Transcript Bond Portfolio Management

CFA REVIEW
Fixed Income
Bond Portfolio Management
•
•
•
•
•
Yields and Term-structures
Bond risk
Duration
Convexity
Bond Portfolio Strategies
– Passive strategies
– Active strategies
– Protective strategies
Bond Properties
• Par, Premium and Discount
If
Price
Bond
sold
YTM >
%COUPON
Price<1000
At discount
YTM =
%COUPON
Price =1000
At par
YTM <
%COUPON
Price>1000
At premium
• Bond prices and yield are inversely related
• Bond prices and maturity are inversely related
• Bond prices and coupon are positively related
The Yield Model
The expected yield on the bond may be
computed from the market price
Pp
Ci 2
Pm  

t
2n
(1  i 2)
t 1 (1  i 2)
2n
Where:
i = the discount rate that will discount the cash flows to
equal the current market price of the bond
Computing Bond Yields
Yield Measure
Purpose
Nominal Yield
Measures the coupon rate
Current yield
Measures current income rate
Promised yield to maturity
Measures expected rate of return for bond held
to maturity
Measures expected rate of return for bond held
to first call date
Measures expected rate of return for a bond
likely to be sold prior to maturity. It considers
specified reinvestment assumptions and an
estimated sales price. It can also measure the
actual rate of return on a bond during some past
period of time.
Promised yield to call
Realized (horizon) yield
Current Yield
Similar to dividend yield for stocks
Important to income oriented investors
CY = Ci/Pm
where:
CY = the current yield on a bond
Ci = the annual coupon payment of bond i
Pm = the current market price of the bond
Promised and Realized Yield to Maturity
• PYTM Assumes that all the bond’s cash flow is reinvested at
the computed yield to maturity (same as IRR)
• RYTM assumes that all the bond’s cash flow is reinvested at
the computed yield to maturity
• Example: A bond yields 5%. It has 20 years to maturity and
pays 20% coupon annually. What is the realized yield to
maturity over a 6 years horizon and a reinvestment rate of
3%?
RYTM=(FV/PV)1/6-1
Where FV= Future values of coupons re-invested at 3% over 6
years (i=3%, pmt=200, n=6) + present value of bond with 14 years
left to maturity (i=5%, pmt=200, n=14, FV=1000)
PV is the present value of the bond (i=5%, pmt=200, n=20,
FV=1000)
Promised Yield to Call
• Callable bond pay the face value (1000) + one periodic coupon and
expire prior to maturity
Example: A 10-year, 10% semiannual coupon,$1,000 par value bond is
selling for$1,135.90 with an 8% yield to maturity.It can be called
after 5 years at $1,050. What’s the bond’s nominal yield to call
(YTC)?
• Note: In general, if a bond sells at a premium, then coupon > kd, so a
call is likely. Then, expect to earn: YTC on premium bonds.; YTM on
par & discount bonds.
How do you make money on a bond?
Annual
coupon
pmt
Current yield =
Current price
Change
in
price
Capital gains yield =
Beginning price
Exp total
Exp
Exp cap
= YTM =
+
return
Curr yld
gains yld
Find current yield and capital gains yield for a 9%,
10-year bond when the bond sells for $887 and YTM
= 10.91%.
$90
Current yield = $887
= 10.15%.
YTM = Current yield + Capital gains yield.
Cap gains yield = YTM - Current yield
= 10.91% - 10.15%
= 0.76%.
What four factors affect the cost of money?
• “Nominal” Rate =Risk-free rate
=Real rate + Inflation
 Production opportunities
 Time preferences for consumption
 Risk
 Expected inflation
+ Risk Premium
+ Risk Premium
k = k* + IP + DRP + LP + MRP.
DRP
= Default risk premium.
LP
= Liquidity premium.
MRP = Maturity risk premium.
Treasury: IP, MRP
Corporate: IP, DRP, MRP, LP
Term Structure of Interest Rates
• Relationship between term to maturity and yield to maturity
for a sample of bonds at a fixed point of time.
• Referred to as the “yield curve.”
• Issues differ only in their maturities--Treasury instruments
• 3 shapes (Normal,Flat,Inverted)
• 3 underlying theories, relating to the different supply and
demand pressures in different maturity sectors:
– Expectation (expected to earn on successive investments in ST bonds
during the term to maturity of a LT bond)
– Liquidity (investors prefer the liquidity of ST bonds but will buy LT
bonds if the yields are higher)
– Market segmentation (yields curve reflects the investment policies of
financial institutions who have different maturity preferences)
Hypothetical Treasury Yield Curve
Interest
Rate (%)
15
Maturity risk premium
10
Inflation premium
1 yr
10 yr
20 yr
8.0%
11.4%
12.65%
5
Real risk-free rate
Years to Maturity
0
1
10
20
Actual Treasury Yield Curve
Interest
Rate (%)
15
1 yr
5 yr
10 yr
30 yr
6.3%
6.7%
6.5%
6.2%
Yield Curve
(May 2000)
10
5
Years to Maturity
0
10
20
30
Corporate yield curves are higher than for Treasury
bond. However, corporate yield curves are not
necessarily parallel to the Treasury curve. The spread
between a corporate yield curve and the Treasury curve
widens as the corporate bond rating decreases.
15
Interest
Rate (%)
BB-Rated
10
AAA-Rated
5
Treasury
6.0%
yield curve
5.9%
5.2%
0
0
1
5
10
15
20
Years to
maturity
U.S. Yield Curve Inverts Before Last Five Recessions
(5-year Treasury bond - 3-month Treasury bill)
% GDP Growth/
8
Yield Curve
% Real annual GDP growth
6
4
2
0
Yield curve
-2
Recession
Correct
Recession
Correct
?
1
9
M
ar0
M
ar9
7
M
ar9
5
3
M
ar9
M
ar9
1
M
ar9
9
M
ar8
7
5
M
ar8
M
ar8
3
M
ar8
1
Data though 12/20/00
M
ar8
9
7
M
ar7
5
M
ar7
M
ar7
3
M
ar7
1
M
ar7
M
ar6
9
Recession
Correct
-4
Recession
Correct 2 Recessions
Correct
-6
Expected yields
Bond Risks
• Interest rate risk dichotomy:
– Price risk or price volatility
– Reinvestment risk or “ending wealth” volatility
• If interest rates are expected to increase; bond price will
decrease and ending wealth will increase.
• interest rates are expected to decrease; bond price will increase
and ending wealth will decrease.
Bond risk…
• As Coupon is greater, Price sensitivity to yield
decreases.
• As Maturity gets greater, Price sensitivity to yield
increases.
• A bond with high yield is less sensitive to a
change in interests than a bond with low yield.
• Bond risk = Price risk and reinvestment risk
• Q: with an expected change interest rates, which
bond would you pick?
Some Trading Strategies…
• If market rates are expected to decline, bond prices
will rise you want bonds with maximum price
volatility.
– Maximum price increase (capital gain) results from long
term, low coupon bonds, low yield
• If market rates are expected to rise, bond prices will
fall you want bonds with minimum price volatility.
– Invest in short term, high coupon bonds to minimize price
volatility and capital loss, high yield.
Evidence of reinvestment risk (8% coupon, 25
years, 8% yield, semi-annual). How does the ending
wealth change if interest rates increase by
1%ANS: ≈+15%)
Approximating Price Risk
n
If p  
CFt
t 1(1  i )
t
2

p
1

p
dp 
 di  
 di2  ...
i
2 i 2
dp 1 p
1 1 2p
   di   
 di2
p
p i
2 p i 2
dp p

Duration
p
p
CFt
1 n
    t 
p t 1
(1  i ) t

1
 i

 (1  i )
Convexity
n 
CFt 
1 1
1
2
2
  
  (t  t ) 


i

2
t
2 p (1  i )
(1  i ) 
t 1
The Duration Measure
n
Ct (t )

t
t 1 (1  i )
D n

Ct

t
t 1 (1  i )
n
 t  PV (C )
t
t 1
price
Duration: the weighted average time to full recovery of
principal and interest payments.
Developed by Frederick R. Macaulay, 1938
Where:
t = time period in which the coupon or principal payment occurs
Ct = interest or principal payment that occurs in period t
i = yield to maturity on the bond
Characteristics of Duration
• Duration of a bond with coupons is always less than
its term to maturity because duration gives weight to
these interim payments
– A zero-coupon bond’s duration equals its maturity
• There is an inverse relation between duration and
coupon
• There is a positive relation between term to maturity
and duration, but duration increases at a decreasing
rate with maturity
• There is an inverse relation between YTM and
duration
• Sinking funds and call provisions can have a
dramatic effect on a bond’s duration
Modified Duration and Bond Price
Volatility
An adjusted measure of duration can be used to
approximate the price volatility of a bond
Macaulay duration
modified duration 
YTM
1
Where:
m
m = number of payments a year
YTM = nominal YTM
Duration and Bond Price Volatility
Bond price movements will vary proportionally with modified
duration for small changes in yields
An estimate of the percentage change in bond prices equals the
change in yield time modified duration
P
100   Dmod  i
P
Where:
P = change in price for the bond
P = beginning price for the bond
Dmod = the modified duration of the bond
i = yield change in basis points divided by 100
Example: Given a bond that pays semi-annual coupons with a
duration of 6 years and a yield of 8%, what will the percentage
change in price be if market rates are expected to rise by 50 basis
points?
Trading Strategies Using Duration
• Longest-duration security provides the maximum price
variation
• If you expect a decline in interest rates, increase the average
duration of your bond portfolio to experience maximum
price volatility
• If you expect an increase in interest rates, reduce the
average duration to minimize your price decline
• Note that the duration of your portfolio is the market-valueweighted average of the duration of the individual bonds in
the portfolio
Convexity
The convexity is the measure of the curvature and is the
second derivative of price with resect to yield (d2P/di2)
divided by price
Convexity is the percentage change in dP/di for a given
change in yield
d 2P
2
di
Convexity 
P
• Inverse relationship between coupon and convexity
• Direct relationship between maturity and convexity
• Inverse relationship between yield and convexity
Modified Duration-Convexity Effects
• Changes in a bond’s price resulting from a change in yield are
due to:
– Bond’s modified duration
– Bond’s convexity
• Relative effect of these two factors depends on the
characteristics of the bond (its convexity) and the size of the
yield change
• Convexity is desirable
P
1
2
  Dmod  i   C  i
P
2
Effective Duration
• Measure of the interest rate sensitivity of an asset
• Use a pricing model to estimate the market prices
surrounding a change in interest rates
Effective Duration
P   P 
2 PS
Effective Convexity
P   P   2P
PS
2
P- = the estimated price after a downward shift in interest rates
P+ = the estimated price after a upward shift in interest rates
P = the current price
S = the assumed shift in the term structure
Passive Bond Portfolio Strategies
Buy-and-Hold Strategy
• Investor selection based on quality, coupon and maturity
• Match maturity with investment horizon
• Modified buy and hold
Indexing Strategy
• Money managers can’t beat the market“If you can’t beat
them, join them.”
• Difficulties:
– Tracking error - difference between the portfolio’s return and the
return for the index.
– You must know characteristics and composition of the various
indexesIndexes change over time.
Active Bond Strategies
• Active management strategies » Interest Rate Anticipation (Valuation
Analysis, Credit Analysis, Yield Spread Analysis, and Bond Swaps)
• Riskiest
– If i is expected to increase, preserve capital
– If i is expected to decrease, make capital gains
• Objectives are achieved by adjusting the portfolio’s duration (maturity).
– Shorten duration if rates are expected to
• Play the Reinvestment advantage card and get Cash flow ASAP (liquidity)
– Lengthen duration if rates are expected to 
• Play the Interest rate card : lower coupons and play on an increase in bond
prices
• Q: What is the duration of a portfolio of bonds?
• A: The weighted average duration of each bond in a portfolio—I.e.,
i n
D   Wi  Di
i 1
Matched Funding Techniques: Dedicated Portfolios
What are they? Bond portfolio management technique used to service a
specific set of liabilities
Pure-Cash Matched Dedicated Portfolio
• Cash flows from all sources exactly match up in timing and size with
the liability schedule.
• Can be achieved by buying a series of zero coupon Treasury securities.
• Total passive strategy
Dedication with Reinvestment
• Cash flows don’t exactly match the liability schedule, also cash flows
received earlier are reinvested at a relatively low interest rate.
• Advantages: (1) Allows for wider set of bonds to be considered; (2)
Lower net cost of the portfolio; (3) Safety equivalent to with pure cashmatching.
• Potential problem: Early redemption
Matched Funding Techniques: Immunization Strategies
• Immunization: Attempt to generate a specified rate of return
regardless of what happens to market rates during an investment
horizon.
• Immunization is a process intended to eliminate interest risk; it is
achieved if the ending wealth of a bond portfolio is the same regardless
of whether interest rates change
• Example: Assume a 6 year strategic asset allocation horizon and market
rates on 6% coupon bonds is 6%.
– Strategy one: Maturity (cash) Matching Strategy
• A manager has a portfolio of bonds with an average maturity of 6 years. The
average coupon rate of the portfolio is 6%.
– Strategy two: Duration Matching strategy=portfolio immunization
• A manager has a portfolio of bonds with an average maturity of 7 years. The
average coupon rate of the portfolio is 6%. The average duration is about 6
years.
– Q: What happens if interest rates increase or decrease suddenly by
Example…continued
Interest rates unchanged or R=6%
Strategy 1: FV=PMT x FVIFA + 1000=60 x 6.975 +1000=$1,418.5
Strategy 2: FV=PMT x FVIFA + PMT x PVIFA +1000/FVIF=
=60 x 6.975 +60 x .943 + 1000/1.06=$1,418.5
Decrease of 1% or R=5%
Strategy 1: FV=PMT x FVIFA + 1000=60 x 6.802 +1000 =$1,408
Strategy 2: FV=PMT x FVIFA + PMT x PVIFA +1000/FVIF
=60 x 6.802 +60 x .952 + 1000/1.05=$1,417.6
Increase of 1% or R=7%
Strategy 1: FV=PMT x FVIFA + 1000=60 x 7.153 +1000=$1,429.2
Strategy 2: FV=PMT x FVIFA + PMT x PVIFA +1000/FVIF
=60 x 7.153 +60 x .935 + 1000/1.07=$1,419.9
For strategy 2: At t=6
years, bonds have 1 year
left of life!
Conclusion:
Strategy 1
1,418.5
1408
1429.2
-0.07%
0.08%
Strategy 2
1,418.5
1417.6
1419.9
0%
0%
R=6%
R=5%
R=7%
%change (-1%)
%change (+1%)
Applications:
• immunize the bond portion of your strategic allocation
• Immunize a future cash outflows (pension funds, insurance
companies)
• Not as easy as it sounds (rebalancing, duration drift,
unavailability…)
Questions
• You immunize a 4-year investment by purchasing a
coupon bond with a duration of 4 years. If interest
rates do not change, is your bond still immunized one
year after?
• What if you purchased a 4-year zero coupon bond?
Matched Funding Techniques: Horizon Matching
and contingent immunization
• Horizon matching combines cash matching and immunization to
Provide protection against unequal interest rate changes
– » Short term end is set up as a cash matching portfolio
– » Longer term end is duration immunized
– » Roll out occurs when the time horizon is pushed out one year
further into the long term time horizon
• Contingent immunization allows for active portfolio management
while assuring a minimal return by creating a Cushion spread (
difference between market rates and the minimum that investors are
willing to accept.)Immunize a specific return; play with the cushion!