Chapter 12

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 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

-5 -4 -3 -2 -1 0 1 2 3 4 5 -50

0

50

100

150

200

A B C D

Initial Bond Coupon Maturity YTM A 12% 5 years 10% B 12% 30 years 10% C 3% 30 years 10% D 3% 30 years 6%

Change in yield to maturity (%)

Perc

enta

ge c

hang

e in

bon

d pr

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Duration - 1Duration 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 Ct x t

t=1 (1+r)t

D =n Ct

t=1 (1+r)t

t = Time period in which the coupon / principal payment occursCt = Interest & / or principal … t

r = Market yield on the bond

Duration - 2To illustrate how duration is calculated consider bond a.

Bond AFace Value RS 100Coupon (Interest Rate) 15 Percent payable annuallyYears to maturity 6Redemption value RS 100Current market price RS 89.50Yield to maturity 18 Percent

Calculation of DurationBOND A : 15 PERCENT COUPON

YEAR CASH FLOW PRESENT VALUE PROPORTION OF PROPORTION OF THE AT 18 PER CENT THE BOND'S VALUE BOND'S VALUE TIME 1 15 12.71 0.142 0.142 2 15 10.77 0.120 0.241 3 15 9.13 0.102 0.306 4 15 7.74 0.086 0.346 5 15 6.56 0.073 0.366 6 115 42.60 0.476 2.856 DURATION 4.257 YEARS

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 Modified Change in yield

change duration in decimal form

Example

D* = 3.608

y = 0.2 percent

P/P – 3.608 x 0.2 = – 0.722 percent

— X

Properties Of Duration–1

1. The duration of a zero coupon bond is the same as it 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 Duration6. 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 Bond1 + y (1 + y) + T (c - y)

- y c [(1 + y)T - 1] + y

C : Coupon rate per payment periodT : Number of payment periodsy : 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.

-5 - 4 - 3 - 2 - 1 0 1 2 3 4 5

- 40

- 20

20

40

60

80

•

Percentagechange in bond price

0

Change in YTM(percentage points)

Convexity20-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.

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.

Formula for Convexity

The formula for computing convexity is given below:

Convexity =Convexity =

nn

tt = 1 = 1

((tt22 + + tt) x ) x CCtt

(1 + (1 + yy) ) tt

PP x (1+ x (1+yy))22

where Ct is the cash flow at the end of year t, y is the yield to maturity, and P is the price of the bond.

The convexity of the bond described is:

Convexity Convexity ==

(1(122+1)x15+1)x15 ++ (2(222+2)x15+2)x15 ++ (3(322+3)x15+3)x15 ++ (4(422+4)x15+4)x15 ++ (5(522+5)x(115)+5)x(115)

(1.18)(1.18) (1.18)(1.18)22 (1.18)(1.18)33 (1.18)(1.18)44 (1.18)(1.18)55

89.5 (1.18)89.5 (1.18)22

= 14.94= 14.94

(12.5)

By using both duration and convexity we can estimate more accurately the effect of interest rate change on bond price changes. Adding the effect of convexity to Eq. (12.3) results is:

% change in bond price = - modified duration x % yield change +½ x convexity x (yield change)2

In the above example it works out to:

-3.608 x (0.002) x ½ x 14.94 (0.002)2

= -0.007216 + .000030 = -0.007186 = - 0.7186 or - 0.7186 percent

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 - 1Capital Value

Interest Rate

Return on re-investment ofinterest

Capital Value

Interest Rate

Return on re-investment of interest

For immunisation set duration equal to investment horizon

Duration And Immunisation - 2May 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

IllustrationAn 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

Bond A Bond B

Par value Rs.1,000 Rs.1,000Market price Rs.1,000 Rs.1,000Coupon rate 12.6% 12.6%Yield to maturity 12.6% 12.6%Maturity period 4 years 5 yearsDuration Less than 4 years 4 yearsRating A A

Exhibit 12.6 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 : Bond A: Market Yield Remains at 12.6%

Year Cash flow Reinvestment rate

Accumulated value

1 Rs. 126 12.6% 126 (1.126)3 = 179.9 2 Rs. 126 12.6% 126 (1.126)2 = 159.8 3 Rs. 126 12.6% 126 (1.126)1 = 141.9 4 Rs. 1126 NA 1126.0 Total 1607.6

Part II Bond A: Market Yield Falls to 10% in year 2

Year Cash flow Reinvestment rate

Accumulated value

1 Rs. 126 12.6% 126 (1.10) (1.10) (1.10) = 167.7 2 Rs. 126 10.0% 126 (1.10) (1.10) = 152.5 3 Rs. 126 10.0% 126 (1.10) = 138.6 4 Rs. 1126 NA 1126.0

Total 1584.8

contd

Terminal Value With Bonds A and BPart III Bond B: Market Yield Remains at 12.6%

Year Cash flow Reinvestment rate

Accumulated value

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

Year Cash flow Reinvestment rate

Accumulated value

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

bond) NA 1023.6

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 Rate Forecasting-2 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 available- 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 adjust 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 interest rate forecasting models.

Generally, firms 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 AnalysisHorizon 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.

Example of Horizon AnalysisAn example may be given to illustrate horizon analysis. A Rs.100,000 par 10-year maturity bond with a 10 percent coupon rate (paid annually) currently sells at a yield to maturity of nine percent. A portfolio manager wants to forecast the total return on the bond over the coming two years, as his horizon is two years. He believes that two years from now, eight-year maturity bonds will sell at a yield of eight percent and the coupon income can be reinvested in short-term securities over the next two years at a rate of seven percent.

The two-year return on the bond is calculated as follows.Current price = 10,000 x PVIFA (9%, 10 years) + 100,000 x PVIF (9%, 10 years)

= 10,000 x 6.418 + 100,000 x0.422 = Rs 106,380Forecast price = 10,000 xPVIFA (8%, 8 years) + 100,000 xPVIF (8%, 8 years)

= 10,000 x5.747 + 100,000 x0.540 = Rs 111,470

Future value of reinvested coupons = 10,000 (1.07) + 10,000 = 20,700Two-year return = = 0.242 or 24.2%

The annualised rate of return over the two-year period would be: (1.242)0.5 – 1 = 0.114 or 11.4 percent.

Riding the Yield Curve

A particular version of horizon analysis is riding the yield curve. An

investor pursuing this strategy buys an intermediate or long-term

bond when the yield curve is upward sloping and the investor

expects the yield curve to remain unchanged. As the bond

approaches maturity it moves toward the lower end of the upward-

sloping yield curve and hence appreciates in value. Thus, the

investor earns interest as well as enjoys capital appreciation. The

risk in this strategy is that the level of interest rates may rise or the

yield curve may become downward sloping thereby causing the bond

value to erode.

Maturity – Based Strategies

A bond portfolio consisting of only short-term bonds earns a modest

yield but enjoys a high degree of price stability. On the other hand, a bond

portfolio consisting of only long-term bonds earns a relatively high yield

but is subject to greater price volatility. Bond investors seeking a higher

interest income without much price volatility can follow a maturity-based

strategy called laddering.

Laddering involves constructing a bond portfolio with a series of

increasing maturities, resembling a bond maturity “ladder.” For example,

consider an investor who has projected financial needs in 2, 4, 6, 8, 10, and

12 years. Such an investor can construct a portfolio by buying appropriate

amounts of bonds maturing in 2, 4, 6, 8, 10, and 12 years.

Barbell

A barbell strategy involves concentrating the portfolio at the two

ends of the maturity spectrum. An investor following such a strategy

invests in Treasury bills (maturing within a year) and 30-year

Treasury Bonds. Such a portfolio produces moderate interest

income with moderate price volatility.

Exploiting Relative

Mispricings Among Bonds

• Substitution swap

• Pure yield pickup swap

• Intermarket spread swap

• Tax swap

Contingent Immunisation-1

Contingent immunisation is a hybrid passive-active strategy. To illustrate,

suppose that the interest rate is currently 8 percent and the value of a bond

portfolio Rs.100 million. At the current interest rate, the portfolio

manager, using the conventional immunisation techniques, can lock in a

future value of Rs.136.05 million four years hence. Now assume that the

portfolio manager has to ensure that the terminal value of the bond

portfolio at the end of four years is at least Rs.125 million. Subject to this

constraint, he is willing to pursue an active strategy. Because Rs.91.88

million is required to achieve a terminal value of Rs.125 million (91.88 x

1.084 = 125) and the portfolio is currently worth Rs.100 million, the

portfolio manager has some cushion available at the outset. So he can start

off with an active strategy, rather than immunise immediately.

Contingent Immunisation-2

When should he resort to immunisation ? To answer this question,

we have to calculate the fund required under the immunisation

strategy to provide Rs.125 million at the horizon date (four years

from now). If T is the time left until the horizon date and r is the

prevailing interest rate, the fund required to guarantee a target

terminal value of Rs.125 million is Rs.125 million / (1 + r)T . This

value acts as the trigger point. As long as the fund value exceeds the

trigger value, the portfolio manager can pursue an active strategy.

Interest Rate SwapsAn 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% 5.95%

6% coupon MIBOR

Swap Dealer

Company A

Company B

MIBOR MIBOR

CAPM and Bond Returns

Since the major bond risks are largely non diversifiable, bond

returns can perhaps be explained in the context of the CAPM, which

contends that security returns are related to non diversifiable

market risk. Yet, only few studies have been attempted because of

data collection problems.

There is mixed evidence on the usefulness of the CAPM for

the bond market. There are problems on account of the lack of

clarity about the appropriate market index to use and the instability

of the systematic risk measure. It appears that the risk-return

relationship using beta does not hold for the higher quality bonds.

Bond Market Efficiency

• In the context of fixed income securities, the weak-form and

the semi strong-form of EMH have been examined

• There is support for the weak-form EMH.

• There is mixed evidence for strong-form efficiency.

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 an 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.