Investments Analysis
• Last 2 Lectures: Fixed Income Securities
– Bond Prices and Yields
– Term Structure of Interest Rates
• This Lecture (#7): Fixed Income Securities
– Term Structure of Interest Rates
– Interest Rate Risk
– Bond Portfolio Management
Part II:
Fixed Income Markets
Interpreting the Yield Curve
• TSOIR and interest rate uncertainty
• link between
» interest rates
» liquidity premia
• Various cases
• possible cases
• interpretation
“TSOIR” and 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” and 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
• f2 ≥ E[r2]
• f2 = E[r2] + risk premium
“TSOIR” and 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[r2] ≥ f2
• f2 = E[r2] - risk “premium”
• Expectation Hypothesis
• E[r2] = f2
(risk premium = 0)
• idea: “arbitrage”
“TSOIR” and 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” and Interest Rate Uncertainty 5
• Example 2
• short term rates: r1 = r2 = r3 = 10%
• liquidity premium = constant 1% per year
• YTM
y1 = r1 = 10%
y2 = (1 + r1 )(1 + f 2 ) − 1 = (1 + 10%)(1 + 10% + 1%) − 1 = 10.5%
y3 = 3 (1 + r1 )(1 + f 2 )(1 + f3 ) −1 = 3 (1 + 10%)(1 + 11%)(1 + 11%) −1 = 10.67%
“TSOIR” and Interest Rate Uncertainty 6
• Example 3
• Zero-Coupon Rates
•
•
•
•
•
12.00%
11.75%
11.25%
10.00%
9.25%
Bond Maturity
1yr Fwd Rate
1
2
3
4
5
spot = 12%
11.5%
10.26%
6.33%
6.30%
“TSOIR” and Interest Rate Uncertainty 7
• Example 3 (continued)
1yr Forward Rates
1yr from now
[(1.1175)2 / 1.12] - 1
= 0.115006
2yrs from now
[(1.1125)3 / (1.1175)2] - 1 = 0.102567
3yrs from now
[(1.1)4 / (1.1125)3] - 1 =
0.063336
4yrs from now
[(1.0925)5 / (1.1)4] - 1 =
0.063008
Interpreting the TSOIR
• Types of yield curve
• cases
» upward sloping (most common)
» downward sloping
» hump-shaped
• Fig 15.5
• 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[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?!
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
Interpretation 4: Hump-Shape curve
• Interpretation
• liquidity?
• Most recent 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
2
[120/1.05] + [1,120/1.06 ] = $1,111.08.
The arbitrage strategy is to:
buy zeros with face values of $120 and $1,120 and respective maturities of 1 and
2 years
simultaneously sell the coupon bond.
The profit equals $2.91 on each bond.
Fixed Income Portfolio Management
• In general
• bonds are securities just like other
• −> use the CAPM
• Bond Index Funds -- NOT Exam Material
• Immunization
• net worth immunization
• target date immunization
• contingent immunization -- NOT exam material
Bond Index Funds -- NOT Exam Mat’l
• Idea
• US indices
» 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: Scotia-McLeod (esp. Universe Index)
Bond Index Funds 2 -- NOT Exam Mat’l
• Problems
• lots of securities in each index
• portfolio rebalancing
» market liquidity
» bonds are dropped (maturities, calls, defaults, …)
Bond Index Funds 3 -- NOT Exam Mat’l
• Solution:
– “cellular approach”
– 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
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 = PV(annuity) + PV(final payment)
T
•
coupon
Par
+
= ∑
t
T
(1 + r )
t =1 (1 + r )
• Yield sensitivity of bond Prices:
• P ↑ ⇔ yield ↓
• Measure?
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 ↓
Duration
• Idea
• maturity ↑ ⇒ sensitivity ↑
• ⇒ to measure a bond’s yield sensitivity,
•
measure its “effective maturity”
• Measure
T
• Macaulay duration: D = ∑ t.wt
t =1
Ct
wt =
t
P.(1 + YTM )
T
Ct
P
1 T
w
=
=
=1
∑
∑
t
t
P t =1 (1 + YTM )
P
t =1
Duration 2
• Duration = effective measure of elasticity
∆P
 ∆(1 + YTM ) 
= − D.

P
 1 + YTM 
• Proof
• Modified duration
∆P
= − D * .[∆YTM ]
P
with
D
D =
1+ y
*
Duration 3
• Interpretation 1
T
• D = ∑ t.wt
t =1
= average time until bond payment
• Interpretation 2
•
% price change of coupon bond of a given duration
• = % price change of zero with maturity = to duration
Duration 5
• Example (Table 16.3)
• first bond is zero coupon with 1.8853 years to maturity
• second bond is 2-year, 8% coupon bond
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
40
36.281
1.5
40
34.553
0.0358
0.0537
2.0
1040
855.611
0.8871
1.7742
sum
964.540
1.000
1.8852
0.0376
0.0376
=36.281/964.54
Duration 5
• Example (Continued, Table 16.3)
• suppose YTM changes by 1 basis point (0.01%)
• zero coupon bond with 1.8853 years to maturity
– old price
– new price
=
=
1000
(1.05)
3.7706
= 831.9623
1000
(1.0501)
3.7706
= 831.6636
∆P 831.6636 − 831.9623
 ∆ (1 + YTM ) 
=
= −0.0359% = − D.

P
831.9623
 1 + YTM 
Duration 6
• Example: 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
∆P
 5.01% − 5% 
 ∆ (1 + YTM ) 
= −0.0359%
= − D.
=
−
1
.
8853
x
2
x



P
1.05
 1 + YTM 


Duration 7
• 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)
Duration 8
• Properties of duration
y
• duration of perpetuity = D =
1+ y
– less than infinity!
• coupon bonds (“annuities + zero”)
– see book
– simplifies if par bond
Duration 9
• 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
T
Ct
1 T
D = ∑ t.wt = ∑ t.
P t =1 (1 + YTM ) t
t =1
» only valid for level changes in flat term structure
• Fisher-Weil duration measure (NOT exam material)
T
1 T
D = ∑ t.wt = ∑ t.
P t =1
t =1
Ct
t
∏ (1 + r )
s
s =1
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 (NOT exam material)
• bottom line: let’s keep Macaulay
Possible Caveats to Duration 3
• 2. Convexity
• Macaulay duration
– first-order approximation:
∆P
= − D * .∆ (1 + YTM )
P
– small changes vs. large changes
» duration = point estimate
» for larger changes, an “arc” estimate is needed
– solution: add convexity
Possible Caveats to Duration 4
– Convexity (continued)
– second-order approximation:
1
convexity =
P (1 + YTM ) 2
T
2
t
(
∑ + t ).
t =1
Ct
(1 + YTM ) t
∆P
1
2
*
= − D .∆YTM + .convexity.(∆YTM )
P
2
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 = 212.4 years
– suppose YTM = 8% −> YTM = 10%
∆P
= − D * .∆YTM = −11.26 x0.02 = −22.52%
P
∆P
1
2
= − D * .∆YTM + .convexity.(∆YTM ) = −18.27%
P
2
∆P 811.46 − 1,000
=
= −18.85%
P
1,000
Bottom Line on Duration
• Very useful
– But take it with a grain of salt for large changes
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
• Who?
• banks
» net worth immunization
• insurance companies, pension funds
» target date immunization
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)
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
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
Target Date Immunization 3
Question:
Pension funds pay lifetime annuities to recipients.
Î Firm expects to be in business indefinitely, its pension obligation ≈ perpetuity.
Î Suppose, your pension fund must make perpetual payments of $2 million/year.
Î 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
Î 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: Table 16.6
•
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
– ex.: perpetuity
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