 10%
– or simply compute the price change from the duration
∆
P
P
= −
D .
 ∆
( 1
+
YTM )
1
+
YTM
= −
1 .
10 .
01 %
−
10 %
1 .
10
= −
0 .
0175 %
Duration 6
• Example
(Spreadsheet 16.1)
• 1 st bond : zero coupon with 1.8853 years to maturity
• 2 nd bond : 2-year, 10% ytm, 8% coupon, semi-annual
8%
Bond
Time years
Payment PV of CF
(10%)
Weight Col.1
times Col.4
0.5
40 38.095 = 40 / 1.05
0.0395
0.0197
1
1.5
40
40
36.281
34.553
0.0376
= 36.281 / 964.54
0.0376
0.0358
0.0537
2.0
1040 sum
855.611
964.540
0.8871
1.000
1.7742
1.8852
Duration 7
• Example (Continued, Spreadsheet 16.1)
• suppose YTM changes by 1 basis point (0.01%)
• zero coupon bond with 1.8853 years to maturity
– old price
=
1000
( )
3 .
7706
=
831 .
9623
∆
P
P
=
– new price
=
1000
(
1 .
0501
) 3 .
7706
831 .
6636
−
831 .
9623
=
831 .
9623
=
−
0 .
0359 %
831 .
6636
= −
D .
∆
( 1
+
YTM )
1
+
YTM
Duration 8
• Example: (Continued, Spreadsheet 16.1)
• suppose YTM changes by 1 basis point (0.01%)
• 2-year, 8% 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
∆
P
P
= −
D .
 ∆
( 1
+
1
+
YTM
YTM
)
= −
1 .
2 x
 5 .
01 %
1 .
−
05
5 %
= −
0 .
0359
Duration 9
• Properties of duration (other things constant)
• zero coupon bond: duration = maturity
• time to maturity
» maturity
↑ ⇒ duration
↑
» exception: deep discount bonds (e.g., 3% bond, Fig. 16.3)
• coupon rate
» coupon
↑ ⇒ duration
↓
• YTM
» YTM
↑ ⇒ duration
↓
» exception: zeroes (unchanged)
Duration 10
• Properties of duration – Illustration (Fig. 16.3)
7

Duration 11
• Formulae for duration
• duration of perpetuity = D
=
1
+ y y
– less than infinity!
• coupon bonds (“ annuities + zero”)
– see book
– simplifies if par bond
Duration 12
• Importance
• simple measure
• essential to implement portfolio immunization
• measures interest rate sensitivity effectively
Possible Caveats to Duration
• 1. Assumptions on term structure
• Macaulay duration uses YTM
D
=
T ∑ t
=
1 t .
w t
=
1
P t
T ∑
=
1 t .
( 1
+
C t
YTM ) t
» only valid for level changes in flat term structure
• Fisher-Weil duration measure (NOT exam material)
D
= t
T
=
1 t .
w t
=
1
P
T
t
=
1 t .
s t
=
1
C t
( 1
+ r s
)
Possible Caveats to Duration 3
• 2. Convexity
• Macaulay duration
– first-order approximation :
∆
P
P
= −
D * .
∆
( 1
+
YTM )
– small changes vs . large changes
» duration = point estimate
» for larger changes, an “arc” estimate is needed
– solution: add convexity
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 issue as for realized compound yield
• Cox-Ingersoll-Ross duration (NOT exam material)
• Conclusion: let’s keep Macaulay
Possible Caveats to Duration 4
8

Possible Caveats to Duration 5
– Convexity (continued)
– second-order approximation : convexity
=
P ( 1
1
+
YTM ) 2
T ∑ t
=
1
( t 2
+ t ).
( 1
+
C t
YTM ) t
∆
P
P
= −
D
*
.
∆
YTM
+
1
2
.
convexity .
( ∆
YTM
)
2
Possible Caveats to Duration 6
– Convexity: numerical example
– P = Par = 1,000; T = 30 years; 8% annual coupon
– computations give D*=11.26 years; convexity = 212.4 years
– suppose YTM = 8% −> YTM = 10%
∆
P
P
= −
D * .
∆
YTM
= −
11 .
26 x 0 .
02
= −
22 .
52 %
∆
P
=
P
−
D
*
.
∆
YTM
+
1
2
.
(
∆
YTM
)
2 = −
18 .
27 %
∆
P =
P
811 .
46
−
1 , 000
1 , 000
= −
18 .
85 %
Possible Caveats to Duration 7
– Is convexity priced?
• Attractive
−> bond A more expensive than B ( why?)
Possible Caveats to Duration 8
– Convexity for callable bonds
Possible Caveats to Duration 9
– Convexity for MBS
Bottom Line on Duration
• Very useful
– But take it with a grain of salt for large changes
9

Immunization
• Why?
• 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
Immunization
• What?
• net worth immunization
» match duration of assets and liabilities
• target date immunization
» match inflows and outflows, immunize the net flows
» match duration to holding period
• Who?
• banks
» net worth immunization
• insurance companies, pension funds
» target date immunization
Net Worth Immunization
• Gap management
• assets v s. 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)
Target Date Immunization
• Idea:
• Price risk & reinvestment risk offset if bond is sold at D
• Example for intuition: 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
Target Date Immunization 2
• Idea : Invest in assets with D matching the funds’ investment horizon, so that yields don’t affect the FV
• Implementing this solution/idea
• 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
10

Target Date Immunization 3
Question:
Pension funds pay lifetime annuities to recipients.
è
Firm expects to be in business indefinitely, its pension obligation
≈
perpetuity.
è
Suppose, your pension fund must make perpetual payments of $2 million/year.
è The yield to maturity on all bonds is 16%.
(a) duration of 5-year bonds with coupon rates of 12% (paid annually) is 4 years duration of 20-year bonds with coupon rates of 6% (paid annually) is 11 years
è how much of each of these coupon bonds (in market value) should you hold to both fully fund and immunize your obligation?
(b) What will be the par value of your holdings in the 20-year coupon bond?
Target Date Immunization 4
Answer:
(a)
•
PV of the firm’s “perpetual” obligation = ($2 million/0.16) = $12.5 million.
• duration of this obligation = duration of a perpetuity = (1.16/0.16) = 7.25 years.
Denote by w the weight on the 5-year maturity bond, which has duration of 4 years.
Then, w x 4 + (1 – w ) x 11 = 7.25, which implies that w = 0.5357. Therefore,
0.5357 x $12.5 = $6.7 million in the 5-year bond and
0.4643 x $12.5 = $5.8 million in the 20-year bond.
The total invested = $(6.7+5.8) million = $12.5 million, fully matching the funding needs.
Target Date Immunization 5
Answer:
( b ) Price of the 20-year bond = 60 x PA(16%, 20) + 1000 x PF(16%, 20) = $407.11.
Therefore, the bond sells for 0.4071 times Par, and
Market value = Par value x 0.4071
=> $5.8 million = Par value x 0.4071
=> Par value = $14.25 million.
Another way to see this is to note that each bond with a par value of $1,000 sells for
$407.11. If the total market value is $5.8 million, then you need to buy 14,250 bonds, which results in total par value of $14,250,000.
Dangers with Immunization
• 1. Portfolio rebalancing is needed
– Time passes
⇔ duration changes
• bonds mature, sinking funds, …
– YTM changes
⇔ duration changes
• example: BKM7 Table 16.4
• duration
5
4.97
5.02
YTM
8%
7%
9%
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
An Alternative? Cash-Flow Dedication
• Buy zeroes
• to match all liabilities
• Problems
• difficult to get underpriced zeroes
• zeroes not available for all maturities
– example: perpetuity
11

Contingent Immunization
• Idea
• try to beat the market
• while limiting the downside risk
• Procedure (Fig. 16.12) – NOT Exam Material
• compute the PV of the obligation at current rates
• assess available funds
• “play” the difference
• immunize if trigger point is hit
Contingent Immunization 2
• Procedure
(Fig. 16.12)
12