Derivatives : A Primer on Bonds

Derivatives: A Primer on Bonds

• First Part: Fixed Income Securities

– Bond Prices and Yields

– Term Structure of Interest Rates

• Second Part: TSOIR

– Term Structure of Interest Rates

– Interest Rate Risk & Bond Portfolio Management

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.

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

Prices vs. Yields

• P yield • intuition

• convexity

• BKM6 Fig. 14.3; ; BKM4 Fig. 14.6

• intuition: yield P price impact

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.

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


• Yield To Call• definition

» discount rate s.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


• Yield To Default• definition

» discount rate s.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


• 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

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

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)

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


• Example

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

• compute YTM:

• 1st year taxable interest

%10)1(

000,1$31.57$

30

y

y

73.5$31.57$04.63$%)101(

000,1$

%)101(

000,1$3029


• 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

Term Structure of Interest Rates

• 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

“Term”inology

• 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

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

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

nn

nn

n YTM

YTMf

)1()1()1( 11 n

nn

nn fYTMYTM

Bootstrapping Fwd from Zeroes 2

• Data • BKM Table 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%

Bootstrapping Fwd Rates from Zeroes 3

• Forward interest rate for year 1

• Forward interest rate for year 2

%8)

11(

000,1$

)1

1(

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)

21(

93.925$75.841$

f

f

Bootstrapping Fwd Rates from Zeroes 4

• Short rate for years 3 and 4• keep applying the method

• you find f3 = 11% = f4

• General Formula

• f1 = YTM1

• 11)1(

)1(1

n

n

nn

n YTM

YTMf

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?

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)

11(

000,1$

93.925$

93.925$000,1$

HPR

y

%875.841$

75.841$09.909$1

HPR

Fwd Rate & Expected Future Short Rate

• Interpreting the term structure– Short perspective

– liquidity preference theory (investors)

– liquidity premium theory (issuer)

– Expectations hypothesis

– Long perspective

– Market Segmentation vs. Preferred Habitat

– Examples

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

• f2 E[r2]

• f2 E[r2] + risk premium


• Long perspective• “long investors” wish to lock in rates

» roll over a 1-year zero at 8%

» or lock in via a 2-year zero at 8.995%

• E[r2] f2

• f2 E[r2] - risk “premium”


• 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


• In practice• liquidity preference + preferred habitat

» hypotheses have the edge

• Example• BKM Fig. 15.5


• Example 2

• short term rates: r1 r2 r3 10%

• liquidity premium = constant 1% per year

• YTM

%67.101%)111%)(111%)(101(1)1)(1)(1( 333213 ffry

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

%1011 ry

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

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

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 = d1 x 30 + d2 x 1,030

» $ 1,053.87 = d1 x 120 + d2 x 1,120

• Solve the system: d2 = 0.8417, d1 = 0.9259

• Conclude ...


• Example (continued)

%810.9259

11

11

11 r

dr

%1018%)x0.8417(1

11

x)1(

12

212

r

drr


• 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

Rising yield curves

• Causes• either short rates are expected to climb: E[rn] E[rn-1]

• or the 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?!

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

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 arbitrageprofit?

Arbitrage StrategiesAnswer:

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 ofzeros with maturities of one and two years, the coupon payments could be sold separatelyfor

[120/1.05] + [1,120/1.062] = $1,111.08.

The arbitrage strategy is to:

buy zeros with face values of $120 and $1,120 and respective maturities of 1 and2 years

simultaneously sell the coupon bond.

The profit equals $2.91 on each bond.

Fixed Income Portfolio Management

• In general• bonds are securities just like other

• use the CAPM

• Bond Index Funds

• Immunization• net worth immunization

• contingent immunization

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)

Bond Index Funds 2

• Problems• lots of securities in each index

• portfolio rebalancing

» market liquidity

» bonds are dropped (maturities, calls, defaults, …)

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

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

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

Interest Rate Risk 2

• Determinants of a bond’s 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

Duration

• Idea• maturity sensitivity

• to measure a bond’s yield sensitivity,

• measure its “effective maturity”

• Measure• Macaulay duration:

1)1(

1

11

P

P

YTM

C

Pw

T

tt

tT

tt)1.( YTMP

Cw t

tt

t

T

twtD .

1

Duration 2

• Duration = effective measure of elasticity

• Proof

• Modified duration

with

YTM

YTMD

P

P

1

)1(.

YTMDP

P

.*

y

DD

1*

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

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

10007706.3

YTM

YTMD

P

P

1

)1(.%0359.0

9623.831

9623.8316636.831

Duration 5

• Example: BKM4 Table 15.3 • suppose YTM changes by 1 basis point (0.01%)

• coupon bond

» either compare the bond’s price with YTM

= 5.01% relative to the bond’s price with YTM

= 5%

» or simply compute the price change from the duration

%0359.005.1

%5%01.528853.1

1

)1(.

xxYTM

YTMD

P

P

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)

Duration 7

• Properties of duration

• duration of perpetuity =

– less than infinity!

• coupon bonds (“annuities + zero”)

– see book

– simplifies if par bond

yy

D

1

Duration 8

• Importance

• simple measure

• essential to implement portfolio immunization

• measures interest rate sensitivity effectively

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

ss

tT

tt

r

Ct

PwtD

1

1

1 )1(.

1.

T

tt

tT

tt YTM

Ct

PwtD

11 )1(.

1.

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: let’s keep Macaulay


• 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


– Convexity (continued)

– second-order approximation:

2* ..2

1. YTMconvexityYTMD

P

P

T

tt

t

YTM

Ctt

YTMPconvexity

1

22 )1(

).()1(

1


– 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.*

xYTMD

P

P

%27.18..2

1. 2*

YTMconvexityYTMD

P

P

%85.18000,1

000,146.811

P

P

Bottom Line on Duration

• Very useful

– But take it with a grain of salt for large changes

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

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

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)

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

Target Date Immunization 2

• Solution• match duration of portfolio and fund’s horizon

– single bond

– bond portfolio

» duration of portfolio

» = weighted average of components’ duration

» condition: assets have equal yields


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 yearsduration 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?


Answer:

(a) PV of the firm’s “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 w the weight on the 5-year maturity bond, which has duration of 4 years.Then,

w x 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.


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 par value of $14,250,000.

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 YTM

5 8%

4.97 7%

5.02

9%

Dangers with Immunization 2

• 2. Duration = nominal concept• immunization only for nominal liabilities

• counter example» children’s tuition

» why?

• solution» do not immunize

» buy assets

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

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

Documents

Derivatives : A Primer on Bonds