chapter 7- bond management

8/13/2019 Chapter 7- Bond Management

1/17

1 | P a g e

-

Financial Mathematics- Actuarial 2013-2014

Jihane Bou Zakhem

Chapter 7

Bond Management

7.1 Macaulay Duration

Definition 7.1.1.

Suppose an investor purchases a n-year semiannual coupon bond forP0 at time 0 and

holds it until maturity.

As the amounts of the payments she receives are different at different times, one way to

summarize the horizon is to consider the weighted average of the time of the cashflows.

We use the present values of the cash flows (not their nominal valuesto compute theweights.

!onsider an investment that generates cash flows of amount Ct at time t = 1, , n,

measured in payment periods. Suppose the rate of interest is i per payment period andthe initial investment is P.

We denote the present value of Ct by PV(Ct), which is given by

"sing PV(Ct)as the factor of proportion, we define the weighted average of the time ofthe cash flows, denoted by D, as


2/17

2 | P a g e

-


Jihane Bou Zakhem

As wt 0for all t and wt= 1, wt are properly defined weights and Dis the weighted

average of t = 1, , n.

We call D the #acaulay duration, which measures the average period of the

investment. Dis calculated in terms of the number of payment periods

$f there are % payments per year and we desire to e&press the duration in years, we

replace tby t/k. 'he resulting value of Dis then the #acaulay duration in years.

Example 7.1.2.

!alculate the #acaulay duration of a -year annual coupon bond with )* coupon and

a yield to maturity of +.+*.

Solution:

Example 7.1.3.

!alculate the #acaulay duration of a -year semiannual coupon bond with * coupon

per annum and a yield to maturity of .* compounded semiannually.

Solution:

Definition 7.1..

!onsider a bond with face value (also the redemption value F, coupon rate r per

payment, and time to maturity of n payment periods. 'he rate of interest i applicable

is the yield to maturity per coupon-payment period.

ow Ct is e/ual to Frfor t = 1, , n - 1and Cn = F r + F, therefore


3/17

3 | P a g e

-


Jihane Bou Zakhem

We have P=(F r)an + Cvn . ence the #acaulay duration of the bond is (in terms of thenumber of payment periods.

Example 7.1.!.

!alculate the #acaulay duration of the bonds in 1&amples 2.3. and 2.3.4 using e/uation(2.3..

Solution:

7.2 Modified Duration

Definition 7.2.1.

While the #acaulay duration was originally proposed to measure the average

horizon of an investment, it turns out that it can be used to measure the price

sensitivity of the investment with respect to interest-rate changes.

'o measure this sensitivity we consider the derivative dP /di. As the price of the

investment P drops when interest rate i increases, dP /di < 0.

We consider (the negative of the percentage change in the price of the investment per

unit change in the rate of interest, i.e., -(dP /di)/P.


4/17

4 | P a g e

-


Jihane Bou Zakhem

and call it the modified duration, which is always positive and measures thepercentage decrease of the value of the investment per unit increase in the rate ofinterest.

Example 7.2.2.

!alculate the #odified duration of the bonds in 1&amples 2.3. and 2.3.4

Solution:

7.3 Duration for price correction

Definition 7.3.1.

We now consider the use of the modified duration to appro&imate the price change of a

bond when the rate of interest changes.

We denote P(i) as the price of a bond when the yield to maturity is i per coupon-

payment period.

When the rate of interest changes to i + i, the bond price is revised to P (i + i).

5or a continuous function (!) with first- and second-order derivatives, the function

evaluated at ! + !, i.e., (! + !), can be appro&imated by 'aylor6s e&pansion as

follows 7


5/17

5 | P a g e

-


Jihane Bou Zakhem

'hus, if we e&pand the bond price P (i+i)using 'aylor6s e&pansion up to the first-order

derivative, we obtain

ence, we can use the modified duration to obtain a linear appro&imation to the revised

bond price with respect to a change in the rate of interest.

ote that in the above formula , as i is per coupon-payment period, D" and ishould

also be measured in coupon-payment period.owever, we may also e&press D"in years, in which case iis the change in the rate of

interest per annum.

Example 7.3.2.

A 30-year semi-annual coupon bond with coupon rate of 2* is selling to yield ).+* per

year compounded semi-annually. What is the bond price if the yield changes to (a )*,

and (b ).2*, compounded semi-annually8

Solution:

5igure 2.4.4 illustrates the application of 9efinition 2.4.3. 'he relationship between the

bond price and the rate of interest is given by the curve, which is conve& to the origin.

'he final e/uation in definition 2.4.3. appro&imates the bond price using the straight line

which is tangent to the point (i, P (i)) with a negative slope of -P (i)D".

ote that due to the conve&ity of the relationship between the interest rate and the bond

price, the correction based on the modified duration always under-appro&imates the

e&act price.


6/17


7/17

7 | P a g e

-


Jihane Bou Zakhem

:ased on the definition 2.3.3 the conve&ity is

5or a bond investment, Ct 0for all t, so that C # 0, verifying the conve&ity relationship.

Example 7..2.

;evisit 1&ample 2.4.. and appro&imate the bond prices with conve&ity correction.

Solution:

7.! Some &ule$ for Duration

We summarize some useful rules for duration.

&ule 1: $%& 'aa*a dratin D a nd i a*wa *& t%an r &a* t it ti& t

atrit n. a*it %*d n* r a 2&r-3n nd.

&ule 2: 4*din5 t%& ti& t atrit n a nd ntant, w%&n t%& 3n rat&

int&r&t r d&r&a&, t%& 'aa*a dratin D inr&a&.


8/17

8 | P a g e

-


Jihane Bou Zakhem

&ule 3: 6t%&r t%in5 &in5 &a*, w%&n t%& i&*d t atrit i d&r&a&, t%& 'aa*a

dratin D inr&a&.

&ule : Fr a *&v&* 3&r3&tit, t%& dii&d dratin D" i &a* t 1/i.

&ule !: Fr a *&v&* annit n 3a&nt, t%& dii&d dratin i

'his rule can be proved by direct differentiation of the price of the annuity,

&ule ': $%& dii&d dratin D" a 3n nd wit% 3n rat& r 3&r 3a&nt, n

3a&nt t atrit and i&*d t atrit i i

&ule 7: Fr a 3n nd &**in5 at 3ar, t%& dii&d dratin i

&ule (: 4*din5 t%&r t%in5 ntant, a nd7 dratin D a** inr&a& wit% it ti&

t atrit n.

Suppose a portfolio of bonds is constructed from ' bonds, with durations D1 D'.


9/17

9 | P a g e

-


Jihane Bou Zakhem

Example 7.!.1.

A bond manager has a choice of two bonds, A and :. :ond A is a -year annual coupon

bond with coupon rate of )*. :ond : is a -year annual coupon bond with coupon rate

of *. 'he current yield to maturity in the mar%et is +.+* per annum for all maturities.

ow does the manager construct a portfolio of >300 million, consisting of bonds A and

:, with #acaulay duration of .+ years8

Solution:

7.' )mmuni*ation Strategie$

5inancial institutions are often faced with the problem of meeting a liability of a given

amount sometime in the future.

We consider a liability of amount Vto be paid $periods later.

A simple strategy to meet this obligation is to purchase a zero- coupon bond with face

value V, which matures at time $.


10/17

10 | P a g e

-


Jihane Bou Zakhem

'his strategy is called ca$h#flo+ matching.

When cash-flow matching is adopted, the obligation is always met, even if there is

fluctuation in the rate of interest.

owever, zero-coupon bonds of the re/uired maturity may not be available in the

mar%et.

$mmunization is a strategy of managing a portfolio of assets such that the business is

immune to interest-rate fluctuations.

5or the simple situation above, the target#date immuni*ation strategy may be adopted.

'his involves holding a portfolio of bonds that will accumulate in value to Vat time $at

the current mar%et rate of interest.

'he portfolio, however, should be constructed in such a way that its #acaulay duration

Dis e/ual to the targeted date of the liability $.

Suppose the current yield rate is i, the current value of the portfolio of bonds, denoted

by P (i), must be

P (i) = V/ (1 + i)$

$f the interest rate remains unchanged until time $, this bond portfolio will accumulatein value to ? at the maturity date of the liability.

$f interest rate increases, the bond portfolio will drop in value. owever, the coupon

payments will generate higher interest and compensate for this.

@n the other hand, if interest rate drops, the bond portfolio value goes up, with

subse/uent slow-down in accumulation of interest.

"nder either situation, as we shall see, the bond portfolio value will finally accumulate to

V at time $, provided the portfolio6s #acaulay duration Dis e/ual to $.

We consider the bond value for a n&-ti& a** %an5&in the rate of interest.

$f interest rate changes to i + i immediately after the purchase of the bond, the bond

price becomes P (i + i) which, at time $ , accumulates to P (i + i)(1 + i + i)$ if the

rate of interest remains at i + i.

We appro&imate (1 + i + i)$ to the first order in i to obtain (apply 'aylor6s

e&pansion to (i) = (1 + i)$)


11/17

11 | P a g e

-


Jihane Bou Zakhem

owever, as D" = D/(1+i)and $ = D, the above e/uation becomes

Example 7.'.1.

A company has to pay >300 million 4.)2)3 years from now. 'he current mar%et rate of

interest is +.+*. 9emonstrate the funding strategy the company should adopt with the

)* annual coupon bond in 1&ample 2.3.. !onsider the scenarios when there is an

immediate one-time change in interest rate to (a +*, and (b )*.

Solution:

Example 7.'.2.

A company has to pay >300 million years from now. 'he current mar%et rate of

interest is +.+*. 'he company uses the )* annual coupon bond in 1&ample 2.3. to

fund this liability. $s the bond sufficient to meet the liability when there is an immediate

one-time change in interest rate to (a +*, and (b )*8

Solution:

5igure 2.).4 describes the wor%ing of the target-date immunization strategy.

$f a financial institution has multiple liability obligations to meet, the manager may

adopt cash-flow matching to each obligation.

'his is a dedication $trategy in which the manager selects a portfolio of bonds (zero-

coupon or coupon bonds to provide total cash flows in each period to match the

re/uired obligations.


12/17

12 | P a g e

-


Jihane Bou Zakhem

B 'he manager may also consider the liability obligations as a whole and construct a

portfolio to fund these obligations with the ob=ective of controlling for the interest-rate

ris%. A commonly adopted strategy is duration matching.

"igure 7.'.3: )llu$tration of target#date immuni*ation

We assume a financial institution has a stream of liabilities 91, 9: 9;to be paid out at various

times in the future.

$t will fund these liabilities with assets generating cash flows1, :, 'at various times in the

future.

We assume that the rate of interest i am fiat for cash flows of all maturities and apply to both

assets and liabilities.


13/17

13 | P a g e

-


Jihane Bou Zakhem

We denote the #acaulay durations of the assets and liabilities byDand D9, respectively.

'he duration matching strategy involves constructing a portfolio of assets such that the

following conditions hold7

3. V V9

. D= D9.

!ondition ensures that, to the first-order appro&imation, the asset- liability ratio will

not drop when interest rate changes. 'his result can be deduced as follows7

Example 7.'..

A financial institution has to pay >3,000 after years and >,000 after years. 'he

current mar%et interest rate is 30*, and the yield curve is assumed to be fiat at any

time. 'he institution wishes to immunize the interest rate ris% by purchasing zero-

coupon bonds which mature after 3, 4 and + years. @ne member in the ris%management team of the institution, Alan, devised the following strategy7

B Curchase a 3-year zero-coupon bond with a face value of >.2,

B Curchase a 4-year zero-coupon bond with a face value of >,+0.4,

B Curchase a +-year zero-coupon bond with a face value of >+00.00.

(a 5ind the present value of the liability.

(b Show that Alan6s portfolio satisfies the conditions of the duration matching


14/17

14 | P a g e

-


Jihane Bou Zakhem

strategy.

(c 9efine surplus = V - V9, calculate S when there is an immediate one-time

change of interest rate from 30* to (i D*, (ii 33*, (iii 3+*, (iv 40* and (v0*.

(d 5ind the conve&ity of the portfolio of assets and the portfolio of liabilities at i

E 30*.

Solution:

'o protect the asset-liability ratio from dropping when interest rate changes, the

;edington immunization strategy, named after the :ritish actuary 5ran% ;edington,

imposes the following three conditions for constructing a portfolio of assets7

3. V V9

. D= D9.

4. C # C9.

Example 7.'.!.

5or the financial institution in 1&ample 2.)., a ris% consultant, Alfred, recommended the

following strategy7

Curchase a 3-year zero-coupon bond with a face value of >3+.3),

Curchase a 4-year zero-coupon bond with a face value of >,3).0,

Curchase a +-year zero-coupon bond with a face value of >))0.3.

(a Show that Alfred6s portfolio satisfies the three conditions of the ;edingtonimmunization strategy.

(b 9efine surplus = V - V9, calculate S when there is an immediate one-time

change of interest rate from 30* to (i D*, (ii 33*, (iii 3+*, (iv 40* and (v 0*.

Solution:


15/17

15 | P a g e

-


Jihane Bou Zakhem

"nder certain conditions, it is possible to construct a portfolio of assets such that the

net-worth position of the financial institution is guaranteed to be non-negative in any

positive interest rate environment.

A full immunization strategy is said to be achieved if under any one-time shift of interest

rate from i0 to i,

(i) = V(i) - V9(i) 0, r i # 0.

We consider the e&ample of a single liability of amount 9to be paid $9periods later.

5ull immunization strategy involves funding the liability by a portfolio of assets which

will produce two cash inflows. 'he first inflow of amount1is located at time $1, which

is 1periods before time $9. 'he second inflow of amount : is at time $:, which is :periods after time $9.

5igure 2.).) illustrates these three cash flows . $t should be noted that all the values of

i0, i,1, :, 9, 1, :, $9, $1 and $: are positive, and 3 is not necessarily e/ual to:.

$n this particular e&ample, the conditions for constructing a portfolio of assets under the

full immunization strategy are7

3. V= V9

. D= D9

"igure 7.'.'

'he above conditions can be rewritten as7


16/17

16 | P a g e

-


Jihane Bou Zakhem

Example 7.'.7.

5or the financial institution in 1&amples 2.). and 2.).+, an actuary, Albert, constructed

the following strategy7

B Curchase a 3-year zero-coupon bond with a face value of >+.++,

B Curchase a 4-year zero-coupon bond with a face value of >3,+D.0D,

B Curchase a +-year zero-coupon bond with a face value of >3,300.00.

(a Show that Albert6s portfolio satisfies the conditions of the full immunization strategy.

(b 9efine surplus = V - V9, calculatewhen there is an immediate one-time change of

interest rate from 30* to (i D*, (ii 33*, (iii 3+*, (iv 40* and (v 0*.

Solution:

'o compare the duration matching, ;edington and full immunization strategies, theresults of 1&amples 2.)., 2.).+ and 2.).2 are plotted in 5igure 2.)..


17/17

17 | P a g e

-


7.7 ,a$$i%e %er$u$ -cti%e Bond Management

A bond fund may adopt a passive or active strategy.

A passive strategy adopts a non e&pectational approach, without analyzing the li%ely

movements of the mar%et. $mmunization, inde&ing and buy-and-hold are passive bond

management strategies.

An active bond management strategy may involve some form of interest-rate

forecasting.

A broader active management framewor% would ta%e a /uantitative approach in

assessing the value of a bond, ta%ing into account all embedded options and structuresof the bond, and includes assessment of the sector of the bond issuer and its credit

profile.

chapter 7- bond management

Documents