CHAPTER 16 Managing Bond Portfolios
Bond Pricing Relationships
1) Bond prices and yields are inversely related.
2) An increase in a bond’s yield to maturity results in a smaller price change than a
decrease of equal magnitude.
3) Long-term bonds tend to be more price sensitive than short-term bonds.
4) As maturity increases, price sensitivity increases at a decreasing rate.
5) Interest rate risk is inversely related to the bond’s coupon rate.
6) Price sensitivity is inversely related to the yield to maturity at which the bond is
selling.
Duration
A measure of the effective maturity of a bond
The weighted average of the times until each payment is received, with the weights
proportional to the present value of the payment
Duration is shorter than maturity for all bonds except zero coupon bonds.
Duration is equal to maturity for zero coupon bonds.
Duration: Calculation


w t  CFt (1  y ) Price
t
T
D   t wt
t 1
CFt=cash flow at time t
Duration/Price Relationship
D* = modified duration
Modified D  D* 
D
1  YTM 
Price change is proportional to duration and not to maturity
%P  D* * Y
Par Value
Annual Coupon
Coupon per Period
Required Return
Time to Maturity
Compounding Frequency
Price
Current Yield
$
$1,000.00
8.00%
8.00%
10.00%
30
1
1,276.00
6.27%
Macaulay's Duration
Modified Duration
Convexity
Bond Value
$811.46
Yield to Maturity
Yield to Call
6.00%
if semi-annual, multiply by 2
#NUM!
Realized Yield to Maturity
#NUM!
6.7708
6.3878
105.5115
% Change in YTM
% Change in Value
New Price
2.00%
-12.78%
$1,112.98
With Convexity
% Change in Value
New Price
-10.67%
$1,139.91
Rules for Duration
Rule 1 The duration of a zero-coupon bond equals its time to maturity
Rule 2 Holding maturity constant, a bond’s duration is higher when the coupon rate is
lower
Rule 3 Holding the coupon rate constant, a bond’s duration generally increases with its
time to maturity
Rules for Duration
Rule 4 Holding other factors constant, the duration of a coupon bond is higher when
the bond’s yield to maturity is lower
Rules 5 The duration of a level perpetuity is equal to: (1+y) / y
Convexity
The relationship between bond prices and yields is not linear.
Duration rule is a good approximation for only small changes in bond yields.
Bonds with greater convexity have more curvature in the price-yield relationship.
Figure 16.3 Bond Price Convexity: 30-Year Maturity, 8% Coupon; Initial YTM = 8%
1
Convexity 
P  (1  y )2
n
 CFt
  (1 y)
t 1



2
(
t

t
)

t

P
 D* * y  1/ 2 * convexity * ( y )2
P

Why do Investors Like Convexity?
Bonds with greater curvature gain more in price when yields fall than they lose when
yields rise.
The more volatile interest rates, the more attractive this asymmetry.
Bonds with greater convexity tend to have higher prices and/or lower yields, all else
equal.
Callable Bonds
As rates fall, there is a ceiling on the bond’s market price, which cannot rise above the
call price.
Negative convexity
Use effective duration:
Figure 16.5 Price –Yield Curve for a Callable Bond
Mortgage-Backed Securities
The number of outstanding callable corporate bonds has declined, but the MBS market
has grown rapidly.
MBS are based on a portfolio of callable amortizing loans.
Homeowners have the right to repay their loans at any time.
MBS have negative convexity.
Often sell for more than their principal balance.
Homeowners do not refinance as soon as rates drop, so implicit call price is not a firm
ceiling on MBS value.
Tranches – the underlying mortgage pool is divided into a set of derivative securities
Figure 16.7 Cash Flows to Whole Mortgage Pool; Cash Flows to Three Tranches
Passive Management
Two passive bond portfolio strategies:
Indexing
Immunization
Both strategies see market prices as being correct, but the strategies have very
different risks.
Bond Index Funds
Bond indexes contain thousands of issues, many of which are infrequently traded.
Bond indexes turn over more than stock indexes as the bonds mature.
Therefore, bond index funds hold only a representative sample of the bonds in the
actual index.
Similar to stock indexing
Except bonds are more difficult to manage
This is supposed to be a passive strategy but bonds by nature mature
Basic strategy
Identify characteristics in the bond index by type and maturity
Determine the proportions of each type etc in the index
Create a portfolio to match the above proportions
Immunization
Immunization is a way to control interest rate risk.
Widely used by pension funds, insurance companies, and banks.
Immunize a portfolio by matching the interest rate exposure of assets and liabilities.
This means: Match the duration of the assets and liabilities.
Price risk and reinvestment rate risk exactly cancel out.
Result: Value of assets will track the value of liabilities whether rates rise or fall.
An attempt to “match” maturity structure of the portfolio
This process attempts to “equate” the impact from interest rate risk.
When interest changes the impact can be divided into 2 parts.
Price risk – as rate go up; prices fall
Bad for the investor
Reinvestment risk – as rate go up; future reinvested cash flows will earn higher
returns
Good for the investor
Use solver to find a portfolio with a specific Duration
We can also use the solver function to establish a portfolio with a specific duration.
Check solution D; if not equal to the amount searching that duration is not possible try
another.
Cash Flow Matching and Dedication
Cash flow matching = automatic immunization.
Cash flow matching is a dedication strategy.
Not widely used because of constraints associated with bond choices.
Active Management: Swapping Strategies
Substitution swap
Suspect temporary mispricing
Intermarket swap
Believe spread between 2 sectors out of wack
Rate anticipation swap
Forecast interest rate changes
Pure yield pickup
Just move to higher yielding bond with same risk
Tax swap