bond primer slides 2007

8/12/2019 Bond Primer Slides 2007

1/71

Derivatives: A Primer on Bonds

First Part: Fixed Income Securities

Bond Prices and Yields

Term Structure of Interest Rates

Second Part: TSOIR


Interest Rate Risk & Bond Portfolio Management


2/71

Bond Prices and Yields

Time value of money and bond pricing

Time to maturity and risk

Yield to maturity

vs. yield to call

vs. realized compound yield

Determinants of YTM

risk, maturity, holding period, etc.


3/71

Bond Pricing

Equation:

P= PV(annuity) + PV(final payment)

=

Example: Ct= $40; Par = $1,000; disc. rate = 4%; T=60

)1()1(1 r

Par

r

couponT

T

tt

000,1$06.95$94.904$)04.01(

000,1$

)04.01(

40$60

60

1

t

tP


4/71

Prices vs. Yields

P yield

intuition

convexity

BKM6 Fig. 14.3; ; BKM4 Fig. 14.6

intuition: yield P price impact


5/71

Measuring Rates of Return on Bonds

Standard measure: YTM

Problems

callable bonds: YTM vs. yield to call

default risk: YTM vs. yield to expected default

reinvestment rate of coupons

YTM vs. realized compound yield

Determinants of the YTM

risk, maturity, holding period, etc.


6/71

Measuring Rates of Return on Bonds 2

Yield To Maturity definition

discount rate such that NPV=0 interpretation

(geometric) average return to maturity

Example: Ct= $40; Par = $1,000; T=60; sells at par

%4)1(

000,1$

)1(

40$000,1

60

60

1

yyy tt


7/71


Yield To Call definition

discount rates.t.NPV=0, with TC = earliest call date deep discount bonds vs. premium bonds

BKM6 Fig. 14.4; ; BKM4 Fig. 14.7

Example: Ct= $40, semi; Par = $900; T=60; P = $1,025;callable in 10 years (TC=20), call price = $1,000

%4)1(

000,1$

)1(

40$025,1

20

20

1

ytcytcytct

t


8/71


Yield To Default definition

discount rates.t.NPV=0, with TD= expected default date

default premium and business cycle

economic difficulties and flight to quality

Example: Ct= $50, semi; Par = $1,000; T=10; P = $200;expected to default in 2 years (TC=4), recover $150

)1(

150$

)1(

50$200

4

4

1 ytdytdtt


9/71


Coupon reinvestment rate YTM assumption: average

problem: not often true

solution: realized compound yield forecast future reinvestment rates

compute future value (BKM6 Fig.14.5; BKM4 Fig.14.9)

compute the yield (rcy) such that NPV = 0

practical?

need to forecast reinvestment rates


10/71

Bond Prices over Time

Discount bonds vs. premium bonds coupon rate < market interest rates

built-in capital gain (discount bond)

coupon rate > market interest rates

built-in capital loss (premium bond)

Behavior of prices over time

BKM6 Fig. 14.6; BKM4 Fig. 14.10

Tax treatment capital gains vs. interest income


11/71

Discount Bonds

OID vs. par bonds original issue discount (OID) bonds

less common

coupon need not be 0

par bonds

most common

Zeroes what? mostly Treasury strips

how? certificates of accrual, growth receipts, ...

annual price increase = 1-year disc. factor

(BKM6 Fig. 14.7; BKM4 Fig. 14.11)


12/71

OID tax treatment -- Discount Bonds 2

Idea for zeroes

built-in appreciation = implicit interest schedule

tax the schedule as interest, yearly

tax the remaining price change as capital gain or loss

Other OID bonds

same idea

taxable interest = coupon + computed schedule


13/71


Example

30-year zero; issued at $57.31; Par = $1,000

compute YTM:

1styear taxable interest

%10)1(

000,1$31.57$

30

y

y

73.5$31.57$04.63$%)101(

000,1$

%)101(

000,1$3029


14/71


Example (continued)

interests on 30-year bonds fall to 9.9%

capital gain

tax treatment: taxable interest= $5.73; capital gain

41.7$31.57$72.64$)1.01(

000,1$

)099.01(

000,1$3029

68.1$04.63$72.64$)1.01(

000,1$

)099.01(

000,1$2929


15/71


Basic question link between YTM and maturity

Bootstrapping short rates from strips forward rates and expected future short rates

Recovering short rates from coupon bonds

Interpreting the term structure

does the term structure contain information?

certainty vs. uncertainty


16/71

Terminology

Term structure = yield curve (BKM6 Fig. 15.1)

= plot of the YTM as a function of bond maturity

= plot of the spot rate by time-to-maturity

Short rate vs. spot rate

1-period rate vs. multi-period yield

spot rate = current rate appropriate to discount a

cash-flow of a given maturity

BKM6 Figure 15.3; BKM4 Figure 14.3


17/71

Extracting Info re:Short Interest Rates

From zeroes non-linear regression analysis

bootstrapping

From coupon bonds

system of equations

regression analysis (no measurement errors)

Certainty vs. uncertainty

forward rate vs. expected future (spot) short rate


18/71

Bootstrapping Fwd Rates from Zeroes

Forward rate break-even rate BKM Fig. 15.4

equates the payoffs of roll-over and LT strategies

Uncertainty no guarantee that forward = expected future spot

General formula

f1= YTM1 and 1)1(

)1(1

1

n

n

n

nn

YTM

YTMf

)1()1()1( 11 nn

n

n

n fYTMYTM


19/71

Bootstrapping Fwd from Zeroes 2

Data

BKMTable 15.2 & Fig. 15.1

4 bonds, all zeroes (reimbursable at par of $1,000)

T Price YTM

1 $925.93 8%

2 $841.75 8.995% 3 $758.33 9.66%

4 $683.18 9.993%


20/71

Bootstrapping Fwd Rates from Zeroes 3

Forward interest rate for year 1

Forward interest rate for year 2

%8

)11(

000,1$

)11(

000,1$93.925$ 11

yf

yf

)2

1(

93.925$

)2

1%)(81(

000,1$

)2

1)(1

1(

000,1$

275.841$

ffffP

%102

)2

1(

93.925$75.841$

f

f


21/71

Bootstrapping Fwd Rates from Zeroes 4

Short rate for years 3 and 4

keep applying the method

you findf3= 11% =f4

General Formula

f1= YTM1

1

1)1(

)1(1

n

n

n

nn

YTM

YTMf


22/71

Yield, Maturity and Period Return

Data

2 bonds, both zeroes (reimbursable at par of $1,000)

T Price YTM

1 $925.93 8%

2 $841.75 8.995%

Question investor has 1-period horizon; no uncertainty

does bond 2 (higher YTM) dominate bond 1?


23/71

Yield, Maturity and Period Return 2

Answer: Nope

Bond 1 HPR:

Bond 2 HPR:

f2= 10%

price in 1 year =Par/(1+f2) = $ 909.09

capital gain at year-1 end =

%81

)1

1(

000,1$

93.925$

93.925$000,1$

HPR

y

%875.841$

75.841$09.909$1

HPR


24/71

Fwd Rate & Expected Future Short Rate

Interpreting the term structure

Short perspective

liquidity preference theory (investors)

liquidity premium theory (issuer)

Expectations hypothesis

Long perspectiveMarket Segmentation vs. Preferred Habitat

Examples


25/71

Fwd Rate & Exp. Future Short Rate 2

Short perspective

liquidity preference theory (short investors)

investors need to be induced to buy LT securities

example: 1-year zero at 8% vs. 2-year zero at 8.995%

liquidity premium theory (issuer)

issuers prefer to lock in interest rates

f2E[r2]

f2E[r2] + risk premium


26/71


Long perspective

long investors wish to lock in rates

roll over a 1-year zero at 8% or lock in viaa 2-year zero at 8.995%

E[r2]f2

f2E[r

2] - risk premium


27/71


Expectation Hypothesis risk premium = 0 and E[r2]f2

idea: arbitrage

Market segmentation theory idea: clienteles

ST and LT bonds are not substitutes

reasonable?

Preferred Habitat Theory investors do prefer some maturities

temptations exist


28/71


In practice

liquidity preference + preferred habitat

hypotheses have the edge

Example

BKM Fig. 15.5


29/71


Example 2

short term rates: r1r2 r3 10%

liquidity premium = constant 1% per year

YTM

%67.101%)111%)(111%)(101(1)1)(1)(1( 33 3213 ffry

%5.101%)1%101%)(101(1)1)(1( 212 fry

%1011 ry


30/71

Measurement: Zeroes vs. Coupon Bonds

Zeroes

ideal

lack of data may exist (need zeroes for all maturities)

Coupon Bonds

plentiful

coupons and their reinvestment

low coupon rate vs. high coupon rate

short term rates they may have different YTM


31/71

Short Rates, Coupons and YTM

Example

short rates are 8% & 10% for years 1 & 2; certainty

2-year bonds; Par = $1,000; coupon = 3% or 12%

Bond 1:

Bond 2:

%98.878.894$%)101%)(81(

030,1$

%)81(

30$

YTM

%94.887.053,1$%)101%)(81(

120,1$

%)81(

120$

YTM


32/71

Measurements with Coupon Bonds 2

Example

2-year bonds; Par = $1,000; coupon = 3% or 12%

Prices: $894.78 (coupon = 3%); $1,053.87 (coupon = 12%)

Year-1 and Year-2 short rates

$ 894.78 = d1x 30 + d2x 1,030

$ 1,053.87 = d1x 120 + d2x 1,120

Solve the system: d2= 0.8417, d1= 0.9259

Conclude ...


33/71


Example (continued)

%810.9259

1

1

11

11 rdr

%1018%)x0.8417(1

11

x)1(

12

21

2

rdr

r


34/71


Practical problems pricing errors

taxes

are investors homogenous? investors can sell bonds prior to maturity

bonds can be called, put or converted

prices quotes can be stale

market liquidity

Estimation statistical approach


35/71

Rising yield curves

Causes eithershort rates are expected to climb: E[rn]E[rn-1]

orthe liquidity premium is positive Fig. 15.5a

Interpretative assumptions

estimate the liquidity premium assume the liquidity premium is constant

empirical evidence

liquidity premium is not constant; past future?!


36/71

Inverted yield curve

Easy interpretation if there is a liquidity premium

then inversion expectations of falling short rates why would interest rates fall?

inflation vs. real rates

inverted curve recession?

Example

current yield curve:The Economist


37/71

Arbitrage Strategies

Question:

The YTM on 1-year-maturity zero coupon bonds is 5%

The YTM on 2-year-maturity zero coupon bonds is 6%.The YTM on 2-year-maturity coupon bonds with coupon rates of 12% (paid annually) is5.8%.

What arbitrage opportunity exists for an investment banking firm? What is the arbitragerofit?


38/71

Arbitrage Strategies

Answer:

The price of the coupon bond, based on its YTM, is:

120 PA(5.8%, 2) + 1000 PF(5.8%, 2) = $1,113.99.

If the coupons were stripped and sold separately as zeros, then based on the YTM ozeros with maturities of one and two years, the coupon payments could be sold separately

for[120/1.05] + [1,120/1.06

2] = $1,111.08.

The arbitrage strategy is to:

buy zeros with face values of $120 and $1,120 and respective maturities of 1 and

2 yearssimultaneously sell the coupon bond.

The profit equals $2.91 on each bond.


39/71

Fixed Income Portfolio Management

In general

bonds are securities just like other

use the CAPM

Bond Index Funds

Immunization

net worth immunization

contingent immunization


40/71

Bond Index Funds

Idea

US indices

Solomon Bros. Broad Investment Grade (BIG) Lehman Bros. Aggregate

Merrill Lynch Domestic Master

composition

government, corporate, mortgage, Yankee bond maturities: more than 1 year

Canada: ScotiaMcLeod (esp. Universe Index)


41/71

Bond Index Funds 2

Problems

lots of securities in each index

portfolio rebalancing

market liquidity

bonds are dropped (maturities, calls, defaults, )


42/71

Bond Index Funds 3

Solution:

cellular approach

idea classify by maturity/risk/category/

compute percentages in each cell

match portfolio weights

effectiveness

average absolute tracking error = 2 to 16 b.p. / month


43/71

Special risks for bond portfolios

cash-flow risk

call, default, sinking funds, early repayments,

solution: select high quality bonds

interest rate risk

bond prices are sensitive to YTM

solution

measure interest rate risk

immunize


44/71

Interest Rate Risk

Equation: P= PV(annuity) + PV(final payment)

=

Yield sensitivity of bond Prices: P yield

Measure?

)1()1(1 r

Par

r

couponT

T

tt


45/71

Interest Rate Risk 2

Determinants of a bonds yield sensitivity time to maturity

maturity sensitivity (concave function)

coupon rate

coupon sensitivity

discount bond vs. premium bond

zeroes have the highest sensitivity intuition: coupon bonds = average of zeroes

YTM initial YTM sensitivity


46/71

Duration

Idea

maturity sensitivity

to measure a bonds yield sensitivity, measure its effective maturity

Measure

Macaulay duration:

1)1(

1

11

P

P

YTM

C

Pw

T

tt

tT

t

t)1.( YTMP

Cw

t

tt

t

T

t

wtD .1


47/71

Duration 2

Duration = effective measure of elasticity

Proof

Modified duration

with

YTM

YTM

DP

P

1

)1(

.

YTMDP

P

.*

y

DD

1

*


48/71

Duration 4

Interpretation 1

= average time until bond payment

Interpretation 2

% price change of coupon bond of a given duration

= % price change of zero with maturity = to duration

t

T

twtD .

1


49/71

Duration 4

Example (BKM Table 15.3)

suppose YTM changes by 1 basis point (0.01%)

zero coupon bond with 1.8853 years to maturity

old price

new price

9623.831

05.1

10007706.3

6636.831

0501.1

1000

7706.3

YTM

YTMD

P

P

1

)1(.%0359.0

9623.831

9623.8316636.831


50/71

Duration 5

Example: BKM4 Table 15.3

suppose YTM changes by 1 basis point (0.01%)

coupon bond

either compare the bonds price with YTM = 5.01%

relative to the bonds price with YTM = 5%

or simply compute the price change from the duration

%0359.005.1

%5%01.528853.1

1

)1(.

xx

YTM

YTMD

P

P


51/71

Duration 6

Properties of duration (other things constant) zero coupon bond: duration = maturity

time to maturity

maturity duration exception: deep discount bonds

coupon rate

coupon duration

YTM YTM duration

exception: zeroes (unchanged)


52/71

Duration 7

Properties of duration

duration of perpetuity =

less than infinity!

coupon bonds (annuities + zero)

see book

simplifies if par bond

y

yD

1


53/71

Duration 8

Importance

simple measure

essential to implement portfolio immunization

measures interest rate sensitivity effectively


54/71

Possible Caveats to Duration

1. Assumptions on term structure

Macaulay duration uses YTM

only valid for level changes in flat term structure

Fisher-Weil duration measure

T

tt

s

s

tT

t

t

r

Ct

PwtD

1

1

1 )1(

.1

.

T

ttt

T

t

tYTM

CtPwtD 11 )1(.

1.


55/71

Possible Caveats to Duration 2

problems with the Fisher-Weil duration

assumes a parallel shift in term structure

need forecast of future interest rates bottom line: same problem as realized compound yield

Cox-Ingersoll-Ross duration

bottom line: lets keep Macaulay


56/71


2. Convexity

Macaulay duration

first-order approximation:

small changes vs. large changes

duration = point estimate

for larger changes, an arc estimate is needed

solution: add convexity

)1(.* YTMDPP


57/71


Convexity (continued)

second-order approximation:

2* ..21. YTMconvexityYTMD

PP

T

tt

t

YTM

Ctt

YTMPconvexity

1

2

2 )1().(

)1(

1


58/71


Convexity: numerical example

P = Par = 1,000; T = 30 years; 8% annual coupon

computations give D*=11.26 years; convexity = 212.4 years

suppose YTM = 8% -> YTM = 10%

%52.2202.026.11.*

xYTMDP

P

%27.18..21. 2* YTMconvexityYTMD

PP

%85.18000,1

000,146.811

P

P


59/71

Bottom Line on Duration

Very useful

But take it with a grain of salt for large changes


60/71

Immunization

Why?

obligation to meet promises (pension funds)

protect future value of portfolio ratios, regulation, solvency (banks)

protect current net worth of institution

How? measure interest rate risk: duration

match duration of elements to be immunized


61/71

Immunization

What? net worth immunization

match duration of assets and liabilities

target date immunization match inflows and outflows

immunize the net flows

Who?

insurance companies, pension funds target date immunization

banks net worth immunization


62/71

Net Worth Immunization

Gap management

assets vs. liabilities

long term (mortgages, loans, )vs. short term (deposits, )

match duration of assets and liabilities

decrease duration of assets (ex.: ARM)

increase duration of liabilities (ex.: term deposits)

condition for success

portfolio duration = 0 (assets = liabilities)


63/71

Target Date Immunization

Idea

Example: suppose interest rates fall

good for the pension fund

price risk

existing (fixed rate) assets increase in value

bad for the pension fund

reinvestment risk

PV of future liabilities increases

so more must be invested now


64/71

Target Date Immunization 2

Solution

match duration of portfolio and funds horizon

single bondbond portfolio

duration of portfolio

= weighted average of components duration

condition: assets have equal yields


65/71


Question:

Pension funds pay lifetime annuities to recipients.

Firm expects to be in business indefinitely, its pension obligation perpetuity. Suppose, your pension fund must make perpetual payments of $2 million/year. The yield to maturity on all bonds is 16%.(a)duration of 5-year bonds with coupon rates of 12% (paid annually) is 4 years

duration of 20-year bonds with coupon rates of 6% (paid annually) is 11 years

how much of each of these coupon bonds (in market value) should you hold to bothfully fund and immunize your obligation?

(b)What will be the par value of your holdings in the 20-year coupon bond?


66/71


Answer:

(a)PV of the firms perpetual obligation = ($2 million/0.16) = $12.5 million. duration of this obligation = duration of a perpetuity = (1.16/0.16) = 7.25 years.

Denote by wthe weight on the 5-year maturity bond, which has duration of 4 years.Then,

wx 4 + (1 w) x 11 = 7.25, which implies that w= 0.5357. Therefore,

0.5357 x $12.5 = $6.7 million in the 5-year bond and 0.4643 x $12.5 = $5.8 million in the 20-year bond.

The total invested = $(6.7+5.8) million = $12.5 million, fully matching the fundingneeds.


67/71


Answer:

( b ) Price of the 20-year bond = 60 x PA(16%, 20) + 1000 x PF(16%, 20) = $407.11.

Therefore, the bond sells for 0.4071 times Par, and

Market value = Par value x 0.4071

=> $5.8 million = Par value x 0.4071

=> Par value = $14.25 million.

Another way to see this is to note that each bond with a par value of $1,000 sells for$407.11. If the total market value is $5.8 million, then you need to buy 14,250 bonds,which results in total ar value of $14,250,000.


68/71

Dangers with Immunization

1. Portfolio rebalancing is needed

Time passes duration changes

bonds mature, sinking funds,

YTM changes duration changes

example: BKM4 Table 15.7

duration YTM5 8%

4.97 7%

5.02 9%


69/71

Dangers with Immunization 2

2. Duration = nominal concept

immunization only for nominal liabilities

counter example

childrens tuition

why?

solution

do not immunize buy assets


70/71

An Alternative? Cash-Flow Dedication

Buy zeroes

to match all liabilities

Problems difficult to get underpriced zeroes

zeroes not available for all maturities

ex.: perpetuity


71/71

Contingent Immunization

Idea

try to beat the market

while limiting the downside risk

Procedure (BKM6 Fig. 16.10; BKM4 Fig. 15.6)

compute the PV of the obligation at current rates

assess available funds play the difference

immunize if trigger point is hit

bond primer slides 2007

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