MANAGING BOND PORTFOLIOS: RISK AND RETURN

Session 10
SESSION PLAN
▷INTRODUCTION

▷RISKS OF BOND INVESTING

▷UNDERSTANDING INTEREST RATE RISK OF THE

PORTFOLIO

▷DURATION AND CONVEXITY

▷MANAGING BOND PORTFOLIOS

PORFOLIO IMMUNIZATION

▷PASSIVE AND ACTIVE BOND PORTFOLIO MGT

INTRODUCTION
Bond market discussion at elementary level largely focusses
on risk free bonds or Treasury securities. A large part is also
non-gilt edged market.

Market prices contain information of

1. INTEREST RATE EXPECTATION

2. LIKELIHOOD OF DEFAULT
BONDS
Coupon or zero coupon
Maturity
Spreads : Difference in yields of similar maturity
Buckets: Differing risk maturity profiles
Issuer of securities
Cellular or Granular approach to classification
RISKS OF BOND INVESTING
1. CREDIT RISK OR COUNTER PARTY RISK

2. INTEREST RATE RISK

3. LIQUIDITY RISK

4. REINVESTMENT RISK

5. FOREX RISK
INTEREST RATE RISK
INTEREST RATE SENSITIVITY
▷Sensitivity of bond prices to the change in the
interest rate is most critical for bond portfolios.

▷Example
PROPOSITIONS
1. Bond prices and yields are inversely related.
As yields increase, bond prices fall; as yields
fall, bond prices rise.

2. An increase in a bond’s YTM results in a

smaller price change than a decrease in yield
of equal magnitude.

A and B bonds, same coupon, B has higher

maturity. B Shows greater sensitivity
PROPOSITIONS
3. Prices of long term bonds tend to be more
sensitive to interest rate changes than prices of
short term bonds.

4. The sensitivity of bond prices to changes in

yield increases at a decreasing pace as
maturity increases. In other words interest
rate risk is less than proportional to bond
maturity.

Bonds B and C same except coupon and can see

lower coupon bonds show more sensitivity
PROPOSITIONS
5. Interest rate risk is inversely proportional to
the bonds coupon rate

Bonds C and D are similar except YTM, D is

more sensitive (lower YTM)

6. The sensitivity of a bond’s price to a

change in its yield is inversely related to the
yield to maturity at which the bond currently is
selling.
DURATION AND CONVEXITY
DURATION
▷DURATION IS THE PERIOD BY WHICH A BOND
HOLDER NEED TO WAIT TO RECEIVE THE PV OF
CASH PAYMENT.

▷ZERO COUPON BONDS = n

▷COUPON BEARING BONDS < n

DURATION
Measure of Portfolio’s exposure to yield curve
movement.

▷Suppose y is bond yield and B is market price of

the bond. The duration is defined as

D = - 1/B * ΔB/Δy
So that
ΔB = -D *B * Δy

▷or
B
= − Dy
B
CALCULATION DURATION
▷3 Year bond, coupon 10% and yield 12%
 EXAMPLE
▷BOND PRICE B = 94.2130
▷DURATION D = 2.653

▷CHANGE IN BOND PRICE IS

ΔB = -B *D * Δy
= -94.213 * 2.653* Δy
= -249.95* Δy

IF YIELDS INCREASES BY 10 BPS = 0.001

= -249.95*0.001
= -0.2500

BOND PRICE GOES DOWN TO 94.2130-0.2500 = 93.9630

PRICE AND DURATION
▷3 Year bond, coupon 10% and yield 12.1%
MODIFIED DURATION

▷When the yield y is expressed with compounding m

times per year

BDy
B = −
1+ y m
▷The expression

D
1+ y m
Is referred to as “modified duration”
MODIFIED DURATION
▷3 Year bond, coupon 10% and yield 12%, with semi-
annual compounding.

With continuous compounding 12% rate semi-annual

works out to 12.3673%
D
1+ y m
This works out to 2.653/ 1+ 0.1236/2 = 2.4985
MODIFIED DURATION
▷Using this equation ΔB = -94.213*2.4985Δy
OR

ΔB = 235.39 Δy
IF YIELDS INCREASES BY 10 BPS = 0.001
= -235.39*0.001
= -0.2353

BOND PRICE GOES DOWN TO 94.2130-

0.2353 = 93.9776
PRICE AND DURATION
▷3 Year bond, coupon 10% and yield 12.1%
FOR SMALL CHANGES
▷DURATION AND MODIFIED DURATION GIVES
GOOD RESULTS FOR SMALL CHANGES IN YIELDS

▷FOR SMALL CHANGES NO DIFFERENCE

DURATION AND CONVEXITY
▷CAN BE COMPARED TO RISK MEASURES
KNOWN AS GREEKS: DELTA AND GAMMA.
FOR LARGE CHANGES
▷Bonds behave differently when there are large
changes in the yield and for that we use Convexity
CONVEXITY
CONVEXITY
▷CONVEXITY MEASURES THE CURVATURE

▷If y is the Bond Yield measured with continuous

compounding, Ci is cash flow at time i, t is time for
which the bond is providing cash flows, B is the
price of the Bond, then

▷Convexity can be defined as:

SECOND ORDER APPROXIMATION
CONVEXITY
▷Convexity of a bond is defined as:

▷Using Taylor Series ( SEE EQUATION G.2)

n
− yt i
 ci t i e
2
2
1 d B i =1
C = 2
=
B dy B

which leads to

B 1
= − Dy + C ( y )
2

B 2
EXAMPLE
▷CONSIDER THE BOND DISCUSSED EARLIER, PRICE
OF THE BOND IS 94.2130, DURATION D IS 2.653,
WE CAN FIND CONVEXITY:
EXAMPLE
▷USING CONVEXITY RELATIONSHIP

▷ΔB/B = -D* Δy + 1/2 * C * Δy^2

= -2.653* Δy + 0.5 * 7.5700* (Δy)^2

▷YIELDS INCREASE FROM 12% TO 14%

▷DURATION GIVES = -4.999

▷CONVEXITY GIVES = -4.856

▷ACTUAL CHANGE IS -4.859

MANAGING BOND PORTFOLIOS
PORTFOLIO IMMUNIZATION
▷Immunization techniques refer to strategies
used by fund managers and investors to shield
their portfolio from interest rate risk
ACTIVE VS PASSIVE
▷Active Bond Portfolios

▷Passive Portfolio
PASSIVE BOND PORTFOLIOS
PASSIVE BOND PORTFOLIO
▷Pure Bond Index Matching

▷Enhanced Indexing

▷Lehman Aggregate Bond Index consisted

6766 issues, 109 Treasury, 693 Federal agency
issues, 3482 credit (corporate and foreign
government), 719 asset backed issues, 1321
commercial mortgage backed sec, 442
mortgage backed issues (pools)
EXAMPLE
▷An insurance company has issued a Guaranteed
Insurance contract (GIC) for $10000. It has 5 year
maturity and rate 8%, Insurance company promised to
pay 10000*(1.08)^5 = 14693.28 in 5 years.

▷Option 1: Bond of 10000, 8% coupon 6 years

▷Scenario 1: Interest rate rise 9%,

CAPITAL LOSS
REINVESTMENT AT HIGHER RATE

If portfolio duration chosen carefully, these two effects will cancel out.

FOR A HORIZON EQUAL TO PORTFOLIO DURATION,

PRICE RISK AND REINVESTMENT RISK ARE
OFFSETTING
CASH FLOW MATCHING
▷WHY NOT SIMPLY BUY A ZERO COUPON
BOND WITH FACE VALUE EQUAL TO THE
PROJECTED CASH OUTLAY.

▷Interest rate risk is zero

▷On a multi-period basis this is known as

DEDICATION STRATEGY
PASSIVE BOND FUNDS IN INDIA
▷ICICI Pru PSU Bond Plus SDL 40:60 Index
Fund – September 2027 (Index of 8 AAA
PSUs and 20 SDLs in 40:60 ratio)
▷ABSL Nifty SDL Plus PSU Bond Sep 26
60:40
▷Edelweiss Nifty SDL Plus PSU Bond 60:40
▷Edelweiss Bharat Bond Index

▷Axis
▷Nippon
TARGET FUNDS
▷AXIS AAA BOND PLUS SDL 2026 MATURITY
▷NIPPON INDIA ETF NIFTY CPSE BOND PLUS
SDL 2024 MATURITY
▷NIPPON INDIA ETF NIFTY SDL -2026
MATURITY
▷IDFC GILT 2027 INDEX FUND
▷IDFC GILT 2028 INDEX FUND
ACTIVE BOND PORTFOLIOS
WHY ACTIVE BOND PORTFOLIO
▷Potentially better returns

(i) Interest rate forecast

(ii) Relative mispricing of securities

When can these be successful?

REBALANCING PORTFOLIO
1. The Substitution Swap

2. The Inter-market spread Swap

3. The Rate Anticipation Swap

4. Pure Yield Pick-up Swap

ACTIVE BOND PORTFOLIO
▷Example of Income Plan

G-Sec
State Development Loans
State Guaranteed Debt
AAA corporate Bonds
AA
A
BBB and less rated bonds

Actual vs Perceived Risk 250-400 bps spread

HORIZON ANALYSIS
▷Select a particular Holding period say 2 years

▷Predict the yield curve at the end of the

period.

▷Given the remaining maturity at the end of

holding price, yield and price can be calculated.

▷Add coupon income and prospective capital

gain or loss to find the total return.
EXAMPLE: HORIZON ANALYSIS
▷Face Value = 1000
▷Maturity = 20 years
▷Coupon = 10 % (annual)
▷YTM =9%

▷Holding period or Horizon is 2 years

▷Expectation of interest after 2 years 10%
▷Reinvestment rate is 8%
PORTFOLIO DURATION AND
CONVEXITY
A GENERALISATION
▷Suppose P is the value of the portfolio of interest
rate dependent securities:

▷D = - 1/P * ΔP/Δy
▷Where Δy is size of small parallel shift
▷Or

▷ΔP/ P = -D* Δy
CONVEXITY GENERALISATION
▷A portfolio consists of NUMBER OF INTEREST rate
dependent securities. The ith asset is worth Xi, and
has duration Di( i = 1,2,3,….n), duration of the
portfolio will be:

▷SIMILARLY FOR CONVEXITY

▷ΔP/ P = -D* Δy+ 1/2C (Δy)^2

PORTFOLIO C AND D

▷The duration of a portfolio is the weighted

average of the durations of the components
of the portfolio. Similarly for convexity.
WHAT DO THEY MEASURE

▷Duration measures the effect of a small

parallel shift in the yield curve

▷Duration plus convexity measure the effect

of a larger parallel shift in the yield curve

▷BUT NOT NON-PARALLEL

STARTING ZERO CURVE

4
Zero Rate (%)

0
0 2 4 6 8 10 12
Maturity (yrs)
PARALLEL SHIFT

4
Zero Rate (%)

0
0 2 4 6 8 10 12

Maturity (yrs)
PARTIAL DURATION

▷A partial duration calculates the effect on

a portfolio of a change to just one point on
the zero curve
PARTIAL DURATION

4
Zero Rate (%)

0
0 2 4 6 8 10 12
Maturity (yrs)
PARTIAL DURATION
▷Can be used to find out the effect of any
changes in the yield curve

⚫ For example, to define a rotation we

could change the 1, 2, 3, 4, 5, 7, and 10
year maturities by −3e, − 2e, − e, 0, e, 3e,
6e for some small e

⚫ SHOWN IN 9.6
EXAMPLE 9.5
COMBINING PARTIAL DURATION:
A ROTATION OF YIELD CURVE
PORTFOLIO IMMUNIZATION

▷Protecting a portfolio of long and short

position in interest rate dependent assets
against small parallel shifts in yield curve
by ensuring its duration is zero. For large
shifts both D and C should be close to zero.
 “
▷KEY IS FORECASTING
INTEREST RATE MOVEMENTS
TO MANAGE RISK
Thanks for Your
time and Attention!