Chapter 13
BOND PORTFOLIO MANAGEMENT
The Passive and Active Stances
OUTLINE
• Interest Rate Sensitivity
• Duration
• Convexity
• Passive Strategies
• Immunisation : A Hybrid Strategy
• Active Strategies
• Interest Rate Swaps
PASSIVE VERSUS ACTIVE STRATEGY
• A PASSIVE STRATEGY SEEKS TO MAINTAIN AN
APPROPRIATE BALANCE BETWEEN RISK AND
RETURN.
• AN ACTIVE STRATEGY STRIVES TO ACHIEVE
RETURNS THAT ARE MORE THAN
COMMENSURATE WITH THE RISK EXPOSURE
INTEREST RATE SENSITIVITY
1. THERE IS AN INVERSE RELATIONSHIP BETWEEN BOND
PRICES AND YIELDS.
2. AN INCREASE IN YIELD CAUSES A PROPORTIONATELY
SMALLER PRICE CHANGE THAN A DECREASE IN YIELD OF
THE SAME MAGNITUDE.
3. PRICES OF LONG-TERM BONDS ARE MORE SENSITIVE TO
INTEREST RATE CHANGES THAN PRICES OF SHORT-TERM
BONDS.
4. AS MATURITY INCREASES, INTEREST RATE RISK INCREASES
BUT AT A DECREASING RATE.
5. PRICES OF LOW-COUPON BONDS ARE MORE SENSITIVE TO
INTEREST RATE CHANGES THAN PRICES OF HIGH-COUPON
BONDS.
6. BOND PRICES ARE MORE SENSITIVE TO YIELD CHANGES
WHEN THE BOND IS INITIALLY SELLING AT A LOWER YIELD.
RELATIONSHIP BETWEEN CHANGE IN YIELD TO
MATURITY AND CHANGE IN BOND PRICE
Percentage change in bond price
200
150
Bond Coupon
A
12%
B
12%
C
3%
D
3%
100
Maturity
5 years
30 years
30 years
30 years
Initial
YTM
10%
10%
10%
6%
50
0
-5 -4
-3
-2
-1
0
1
-50
Change in yield to maturity (%)
2
3
4
5A
B
C
D
DURATION - 1
DURATION IS A MEASURE OF THE AVERAGE LIFE OF A DEBT
INSTRUMENT. IT IS DEFINED AS THE WEIGHTED AVERAGE TIME
TO FULL RECOVERY OF PRINCIPAL AND INTEREST PAYMENTS.
USING ANNUAL COMPOUNDING WE CAN DEFINE DURATION (D) AS:
n

t=1
Ct x t
n

t=1
Ct
(1+r)t
D =
(1+r)t
t = TIME PERIOD IN WHICH THE COUPON / PRINCIPAL
PAYMENT OCCURS
Ct = INTEREST & / OR PRINCIPAL … t
r = MARKET YIELD ON THE BOND
DURATION - 2
TO ILLUSTRATE HOW DURATION IS CALCULATED CONSIDER BOND A.
BOND A
RS 100
15 PERCNET PAYABLE ANNUALLY
6
RS 100
RS 89.50
18 PERCENT
FACE VALUE
COUPON (INTEREST RATE)
YEARS TO MATURITY
REDEMPTION VALUE
CURRENT MARKET PRICE
YIELD TO MATURITY
CALCULATION OF DURATION
BOND A : 15 PERCENT COUPON
YEAR CASH FLOW
1
2
3
4
5
6
15
15
15
15
15
115
PRESENT VALUE
AT 18 PER CENT
12.71
10.77
9.13
7.74
6.56
42.60
PROPORTION OF
THE BOND'S VALUE
0.142
0.120
0.102
0.086
0.073
0.476
DURATION
YEARS
PROPORTION OF THE
BOND'S VALUE TIME
0.142
0.241
0.306
0.346
0.366
2.856
4.257
DURATION AND VOLATILITY – 1
D* = D/(1+y)
D* = modified duration
D = duration
y = the bond’s yield to maturity
P/P  – D*y
Percentage price 
change
—
Modified
duration
Change in yield
X
Example
D* = 3.608
y = 0.2 percent
P/P  – 3.608 x 0.2 = – 0.722 percent
in decimal form
PROPERTIES OF DURATION –1
1. THE DURATION OF A ZERO COUPON BOND IS
THE SAME AS ITS MATURITY.
2. FOR A GIVEN MATURITY, A BOND’S DURATION
IS HIGHER WHEN ITS COUPON RATE IS LOWER.
3. FOR A GIVEN COUPON RATE, A BOND’S
DURATION GENERALLY INCREASES WITH
MATURITY.
4. OTHER THINGS BEING EQUAL, THE DURATION
OF A COUPON BOND VARIES INVERSELY WITH
ITS YIELD TO MATURITY.
5. THE DURATION OF A LEVEL PERPETUITY IS:
(1 + YIELD) / YIELD
PROPERTIES OF DURATION
6. THE DURATION OF A LEVEL ANNUITY APPROXIMATELY IS:
1 + YIELD
NUMBER OF PAYMENTS
YIELD
(1 + YIELD) NUMBER OF PAYMENTS -1
FOR EXAMPLE, A 15 YEAR ANNUAL ANNUITY WITH A YIELD OF
10 PERCENT WILL HAVE A DURATION OF:
1.10
15
=
= 6.28 YEARS
0.10
1.1015 - 1
7. THE DURATION OF A COUPON BOND APPROXIMATELY IS:
1+y
(1 + y) + T (c - y)
y
c [(1 + y)T - 1] + y
WHERE y IS THE BOND’S YIELD PER PAYMENT PERIOD, T IS THE
NUMBER OF PAYMENT PERIODS, AND c IS THE COUPON RATE
PER PAYMENT PERIOD.
DURATION OF A COUPON BOND
1+y
(1 + y) + T (c - y)
c [(1 + y)T - 1] + y
y
C : COUPON RATE PER PAYMENT PERIOD
T : NUMBER OF PAYMENT PERIODS
y : BOND’S YIELD PER PAYMENT PERIOD
14% COUPON BOND, 8 YRS MATURITY, PAYING COUPONS
SEMI-ANNUALLY YTM = 8 PERCENT PER HALF-YEAR PERIOD
1.08
(1.08) + 16(.07 - .08)
-
.08
.07 [)1.08)16 - 1) + .08
= 9.817 HALF-YEARS = 4.909 YRS
NOTE : MAINTAIN CONSISTENCY .. TIME UNITS OF PAY’T
PERIOD & INT. RATE
CONVEXITY
• IF THE DURATION RULE WERE AN EXACT RULE,
THE PERCENTAGE CHANGE IN PRICE WOULD BE
LINEARLY RELATED TO THE CHANGE IN YIELD.
YET WE KNOW FROM THE BOND-PRICING
RELATIONSHIPS DISCUSSED EARLIER THAT THE
ACTUAL RELATIONSHIP IS CURVILINEAR.
• THE DURATION RULE PROVIDES AN
APPROXIMATION WHICH IS FAIRLY CLOSE, FOR
SMALL CHANGES IN YIELD. HOWEVER, AS THE
YIELD CHANGE BECOMES LARGER, THE
APPROXIMATION BECOMES POORER.
CONVEXITY
20-YEAR MATURITY, 9 PERCENT COUPON BOND, SELLING AT AN INITIAL MATURITY OF 9
PERCENT. MODIFIED DURATION IS 9.95 YEARS. THE FOLLOWING EXHIBIT SHOWS THE
STRAIGHT LINE PLOT OF -D*y = - 9.95 x y AS WELL AS THE CURVED LINE REFLECTING
THE ACTUAL RELATIONSHIP BETWEEN YIELD CHANGE AND PRICE CHANGE
Percentage
change in
bond price
80
60
40
20
•
0
-5 - 4
- 20
- 40
-3
-2
-1
0
1
Change in YTM
(percentage points)
2
3
4
5
CONVEXITY
• THE TRUE PRICE-YIELD RELATIONSHIP IS
CONVEX, MEANING THAT IT OPENS UPWARD.
• CLEARLY, CONVEXITY IS A DESIRABLE
FEATURE IN BONDS. PRICES OF BONDS WITH
GREATER CONVEXITY (CURVATURE) INCREASE
MORE WHEN YIELDS FALL AND DECLINE LESS
WHEN YIELDS RISE.
• SINCE CONVEXITY IS A DESIRABLE FEATURE, IT
DOES NOT COME FREE. INVESTORS HAVE TO
PAY FOR IT IN SOME WAY OR THE OTHER.
PASSIVE STRATEGIES
TWO COMMONLY FOLLOWED STRATEGIES BY
PASSIVE BOND INVESTORS ARE: BUY AND HOLD
STRATEGY AND INDEXING STRATEGY.
A BUY AND HOLD STRATEGY SELECTS A BOND
PORTFOLIO AND STAYS WITH IT
AN INDEXING STRATEGY CALLS FOR BUILDING A
PORTFOLIO THAT MIRRORS A WELL-KNOWN BOND
INDEX
DURATION AND IMMUNISATION - 1
CAPITAL VALUE
INTEREST RATE
RETURN ON REINVESTMENT OF
INTEREST
CAPITAL VALUE
INTEREST RATE
RETURN ON REINVESTMENT OF
INTEREST
FOR IMMUNISATION SET DURATION EQUAL TO
INVESTMENT HORIZON
DURATION AND IMMUNISATION - 2
MAY BE DEFINED . . PROCESS … FIXED INCOME
PORTFOLIO IS CREATED HAVING . . AN ASSURED RETURN
FOR A SPECIFIED TIME HORIZON IRRESPECTIVE OF
INTEREST RATE CHANGE. MORE CONCISELY, THE
FOLLOWING ARE IMPORTANT CHARACTERISTICS
• SPECIFIC TIME HORIZON
• ASSURED RATE OF RETURN
• INSULATION FROM THE EFFECTS OF POTENTIAL
ADVERSE INTEREST RATE CHANGE ON PORTFOLIO
VALUE
CAPITAL CHANGES
BALANCE
INVESTMENT RETURN
ILLUSTRATION
AN INVESTOR WHO HAS A FOUR-YEAR INVESTMENT HORIZON WANTS
TO INVEST RS.1,000 SO THAT HIS INITIAL INVESTMENT ALONG WITH
REINVESTMENT OF INTEREST GROWS TO RS.1607.5. THIS MEANS THAT
THE INVESTOR WANTS HIS INVESTMENT TO EARN A COMPOUND
RETURN OF 12.6 PERCENT [1,000 (1.126)4 = 1,607.5
THE INVESTOR IS EVALUATING TWO BONDS, A & B
Par value
Market price
Coupon rate
Yield to maturity
Maturity period
Duration
Rating
Bond A
Bond B
Rs.1,000
Rs.1,000
12.6%
12.6%
4 years
Less than 4 years
A
Rs.1,000
Rs.1,000
12.6%
12.6%
5 years
4 years
A
EXHIBIT 13.4 SHOWS WHAT HAPPENS WHEN THE INVESTOR BUYS
BOND A AND BOND B UNDER DIFFERENT ASSUMPTIONS ABOUT
MARKET YIELD
TERMINAL VALUE WITH BONDS A AND B
Part I :
Year
1
2
3
4
Part II
Year
1
2
3
4
Bond A: Market Yield Remains at 12.6%
Reinvestment
Cash flow
Accumulated value
rate
Rs. 126
12.6%
126 (1.126)3 = 179.9
Rs. 126
12.6%
126 (1.126)2 = 159.8
Rs. 126
12.6%
126 (1.126)1 = 141.9
Rs. 1126
NA
1126.0
Total
1607.6
Bond A: Market Yield Falls to 10% in year 2
Reinvestment
Cash flow
Accumulated value
rate
Rs. 126
12.6%
126 (1.10) (1.10) (1.10) = 167.7
Rs. 126
10.0%
126 (1.10) (1.10)
= 152.5
Rs. 126
10.0%
126 (1.10)
= 138.6
Rs. 1126
NA
1126.0
Total
1584.8
contd
TERMINAL VALUE WITH BONDS A AND B
Part III
Bond B: Market Yield Remains at 12.6%
Reinvestment
Year
Cash flow
Accumulated value
rate
1
126
12.6%
126 (1.126)3 = 179.9
2
126
12.6%
126 (1.126)2 = 159.8
3
126
12.6%
126 (1.126)1 = 141.9
4
126
NA
126.0
4
1000* (sale of bond)
NA
1000.0
Total
1607.6
Part III
Bond B: Market Yield Falls to 10% in Year 2
Reinvestment
Year
Cash flow
Accumulated value
rate
1
126
12.6%
126 (1.10) (1.10) (1.10) = 167.7
2
126
10.0%
126 (1.10) (1.10)
= 152.5
3
126
10.0%
126 (1.10)
= 138.6
4
126
NA
126.0
4
1023.6** (sale of
NA
1023.6
bond)
Total
1608.4
* (126 + 1000)/ (1.126) = 1000
** (126 + 1000)/ (1.10) = 1023.6
CASH FLOW MATCHING
CASH FLOW MATCHING INVOLVES BUYING A
ZERO COUPON BOND THAT PROMISES A PAYMENT
THAT EXACTLY MATCHES THE PROJECTED CASH
REQUIREMENT. IT AUTOMATICALLY IMMUNISES
A PORTFOLIO FROM INTEREST RATE RISK
BECAUSE THE CASH FLOW FROM THE BOND
OFFSETS THE FUTURE OBLIGATION.
A DEDICATION STRATEGY INVOLVES MATCHING
CASH FLOWS ON A MULTIPERIOD BASES.
ACTIVE STRATEGIES
• HENRY KAUFMAN, A RENOWNED BOND EXPERT,
ARGUES THAT “BONDS ARE BOUGHT FOR
THEIR PRICE APPRECIATION POTENTIAL AND
NOT FOR INCOME PROTECTION”
• MANY BOND INVESTORS SUBSCRIBE TO THIS
VIEW AND PURSUE ACTIVE STRATEGIES. THEY
SEEK TO PROFIT BY:
• FORECASTING INTEREST RATE CHANGES
AND/OR
• EXPLOITING RELATIVE MISPRICINGS
AMONG BONDS.
INTEREST RATAE FORECASTING - 1
IRF1
 MODELS BASED UPON FORECASTING EXPECTED
INFLATION
 EXPECTED INFL’N .. KEY DETERMINANT
 SOLID EVIDENCE .. LINK
+S
 RELATIVELY SIMPLE APPROACH
-S
 MAY NOT HELP IN SHORT TERM FORECASTING
 EXPECTED INFL’N .. NOT EASY .. PREDICT
 MODELS THAT FORECAST INTEREST RATES BASED UPON
PAST INTEREST RATE CHANGES
 THESE MODELS EMPHASIZE THE TIME SERIES
BEHAVIOR OF INTEREST RATES & USE DISTRIBUTED
LAGS OF PAST INTEREST RATES IN PREDICTING
FUTURE INT. RATES
+ SIMPLE .. INFOR’N AVAILABLES
SHIFTS .. FUNDAMENTAL FACTORS … BREAK IN
TRENDS
INTEREST RATE FORECASTING - 2

MODELS THAT ASSUME THAT INTEREST RATES MOVE IN
A NORMAL RANGE (WHICH IS KNOWN)
 MEAN REVERSION
+ IF THE NORMAL RANGE .. KNOWN … SIMPLE TO BUILD
… ONLY SPEED ADJUSTT FACTOR
- THE NORMAL RANGE .. MAY SHIFT OVER TIME IF
FUNDAMENTAL VARIABLES (LIKE INTEREST RATE SHIFT)

COMPREHENSIVE MULTI - SECTOR MODELS OF THE
ECONOMY THAT ATTEMPT TO PREDICT INTEREST
RATES
 MODEL ALL FLOWS .. ECONOMY
 S & D OF FUNDS
 IMBALANCE
INT. RATE CHANGES
+ COMPREHENSIVE … FUNDAMENTALS
- NUMEROUS INPUTS .. ERRORS IN THESE INPUTS …
ERRORS IN INTEREST RATE FORECASTS
INTEREST RATE FORECASTING - 3
PERFORMANCE
OF
FORECASTING MODELS
INTEREST
RATE
GENERALLY, IRFMs HAVE NOT PERFORMED WELL
IN FORECASTING SHORT - TERM MOVEMENTS.
THEY PERFORM BETTER IN EXPLAINING WHY
INTEREST RATES HAVE MOVED & IN PREDICTING
LONG - TERM MOVEMENTS IN INTEREST RATES.
HORIZON ANALYSIS
HORIZON ANALYSIS IS A METHOD OF FORECASTING THE
TOTAL RETURN ON A BOND OVER A GIVEN HOLDING
PERIOD. IT INVOLVES THE FOLLOWING STEPS.





SELECT A PARTICULAR INVESTMENT PERIOD AND
PREDICT BOND YIELDS AT THE END OF THAT
PERIOD.
CALCULATE THE BOND PRICE AT THE END OF THE
INVESTMENT PERIOD.
ESTIMATE THE FUTURE VALUE OF COUPON
INCOMES EARNED OVER THE INVESTMENT PERIOD.
ADD THE FUTURE VALUE OF COUPON INCOMES
OVER
THE
INVESTMENT
PERIOD
TO
THE
PREDICTED CAPITAL GAIN OR LOSS TO GET A
FORECAST OF THE TOTAL RETURN ON THE BOND
FOR THE HOLDING PERIOD.
ANNUALISE THE HOLDING PERIOD RETURN.
EXPLOITING RELATIVE
MISPRICINGS AMONG BONDS
• SUBSTITUTION SAWP
• PURE YIELD PICKUP SWAP
• INTERMARKET SPREAD SWAP
• TAX SWAP
INTEREST RATE SWAPS
AN INTEREST RATE SWAP (IRS) IS A TRANSACTION
INVOLVING AN EXCHANGE OF ONE STREAM OF
INTEREST OBLIGATIONS FOR ANOTHER.
KEY FEATURES
• NET INTEREST DIFFERENTIAL IS PAID OR
RECEIVED
• NO EXCHANGE OF PRINCIPAL REPAYMENT
OBLIGATIONS
• STRUCTURED AS A SEPARATE TRANSACTION
• OFF BALANCE SHEET TRANSACTION
INTEREST RATE SWAP
6.05%
Swap
Dealer
MIBOR
MIBOR
5.95%
Company A
6% coupon
Company B
MIBOR
SUMMING UP
• Interest rate risk is measured by the percentage change in the
value of a bond in response to a given interest rate change.
• The duration of a bond is the weighted average maturity of
its cash flow stream, where the weights are proportional to
the present value of cash flows.
• The proportional change in the price of a bond in response to
the change in its yield is as follows:
P/P  D* y
• The two commonly followed passive strategies for bond
portfolio management are: buy and hold strategy and
indexing strategy.
• If the duration of a bond equals the investment horizon, the
investor is immunised against interest rate risk.
• Those who follow an active approach to bond portfolio
management seek to profit by (a) forecasting interest rate
changes and /or (b) exploiting relative mispricings among
bonds.
• A wide range of models are used for interest rate forecasting.
• Horizon analysis is a method of forecasting the total return
on a bond over a given holding period.
• An interest rate swap is a transaction involving an exchange
of one stream of interest obligations for another.